An Introduction to Diagrammatical Methods in Representation Theory

VORLESUNGEN Fusdem

FACHBEREICH MATHEMATIK der UNIVERSITAT ESSEN ..

ii

'

Helt7

il-

V. B. Dlab

TO DIAGRAMMATICAL AN,INTRODUCTION METHODS IN REPRESENTATION THqORY' ,]

'

,Ausalbeitung: Riüard Dipper . ,,

l

I

e i f",

,l

1981 ,'

u/

+'' i/ 4 uu.iqi't i( -

/'t.< t '"

.1,: :i t'

r

1,,

VORLEST'NGEN aus dem FACHBEREICH UATHEMATIK

der UNIVERSITATESSEN

Iteft

7

V. B. Dlab

AN

INTRODUCTTON TO IN

DIAGRAMI.,IATICAL MEIHoDS

REPRESENTATION TIIEORY

Ausarbeltung:

Rlchard

19 8 1

Dlpper

Acknowledgements

These notes lectures

contaln

addressed

the materlar

to

the

graduate

presented students

rn a serres in

Algebra

of

at

the Essen durlng the surmer semester 1g7g. T should lLke to thank prof. c.o. Mlchler for hls klnd lnvltatlon to vlslt Essen, and the audlence for their partlclpatlon actlve ln the lectures. unlverslty

to

of

The alrn of

thl-s

some baslc

results

and conseguently tensor

algebras

brief of

the

course the

content

was to

expose

the

representatlon

theory

was restrlcted

to

and a few slmple

audience graphs,

of

a study

1llustratlons

of

of

the

theory.

As a result,

the reader finds a major overlap wlth the Me_ the Amer. ltath- society No. 't 73 and an omission of general theory of M. Auslander and I. Reiten.

molrs the

of

My special written tails

notes

are

due to

wLth great

r should

Department

thelr tatlon

excellent of

the

of

lr-ke to

Mathematics

typlng,

p.ichard

Dr. care,

and in many ways improvlng

Pinally, the

thanks

up the

of

contributed

supplylng

the

thank

Digper

original

alr

those the

de_

expositlon. secretalies

Essen University to

who has

missing

formal

who,

of by

presen_

notes.

Vlastimil

B. Dlab

CONTENTS Chapter

f

Valued

graphs

s1

Valued

52

The roots

Sg

Graphs wlth

Chapter

II

PAGE

graphs, of

1

Dynkln

and Euclldlan

graphs

graphs

valued

7

orlentatlon

Reallzatlons

of

1

11

valued

graphs

and its

20

repre9entatlons

s s s

1

X-reallzatlons

2

The Coxeter

3

Prejectlve

Chapter

ffl

Graphs

of

and representations functors

30

modules and extenslons

finlte

and of

The representatlons

of

s2

The non-homogeneous

representations

Dynkin

Euclld1an

Appendlx

graphs

56 of

60

of

85

graphs

'Blmodules Graphs

3f

graphs

The hornogeneous representatlons

s4 Ss

47

tame type

St

Eucll-dlan

20

of

Euclidian

and bimodules

type of

wild

100 type

ru)

A

Applicatlons

118

1

Algebras

118

2

Normal

3

Further

Appendix

B

Exanples

173

Appendlx

C

TabIes

180

form

problems

applicatlons

139 171

-1-

Chapter S 1

graphs,

Valued

graph

A valued rrith

Valued qraphs

r Dynkin

(I,d)

is

and

a finite

a set d ={(dii,dii)lar.

graphs

Euclidian

€N

u

(of

f

set

vertices)

together

€ I}(of

{c},i,j

vaiues)

L J

J L

satisfying i) li)

dlt=oforalli€I For every utj

= di1 fi

tj

Notice that join

€ f there

i

d, : i rJ

exists

for all

o L f a n d o n l y- i f

, If

d,. rl

j 1 Another valued graph ( l',d') (t,d),if Veltices,

( l,d)

is

a sequence 1 = ior that

the fi,

scalar

d:. )L

I o. fn this

i1,

multlples,

if n} c

For the rational

with

ft

g t

is

a subgraph of

(xi)l*iee,

s!'nnetric

bilinear

form

(1) :=

-f .i 'x. . y .

i,j

€ t

, there

ik = i of neighbours in

is

ccnnected.

-

i€r}withn= Bf

- I .

quadratic (r,r)

"l.

all

f

determj.ned up to

Of course we can nrite

=

definethe

lrl

by r (r,j)€f"f

d..f .x..;. rifixiYi

for

y I , I \ , .€ Q '

form tr*r'

,ä.

-

dijrj*i*j .1. ij

,

rvhere in edges

the second term the sum has to be taken over aLl (f,d) - clearly in Br and en der:end on i-;

the choice of the fj.'= , i € f , factors, j-f (i,d) is connected.

is

. Notice

space

{x=

wLth associated

for

neL_qhbours. A

N fo sone n €N.

vector

r ier

if

.i ,d.i i )

:

above are uniguely

A':=0"=

(x,1) =

*!

and wrlte

= 1 = tt... l1

....,

( frd)

case, we (di

(dii,dii)

connected,

I € f, 1n ii)

| = {1, ....r

Qf

-

d t.t l, = d , . ft ot r af lol ri , i a€ l l i , j € f lr c f . _. which are joindd by an edge, are called

valued graph

Br

€ f

i'j

i and j by an edge of value

or simply

€ N such that

fl

but are unique up to scalar

-2-

Eor k€I

let

= (k.)

k

€ 0f

definedby

k,

= o for

k I

I

€ f

and k* = 1.

1.1.1

Definition:

(trd)

a) Let

be one of

the

following

valued

sraPhs (lrl= n)' Arr: *o......a.1 1, 2 ) Brr:ffi......H

(2,1) C: n

o--O......H

\ )--......a---a

/ ? n=6

E.: o

E7'

? n=8

3r;

n=4

tt.

I

( 1, 2 ) '4'

(1,3) Gr: o---<

n=2 then

(f,d)

b) Let n=2

is called Dynkin (graph).

(I,9)

be one of the following (1,4) H 7-: tl

valued graphs. (lfl

= n1'

-3-

Är,

n=2

n>

3

( 2, 2 ) F

ärr-t

n:3

!rr-t

na3

7-n - 1

n>3

ft'-t

n>4

Sn-t

n>4

örr-t

n> 5

örr-

t

( 1, 2 1

(2t1)

(2r1)

( 1. 2 ' , )

( 2, 1 1

(2t1)

.E---{-_o.....H

\

-7-n=7

E6

n=8

Ej

-/'

\

-4-

n=9

-8

n=5

x "41

n=5

-F 4 2

n=3

dr.,

n=3

"22

( f ,d)

Chen vre call

1.1.2

Lemma:

( f ,{)

Let

of the followlng

(rr,4r,,

i)

(i,j)

(r

ii)

Proof:

a

If

i,j

( | ,d) contains

one

dijdji

> 5 for a pair

x I

a triple

(i,j,k)

Euclldian

graph.

there are i,j

€ f.

Then

graphs as a subgraph:

€

ijk

d . . d . . -> 4 , rl lr.

Ä,,.,' or Ä,,r. thus F I

= 3 for

lä) ,

€ fxlxf

€ I with

(f1,d1),

Let i,j

,@w*h

= 2 for t r(resP.

= 3 and djrd*j

a subgraph of type for all

be Dynkin.

(resp. ( | i,gä)

urjuj,

iii)

not

.

:&+'i.h

€ f

2,9)

for

(graph)

Euclj.dian

. . i ! L W f L r r

I I u . . s . i i ' )

-J -

2 J .

( f ,d)

contains

a s s u m e d .r l . d .1 1 . -< 3 Because (f,d)

is

-t-

not

Dynkln,

i or j must have a nej_ghbour k € I

above dildkj

< 3. It

graph of type utjdjtg there

is

( tz,lzl

S t

a subgraph of type Et, all

{1,j}

with

€t

Ct or 60a,lrl

;

{k,f}

g I

F41 ot Fnr,

or it

must havg 6 branching

ls

of

type 0a

l,j

not Dynkln.

€ I

type Äa,

rrr

If

or !r,

because it

has only

branchlng point,

A bilinear vector

it

Leumas tet

varued graph

Proof:

(r,g).

Clearly

and take x'=

lt

Brr

(f',d')

is

the vector

it

a subgraph type agaJ-n because

lt

if

contalns

positive.

lrl

2 t €N.

if

O(I)

all

I

> o for

aII

€ en and positive definite.

the connected

then Brrispositivedefinite.

Assrxle B.

minimal hrith this

defined

E,

form o on the rational

a proper subgraph of

is

one branching

more then one

but not positlve

B, is positive,

ar1

a subgraph of type Er,

> o for

rf

a subgraph = 1 for

at reast

quadratic

g(x)

positive,

€ Of'

= 1 for

contaj,ns a subgraph of

is called. positive,

(f,d)

dttdtx

a subgraph of type ö,,

5 (xi) € QI' b" with er,(x,) z =(?t)

If

associated

(f',$')

contatns

contains

contalns

1t contains

contains

definite,

lf

it

not Dynkln.

space en, n €N,

senldeflnlte,

1.1.3

one,

form B wlth

€ 0n, positive

. rf

2. rf

( I,d)

Thus assurne drjdj,

a circle,

Otherwise

is

now

polnt,

( f,d)

case

t € N.

contalns

t €N.

I

Po1nt.

it

lf | :

( f,4)

If

thus.assume

, our graph must contaln

fn the latter

or döa,

a sub-

= 2, then t €N

By the

contains

withdlidir=

dXtdtt

I

it

I

Leti,j

, say j.

(f,{)

Er., or örr.

i,di'),

e r

{k,1}

I

( f

,

2foral1l,j 1s {i,j}

easy to see, that

is

not positive

property.

Let

= o. since for arbitrary by z., = iyrl

for

all

definite,

I

€ eI,

i € f,

-o-

always

satisfj-es

for

i

f

all is

Qr, (z) < er, (1)we can assume

€ f'.

In view of minimality

connected,

d., 'oJo I

there

o. oefine

d. ].'l o-o x+ *.. ro2'or

x

e ef Uy

l',

f'

all

xi

+ fi -o

-.

x?

-o

*i

for

dr.

f.

x.x.,

with

€ I,,

i

E

-Jo

Jo

a

Jo

'

di s f* x] lo lo

> o

I o. Because

and io € lr = *i

,a-oJo i€f

-

xi

= o othenrj.se.

and x.

Then Qf (x) = Qr, (I')

jo € f -

exists

= (*i)

of

that

= f., (x. d' .i f+ x., X+ roJo Jo r.o Jo Jo Jo

d. d. ro lo

i

39

x' ,2Lo

2

!

_r

' o 'i o * ?

=-


acontradiction.

4rot 1.1.4

Proposition:

Let

(f,{)

a)

(f,d)

is

Dynkin'

b)

(f,d)

is

Euclidian,

be a connected va}ued graph

and only

if

if

if

and only

Qa is if

positive

Q.

is

definite-

positive

semi-

definite.

Proof:

First,

calculation

shows, that

Now let

in

(I,d)

defined as in rf

(fl,d.')s

*i

= 1 for

is Qf

(The quadratic

semidefinite. are listed

( f,d)

if

tables,

Dynkinrresp. is

positive

Euclidj.an, definite

forms Q, for

a direct

resp.

positive

graphs

Euclidian

Appendix C).

be not Dynkin. Let

(t1,d1),

(f2,d)

and (ri,dj)

(1.1.2) . (l,d)

{i, j}

=

definex=

(xr)i61€Q^

f ,, and *k = o othervrise.

r

byx.:

= ä 1d .i j ,

f

*2 rc

-7-

rhenQ. (x) = rr - lrurjtjuj, * u3r * t) = ujrtr,

because utjtj If

(tZ,d)

*i

= diidti,

= 2dki,

*j

dijdjr

"td wlth I,

(f,d)

c

=

* rr(a diidii) < o,

2 5. Thus e, is not positive

= {i,j,k}

S I define I

*k = 1 and xl

= o otherbrise.

Qr(x) = ridii dxl * lrra*j * rk - 2diiridiidrjdrj = fk(3dkjdjr = f*fi Thus Qf is

-

* 4 drjdjk

+ 1 - udjrd*j

not positive. (r;,qä)

Euclidian

subgraph, e, is not positive

= (f,d).

Of be posltive

(f,d)

is not Dynkin. er,

Let

The roots (l,d)

)

and (l,d)

"X IrI

Bl is

(f,Ä)

cannot

(ri,dä).

Thus

hence (f,d)

=

is Euclidian. This proves b).

graph

valued

ef . For k € f ,2Br(x,k)

= x --Effik

(s*><.)i = xi

= t.

groves 6;.

This

subgraph 11,,g, ), by (t .1 .21 , because

the

with

linei .tt

for

k I

invariant

bllinear *un

sk

form

I : @' -

Bf and P Q' defined

for x e Ql is called simple reflection

( f o r k € r ) . v r eh a v e f o r x = l x r ) e q r , and

or

j.s not

ef

contaj.ns a

definite.

(t2,d)

that

of valued sraphs

form

:

(r,4)

if

is not posj.tive definite,

be a connected

quadratic

by s1x

Euclidian

by (t.1.3),

- 2djkdkj)

As above,

a subgraph of type(It,d1),

contains a

S 2

clearly,

sernldefinite.

(I,d)

(f',d')

Then

- 2dirrrdri

The same argurnent shows

if

contai.n

UV

< o.

\idis)

Positive,

Norr let

€ @r

i

€ |

under's*,

(s1{)

1

particul.,,"

fn

i.

e. Br(x;1)

= _*k * ,!r n"a

s*k

= efs*!,

r € Q:.

The group tti of automorphism of @I, generated by the sk, is calledtrlevl groupof

f

. ilet Itl

= . . . n . a n d I '_ { k t ,

:,.,

dit*i_

= -

k and

sif)

foi"

k € f, krr}-;t .,

-8-

Then c = su. .^n^nrl

of ql'.

If

su

s. € lv-- is (xe

wx=x

x = (xr) € d

called

forallw€

d)

Coxeter transformatlon wF

, thenxls

stable.

- l

is positive

x. > o resp. xi5

(x

o) resp.

;

o for aLl i € f

(x S o),

neqatlve

. For x,I€

QI let

I:

lf lf

Ir

x-r>o, = iwllw

The set Rf t;

the si:nple posltlve

1.2."1. let

€ Wf, k e f) ale

statenents

are

x is

ii)

cx=I

v)

transformatlon

= o for

Bt (x,1) ff

moreover

Let

f = {k1,

alr

of

k € I.

th"n

Qr

the

follonhg

...,

and c = s* .

kr}

I € f.

follows

With thls

using s*x = I equivalence

-

implications

then the above to Qp (x) = o.

are equivalent

'"*1O,*.

The other

1 € 0' positive,

Q, is

xka = (c:!)*a = ("k;

fhe

of

stable

statements

iii).

the k with

roots

skä=Iforallk€I

iv)

sr,

the set of

equivalent:

i)

iii)

of

is

Leluna: Let (t,d)be a connected v a L u e d g r a p h , x € Q f , a n d

c be a Coxeter

Proof:

roots

c e'

in

for

...

15

' sk1. Nort

t s n by definition

inductively,

that

(i) C)(fi)(=)(1i1)

11) jrnplies are tlivial.

28- (x,k) diEE

(iv) (+

(v)

set Nf of vectors

( ' l . 2 . 1) , i s c a l l e d

k , we see (iii)

(=) (iv).

(rrtith QI pos j-tive) in Q' , which

the radical

of

satisfy

(f,d) .

is

rhe last

well-known. the conditj.ons

of

-9-

1.2.2

Lema: If

(f,d)

is Dynkin, then Na = (o).

b)

If

(f,d)

is

integrer

The proof

of

in Appendlx

C.

also

satisfles

, and nf

this nf

Notice

Euclidian,

for

that

the

set

the definrng

Rn generat,es Of .

root

rf

. *rr3*!()

x € R- ,

them equal

to one.

(1,y)

then

A list

t23l

of

but

in

a valued

the

the

graph

finiteness

of tables

( frd)

conditlon

:

Rf mapsRrintoR'

is

an inteser

.

2x ( Ra .

This

is

easT to prove.

that

root

is

a linear

every

As a

combination

l.

!,s

Furthermore,

e e',

result,

This

well-known

| 22 , 23

I

k € l,

with integer coefficients.

as an inEnediate consequence of

general is

roots

[ 3 ]

forl€

consequence \^/e get, the

positive

Ls contained

1f,dy

Rf of

system of

28. (x,I) I-x--re;;"

iv)

chosen with

one of

properties

1)

For I,I

can be

least

Euclldian

a reduced

iii)

€ NI

one_dimensional;,

lermna can be found in

for

li)

then Nf is

componeits,at

the vectors

to

be a connected valued graph

a)

Nf =
,

(I,d)

Let

one of the

x or

and can be proved

We will

E. Scherzler.

roots

give

-x using

the

is

following

always positive.

Kac-Moody Lie

a ne\d, pure combinatorialr

argebras

proof,

due

of

-10-

1.2.3 ä

Lemma:

t € O-

or

-

x

(f,

Let

1s a positi.ve k

root

and k € J.,

valued

then

graph.

either

If

sk I

O

:

(and s,-x = -k). K_

3 For x,I € gr with B'(I,I) I O let <x,L>: = -rff#.

Proof

Thus skx = x + <xrk> b Furthermor" Let

be a connected

9)

I

qi,i>

e Qf, k,i

€ l,

by direct

calculation

(1)

- -2Yk *

(2) sr

"i =k si

I

= I

and either

= -2,

and

t x or

I "k = dik

€ N

for

i,k

L * k. We get the following from the definition ur* yjr

rä, +

= s. sk if

sk x < x.

i

if

+ dik

formulas

o, )k;

= dki

i

I k. easily

Bf,

hence for 1:

+ (.I,kr

and only

of

€ r,

+ 2yk > o hence

= O

(3) sr"ksi"kL-sksiI= (.1 ,i> <1,k> (4) If "i

1) (.I,\>

d.* = d*. = 1 "k

I

"k

I

k +

+

i



i

then

= (

+ <1,k>)i

"k"i"kI-si"kI=k - sk =i sk - <1rk> i 1 = "j. "k "i "k I (5) From (3) we conclude easily: rf

sk I

f

yor

sil

I I,

only if dir = tri Define

"k "i

I

= 1.

Ru S Rf , -1

now

then

I

v € Z,

= "i

"k "i

if "k L

and

by

R_1,={-xeerlk€r} Ro,=ireErlk€r] Rv+1: = Rv U {skIlx Let v > O, I an I

s1ts>I},

€ Ru, x e Rv_1. Then, by definition

€ Ru_,|and some

O < x - I

€ Rv, k € f,

=

i,

i

€ f , such that

hence

> O.

* =

"i

!

v > O. of > I,

Rv, there i.e.

is

- 11 -

Let

i

f k € r

o t

sk x - x =

on \,, that trivj.al. of 1.

So 1et lf

by lnductlon induction

v:1.

= (
sil

x*

Norr s.

on

v,

that

again.Ifv=1then =*tI=I.

y=f

X =

i>
k>)k,

show now by induction v = o this

changes only

the

i-th

o.

Thus il.*

Next

we will

forsome

ls

and

by

trj.vial

LfI,

inplies

]="kI..y=l "f * dff i > k,

5 1

show again

I€f

is

component

:: 1 and drr

€ Rv_1. For v = O this

sk I

Nord

+.I,

kt

For

< I.

s* I

(2),

Then, by

I o.

yk I

o also

I

because

"if Because si

sk x < x.

I . L. We will "k 5 1 and d*, < 1, if x*

dif

Thusr

-

L

"k "i !> < O and


hence

be such that

because

k=1.

we get dki t O. On the ! = ! "i other hand, by above, hence dti = dik = 1. Now by (2) tf:1, sk k =k + i - k = i € Ro. So let v:2. "k*= "iI= "k "i By the above di:. = O = dki_ or dik = 1 = dX:.. In the first case = hence sk ä = I = L. By induction "i "k "k "1, "k "i "j. "k s* I € Rv_2.Thus -k I € Rrr with u = v- 3 (> - 1), I = "k "i if s. s* I < sk 1by induction, u = v - 2, Lf s. sk I = skI and p = v- 1, if y t I. Thus, in every case, "i "k "k s* x € Ru-r. Let

now dik

= 1 = dki.

o > skl

"k "i this implies O > "i again by lnduction,


"i

"k s.

E> < O we conclude "i

"k

I

"i I

By induction -

f !-, thus

€ Ru_2. Furthermore

+
i>)

I = ( +
I

frorn

I t =k =i "k "i "k =k I € Ru_2 atd =i = \

= (
I

"i -

s* I

we get

(4)

sk I, =k

k by \>)

i,

i> > O,

that "i

"i "k frorn (5)

=k I

Therefore

I € Ru_1. of

(4) and

course

hence

-12-

=

sk I

=

si I

"k

"i

Let now n: = V

"k "i

€ Ru_l, as desired.

I

"k

R,,. By construction

R = R. and all

v>ov-l

positivel

R are

Now every

... x = s. t*t1{ now by that

s.

i

induction

either

with on m, -

Ä or

x

root it

I

€ f,

that

O j

either

is -

€ R or

From this

in

as

be written

< m. It

t I

positlve.

ls

can

€ Rf

roots

easy

ä € R,

prove

to

hence

lemma follows

the

easily. '|.2.4. bilinear l*n/ H r _

(f,d)

Lemrna: Let from induced

B

. Then

be a connected group

the

fr,

of

all

is

finite.

P r o o f : W e c a n a s s u m ef = t 1 , . . . , n 1 .

l,et

by

elements

of

wr

graph

valued

positive

with

of

transformations

I

M: = {x e Q'I Qr(x) < 1, *i there

tauschsatz {nr, k

h.z

2 S k < lrl.

matrix

i

a basis

2 < k < lrl,

generated

of

the entries linear

of the

i =: v and Q.

fr, . v

€ v. thus Qr/".

product,

transformation

combination of n, and the 1<,

The radical

by !,

of ? and tr ? is

also

is

v/;,=,

c-|

hence atot*i.l

if

Ze

M. Now it

exarnple a suitable

linear

an integer

nultiple

of

it

will

follow

V,

on V. ror L.

combination

is well-known that

From this

hence, r.rith

rational,

e ? denote the image of x in V. rhen 8, 2.

induce the sane topology. for

the base change is

form on v.

and Q, induces and scalar

is

of

q r c a n b e e r n b e d d e di n i l f

clearry

one dimensional,

Aus-

say k = 1, such that

(z€11) is an integer

a quadratic

x € v let

k € f,

conmon denominator

h e 7, the

induces

a

By stelnltzr

is a basis of (D'.

I z < i < n}

The transformation

matrix,

is

€ a (i e r)}.

all

lrl,

of the [,

norms on V easilyr

using

the maximum normon V, that

- 13 -

h maps M nodulo fff (injectively) hence f, = M + N"/p r

lnto

must itself

ttF

a finite

be finite.

subset of

i

,

By the construction

I

of Bf and the ü tnto

itself.

operates of the

of

€ Qr,

of B, under W, r the

ü contains

= I then

element

ä. x is

of x

and if

x € Q has integral

(wlth

€ or deflned

respect

i)

for

quadratic

all

cx I o if

-1

c

I

'x

*

O

er ls positlve

and is

ac x € n.

called

linear

the defect

: =

folloers

the

The vector

€ r 1s carled

I (l^)., x., i€r easily that

Let O S. x e eI

be a root

be a Coxetertransformation.

and only

= 91.:= sn --t

means for

and by

be a connected valued graph with

fonn er.

if

i

the

x € Al.

(t,d)

s,- € W K1

be the orcler of

ln fra. This

= (ac, I) it

positive

Of course äc , ef - e is

= aci for

and only

....-

and

Then

if

2 -. .- .-.-.-x = -DK. t ! = Sk, ]- -sk, _ ii)

graph with

coeffj-cients,

bv (3c)i:

L e r n m a :L e t

c = Su ... .. ^n

as a subgroup

some ac x € e. If

to c).

of Nf under c,

ä" (x) = ä. (cx)

positive

finite

unlqueJ.y determined

Clearly we have ä. I

invariance

1.2.5.

c € t{I

+ (ä. x)n for

defect

the

therefore

be a connected valued

Coxeter

c* I

vector.

of V, the group fia

form Qr. Let fta as above and n the

group fr, transforms

Aroup on li.

(r,d)

semldefinite,

i"

a basls

on !i and is

sltmmetrlc

quadratic

I

since

falthfully

Let again

lnage

invarlance

-sk, a _ t k a f o r s o m e 1 < t < lfl.

if

tt*lka

for some1 :

t:

Itl

-

Proof: tttn

(1'2'3), ii)

< O if

1) c I

t*a-.,

14 -

and only if

sk1 5 > o and

""'

sk1 x = kt,

.....

"*a_,

=

some 1 < t < lrl

is

"

x < o' By

""'

"*a "n., = ---.. hence x "o., "na

"*a_.,Ea.

Similar.

(with

The notation

pkt,

c € Wr) will

be kept throughout

1.2.6.

Lenma: Let

c = sk

.. ...

sk

^1

gkt

(f,9)

Proof:

< n,

have

is

N, cI

=

-u

by I.

the

is

1.2.4). Hence there

.

cr-*r,

,

roots

smallest

(I,d)

Then

r*r,

positive

all

n, a.ta

,:k

-9. -r 2. , . .

Because

saym€

Dynkin graph,lrl=

=K2"

of

transformatj.on

paper,

this

-a^.

c- IK_l

a list

be a

to a Coxeter

!*r,

-a" I ^r:^z^^n^ =*1 ,

'| < i

respect

€ Wf be a Coxeter transformation.

&., ,

is

there

of

integer

Dynkin, Thus for exists

(l,d)

where a.

such

t,hat.-1.-"i

c is

of

< m -

nn,

order

finite

O < x € Rr andy ' 1 < r

€ N,

1

mh E c" x h=l such that =

rre

-

r-'l

= pf<. for

c-'I

some 1 < t

_:,

< n and therefore

x = .-(t-t)n*a

_fe

=c

Letnowa'pk. .*",

1 < s,t

" r*_, '-s

"t o ..2.

15 -


o < 11 Sä1, .-t

and assume 11 Z 12. If

r1 I rr, then O . .1 - 12 - 1 < äar and therefore -f. *r.+1 | ' O
= 12 and so pu

rj

irnplier

= ks.

kr

1'2'7'

Lemma: Let

natural

number 9,

that the

(f,d)

of

the

Io

a posltive

is

x € ef of

easy to

shoe, that

Iemna.

;.orrJ"an.

be Euclidian.

roots

Then there the tier

(f,d)

exists

a

Layzo.Lt

number),

with

such ?uJ_ra.U v ä^ ä = O.re

form

ä=Io+r9gl \rith

It

1 < 9 < 3 (called

the positive vectors

rhi;

= p,_

,r€N root

x

of

u{o} (I,d)

of defect

zero satisfying

. 9 .lr.

t23 l

1.2.9.

Lemma: Let

be a Coxeter if

(r,d) i)

is

(r,d)

be Euclidian

transformation.

and c =

Let c be not

"*r., conjugate

.....

s*,

to sl

€ w,

. .. . sn

of type Ärr. then

the vectors positive to c),

.l = t . n, are just all , O ( r, c of (f,d) of negative defect (\rith respect

.--&.

roots

- 16 -

ii)

iii)

t,

positive

of positive

c*rq,-, O : +t roots of (t,d)

the vectors described all

from i)

positive

mod N, of c.

and ii)

(1.2.7)

in

roots

Let O < x € n, with

Proof

Then for

T h u s x = c - r p-K., -

and the vectors of

zero) form a list

< O, and let

ä" I

m be the order

s € N

all

U toi

with

f o r s o m et € { 1 , . . . , n }

crx>O

and cr+1x < O.

by (1.2.5).

t o r s o m es € N

u {oi,

similarly,

if

t € {t....,n}.

1 < t < n we haveä^(g, ) = ä^(-c p,. ), so we get "^t"*t positive roots of (f,d), if we show r.(p**) . O

Of course for a list

with

a1l

(t,d)

of

o, we gät Ä = cs g*.

t

defect,

together

(of defect

""^I=x+s(ä.x)nr. Therefore there is an r € N

ä" I

< n, are just

1 < t

the vectors

of all

foralll
in wf to the Coxeter element cr

of type An to one of the coxeter in Appendix

C in the

following

s. and s. are adjacent

if

.....

^t-1

sequence sr tt

of

.....

assume thaL ä^,(s,u

^.

-x )

S,

.....

sr

t1

tn

operations

sr tn

Irre get

as c'.

= sk sr t1 k'

cr

= ä^(x) u

is

in the tables

S t - . A s s u m ei , j t1

€ I are not

s. and s. in c doesn't in c, and corresponds to

and s. = s, s., if s. = s. s. = s. s = s. *1 . k1 *., kr, i I I I into can be shown that c can be transformed

c o n i u q a t i o n b 'v s i 4 in it As t I ö = s.

(r,d)

way:

..... Let c = sk ..... s. , c'= s, tr, K1 n joined by an edge. Then interchanging change c,

given

elements)

(if

for

s,

tt

-

comparing

si- = s. *1 K2

..... x € R. r

.

Now bl'

above.

described Therefore

by a

f o r s o m eL € { 1 , . . . , n }

c

Hence if

conjugation

c and ct

s. . Then *'r

-

r.rith

we can

-17-

(pta)

a"

1 < t

< o for

< n and of

(c = sl -n

.....

p, -n

= s. *r

c

if

ä.,(Pf. --t

)

enough

course

a^,

Thus

O for

all

the

the

,

a_ (p, c._k*

= gX

craphs with

An orientation

if are to

i

is .

+.

joined

.

o tor

alt

< n -

1 < i

1 and

each edge

lt

in

(1 < t

is

the

< n)

and 12.2.12) that for

tables.

given

it

arbitrary

s < n) implies

define

arrows

io

resp.

i

o-

there

ls

of

j).

all

resp.

For

in

is

in fact

valued graphs

that

given

(frd)

a given

edges

it

must be of

is

for

an order orientation 0 of

containing

k.

with al1

An orlentation an ordering

by

sk

a source o k

k by an edge. if

-.j

a new orientatj.on

along

k

(f,d)

graph

io ----rr

be a sink

be admissible,

above,

and only

now.

< O. Furthermore,

to

with

< n if

elements

ä"(t)

be proved

(2.2.11)

a valued

and k € f

said

by the

using

) < O can

by an arrow

a1l

So,

1 < t

all

orientatlon

Q of for

by reversj.ng k € f

n - 1.

Coxeter

(some r > O, 1 5 t,

"-t ES Dynkin type.

Prescribing,

the

calculation,

ä. (Lt)

(f,d)

px.)

) .'-Ä.;:-,

for

< n.

lernrna for

enough to prove

0 of

.-t

1 < t

see later in

(indicated

(sL

Oc, ("k.

< O for ä"(pt .-t )

Remark: l{e will

S3

then

= s, pL ^1 *i+1

-Dk-1 -

direct

pk.

n,

-1

By an easy tables

(

1 < t

) pr ^i " t- 1

prove

to

aIL

A vertex

respect

vertices 0 of

k1

(frd)

to i. €f

(t,d) .,

n, , which

is

kn of

said f

such

-

that

a sink

...

s. *i-1 is

is

k1

with

called

admissible admissible

and if

...,k'

k1,

l. 3 .1 .

(i) ii)

+

-

e

is

iii)

(iii)

(ii)

t

i

,

k,

It

sinks,

also

is

for

easy

sources, kr,

,..

to

show

connected

= {1,...,n}.

graph

Then the

\rith

following

cj-rcuj.ts

in

(t,d)

n of

inplj-es

n of n(i)

{1,...,n}

such that

< n(j).

: easy : Choose an admissible

n bY n (kr) :

It

is

= i'

i

easy to

sequence k1r...,k'

= 'l,...,n. see,

that

n(1),..,,n(n)

is

sequence.

alfows

an admissible

orientation

seguence for

sources.

j-s a permutation

(ii)

admissj.ble

(1.3.1

be defined

can also

no oriented

o +o

and define iii)

.....

admissible

There j

Proof:

are

o is

iii)

case k1 ,

equivalent.

There

ii)

respect

with

sinks).

be a valued

and let

are

i)

( f , d)

Let

fl,

statements

this

In

a sink

is

s. 0 = a. K1

I,erq!4.

orientation

(for

for

, and k.

1.

an admissible

sequence

.....

s, *r,

a

1 < n -

sequences

is

an admissible

that

-

to

sequence

Of course,

is

respect

1 < i

n, s. K1

|8 -

us,

to

sequence (f,d).

order of

f

sinks

i-n such a way that for

the

fj.xed

1,...,D

admissible

an

. k.l

to

_ 19 _

Note alsor if

k1,...k.

sequences for

the same admissible

C =

S,-

for

any pair

*r,

...

Ref erences:

sr_

*1

-

Srt

*r,

and kir...,k,

...

of vertices

't I I ] ,

t22l

Sr_r =

ki

i,j

are two adrnissible orientation

Cir

since

s. -i

e, then a---nd

s. -j

c-----ommuce

€ I which are not neighbours.

and [ 23 ]

Realizations

2

Chapter

S 1

K-realizations

2.1.1

Remark: Let K in

a

: =

Proof:

\J

'r (d.,dr) D is

i J

t' \ \.\

the

= t

(drd, )

= tr

dr)

tr(d)

= k+ o,

projection

. Then

€ K1.

the f

"r tp

HomK (

K-Linear.

by the

above property

O +

r

by tr

o and with

d€D

the

by

followed

cases, o + r

both

._HomK

of

DDD)

is

, it

t

(DMF, K)

by

easy

to

of

map is

. This

and'

an F-D-bimodule

is

K)

DMF,

of

HomD (

, f

must be injective, the

(DMF, K)

Thus let

DDD)

DMF, oDo)

see,

are

f

that

Thus Homo (

K-dimensions.

hence

as F-D-bimodule,

also

the

by

DMF, DDD)

Hom" (oM",

similar

and

surjective

bijective

therefore

HomK (DMF, K) as F-D-bimodule. Honk

dZ€D.

K1

tr:D*

D and

of

Hence, in

D onto

of

P 1691 tr+

HonD (DMF,

elements

of

projection

r

then

f inite,

Property.

all

equality

K hrith

homomorphism.

an bimodule

since

dt'

Otu,

19 €

for

K is

126]

Then define

: Horno (

course

is

for

desired

If

center

lsee

K1 on K'k
a map with

: rp l-

be the

Kl

r

be the

t

: D -

naP

d, € D:

Then

as F-D-bimodule'

Homo{DMF'pDD)

K-linear dr,

(d2 dr) k

of

Now define f

let

trace

trtd,

for

on it.

centrally

!

and we let

Other$iise

reduced

is

a nonzero

is

coNnutative

K '1 cD.

\

K acting

Let

K.

over

dimensional

and finite

Homu (oMu,"Fu)

There

containing

D and F skew fields

K be a field,

D-F-bimodule,

( pMr)r

U

and representations

theücenters,

n$f

val

of

:

rFf)

:

HomD (DMF, DDD) =

Hom" (OMU, FFF ) as F-D-bimodule. Let

K be a field,

A K-modulation

and tü of

( f ( f ,

, d) )

be a connected is

a set

of

vaLued graph.

division

rings

Ft'

-21

! € f

, together

nith

an Fr-Fr-bj.module

Ft - Fr-birnodule

.M, for

1) K is contained

rl

F, 1s finite

li)

11i)

-= *

jMi

every edge

the center

dimensional

K operates

centrally

and an

lMj

such that

i.-.ir

of Fi tor

each i €f

airC

over K

on every

as bimodule

iMj

-

for

j

i,

il{i, each

edge

€ f.

.-.

,

. in

(f

, q

)

(compare (2.1 .1) ) . iv)

(ttj)r,.

din

= dij

-)

By iil)

= dinr.

we have also dii I_---..-.--'_choosen byl ff = dlrrk ri

( jMi ). The fr,

1 and

dtj

get

we

i €

= dij

tj

f

can be

= drro*

fi

= dr*x jt;,- l, j € r. A K-realizatlon of

( l,

let

in

d ) the

together

valued

such

1 , .. . ,n

A representation is

a set

together

of

or

finite

with

there

is

Also Lre omit xl a

(I

is

a modulation

ori.entation

, Q ) and let

module

=

X

dirnensional

f

sequence (xi,ro,

right

)

M

n

0 ) be a K-realizati.on

an admissible

Fr-Iinear

jgi

(If

graph

, d )

an admissible

M ,

(

is

(f

e ) of

with

folloning

connected that

M,

(

of

. Thus a

= {1,...,n}

(i€

ordered

sinks

of f)

F.-spaces

(1 .3.1 ) . (

[],

i

€ f,

of Xi,

e )

mappings

t xi "",

*

ttj

*j

for

all

arrows

i

.4.

-J

no arrovr from i to in the following

j F'

(i, in

j € |

) we set

the tensor

irj product:

i.

= O.

iMj. ) X = (Xi,

For two representatlons morphism mapPings oi

( cr)

rMj

: I

: X. + Yr,

*

I

i € f,

and y = (y' jqi ) . ier) is a set of F, - Linear

such

that

the following

diagran

-22comnuces.

oiol ia

M

io

iMj

Il

I

II I

II

.o, l1

II

.tt.

I'r

I

I ,

{/

*j "j

oj

Vte denote

F, = (X1,ior) be the

Fori€flet defined j,

=Fr,Xr=Ofor

xi

by

of

representation

(M,

0 )

rsrth

the

andlqi=Oforall

i+j€f

k€ f.

just

morphisms +L ILI Ii J^

^E Vr s' ^t I' r '

^hiFcf

up

unique

objects.

Proof:

the

is

algebra

follow

as wefl.

Let

the

in

Also unless If I

to

easy

category

Artj-n

let

fr,

lenflth

finite

over

K,

the

t (

€ f

otherwise = 1 , we will

be

stated. also

exclude

iS

Fi,

i€

f,

the

alvrays all this

the

) over

equivafent

is

a certain

proposj-tion

category as dim-

just Fi

in

the

will

defined. (i€

becomes trivial, case

But

directLy-

modules

this

Because

In

indecomposable

l(M,0

that

M , Q )

chosen

ohieet v v J v v

sum if

proposition

from

L ( M , Q) .

objects.

generated ano

p r vr r er Jr v e

anri

sirnple objects

(AppendixA), of

together

category

q r r u

a direct

injective

prove

follo\ring i

=.t

lft

resp.

see later

wewil.l to

Q) has

projective

It

finife

j-somorphisms

to

t(M,

n

and

häs

v u J e v u

an abelian

build

defined

( rll , Q )

of

The rePresentatJ-ons

2.'l .2 Proposition:

-

Y by Hom (X, Y).

set of morphisms X +

the

f) if

following

_23_

2.1.3

Proposition:

In

a)

If

k € f

b)

If

k € l. is

particul-ar

Proof:

of

ßi = O for

then

(cr):

be an epimorphism Xk.

a source,

F*

f beasink,o= in

Bk : Fk -

then

and Fn is

be an epirnorphism

by

a sink,

projective,

F1 is

a) Letk€

is

f

(

M, O ).

vector ß =

Let

k+ l€f

Then

Then for

-

: It j

i,

F*

is

X

-Ef

ok

: Xk r

(>r= (Xj-,jei)),

€ f

, the

Fk

of

defined

I

injective.

injective.

splits

spaces,which

( Br)

projectlve.

is

must

course,

say

by

diagram

81 a1 IF -K'

nitj

\ I

x.

o

I

I I

il

M.

I

.(r. Ir-

lI o I

I I

$ (I x)j

€nmutes: are

This arrotrs

only

diagram then

trlvial,

to

i +k

and

therefore

k,

if

that in

i+

k +j.

the

j- = k.

case

1 = O, and the

ß, O

splitsr

( U,

L

and F* is

Because k is = O for

means jqk

a morphism in o

x. l

commutes aLso

Hence B is

b)

is

'1

Final_ly,

diagram

0 ).

all

j

a sink, € f

if

j

and the = 1,

commutes in

Of course

there

this

car_c.

Sg = ig

Ik,

Fk is

always

projective.

SirnlLar

Remark:1.)

The proof

a submodule of

X =

of (Xr,

(2.1.3) .q.)€

shows also, L(M,n),i.f

that k€f

isasink

JL

and xk k €| 2.)

is

+ (O) . Sijnilar a source

Simil-ar

Fk is

and Xn

arguments

always

a homomorphic

image of

X ,

+(o).

as t h e s e

Qne used in

(2.1.31, show that

it

-24-

k j.s a sink

respectivöIy

respectively

injective

Themap trith

= din

of

X,

0)

into

therefore the

OP

Let

ao

for

x =

all

arrows

Of course, but

5 i€r

(xr),

1

I,

I

€

the

slgE&l

(9im

X),

is

just

the

isomorphic

to

F,

in

can be considered

=

o BI""

y, ^

(Vr)€

qf (f

in

depends

L( [{,

e! taim X,dim

Y) ,'

respectively

Qt

-

0),

again

(with

Q^

chosen 8"0

xi

to

choice

of

f.

= dim*

( x,

Y )

Otp

Yi

sum is

respect

on the

Y) und

fi

second

, d ) with

we write

B (x,

d..

E i.-r.j

, where

we have

that

of

).

by

x,

f;

as map from

form

bilinear

number

a composition

L ( M,0

group of

crothendieck

i.--t.j

remember,

x

dim

0 ) defined

o Br""(x,1) = ^

called

is

0)

x:. =

(non-slrmmetric)

Br'" be the

.."n""a

of

projective

Fk is

if

QloefiniedbydimX=(xi)

L( M,

e

course

factors

of

L ( M,

x

0) . Of

composition

series

rf

(xr)o , , ri

L ( M,

map of

*

l(M,0)

dim:

x. .

a source,

taken

over

0 the

F,

fi,

for

i€

f

i- € f

instead B (dim X, dim Y)

t"r

(dim Y ).

!-

2.1.4i

Proposition: 'o1 (x, Y) iB-'" lrAProof:

Let

=

dittt j X =

Y € L( M

X,

Let

Hom(X,Y

(Xi

,

igi),

rJIJL

Then define Ä*,y t

.9.

aci

)

, -

Y

0) .

Then (x,

di-ttL, ext =

(Yi

,iüi).

(Xr,Yr)" o * " .t

tet

-a =

I i€f

o","

cr. € e i€l

(cr) = ö=

HomF. (Xi,Yi) r

,.1.r

:6i

with

€

,*t O iMj,Yj)

o i't'j

as follows: If

Y).

cj '

" o * " .) € HomF ^i

,.3.j ""*rj

(Xi,

Yi)

(xi e il'{j,Yi)

_25_ with j6i = jtr Now t=

( c. o 1) - oj jr,

oi € ker o","

,ä,

Letö= an

E iI ö i '€ i.-.i exact sequence E

6 + (Y with to

,,

O -.j ( ö ) by i.

canonical that

every

extension

and U and e

are morphisms in

Now E([

) and E( 6'),

ir,{., "j). is ln Hom (x, y).

(X.e.M.,y.).Define ----. r l' l''

is

they if

!

is

(

+

\Yio

6' =

term

xi'

€ . It

is

E

-(,

@ X1r

ri:YiOx.+YiExi

" (lll,:;l

rdith

Because in & ttre squares

by such t (,U , 0)

* ö r .L a r e e q u i v a l e n t ,

same eläment

in

Extl (x,

y),

if

diagram

('l: :l: ) + rr 1= (ri)

(irr,,u'i)\ \o tai //

-X

can be written

.{ä'l

in

easv

J

j

J

il

given

is

o

t ( M,A )).

a commutative

\-

tl I

the middle

defi-ne t h e

there

( Xr ,,ar ) -

y by X is

i.-.

and only

O-

HomF tj

of

(Obviously

@

ana onty ot i

.U and projection

injection

a sequence.

O-

"o\j

, j v i ) - I r - (" , ' *r, (3*o ;;i l )-

see,

(that

(xl o

a

x+ il tl :

+

in matrix

form:

'' 1( 1i )

Yi-Yi

(i) "' 1 2

"'i

-' 2( 1i )

Yi._xi

(i) "' 2 2

: X 1. + X .

conmute,

it

follovrs that

-Yi

1

-26-

?r.l\

lt

t- t = \-o |

be a morphism

Now r must the

' ' i 7' X i - Y i

i)

diagram

following

(Y. o Xi

- - ))

l-

o

L ( M ,

in

means,

Q) that

that

colunutes :

(Y. O xi)

iM.,

irj

"

('o

tl

tt:,

I

tl

\o (Yi e

M i ".i

o

)e(xt

(Yi e

iMj)

I,

II ti,*' o i\

t.

l

,tr)

\(r, jui\

:*t l

\

o x. /r

nontriv

jVi

( ?:- o

are

equivalent,

This

If

is

term

iai

1) +

=

Extl

r t.-.1 (x,Y):

nov, dim X = 5 = (xr),

x

I

-i " i l I o.e,/

-tt' 1/\

/\/\"l th respect

to

n.

is

and only

uch that

proves

,,,)

"

I ' i l \l i y/ i, ' .

.' t/ _ - \to

=j6i*?j

j6i if

-q- _q'

\rt/\ \ /. I l'

i.+.jinfwi

ri:Xi-Yi" that

)

\

,',/\" .l '

[ , V^, that means: l' \o \

The only

?'

j I tl j -^ , i\l l/,Ä ;' i \l I', \o

arrows

iMj

Y . O ) {j l-

l

for all

e (xi o

iMj)

.to.. Thus, E(6) and E(ö')

t1

if

t here exists

a family

?= (?i) ,

o 1) -?r't'or, :*r(ri (.6.-.6: ) =a.,(i),i= tji. r, j6i

-

j6

i

=

I-

i€l

coker \ -x,y,'

dim Y = a = (V1), then the K-dimensicn

- zt of

O Hom* (X., yi) eguals whereas the ti : ,I- fi *i yi, i€f i€f K-dimension of O Homt,. ( x r o Yr) is given by -i rMj, i€ r E i'--'j of

dij

-.

the

fi,

f1 xi i

€f

( here we need the

Yi,

special

choice

).

Now we have y)

Bao {x,

= climx ( domain of

= din*

(ker

= dim*

and the

2.1.5

proposition

Corollary:

Let

i,

A

Hom ( X , Y )

proved.

j€

f

i+

,

-

)

X,Y

is

on,"

j.

-

)

dim^

dim*

-

(range

dtux

(coker

uxtl

o","

of

)

1x y1

Then

1 t'l

i fl-1 I t

url =

dimExr'( F'

di:. =

dim". I

Proof:

This

o Bt"" (Fi,

Ei)

a) X

Then

(s*

b) has

= -

Let

F.i) '

from

drjfj

Hom (F.,

2.'l .6 Corollarv: that

Ext, (Fi, -

follows

because

E-i . . )1u .

.)

(2.1.4)

and

= -

fi

dii

= -

f be a sink

k€

has no direct

and X €

l(

summands isomorphic

i , l, e

)

such

ao Ek.

(dirn x

= dim",- Extl (X, Ef ) )) * f be a source e ) a L( M, ""ä a no direct summand.s isomorphic ao If. (dim X ))u

o

= dim"

Extl

such,

that

( -Fl k , X )

"k

Proof:

a)

Fi),

F'r) = O.

r,et k€

Then (sk

(Fi,

dim* Extl

By

(2.1.3)

Fk is

projective

slmple.

Therefore

x

o*,")

-28H o m ( x | F ' -^ -)=o,becauseXhasnodirectsumrnandsisomorphic to

So we get

F-.

A-I

X = x =

dirn

fk *k -

{x, rO) =

Br"

with

(x,) :

= -fn

diL fk *i

E

(-xn

:

1.r.1 = -f,-

(s,-

^ ^ ^ ^ - ^( d i m

x

= -

)),-

E

*i)

.dit

I.+.K

(X , F,-) and the

dim,- Extl

resuft

follows. b)

Sirnilar

Remark:

If

(dim x then

of

k €f

is

a sink,

) ,*- = d i m o .k

then

H o m ( -! [u ,

x),

course and if

= di^f,_

(dfun X)k

course

of

k€

Hom (X,

|

is

a source, (see also

F*),

(2.1 .3) ) . The following

is

simplicity,

b-"ß.

in

abelian

we formulate

(M,0)

X€ t

valid

is

called

it

for

categories, the

a component,

category

if

for

but,

End X is

L(M,O). a dj-vision

r r-ng .

v.-\L1o*J

Tv,ro components

X,Y

L( M ,

€

0 ) are

incongruous,.

called

if

Hom(x,Y)=Hom(Y,x)=o. Let

XqL

X€

t( M,0

( lvl,0 ) be a set

asequence such

2.'1.7

that

Theorem:

is

)

9=Io

Let

all

X < t

The simple

Proof: Let

of

Y€

X

for

L(

gIa_f

have

an

j-n L 1 X )

L ( X ) X )

and

is q

there

is

Io€L(^l,n),

components, L ( X

X-filtration, closed are

closed : X

if

SId=I,

of

subcategory

e ),

incongruous.

1< q< d.

all

full

M,

mutually

X-filtration,

be a set

which

objects

Obviously x,

c

(M,n)

L (

components,

have an

cx1

Then the

objects,

subcategory

to

X-ly 6 : ::o- 1

incongruous. of

said

of

-

with just

under t

L( M,O )

) of is

respect the

mutually

a ful-I to

elements

extensions.

be a morphi.srn.

exact

extension.;. of

X

_29_ Let

o=ä.Ir

c

X -flltrat1ons on d that tbe

the

is

o. Let

we get Yi = tor

1 5i5

i

Y

Let

of

If

and yi,/y. -:i-1

because Xl

-

S :i_t.

belong

to

X

.

= o(x,1) + yi_t

settins "j i

O=I;

+ 1 <j:n =

Ii

we ger

c

the morphism induced

!/y2

9!m_tcy-=y

by q. Then, by

- l

t( X ) . Because im q is

\de get

!n a€

an extension

r( X ) . Because coker ä = coker

X., c ker q

we have by in induction ker q €

then,

as above, we can assume q(Xl)

Äf = !f , and c restrj.cted

to xl

is

contained

in

X.,f1ker q = (O) . But ker s = ker är{ä

t(X). = y.,, hence

E n d ( X 1) , as above).

we ge: again by induction ker o € t(X). It in

is

now obvious, l(X).

q,

coker q € l( X ) .

ker q >11,

therefore

o(xt)

a(X.) = Y:

in ä by Ir,

Now if

we can assume, that

canonlcar projection " rrr,uI r _ t homomorphlsm from 11 to therefore lrl!r_1,

i:n ä e

we get also

For d = o

( with

and Yj = Y, for

- l

induction,

Syn=I

by induction

q (x1) 5 !1 ,

such that

a (Ät) o !r_t.

witir

g , I/r.

S .,.

x -firtratlon.

has an

9

anotherX-filtration of

I

prove

By induction

1s 15 n

_r ". 1s a nonzero )

an lsomorphism,

o =%SIr

We nill

a also

trlvial.

j-

Then _ x ,r

X resp.

image of

state$ent

q (It)+

!

of

=Iand

cä

...

that

X is

just

the set of

simple objects

arrd

-30Let X€

2.1.8 Proposition: L(x) : =

L( M,n ) be a conponent, and ( 2 . 1 . 1 1 ) . S u p p o s eK t o b e

) as in

L( {I}

Then

lnfinite.

, =

Qr(x) = Q" (dim t

> O

r

L(X)

is

semisimple

= O

€

t (!)

is

uniserial

\;

1^

< o

r

t,

It

Proof:

if

object,

must

@

(!)

then

on both

sides.

if

in

true length

full tt

-!.

E" + E^lq tz

nonisomorphic

with

is

x)

is

The Coxeter

(Xr,

respect

(X,X),

Extl

lengths

2

a subcategory'

forn of

not

finite tYPe.

an End(X)-binodule.

(X,

This

x)

-having

is

E1,

over

({)

E2 are

of

End

€ End

(x)

(x)

}

two

is

indecomposable

all

on

the

objects of

over

be uniserial.

to

set of

This

Ext'(X,X) is

also

L (X)

of X by

is

with

X linearly

then

left,

a infinite objects

dimensions

the

(X, X),then

extensions

say

If

simple

one dimensional

some m > 1.

for

subcategory

11

one

only

L (x)

implies

End(X)-binodule

the

semisimple.

is

Extl

End

over

j-ndependent

LetX=

(x,

End (x)< dimK Ext'

dr\

the

-

(2.1.7\

equal,

are

rn-dimensional

S 2

l( X )

of

representation

B (X)

and

be zero,

narnely X

FinaIIy,

{s

in which

Ext'

(x) > dirn*

dirnK End

Ext'(XrX)

End

that

objects

, ef (x) = Br0(x, x) = di^x End(!)- dim* Extl (x, x).

Hence,

in

known,

the

(2.1 .4)

No\"r, by

I

nell

is

and

non-serhisimple

of

length

nutually 2 in

t (x) -

functors

(f , d) i.---+.jbeanarrowin € t( rll , 0)and rOr) to Q . By the definj-ti-on of ,t{ we have jMi = ,"i =

- 31 H o m E , ( i M i , F i ) . B y trne he w werr e l l kKnown n o w n iisomorphism s (rtjrxj) g no\. _j , = xj qFj Ho*Fj (rtj'"j) ! x, e, u, and rhe adjotni.ness of rhe tensor product and the Hom-functor we get , gi Mj, Xj)g "orr.(Xi ! Hom". (Xr, ttorn,. (iMi, Xi)) : HorrF 4r Er(*i'*jQrMr)canonicalrY' r r I Thus,

for

attach

every

Fr-linear

canonically

and conversly an unique

a Fr-linear

to an Fi-linear

Fr-linear

map

6 =.0 .na F = ,f,. Especially .+*j

, *i-

j6j

map tp : xi

tr.

oi

Mj - xj

nap 6' : X. + V, Xi .+

map

ü , Xf ei for

we can *j

Aj ,i,

Uj tj corresponds "j X' such that

.+ "j , Xi Ei Mj ._xj

,o, Thj,s notation

wlll

"j

tire qet

be used throughout

paper.

this

Let k € | be a sink, X = (Xr, € L( M, n ). Then define ior) + = st I tYi, t ( rq, sn ) as follows: iVg)€ " First, if i+ k +j, i, j€f , then we set y. = X. and jvi

=

jei.

Yn is

ilefined

o i€f

as the kernel

xi 6i LL J r ^

enbedding

E j€f

of

the Fu-Iinear

.o. ^ I

, xk

i-n

O Mk. Let j€f "j "j by the projection tto.

U

u followed

with

j{r<

=

xj

.fl irr.). Then define l€f' " -t tt tj Yj = X, for "k

jix

an arrow in arrow in

If

(sj)

( f,

=a

(f

,d)

with

respect

to fl

d ) $rith respect to su e

: X

-Ir

= (Xi,

iai)

natural

t yk .+Xl

1rk rp,

(Thus p =

map

tt

"I onto xl

k:+ j€f

(rf

"j

, then

k.-J.j

Mk ol

be \,

j . --r. k is

l€f

is

an

(see 1.3)).

is amorphism in

L(.M ,0),

-32define and (sl

s)

is

*

u

.

tl ':?, *t

** - -

"

becomes a functor

Zi = Xi

=jei

, jni

For zk hre take the cokernel TI i€r

.O. I rK

let -r?,

xI s t{k . set

=

kti

Ifo:

nn,

.2.2.1

Lenma:

+

j€f

o'

by

followed

n (r =

,n*

,ä,

(

Zi = t Si x')t

-

,

"ru

i€f

q 1. In this was sf, .!_ oi i€f !6 l(M,s*0). rt is easy to see, ttrat sl right

11g,sno).

the diagram

Mk

"i

i,

jti ) isagainamorPhismin "i, then define sf S = ß = (ß1) by ß, = c, for

and ßn : Zn

and sl

i+ k+ j,

injection

for

[+I'=

q = (si)r

tr

= (zr'jni)€t1rtr,s*a)

= !

be the natural Xi

kni

in

for

uk gzk

xk .--.....-"..-.-_r3,xi ti

For I€f

k +i€f

r ?, "i "j

L ( M, n)into

from

Let now k be a source. Then define s; I as follows,

for

):lr*irj*x.

way s[

rn this

o )r= o,

tsl

of

the restriction

j ?.(oj " royk

i.e.

by restriction,

tig

L(

M, Q ),

i+ k

to be the map induced by

also becomes a functor

and sf,

are additive

from l(M,O)

and sf, is

left

exact.

a) Let k€f

Tiren canonically sum of copies

be a sink. Let o + X € 8( M, x = a* ti I

o1 If.

e !,

More exactly,

0 ).

where P is a finite' if

either I = Ex (i.e. s; ti I = o) or x:

x is

direct

indecomPosable'

then

s* sl x+o,(i.e. si x+o,

- 33 -

and lt

indecomposable).In

(sf x) = nnd (!)

rnd b) If

the latter

case also

x) = sk (di.,n x ) .

and dilu (si

k is a source. we ger Ä = Si S; ö O f,

finite

direct

of a) will

Proof:

is

sum

of copi.es of F*.

The

j.s a

where.r

analogous statements

hold also.

a) It

is

definition

enough to consider

the k-th

component. By the

of Sl and Sl we have the fotlowing

(notation

as above):

o+(s;I)k

-J-

o

exact

dlagram

*j e jtr

j€f

Il '

7

.t

/

-+

(sk sk x )k

I

I

o with

o: =

we have ker o= ker t(= imu), kej. .,ta, there exisfs pk Xkhrith p1t = o. tS* Si I)k -

Let a € fs* si { )* wirh ek a = o and rer o.:?, with

t(b)

define

then p = ( or) Now, if that

=3= q. 5o = . gi

tU", r

ti

ti

X +

o is not surjective,

I=LOP,

r,rherey=

for k +i € f , P€ L (

": " :\

= a. Then o(b) = pkr (b) =pk (a) = o, hence

b€ im U=11sr ,ire.t(b) k + i€r

so

t xi

= (si

X is

a monomorphism.

then it (yi,

Yk = i*o,

M ,e ) is a finite

pk is monomorphj-c.For

direct

=

-+

Xi,

can be seen as in

jti)€

jüi

s; I)i

jQi

t( (i,

(2..l.3)

r 1 ,{ A ) w i t h y i = X i j€ 1), and

surn of copies of Fn, in

34 -

particular

projective.

is

, then

-K

indecornposable,

K

then

of

composition

=o.

o=oandslsln. o

has

any

two

maps

be surjective

is bijective,

is

I

Finalfy,

ls

rhus trre

+

ena(si s* si lt Iil snd (S-x)

e n c l( x )

ti

-K

in

K' E"d1s+x1 -5- nnd(si si ll-5 Iil

is a Iocal

hence

and

s[s; sf, I.

{l:

qcq

Endx

rfx+F.

K-K

to

x.rhen we have arso

ti

l:ti

p= Y,

im

op.

x = y o p d t s f s l^ xK -) I f x =_ F .

By construction

End x !

therefore ring,

nrrd (s+ x ) . Nott End (x)

and so the same is

true

for

nnd tsl

and

!)

indecomposabre. the exact

sequence

.----_ x,- x )t ---------.9_ (xi. a.r Mn)r ^ tk K

o -tsf

o

i€f

yields

dim

the

( ( s i -I ^

formula

) k ) F' k. ^

=

L i€f

=

_

dim

dim

(x.

(Xr.)r.

.- -k

Ei

+

Mu)^ !',.

E

i€r

ä di in^

ilim

(xk)F. " 'k

(I i "Mx )\ r i . . roi n*

=

,(!x i ) F i

(sn ($m I ))k, h e n c ed i m ( s i I ) = s k d i m ( x ) . b)

The proof

will

be analogous

/l

to

a)

considering

the

diagran

/xx

/l o -(si

t/+ s; I )n -r:r*i

Ei Mk +

(s; I )k -

o.

-35-

Remark:

It

is

easy to

indecomposable

prove

brith

as above,

that

Stx + O + Si y

for

there

y€

I,

[ ( M , e )

holds

Hom (X, y) = Hom (StX, Sry).

2.2.2 Lernna:

a) Let k€f

be a sink,

X, Xi €

indecornposable, and assume Sf, X, +1O1 Then S[

induces

sl

X, ? t*1 .

extltsl x, si x,).

be a source, X, I,€

indecomposabte, and assume s* I sf, induces

(i.e.

an isomorphisn

sf,: extl (X,x, ) b) Let k€f

L( M , e ) be

t(

M ,

+ o (i.e.

0 ) Ue I

* f*1.

Then

an isomorphisn

: extl

(x, X, 1 ....-..._

(s; I,

s;gl

s; I,

).

X = Ek , then X is projective by (2.2.1), so Extl (x,x')=(o)= Exrl (s+x,ri",), because ,i I = o by (z.2.11. so let s[ 1 +o ). let

Proof: a)(If

Ue O-Xt

irith

U =

+E

(Ur),

Obviously,

e =

rde need

(e.)be only

r

+O

X

an extension

to

verify

that

of the

X by X,. k_th

component

of

'\a,r; s l r ' ) * . " * " , * , ( s ;E ) n r)k tcl

o -+(

,.t

is exact.

Let x = (xi, jpi), I'=

Then

the

we get

following

conmutative

,^+

o

(xi , jei ) and E = (Errrtr). diagrarn

o

U

I

+

o

I (s; I)k -

(s;I')k e

O -'O

xi r

acl

I

U. q 1

siMr

: ä9 r- -*-j j

I

I I

I

.t

o-

xi I

I

and the

third

first

to

see

middle

that

exact.

Finally,

shows

ttrat

are

the the

middle

has to

tsf, e)*

and the

column of

the

Nol.t it

uPPer

row

b)

with

f

the

the

similar

diagram

right

= t1,...,n of

sequence

dimensions

involved

Extl(six,si

x'),

map.

exactness

of

the

sf, is

vtith

resPect

L(Il

c-:=s]sl

...s":

L(M,0)-L(

functors.

and C-

sequence particular

Simj.lar,

do notdepend with

on the

respect

to

of

the

choice

as

1,...,n (1.3.1).

(1.3)

specj.al

Q . obviously, bimodule

Sf,

and

is

an admissible

Then the

functors

Il,o)

,Q)+L(

in

of

proved.

easily

e

to

...si:

Coxeter

exactness

left

ordered such that

)

sinks

above,

c+,=tltl-.'

C-

are

Stunilar

Remark:

Let

is

be surjective.

an inverse

S-* induces

are

).

rn this way, sf, inducesa mapfrom Extl(x, X') and obviously

The

exact.

X and X'

since

(see 2.2.1)

X

caLculation

are

rows

exact,

+ 9 n s;

I

+o

o

Iower

and the

columns

and si

indecomposable easy

II I

n

the

o

I

.t,

t

-

ojtj

J x,x

By assumption,

o

M,Q) it

can be

choice they

mappings

of

arecalled

seen.

that

an admissible

depend

on a

-37-

Itonr, (iMj, Fl) =

jni:

hosever, '

(itj,

the

cholce

For .l

theory.

:t_<

L( g, sa...

of

these

n , I*

is

sn a ) = [(

s!.mple module in

= t(

sa...s., 0 ).

M,

implies

sa...srr0

"t_l O= st

... s' "t_1 (1.3))r(t-1)€ r is a sink,

Dr_t rnen sr_2 etc.and a module in

'(

SturilarIt,

= s;

This notatl_on will following

c woll

c = sn...sl

b)

...

, = s;

O ) "r, = sn ...

denote the Coxeter

L(M,

sr,0) =

this

sl_r

It, g( M , n ).

B( M, 0 ).

paper.

In the

transformation

O ) beindecomposable.

statements

X 3 P* r-or some 1< t< n.

iii)

c+X=o.

lv)

c ( d t u nI

Dually X is

t{

some 't< t
c-I=9. -1

equivalent

injective for

are equivalent:

o.

rde get the following

X = It

(dim x) I o.

"t0

e = c-1fl

...

is a nodule in

ii)

lv) c

!t

st_1 st

X is projective.

iii)

...

e ) . rn st ... srrn (Uy "1 therefore we can apply first

,i)

if)

=t Q) and an

s1 0 and sirnilar

be used throughout

LetX6

the

...

r1*r I.

Then the following

i)

...

€Stl.

2.2.3proposition: a)

...

for

simple module in

"t_2 L ( M, sa*'

get finalty

l/, s1 ... ...

a projective

but

irrelevant

"t+2 (Notlce, that 0 = c e

=

implles

mappings ls

M , st_1

lnieccive

( 2.1.t),

rjl

"o*,

statements:

-

-

Jö

We prove b):

Proof:

-

ii)

iii)

= s:

t I

...

c x = s;.-. ."ri ..... + ii)

iii1

r s;

t"

...

tl*,

"i*,

imPlies

I.

= o implies that there is a 11 t1 n

I

... t" I = sl I = o.

with sl*, ... s" x = y +9 uut sl ti*, Hence by iii)r r

ii):

they

are

i!

I.a ,

11 t

annihilated

by

All

indecomposable

by

vre get

modules,

hetrce all

...

11++-1

Extr(x',rr)

-

...

Q.2.2) 1++

we sho\rl, that

If

inlective

Ea.

because

non isomorphlc,

functors.

different

are

AIso

they

this

modules

are

injective

indecomposable

indecomposable

L

s;

(

M,

+ o.

I'

nonisomorphic

Sn

-X ' =

First

Q ) be indecomposable. Then we get

= Extl(I', sl ...ri*, is lnjective in

b e c a u s eI a tft

pairwise

n nonisomorphic

n and X'€

assurne s!*'

S:

are

ti*r

...

(2.1.21 .

modules by 1 sts


Q.2.1).

injective,

Let

X = si

= Et , and therefore " Obvious by (2.2.1).

12.2.1,

(iv):

that

l. = ri ... s. Et = 9 by (2.2.1).

(2.2.2)

"t+l...sr,

for

and

(2.2-'l);

nxtl (s.*r...s; I',Fa)=o

r.)!

L ( M,

O . t h e n X = -I^, -

by

0 ). If

some n> k >t.

Again by

and (2.2.'l ) we get

E x t r ( t ; . . . s l * r l r . , s i . . . t l * r E . ) ! e * t 1( E r . s, i . . . s l * . , r . ) , Er., ti

...t1*,

Ft€ [ (M,s1+1...sn Q). rr

source. of course, 11 is not a direct

"k*1...s. s u m m a n do f s l

(2.2.i\ rherefore by (2.1 .6 b) ) din

r*+1 re

(x',

2.2.4

Proposition:

"*

...s|*f

It)

= o and rt

Let

X€

_ I

a)X=CC'xeP,whereP€

L (

is

injective.

M , 0 ) . Then

L

(M,Q)isprojective.

Ea

sl+1 !r)

s ; + 1 E r ) r=* r " n - f 1 . : = . . ; ; - . ; - " . ; ; - =

= ( s k ( d t usn; . . . Hence Extl

Q,k is a

"

=

-

Thus,

lf

1< t5i n,

39 -

X ls

lndecomposahle,

i.e.

c* I

1 < t
Proof:

2.2.5

either

i.e.

C- X = O, or I

= C* C

Ext'

Proof:

a)

( c

b) If

If

k €f

=

a sink,

t*

(dirn X ) .

F* as bimodule.

enough to prove a).

choice

of

Because C- döesnrt

an admissible

(2.2.4).

Also C- lf

(2.2.3),

and. it

because 1 € f By (2.1.6)

is is

sequence with

k = 1. Clearfy

C

* O, otherrrise

refrections

of

in

f

sequence for

,It

respect

f.,,

{

depend to

0

oth.rwise

( 2.2.3)

and

F1 vrould. be injective this

is

impossible,

>1 by general

assurnptlon.

E x t l ( C - F 1, F 1 ) = ( s 1 ( d i m C - F 1 ) ) = 1 ("2 ..."r, ( gjr F1))., = 1, s; F1))1 =

d\,

because the a vector

and lll

El

= O by

easy to see, that

a sink

( dtun (s;...

O

X, and then

then

we woul.d have F1 = C' C= C* E1 Ir

exact

some

r* t* r* as bimodure.

,*

we can assume, that

(sl

for

It

is a source, then

is

on the

=

I

(C- X) = c-1

is

=

(Fk, ctk)

It

) tsinjectlve.

(2.2.1'!.

Et'Et'

k€f

1a

l v l, 0

indeconposable,

Lemma: 1-

(

t

some

(dirn x )

X is

Inunediate by

Ext'

€

= End X and dir

End C- I

I

) = c

(c+x

b) X=C'C-XOf,whereI if

=

!a for g = o, or x c-c+ x,rand then

End C+ )i = End x and dim

Thus,

either

for

s,

donrt

2< L< n.

some E €

+E

L(

change the

first

component

Thus \"re have a non split l/,

0 ):

+ "-1Er

q.

-40and we get an exact Hom (E, It)

sequence of F.-bimodules.

-

Hom (F1, Et ) -

Extl

But Hom (E, 11) = o, because the extension

split

doesn't 't

pro jective.

and

Fl

is

and

by

dimension

this

-'r

r Ext'(

C

Et,

)

Et

=

said

Hom(F1 , F1 )

must -F' .

be an

I'

c-4

Ext'

(C

- 1' F l

, It )

Thus

isomorphism.

as bimodule.

Er

L ( M,

Let x € X is

Thus

F1).

,q- Ef,

0 ) indecomposable. Then

to be

a) präprojectir.'e s o m er > o

(indecomposable),

if

x = c-r

Pr for

x = Cr I*

for

and 1
b) preinjective

(indecomposable) , if

some r> O and 1< t< n.

a) If

2.2.5 Proposition: 1< t< n, and C

1 (

M, n )l

c+ x+ o,

of

FaFt

F*

x = c+tJt,

as bimodule r>o,

13t

5n, and

then

nxtl

This

Proof:

= C-t 31, 12 or

X + O, then

extl ( c- x, x) ! b) rf

( M,Q )l I

L

(X, c+ x ) =

statements

Ft F. r .c c

as bimodule.

are straightforvtard

consequences

(2.2.3) and (2.2.5).

Let r:'O,

t (

M,

0) ) x = c-r

Pa (or

1 1t< n. Then the position

by : pos ({)

= n.r

+ t.

If

x is

cr ra ) with pos (X) is

a fi.nite

defined

direct

-41

of preprojective

sum

nodules

(or prelnjectl_ve)

[ ( lr,l, 0 ),

Xt €

preprojectlve

-

(resp.

then X will

preinjective)

indeconposable be called

also

and pos X is deflned

by pos (x) = max pos (Ii)

2.2.7LeIuta:

a) LetX,

Y€

Then Y ls preprojective €

I

of position

L ( M , A ), I

preinjecitve

of positlon

a) First

and Horn (!,

let

< p.

preinjective

of posltion

p,

< p.

X be indecomposable,

o< t
X)+ O.

and Hom (X, y) +O. Then y is

Y indecomposable,

Proof:

e ) andXbepreprojective

p. Let y be indecomposable,

of position

b) Let I,

L( M,

X = a-tgt, Hom (c*ry,

p*)

c+r y +o .

N o w -PE. = s :

- .. St-.! Ig, thus we get O+ Hom (Y, X) = , n o r n ( s - a _ . , . . .s ; c*t !, Fr), in [ ( M, sa...sr, 0 ) (in which Fr is projective

simpte) , if

But then every homomorphism is

si_r

...

an epimorphism and splits,

hence - by the indecomposability

of y - is

and Y I

= o, then ir,s

that

x.

rf

s;_l

...

S;

c*t I

Y has to be preprojectlve

For arbitrary

preprojective

S; c+ry + o.

of position

X the proof

an isomorphisn easy ro see,

< pos (!).

is

now obvious.

b) Analogous

2.2.8

Corollary:

a) Every subrepresentation

representation

in

b) Every quotient is

t

( M,

e ) is projective.

of an injective

ln jectj-ve.

Proof:

of a projective

Obvious by (2.2.7)

representation

in

L( l.{,0)

-42-

2.2.9 Proposition: for

only

a) Let x €

a finite (M,

modules Y€L b)

Let

t(

X €

number

of

we prove

(2.2.4)

Let

c+qI+o

€lN

0 )

nonisomorhic

Then X is

t

a).

and Hom (X, Y)+ O onLy for indecomposable

clearl-y

X be not all

für

such that

q,

is

r

a finite L (M , Q ).

Y €

modules

a sink.

PreProjective. q>o.

Changing

€ IN, q >r

x to

be inde-

Then by

(2'2'3)

vtecansuPPose

Therefore

(dj.n c+qx)a+o

Coxetertransformation Let

indecomposable

preprojective.

for

there

exists

q>k'

all

and

1
Nor"

in L( M, st-l...s., Q ), in

o + c + c + 9 xi m p t i e s t l - . , . . . t i " * n l + o which

0 ) and Hon (Y, I)+

preinjective.

composable.

o
M,

0).

nonisomorphic

Then X is

Proof:

number of

Al,

t(

orientations

we may assulD.e that

t

= 1'

c-lr.,

:

c-tqt'

and assume that

the

and analogous

Then

t g = c'Et Z g- r9-r- t +

N

-ln-r-l)

b y ( 2 , 2 . 3 ) . T h i s s h o $ r st h a t "., g = c - * F t + o b u t C z = o f o r s o m em € I N ' B y ( 2 . 1 . 3 ) a n d +k+1 +k+1 o' Let o+ d € ( 2 . 2 . 1 ) H o m ( z , c ' - I ) ! u o m ( ! 1, c - ^ - ' x ) * + L +l Hom(z,c*^t' I). By (2.2.3)z is injective, hence also being injective c*k*lx must be itserf im a < c*k*1 I, "o indecomposabl-e. But c-c+k+'l x = c*kx *o, a contradiction ro (2.2.3). so{ lc-qr, I q t x} is a infinite

set of pair\tise

non

iso-

I nom(c-9p,'x) morphic modules. Furthermore o +Hom(Fl, c*9 x) by (2.1.3) and (2.2.1' for q:

k. This proves the Proposition'

-43-

2.2.'lO Proposltlon:

LetX€

a) Then the

following

X is preprojectlve.

li)

There is a g>O

indecomposable ! the

are equivalent:

C+q X = O.

such that

Horn (Y, X)+ O only

Dually

).

statements

i)

ili)

b)

l , l, 0

[(

for

(M,

€ t

following

a finite

number of

n ).

statements

are eguivalent:

1) X I-s preinjectj-ve. ii)

= o for

c-ql I

someo< g.

Hon (X, Y ) + O only

iii)

for

a finite

number of

inde-

conposable Ye L (M'0).

Proof: i)

ä) We can assume that r

ii):

iilr

i)

Let x = c-r

'

preprojective

are only

modules of ii)

i):

2.2.11 Proposition:

pt,

r:o,

= nr+ t = : p>O.

then Y is there

+ 1 will

Et for do it

some o< r,

by (2.2.3).

: Trivial

i) riii) pos (I)

x = c-r

By definition

n. Then q : = r

1 :ti

X is.indecomposable.

finitely

position

of

1
By (2.2.7),

if

H o m ( y , Ä )+ O ,

positj.on < p. Of course

rnany non isomorphic

< p in

preprojective

! ( i,{ ,n ) .

(2.2.9)

a) Let,

for

some r >O and 1< t< n

= o for all 1< s< n. 3t = O. Then C-(r+n-l't" "-r b) Let, for some r > O and 1
-44r+h-1

C- "

'I_

= O for

_S

Proof:

all

= (t;

...

exists

11 s< n be a neighbour of t

= (si

then (!")t

-f

such that x = C-(r-1)pa+ o, but

a) Choose r>o

C-I = q. Let

in

11 s 1n.

...s;_1ls)r+

tr.,t., ...

o, if

t"_., E")a+ o,

e€ {o,1}

in |

t> s, i.e.

if

, (c-ps)t=

t< s and

there

(C-egs)t+ O. By the remarks

such that

(2.1.3) and (2.2.1) Hom (par c-t!")+ o. Assume F - I' 3 " * o. with the remark in (2.2.'l) we get

r-1+ C '- "

t'!",+

Hom (x, c-(r-1* and therefore

o. NowX is i.njective (2.2.3\,

every epimorphic

every image of X in i.e.

e)n.=g-a-(r-1*t

indecomposabre. This implies a-(r+ by (2.2.3),

it*1)!"

thus in every case c

proceed with

hence

must be injective, "-(r-1* *,r"a be j.njective, because it is

'!"

C-(r-1+e

image of X (2.2.8),

t'E"

s instead

This shows finally

t considering

c*(r+n-1't"

'r"

= o. Now

neighbours

= O for all

of

s in

1< s1n.

b) Similar.

2.2.12 Corollary: and J=

( f,

b)

P=

c)

statements

1
Jc [(

M,

Ä . ,{ 0

)cL( O )

)

. Then the

are equivalent.

J is

)

d)

t ni

e)

Pc J (resp.

are

< t
rtO,'l

d ) is Dynkin

P (resp.J

Proof :

P= tC-3rl r>O,

{C+r ltl

following a)

Let

linite

+ Q

wj-th

J cP )

(2.2-11')

equivalent,

irnplies

a*tf"

it

is

becaus" = o

for

easy to a-r

Bt

see, = o

b)

= o,

c)

(some r>

some 1< s< n and vj.ce

d) o,

and e) 1< t-< n)

versa.

f

.

-45_

11.2.6)

shows, that

e) r a)

: Let

(f,

( f

a)

implies

e) .

, d) not be Dynkln. Then, by (1.1.2),

d) contalns a subgraph ( f,,

Euclldlan

or one of

the

following

(dt2,dzt (r1 ,91 ) = 1t -

r)

(12, 92) =1.

ii)

d.tZ dZ.t = 3

)

'3

(d12' d21)

whlch is

graphs:

wlrh d1

zdz.t25

(d23,d32)

. ,

.,

wirh

d32 = 2

and dra

(d12'd,1).

(13, g:

rir)

d'),

1 =',.

(dzy

dgz).,

r.

with

dtZ dZt = 3=d23 d32

Let Q'be

the restriction

be the restrictlon L (M there

"

1s a full 0')

(*, (e x) . =1 ' '

e

exact +

L ( M ,0

is

true

0').

clearly

L ( l/',

We will

1gi.-.j

) rr , , . o . =G l'1

0')

e and consider

L( M', 0').

inf,

otherwise

/o L

and similar

for morphisms.

L ( M' ,

as subcategory

show, that

modure, which for

(M,,

and

) by setting

otherlvise ,

).

d,)

( Ä{,,0,)

ernbedding

fori€r'

a preprojective this

( f',

ft A ) to

L(l{,0

X = (Xr,ror)€

Thus we omit of

( 1,, d'),

to

Q',) the rnodulecategory of

e: L(M',

for

(

of

of 0

is

O' )

L (M , e) contains

not preinjective.

Then, by d),

Assume

no preprojective

- 46 (M',

[

module ln

(l

be the Coxeter element of

, d) with

1 is a sink with projective

respect L'( lll',

in

F1 is not preinjective

resPect

to

and'

O

we can assume u = 1. Thus

ay changing orientation

u € f'.

Let c = sn...s.l

is preinjective.

0')

and

to

0

0')

and

in

L ( M ',

and F1 is

Q',

L( l\{ , 0 ) . By assumPtion, by

therefore,

0'),

(2.2.1O) Hom (Et, !) + O for infinite many nonisomorphic modules Y.€ L(l'l ', Q'), hence the same is true in L( M, 0 ) and F., is not preinjective by (2.2.10).

Therefore

Thus hre can assume ( f, the projective

that

= Ct Ia

assumeIt

by (2.2.3).

(1.2.7)

"-r1 given

all

in

the tables

(f

For x = (x.,, xr)€

formula

(s-(r'+1){,)), r foraU.

{o

for

i = 1,2,3').

di),

the Euclidian

graphs, that orj-entation

d1). Let l € f be a sink, c-1 =s1s2.

O2 we get

it

1)x., - d'rax., dt2x1-x2). is

(c-r(t))r>

easy to prove by induction, o for all

r > o,

that

hence c-r !

r:o.

Assume ( f,

d) = ( t 2 ' 42) wittr

(d23' d32) = (2,1\. source.

t

in Appendix C) .

'(1) = (( dztdt2-

Vtith this

, d) = ( f i,

(with the special

rio

d) = ( ft,

Assume(f,

Thus

we can choose a special orientation

we have seen for for

show

is not preinjective.

is enough to prove c-r

(1.2.8)

{ o

we will

d').

some r> O, 1< t< n. Then c-(r+1)1< O

for

(of course also, if

In

c

d) = ( f',

module Fl

Hence it As in

r:O.

Q

is not contained in J also for

P

Q).

l(M,

all

L (M ,0 ) again

in

Then

1€l

(d12'

d21) = (3 ' 1) aPd

beasinkand3€f

a-1 = - 1 s, s, and

bea

/o

-47(x) = (2xt * *2

c

xr, 3x1 * *2 - x3,2x, - x3),

I = (x1 t x2t x3) € e3. now it .

(c

-1

this

x)3 1t

<

(c

of

easily

let

if

modules in

cases

. Call

is

to give

Let x,

fleld,

and ( M , e ) be

a connection

( f,

d) with X€

l(

M,9 )

Our purpose

between the additive

as dirnension types of

subject

between the corresponding crucial,

of argebraic

z et

groups. (for

lemma we need some For this

ex. [2O] or

sre refer tll).

( It{ ,0 ) be indecomposable prejective,

and dirn Z = a::n X + k. k € f

is a sink, o-IL-z

b) rf

<

similarly.

or preinjective.

For the first,

to the standard books in this

a) If

O<1
proved

( M, e ) and extensions

of

arguments from the theory

k € fr

are

From

a representation

considered

| ( M , O ).

2.3.1 Lemma:

that

connected graph

X is preprojective

of the roots

representation

induction,

of a valued

paragraph

structure

x3 <xr
K be an lnfinite

orientation

prelective,

if

modules and extensions

a K-realization

in this

by

The other

paragraph

adnissible

<2(c-'3)1

x),

roots.

S 3 Prejective

ls easy to show, that

-.1

follows

sequence

In this

-1'

k€ f

then there exists -X

an exact sequence +

is a source, then there exists O

-X-Z

o an exact sequenc,-

t Ek.+O

-48-

a)

Proof:

=

Letlfl

n and 1 €

(yi)€Dln.

variety

affine

-) , . . H o m " .( F r - o i M j , F j

n-.-Yi^Y;

v y = i.

Then every v = (rer) = i.l.j i.dentified

,

course

because grouP c =

.fl

of

ditnension

as variety

irreducible over

space

size

of

Fi

GL(Yit

Yi

acting

( direct

Fr)

GL( Yr,

so open

K' Let

infinite.

K is

over

can be

€uX

jei

tyPe

I.

K being

over

a

I

vector

full

t ( M r0)

i(pj)€

is

U

=

,iiläil.,.

the module

wlth

v (Fi

Of

the

consider

sets

ln

UI

are

be the full Y. and let on F '' Fr)

on

acting

Product)

dense' linear

'y

as follows:

(si)

r,wi for

= si

jVt)

.

in

Y = (Yr,

with

j

(ei s 1)-1r ( i'

an isomorphism between (FiYi, [ ( M , o ).

,or)€ ident j"f ied wl-th (

modules in

,,0,

(j{,,i)€ur

in l) '

Vy, g = (E.)€ G'

( jai) €

Then g defines (F'i,

=

'(jei)

i9r)

ancl

on the other hand

11 l'{, 0 ) },ith dim Y = I

can be

describes all U e Uy, "o I ,tor) of di:nension tyPe I' Furthermore' L( M , 0)

Y , x € L ( M . ; ,0 ) , 1 (dirn 4 = djrn !

= !)Fre

(Yi,

191),

isomorphic

= (xr,

x if

1 Ü i )'

and only

if

_49_

(ta.

(

U--and

)€

€Uv

i ü iJ) L

.x.

under the action

of

l1e in

the action

of

G and the

modules y €Lttt{ , 0) with Let fl

be the

extended

action

between the orbits

9f

isomorphi_sn classes =

dim I

algebraic

by base extension

of U v ! ,. , vr

G.

So we qfet an 1-1 correspondence under

the same o Y -rvb - Li t

X.

closure

Uv

of

I. of

K let

of

(, and G. Then [. LY of ö on 7..*, defined "."

tu,

E, defined

ana G and the over K.

F

Let

v € Uy

ö v is

be K-rational.

defined

open in

itrs

over K. It closure,

Then by [Humphreys, p. is

well

known that

hence ö v n U' is

öv

21gl

is

open therefore

dense in

U.,,if the closure of ö ,r-i" V,r. this happens, rof course, if dim ö v = dim 7rr. Furthermore if v = (iei) €q,, L JL )L then ö.v ftUu - G.v. This can be seen in the following way. Let

M(k,F)

i.r,ot"

enbed G in M =

the kxk-matrix n

i€r

I'{(yifi2,

of F. over K. Letn of

c in Ivl is

over

a field

K) by a regular

EK M and notj_ce that

the closure

space of

(which also

defin.

d r., M). For

jti

( m i o ' l) f o r a l l

€ ö.v

F and

representation

a subvector

polynomials, w = (;t.)

=?

ring

M defined

d

by linear

tet

JL

t

= {rn= (ri)e

then Norr i

the

üt m, jet=

transporter

Trans6

j,s a subvector

et €G. v implies subvector nt(]),..., the

v to v/ in ö is

by the equations

inf}, fn ö.

by the

linear

*j

=

j"i

,rlr(mro are contained in K. because w€U . Now i Transc (v,w) +9, hence tnö :.s a nontrivial

space of ü containing m(1)e fi ot tnö

defining

of

space of fi defined

polynomj.als determined whose coefficient

(v,w)

i .-.j

linear

- \ ,T r a n s F ( v , w )

, and a basis

can be choosen in M,

polynomials

because

of i n ö have coefficients

1),

,.

-50-

polynornlal in Transc

*

det(1., .(1)

in K. consi-der K

t11,...,

(v,w)+ 0,

thts

rrlcR

polynomial

+ ...+

det(t1mut rl t

t1*"' i.e.

ll l

trm''')+ al I

+ ...*

alrt''

. Because zero.

cannot be identically

so

o, becauseK is infinite. G = Trans"(vrw),

€ Trans6(v,w)n

w€G.v. E f. j.€f

NowdimG-dimU.,= L

e!

trl ]

[ 1 . ,, . . . ,

K such that

Therefore there are t1r...rtl€ , r\

+ l,rn(r) ) as

...

= Or (I) ,

t1,1)

so let

yr2-

(dimension as varleties)

Y = (Yrtror)€

and indecomposable.

and the definition

of prejectives

End I

Of course

Assume

Then by (2-2-1)

there

such

1o€l

is

= ft = Fi Iclentify Y with (iqi)€U and - o ' - o J r IQ . ( 1 ) the unit group of End (!) is nothing else the (Y ) of Y j.n G, and there C^ g-

centralizer of varieties

1f Y ind

orbit

)

.

dim Y = y.

with

L (M,O)

Y to be prejective

that

f.dliyqyi= ' rr

E i.-.j

"/c"tg)

+

is

an isomorphism

G 'Y, where G' Y

denotes the

Ur.

Hence dim

G/c"{Y) = dim G . Y as varieties,

i.e.

dim G. y = dirn c - dim co(Y) = dimK G -di\

Fi -o

- fr + dimU., - fi 'oL'o = dim u!, thus the closure hence dense in

of G

Y in

U" is

U". In Particular

t/,

and G

thj-s is

Y Is oPen

true

for

Y in

Ux

the groups also write in the V To distinguish z. following G = Gy. z,x Now write tn * Tr the natural projection -.=f'n'k e f'O and let z'-x n for + , Vz -Vx o. Define o rkk n : F*

and Z in

v = (ator) €U" Jr4)L

by o(v) = w = (iVi)€

U*, where

-

=

jti and in

f,(

" iMk we represent

if

with

respect

l'*xk

= F*xk n F*,

to bases of acting

f and j +k

in .-

Fkxk, for (ror)

(ie

from the

left,

to

.k

i.-

as family

the f.zi

the rrow corresponding

l)

of matrices

conFatible

with

then we

the direct

summand

Fkzk = Fkxk o F*).

Fk of Clearly

c is

epimorphism. q

-.j

i.

: F.xi

i.e.

delete

if

jei

=nipi

*ti

51 -

a morphism of varieties, Consider

is continuous,

in particular

the preirnage

o-1 {G* x;

GU z n o,

in

fact

(cx.

a-l

an

X)c [/r.

is open h"r,.u d.rr"e

-' -{ c *

I)

+ e,

L.e.

there

Va,

ln is

some

= trli, L ( M,o) fsornorphic to z such rhat jai); fu = (Fii', q ( is i.somorphic to x. jqi) ) € L( M , a) Of course we can assume I

= Io

B = (Bi)

. L-

= id(Fizi)

Bk = n.

Let

that

X

Uy 8i

i'.+.k

followlng F.zi

a

.

|

id

.Mu

^

,

o

l*'

of

foand äin

k €f

is

a sink,

L (il , 0).

j. + k and

I and remember

Then for

i.+.k

in

diamgrarn F.xi lI

e .\

Ja(*or)

',*

tn*k

by the definition the

o .t"t, = F.zi 1 K

I

-

t*'*

commutative

for

be an arrow in

| . (r.

is

L

and Z = go. Oefine

we assumed k € f to be a sink.

f the

Because

isomorphie

this

Of course

class

means that ß_is

of

c , and the choice of

B is

Z resp.

X. Because

a morphism in

a epimorphism with

kernel

T h u s w e g e t t h e d e s i r e d e x a c t s e u q e n c eO - E r - Z - \ - O .

FO-

-

b) Here we consider

the variety

H o r n o (. F i Y i , -i

ul = II I i.-.j where G" acts

on U!

(FiYi,

t(M,0

jei)€

with

(

modules in

betvreen the

its

module

I

in V*, I

isomorphic

of

classes

and the orbits is

is dense

= dim Y.

. -x, i : Fk^k

o rx

and define

with

and jüx = jqf

embedding

wj.th natural

' -Fzk ' k

c* (ror)=( j,li) in f,

€ u * a n d \ , r eg e t a g a i n a n

L (M , 0 ) the correspondj-nq orbit

ir

rf,k = rfr.

wrl-te

identify

an indecomposable prejective

and for

closure

s;l

of Gr. As in a) such an orbit

t/i under the action

open in

2.2.)l

,oal,t"..

(2.2)

O) of dirnension types I

L(M,

I

jt,

= (si e 1)jai

iüi

) vrith(j(9i)

1-'l correspondence

i

=joi ,ü1 : F*xk ._

s*2 V* for Fjxj

-

UI j,

i'+'

bY

i+ k

a jMk for

k'-...._'j

.

inf

q* is a surjective

Again

as above go = Zo

jtti)

the convention of the beginning of

with

of

FjYj I

by the rule

= (rtr)

s.(jai)

-

32

=-

Z, Io

:I

so we find

morphism of varieties,

1trzL, jqi) , Io =(Fixi,.Ü.)€ (pt)

o*

such that

= (;E)

L( M, Q), . Asain we

can

assume X^= X and Z^ = Z. Now define ß* =(gt): and Sfi. = i diagram

: F*xk -

Snxp

is cornmutative,

rrecauseo*

(]l)

z by Si = id(F.zi)

+

x

for

= (rVr).

"

Fk = F*zk. ali'

for

i+ k

Then the following

arrows lq '+'

i

in

f, again

-53-

t*"*

t***

I

I

t_ t**

t_

"'r"*'

l

|

Y

^I . z . I

E

Because k € f is homomorphism in exact

-;-

t\

rrxi

a source this fact

B* is

e ,n*

lmplies

injectlve

that

ß*

and we get

is

a

the desired

sequence

o-

Let X, y, Z€

2.3.2 Proposition: indeconposable, Then there

-Fk---€.

\-L

and assume dim Z = dim X + dim y.

exists

O-X+Z<

an exact

Y+Oor

Proof:

L ( M, fl) be prejective

sequence 9-y-?-

I-<1. Let y be

We may assume pos(I) <pos(X).

preinjective. r]

= Then Sl IX for suitable ^ r . . . S'r1- I ' l < O, iu € f , is a source in u <'r, where k €f

s1.... -r 9+X' Si..... -P tiu' * si.. si...

-u

= : 0 r . B e c a u s ep o s ( y ) < p o s ( X ) ,

" i-_10 t = s;

...s, rr

S . ,! * -1

9

ti., ! ...

dim

such rhat

Z = O. Then

S.

henQe iu+r =

(din Y ) , whi-ch is

'1

-1

but Sr ." S, ... 'U+ | Iil,+l,

s.i

s.-

. x. Let u € {1,...,r't

X >O and si

...

u

.. . s., (dim x) "ru impossibJ.e, because s.

y

^1 -

> O. Thus

Z ' = s-;r . . . s-]1 z + o . Consequently

(2.3.1)

yields

an exact

sesuence

-54-

O+4,-

sl- ... ^1

si,by -r

and,applying

(2.2.112

Z_ +1.

Comparison of Iast

q

exactness of s1 (i €r )

refr

-X

I

-

!'-Ek

the invoLved

dimension

the

shows that

thus we get the exact

homomorphism must be onto,

sequence

-L

9

If

preprojective,

Y is

-

o

-

we get

X +

I

L-

References:

Most of

$ 1 is

The Coxeter

functorshave

I.M,

valued for

and V.A.

celfand

all

graphs i.

algebraically 5 3 is

(f,

-.j

9.

been defined

Ponomarev in

[1]

vrithoriencation"l,

in

i), field

Ringel

done by C.M.

d)

closed

done by V.

simiJ.arly

where K.

the

I.N.

by

for

quivers

modulation

Rj-ngel in

t251.

Bernstein,

where d..

Furthermore

Dlab and C.M.

in

for [15].

(j-.e. = t

j.s given $ 2 see

= dli by an [11]

-

Chapter

fII

Graphs of

An abelian tion

category

series

A

A

and is

has only

objects, (il,o), if

for

then

of all

is

for

Artin

K',

K'

finite

to the

L(M'o)

denotes the category

type,

generated

exact

to

the

is

free x

assoy.

and if

A

modules over

an

so,

subcategorles

them .

l(M,n)

type,

embedding

one has to consider

also

of those which

L( M,0)

If

said to be of

if

in this

(f,d)

and only

(f,g)

again that

Our purpose in this q'peif

finite

and only

full

typs,

is

neither

tame (repre-

tvpe.

valued graph.

if

(representation)

modules over

finite

in-

K-realization

would be too special;

of

surn of

indecomposable

turo non commuting variables

Assume notr in the following

is of

direct

composi-

(representation)

exact

mod* Kcx,y>

equivalent

nor of wild

sentation)

unique

finite

a full

dirnensional

mod* Kr<x,y>

are representatj_on

is

definition

in addition to

has a finite

said to be of wild

example the category

isomorphic

of

said to be of

where

in

above, this

ring

object

isomorphisms)

there

finite

K'-algebra A

every

number of non isomorphi.c

is

-L(M,A),

over

tame tvpe

be again a valued graph with

sarne field

ciative (For

a finite

L(i,{,0)

mod* Kr<xry>

in which

is

(r,g)

Let

and of

an (up to

decomposable objects, if

finite

55 -

if

chapter (r,g)

is Euclj-dian.

cases more in detail.

is

is

a connected

to show that

L(M,n)

is Dynkin and of tame type

Furthermore we will

describe

-55-

s1

The representation

3.1.1

Theorem: Dynkin.

of Dvnkin graphs

L ( / l ,, ln )

Moreover

is

sable representations (f,g)

(f,d)

lndecomporoots

and the PosLtive

is not Dynkin, then, by

non-isomorphic

preprgective

non-isomorphic

preinjective

f (M,n)

therefore (f,g)

Let

L (M,0)

of

is

of

given by the dim function.

If

Proof:

between the

a bijection

is

there

(r,d)

if

and only

type lf

finite

of

qfldimX=x2o.

(M,0)

modules) in

c

the

is

set

of and

infinite' tyPe'

rePresentation

finite

x € L (M,0)

Let

the set of

(and equivalently

modules

cannot be of

be Dynkin,

Q-2.12)

be indecomposable'

Then

m-

haveorder

--,1

r . l l r f

hence I = o by c' under is invariant E crx € 0t r=O tt*1* ' o (1.2.1). Therefore there is an r > o with "t*' is projective' for some 1 < t < n = ltl crx ! 9t By (2.2.3)

Then

v :=

x=c-tBt

hence L{M,n)

is

is preprojective.

Again by Q.2.12)

L ( l ' ', lo )

preprojective,

nunber of non-isomorphj-c

a finite

module in

Thus every

preprojective.

has only

hence indecom-

posable modules. Now the second assertion

give

we will (f,d) of

will

now some applications

be Dynkin,

K

an infinite

by (1'2'5)'

follow

of

Q.3)

field

and

to Dynkin graphs. (M,o)

a

so let

K.realisation

(f,d) .

3.1.2 Proposition: in

L ( M ,o ) ,

Let

{t,-..,Ia

such that

and

Z

be indecomposable modules

-57-

J

dimZ= Then there

is

a sequence

o = Zo c 21 c tion

n

of

-..

3 x- T r ( t )

t !t-1

there

is

{k1,.

Proof:

Lt,

all

for

...

s kd_t S kd.

ld s 1d_t

, 1 < t < d.,

/2. / lkt

are indecomposable

d ='1,

assertion

the proposltion for

to € {t r...,dl

aLl natural ,

rrithout

dim Z - dim X1 = 1 indecomposable with Then, by

of root

is

system of Dynkin graphs says, as sum of roots, also

trivial.

numbers


hence

By above we find to = 1,

holds

d I dim Xt, t=2 Y, i.e. there

O = Yr c

r

of

t2,3,...,d,)

y € l(,!{,n)

a

is an exact sequence O - Il

dim Y = for

such that

dim Z = dim y + dim X1

O r Now

Y a c- - {. . .

hence, is

by assumption,

Y the

+ I + t + e -

Z a X] +O

propositl,on

a sequence

c

such that

Y,

= 7

lr/\-r

is

Assume we have proved the

Therefore there is

di-n y = I,

there

a root.

Less of generalety

is a root.

(2.3.21 there

and

= {O,1,...,d}.

.,kd,11,...,Id}

if

an

and a permuta-

1 < t < d;

all

= is root \"rritten " afrla I S to S d such that is Z-%

that

Z

such that

An elementary property

If

of submodules of

!

a sequence

o = kt . k2 . such that

=Za=

{ 1, 2 , . . . , d }

lr/_ moreover,

I dimX. t=1

:nd

!

a rermutaLrcn

x;(t)

2 I r <,l

or .

-58-

and

a sequence

o = Ez s [3 s ,.. = [a < Td < ia-r = ... s T, = 6 vl =Ir/y.,

that

such

2 < t

< d

are

If

indecomposable.

there

is

Kt an exact

o = n:

% azt

E...

tl,...,d]

-

O = kl

O -

sequence

.-zd=

3 < t s d.

If

then define l sts

there

is

O s t

r-'(rt), 13t < d, ?-o= o, Lr= = ?, and n(1) = '1, nl{2,...,d}

3 < t s d, lt

< 12 '

r

d - ' 1, 4 = 2 ,

and

k1=O,kt=[a,lt=]a

see,

that

by

by

d, la = Ta, S z - x.,, - 9,

= 9, Zt = v(Yt+1),

lo

for

definitions

O - I

n(d) = 1, n(t)

by

11 = d

= IZ =iZ=

sequence

an exact s d

define

then

9,

s kd < Id s dd-1 s ...

Zt,

this

by by

1, kt = [a,

kZ=

Z 1 Y -

z

{1,...rd}

S k2....

k1 = o,

-

11

2

=?(t+t),


d.

the assertions

satisfy

1st
Itiseasyto of the ProPo-

sition.

If

we apply

a root

j-n particular

the proposition

as sum of

3.1.3 Coroll-ary:

simple

roots,

to the rePresentation

of

lre get:

Every indecomposible module

Z €. L(M,a)

has

series

a composition

and

O=Z_cZ.c-..cZa=Z n--l--ru

a sequence O = kt . k2

...

< kd .

Id .

1d-t s ...

{kt,...,kd,11'...,ldJ

= {o,1,...,d}

for every

1 < t < u, Lrrl

such that,

posable module in

L(M,0)

of length

Ln. t.c

is

s 12 s 1.,= 6

an i.ndecom-

- 59 -

Applying

this

to

the

3.'1.4 Corollary:

Let

decomposable series

root

,ä"(Ut* module I

(t,d),

of

Z €. L(M,n)

(i.e.

module,

be the is

Z), in

\de get

largest

maximal).

L(M,A),

k2 =

=

.,kd,11,...,ld}

{k.,,. /

tt s t

,

z,

s

iO,1,...,d}

for

exist

of

< kd < 1d < ld_t

...

indecomposable

Then,

there

o=%c...-!a-r-%=Z_

O = kt

-2t t1/

maximal

Z.

andasequence

such that

s d,

are

indecomposable

,

for

a suitable

t1

* |

So we can tor

of

the

Remark: tion,

find

that

?r,,

every

/,

-K- r

indecomposabre

indecomposable

We will K

see has

to

an example with

finite

(3.1.2)-(3.1.4)

fails.

module

later be

modul-e in X € l(il,A),

[Appendix infinite,

K'

in

B], is

which

that

real (2.3.1)

= d,

all

modules

-K!

and

in_

a composition

= 12 .

...

every

in

t(M,n)

t,.

L(M,a) with

in

as a subfac-

highest

(3.1.2)

necessary,

dimension.

the

r.,e will

and therefore

as'sump_ give

arso

-60-

The non

S 2

homogeneous

(r,d)

paragraph let

In this

K-realization

(M,n).

the defect of

X

3.2.1

The dimension

function

representations

non-zero

defect

positive

and all

defect-

Moreover

jective

modules in

Let

X €

a(X)

< O.

order

definition

Let

ax < o

(f,d)

the

indecomPosable

of

non-zero pre-

N -

c Q'

be

Q'

= A'ln.

non

radical

the

zero

(2.2.4)

Then, by

I

of

defect, and

and the

defect

the

di.m(c+r\) Because

of

having

b e indecomposable

on

c

just

are

nodules

with

L(M,e).

L(l{,4)

of of

this

t(M,n)

of

roots

bet-

a bljection

induces

indecomposable

the

a(x) = a^ (dim X)'

we rrrite

X € L(M,a),

rveen all

say

graph rrith

be a connected Euclidian

(Fee 1.2).

Proposition:

Proof:

m

If

graphs

Euclidian

of

representations

= din x + ra (x) . n, an

is

there

to

t

o

c*tmx y o.

if

such

that

!^R

(dim X)

c' r

such that c * t 1 x- - K = B u projective

C

'x

,"; of

O,

I

hence there r

I O ";.

and

an

)

r.

O

rl

X = O,

C

k € rr,...,.r,

dimension

is

that rhus

is by (2.2.3) tlgr * = . is pre-

-1pk. a f)

similar, if preinjective clear

with

a(x) > o, x = c'rI r -^ _ 1 .

of

dimens:on

(1. 2. 8) .

.

(b € {1,...,n}, Now the

rest

of

some r, > o) the

proof

is

-

Now it

i-s crear

L(M,a)

that

arl

have defect

describe

3.2-2

such

indecomposable

The next

representations

proposition

says,

in

how we can

modules.

Proposition: are

other

zero.

-

ol

x € L (M,n).

Let

Then the

following

statements

equivalent.

(i)

X

is

of

defect

aI

= o

and for

aI

= O

and

(ii) (iii)

a direct

surn of

indecomposable

representations

every

subobject

X,

of

every

epimorphic

image

zero.

for

X

holds

ax'

s o

ex'>o Proof:

i)

Because

-

ii):

c

is

(with

hence

crease

r

j.i)

would

iii)

the

note

by

t,

preserves

it

embeddings

. n S airn{C+r\)

= din

dim(C+rmx,)

which

is

X

s dim X

would

in_

impossible.

-

SimiLar

i)

resp.

proposition R(M,a)

representations

-

iii)

x € L (M,a)

A representation in

ä

be positj_ve,

increasing

Remark),

o_n 0'l*)

aX' < O.

Thus i

of

= dim X'+räX,

aX' for

(2.2.2

exact

m = order

dim(C+mx')

Now, if

reft

above is the

.

full

i)

is

which said

to

trivial

satisfies

by additivety

the

be a regular

subcategory

of

t(M,a)

of

properties

described

representation. of

all

a

regular

De-

-62-

3.2.3

R(M,o)

Proposition: t (M,a)

Let

Proof:

of

closed under extensions.

O-X+Z+Y+

a! = ax+aY,

Then

zero defect,

of

thus,

beanexactsequencein

if

any two of the representations

are quotients

of subobjects

of

subobjects

X

of

Z

of

of

Y

of

by subobjects

quotients

L,

Z- are extensions

of

and subobiects

the

so ist

are subobjects

{

will

and the assertion

äz = aX+aY and (3.2.21. In particular

follow again by

are

any two of the

if

one. In fact,

in the above sequence are regular,

one; for, I

l(Mra).

O

the third

so is

representations third

subcategory of

exact abelian

a full

is

R(M,0)

i.s closed under extensions. X,Y € R(M,a)

Let nort the exact

and

a : X + Y

sequences and

O-kerc+X-imO-O, O-im

by

that

im q

a(im is

rn

im c

the

following

detail.

For

(3.2.2). of

Y,

non

it

(3.2.2.

we divide

presentations.

First

R(M,0),

in

the

called we will next

the

homogeneous treat

Tbis

the

paragraPh'

and non

Therethe

R (M'0) into

part

homogeneous one.

proof.

more two

homogeneous

non homogeneous the

im o

defect-

once more

Thus

s O.

o)

comPletes

category

R(M,a)

hence

!,

submodule of

Positive

i1-).

describe

Purpose so

has

of

a (in

hence

thus

to

the

y,

of

we show,

if

regular,

a quotient

is

every

we v'ant

subcategories'

are

Furthermore

by

regular

is

this

then,

im s

but

regular;

a submodule

q

and coker

and a submodule

= O

fore

q

ker

is

2 O,

q)

O-Y+COkerO*O.

above,

Then,

ä (im o)

be a morphism. Consider

reof

in

-

Notice

that

modules,

now simple

63 -

modules,

decomposition

series

decomposition etc.

as objects

seri-es

etc.

of

abelian

the

means simpre catesorv

R(M,o).

First

us give

l-et

is

called

in

lt's

Nf

of

conpositlon Bf .

=

X

l (M,A)

for

X €

acts

naturally '

=

(tr)

easy

to

see that

a coxeter the =

hrith

A regular

ii)

= x}

is

0",

and

= Nt,

the

to

of

simple the

radical

a1l, homogeneous

H(M,a).

of

the

all

dual

linear

forms

basis

{i*l(i

=

I

, ef

e rl

of

x =

(wx) (x)

rule

= x1w-11)

k€f,f,,=(Xr)eef*

= x.i

a one dimensional {x e efl

(cx) (x)

radical

of

E € R(l/,0)

r.

S ef*,

= x(x)

w € Wr, skx=E=

Al-soitis

where space

c € tl]f

generated for

all

(

O

for

sorn

X € R(lt{,O),

Ä€ef*J

ei.

is

said

to

possess

an equatJ-on

then

there

is

an embeddins

E9I iii)

if

nX

phism

( X *

O

for C-E.

sorne

X € R([{,a),

is

by

> O nX

e

Wl

for

weget k+i€

=,t.

-

(xr) € ef .

for xi"., ! f ' ' r. € ( d i m X) . The !,Ieyl group x

,*

factor

if

nE if

{-L g eFlcx

representation

.!. € Qr* i)

For

representation

every

belongs

61=-x1,6i=Xi+dikxk,

vector

{x e Qflc*

to

XX ,=

transformation,

defect

by

of

R (it{,a)

of

0f"

by the

eft

type

R (M,a) )

i.".

Or,

rr X€Q'-.

eQI'

space

we write an

and

(in

respect

of

A regurar

dimension

be denoted

with

basis

the

subcategory

vector

(Xi)

defi.nitions.

series

will

the

natural

x€0'

if

The full

Now consider

the

basic

homogeneous,

representations

wrlte

the

then

there

is

an epimor_

=

-64-

equation

be non homogeneous possessing

E € R(M,o)

Lenrma: Let

3.2.4

L>2

and

i)

c+rE :

o < r < r

c+rE,

and all

E

the

of

orbit Then

be 1'

c

transformation

under the Coxeter

dim E

in

the elements

the number of

Let

n.

an

non-

are mutualry

isomorphic,sirnpleregularnon-honogeneousrePresentatj.ons. nX = O

il)

for

and x7c+e.

x7E,

Extl (C+rE, x) = o

iii)

with

there

Because

only

I

- {,

Let

I

this

q

has to be simple

E g Nf,

din

= n (E) > o,

there

is

that

there

is

E + X,

which must be an j-somorphism, because

By (2.2.2) x I

E, I

nX

(O

implies

we can assume, that

7 C*E

and consider

obvious. by defi-

a monomorphism

nX ) O

simil-ar

is

thus,

nition,

regular;

implies,

the orbit

by (2.2.'t\,

Now i)

representation;

regular

thus

a monomorphisrn

E) = dim E

must be an lsomorphism.

be a simple

!'

elements) '

number of

a finite

of

of an equation for

hence

not homogeneous,

factors'

(as element

q

of

Now dim c+le = cr(S

therefore

iii)

factor

on composition

(CIearIy, because E € R(itl,a)'

n (c+In)

Because s - c+lg.

ii)

additive

so by definition

is

E'

by (1.2.1).

can contain i)

is

n

must be a monomorphj-sm E

regular. I > 2

and

) o,

nX

representations

simpre regular

all

for

must be an composition

R(M,a))

representations

si-mple regular

all

x7c*t*ln-

and

nE ) O

Because

there

= o

Extl (4,c+rs) x7c+rn

Proof:

for

x7c+r+1n-

xlc+rE, iv)

with

X,

representations

regular

sinple

all

I

is

XsC+E.

r = O.

Thus assume

an extension

simple

_ 55 _

-O.

O-I-3-E and

Z_ contains

Now

X7y,

E 3 Y lv)

hence

a subrepresentation

because

Z *E

nX = O,

By ii)

is

the

identity

isomorphic

I

XlE,hence

=nX+nE=nE

nZ

Xfly=O on

E,

to

)O E.

andso

i,e.

the

sequence

splits.

Similar.

Now given is

which

E,

an equation

orbit

of

of

possesses

C+rE.

dim E,

an equation 2 < l- (

If

-

is

!r

observe

the

cardinality

crn

that of

the

put

n Kercrl=KnO
By

A finite is

said i)

(ii)

set to

the

tE

iii) iv)

(t)

Kn r

I 1 s t

be a generating

orbits

of

and fi-nite, ii)

always

dim

i.e.

any two

dim n(t)

distinct

orbj-ts

each

E(t)

there

exists

of

following theorem

3.2.5

set

possesses a source

the

acti,on

regular,

in

for

of

c

are

nontrivial

t

t,

I

belong

ro

01.

an equatj,on i

( j ! t, n )

of

non-homogeneous.

dim E(t')

with

property

lemma provj-des

representations

if

is

and c

of

under

E(t)

each E(t) has the /1 rvr l v = Nr . ) 1< t < h K n ( t )

The

s hl

(t)

n

of

Nr.

n(t)

respect

to

it

such

that

= o.

that "lt)

now the

final

argument

in

the

(3.2.71 :

Lemma: Let

i

X € L(M,a)

be indecomposable

be a source

srith

respect

such that

to

..

Let

C-X + O

and

proof

-66'

= O.

Then

k € f

exists

thele

such

that

"i End X = Fo,

Proof:

Of

course

subcategory In

we can assume t (M,a)

of

= o

(X,x)

Extl

= n.

i all

of

Extl

and

x

X

that

such

rnFkr* full

b e the

Ln(M,a)

Let

modules

=

1x,c+x1

= O.

n

particular

: = sl-r ... til € Ln(M,a) is

C-X

and

of

an extension

a dj.rect

by

I

Z_ of

sum

copies

o f F -n O+Y+C+X-Z-O.

So we get

the

exact

1"1 Ext' (x,Y)

Hom(X,Z) -

in

which

= o,

Xi

the

+1 Ext'(X,C'X)

-

first

the

both,

because

sequence

and last

second

+ Ext'(X,Z)

vanishi

terms

injectivity

one by

the

first

of

Z-.

one, Thus

1+-1

Ext' (X,C'X) = Ext' (I,I) Now, there

is

a canonic

isomorphisrn

:

Ln(M,a) -

L (M',Q')

T

( M' ,0' )

where

the

is

(I',d'),

whereas r'

observe,

that

Now

hence also

I,

same is

true

(f',d')

of the

is

a disjoint

i.e.

one connected

resul-ts

of

is

union

(3.1)

TX

to

graph

Dynkin graphs.

of

(2.2-1 ),

and

component

the

induced valuation.

the

indecomposable

TY,

and

g'

and

are

I,

Tx

for

representatrons we can apply

= i\{n}

( M,0)

of

restriction

TY

of

combined with

are

thus in

( | ' , d' ) , (2.2.12).

the fact and In

-

particular

t+

-

with

L(M',Or),

= sl:,. . sf,

C

for some k € f' " ( 2 . 2 . 4 ' ) ( 2 . 2.5) we get By and

r(Y) =c'r(x).

E n d -XK = F ,

(2.2.2)

and the

1

= O

Ext'(x,x)

nxtl{F*,F*) = o

=

fact,

that

2k

a n d t h e Iemma is

3.2.6 Corollarv:

E € | (M,A)

Let

possessrng an equation € f.

i

cannot

Thus there

is

an

(3.2.4)

(3.2.5).

Theorem:

Let

1g(t)

[(M,n).

Then the

H (M,0)

and

h

o(t)

-

orbits equals

the

R

in ordered

(t) in

the

action

of

its

simple

socle

and

are a

C':

going each

and

its

of

R

is

(tl

X = C-E.

if

the

set

product

corresponding

< h.

with

whil-e of

composition

them is

of to

the

r_+.1 in

factors

correspondlng

fu11y

wltich

c

in

OI

The indecomposable

down sequence

length

some source

be a generating

1 < t

serial

for

and only

1e(t) ,c+E(t),...,c-'t"{t)r, j,n the orbit of elements

objects 0

if

number

dirn f(t)

(t)

^ tk

R(M,n)

subcaregories

containing

Thus

and Exrl (E,x) + o

( F u) , "k

s h]

category

fr.

non-homogeneous

= O ", such that

k € f

I 1 < t

for

be a root

Assume

X € R(,U,4)

module

and

Proof:

3.2.7

a simole

by

proved.

E n d( E ) = F k , E x t l 1 n , c + r , c + e 1 = for

(Fk) E,r'. ^K

( F k € L ( M ,, n ' ) ) ,

be regular

n.

and

r*(Fk)ru ,

nxtl (x,c+x) = Extl (x,Y) =

and

(Tx,rx)

Extl

rxtl (t(x),c'+T(x)) =

= Fk,

End t(x)

Finally

the coxeter functor on

''Ix

Itr

T(x) =

we have tf

hence

-

o/

both

determinded

from co by

can be arbitrarv.

68-

,rrdlf* Remark: An abelian sition

serj,es,

is

called

serial,

has an unique composition categories

Proof: Let Then by

if

series.

every indecomposable object of such

A characterization

t19l).

X € R(M,A)

be slmple and assume, that

ii).

1
forall

"rn(t)(dirnx)=o

(:.2.4

Hence

.,.a.nKn

dim X €

sJ.rnpJ-eregular

ti.on is

O(t)

contained

Let now of all

i.n

be the fuII

R(t)

regular

1 s 1

1 s t s h

is

that either

and uniserial

every

= Nr,

R(lr{,0)

of

regular

x € R(r',t ,n)

Let

composition factors be simple.

(t)

0

in

factors

composition

R(t)

in

O

from

some (t)

to the action

of

be indecomposable non homogeneous rrith (t) E,...,c+18, (n e 0 and let ), I € R(M,n) n! t4,11 = dimx Hom(X,Y) - dimKExtl (x,Y)

Non calculate

lo4r

=

I

(2.1.4).

B;(C'-E,Y).

' r=O and only

if

zero if

I

= E €

!

E(t),

() (t)

Nos,

.

k € r,

all

terms

in

the

sum are

(3.2.5).

Fn-algebra,

sition

Floreover

if

by (3.2.6). Hence Ext'(x,Y) = o in this (t) Y € 0 . Let Fk be the endomorphismring of

Then the endornorphismring of an

Hon(X,Y) = Hom(E,Y) + O

g O(tl

case. Thus let

factor

I

is

also

Fk,

and

End(X)

is

because the endomorphj.smring of every compoof

X

.

module of compositlon

in a going down sequence corresponding

ordered

is

consisting

compositionfactors

homogeneous or contained with

x

i'e'

1 < t < h.

subcategory with

o
nonhomogeneous representa-

some

representations

Assurne by induction Iength

for

C U lstsh0(t)

I

andaII a ' -+' l

homogeneous. So every

C'.

has a compo-

1s due to T. Nakayama (in the case of a module cate-

(see e.g.

gory).

in which every fobject

category,

is

Fk.

By definition

[Fn : XJ = fn

_ 69 _ and by (3.2.6)

nftc+ts,v) - ft = -aftc*t-le,Il

y 3 c*"s

for

(1 s r
Y!8,

aft4,It otherwlse

= af te,gt - fr = dinK Hom(E,!) = dj.m* Hom (x,y) aftx,Yl

= o = d\

Hon(x,y)

hence, in both

cases, Extl (x,y) = o 2.) case: y3g+l+1". Then, if

Y3E=

fk - dinK Extl(x,y)

= dhx Hom(x,y)- di\

=

nxtl{x,I;

a ! t x , v l = s l t e , c r + 1 g l* " ! { c l e , c l * 1 g ) = f * - f = o * If

y 7 g,

then, because uorn(x,y) = Hom(E,y) = o,

-dim* extl (x,y) = sf t4,gl = slrcru,cr+1e = -fk. ) Thus, in both cases, extl(x,y)

=

.

f*(Ff)f*

Thj.s shows, that there is just one extension slnple regular module, and U € R(t).

q

of

I

Thus R(i,{,n) is the

direct product R(1), ...rR(h)xH(M,o),i.e.

there are no nonsplit

extensions betvreen indecomposable modules of different By Nakayama's theorem R(t) there are modules in structure.

0.

must be uniserial of arbitrary

factors.

and by above

length of the descrlbed

This proves the theorem.

Let no\"r (l,d) tation

R(t)

by a

b e an Euclidean

In the p r o o f

of

valued

(1.2.8)

graph with

adnissible

we showed, that

orien-

the Coxerer

-70-

c

transformation

the Coxeter

to one of (Appendix there is

in fact

S r,

"*, Let

in the tables. S = S;

,-,|

...

S;

(2.2.1)

With

..,

is

morphism bethreen

R (M, a)

0 = 0r,

of regular tion

M

(f,d).

of

tractlon, graphs.

is

in

outline

(M' , Q' )

A reallzation a contraction

and

T

and

induce an iso-

S

(f,g)

graph

for

general,

o = o'

with 1r(t) 11st

set

s hJ

K-modula-

an arbitrary

if

{vertices

for

valued

arbitrary

(I,d).

of a valued graph

of a valued graph

(M,01,

of

"1Q, described

purpose we need the concept of con-

(M,n) be a realization

So let

.

L(M,n')

a generatlng

L(M,o)

in

For this

which we will

"*r_' the orientation

R (i,1,o' ) . Thus \de can assume

there

representations

.

in

the corresponding functors.

each Euclidian

that

shovrs that

of simple re-

€ f

ki

Wf

tables

in the

= L(M,fl) sl *1

L(M,o)

and

for

under

look to the proof

easy to see, that

and we have to prove orientation

-

conjugated

described

is

Sl ... *a

: L(M,0')

ist

= Q'

s*r0

T =

is

"*r,...,"*., ki_ is a sink

such that

and

$

A closer

a sequence

Wt

in

to

transformations (f,d).

for

C)

flections 1 < i

corresponding

of

(l' ,d' ) rr}

c

is

said to be

{vertices

of

I}

and i)

If

krl

€ l',

and there

- in = 1

k = io - i1 -... n

where in

ii)

If

is a sequence of arrows

i1r...,Lq_,'

(q > 1)

g t',

iii) iv)

(r,q)

with respect to

then there is an arrow

k + I

(f,,q,). k..+

.1

in

(f',d')

with respect to

an unique sequence of arrows in

in

L*,

0',

= (k = io -

then there is il

*

...

-

in = 1)

(r,d).

Lkl n Lk,lt Fk = Ft< for

= 0

for

every

{k,IJ fl {k'rL'} k € l' ,

and

= Q,

(1,kr1'rk'

€ l')

-

v)

=

kMi

...

"k M-i 1. "

(f' ,d')

in

with

L*t = (k -

Now, for

i1 -

each

for

" i- q - 1- M-1

-

tt

every oriented

orientation

where

- 1).

...'in_'

k € f,

e, ,

define

+* := {i € Ili=

-...

io-i1

-in=k

with all

r*

:= {i€

flk=

-...

io-i1

-tn=

Let

X = (Xr,rOr)

be an arbitrary

a contraction

R(x) = u = (uk,t

t

with

for

ipg

r'

for

O
representation

( M , A ).

of

of

(Ä.{,a) and

Define the representatron

where

k),

Uk = Xk

for

all

k € r'

I

lvk

defined

t UX skMI - Ul

for

k -

1

'e; ....'ei '2

- q - 1- " ' i

L*t = (k = io-

^p+1 -p

-

i1-...

(pi rp+19i =i tp 'p+1 'p o I s,.. t"

-

On the

all

(wj.th respect

Q')

to

is

by

r ü r = i ,^eqi

of

f,

k€+k0tk.

(M',O')

and

ipF

o
all Thus

k + 1

edge

other

U = R(X)

Xi - p +.1-

hand, with

ri

if

-p+1

where

' 1 '.1, o rl o ',

ip - tn*., + ...

+ in = l)

and

6.1 : x. 6i M, 8...8. t p " i M^ p t q _ l M,,q_ , i *p+1 -p+1 ^p+2

Mi -p+2

V € l(M',a')

X € L(M,A),

E ...

a. .q_1

such

Mr rq

that

we can define

for

V

is

a]t

O
a subobject

canoni_callv

rwo

-72-

subobjects

of

T (V)

T' (V) .

and

X:

given

first,

=

v

(v*,ftX)

€ I (M',Q')

introduce

the

= io'i.l

-

following

notation. ff

i

€ f

belongs

a path

is

satj-sfying

Vi r. '"k

as

the

mapping

the

ip

inverse

LX =

and g lr

image

for

(i

Vk oi

of

- l

...

-i q - 1

where

-i P

-, p, + 1 6 '- p

(i =iMi Mi -p -p+1 ^p _ l

Mi e tq- _ 1

- l

..

-P-1

: Xi -p

-

i

(0

', o

-

! X, 'o

X. 'q

-

...

q,

. . . oi t . l- Mti o

e

= k)

in

define

...

under

e. M: '1'o

as in

(2.2)

€. M, 'p 'p-1

e

X, Mi ei ^p+1 'p+1 ^p

and

'p+1 )

(pi (! 'l e ^i p + 1 ^ p

=

-i p + 1- 9 -i p

(9

O < p (

aLl

q

- l

;- q I

{k

to

...

: X, -p

e1

-

X, tp*l

...

o,,Mi -o '

+

ei r p + 1l-{-i r p e- ri n -M- ri n - 1 e . . . i . , M i o , (Osp
put

y..+ = fl v. r.K r, tk "k

possible

paths

Similar,

if

i

is a path with the,image -i q 9 i- q - 1

"'

of

S Xi,

Lk

where the

described

€ f

uT oto"t.,o

run through

tk,

and

O ( p.

for

al1

"'

orr_r"rn

l,k = (k = io-i.,+...+in q,

define

Ua"O

under the mapping

(p. -i 1 ' o

(., e' O ( -p *p-1 ,

p < er

Uri

defined

= t-V._t iL" Lk

all-

above.

belongs to

ip g f'

indices

!

as

above).

xt

where

And again,

the

indices

put

run

as

= i)

-

through

possible

all

Now, we def ine v

for

all

k € fr

exists, by

=

t(!)

= Xi.

where

€ +k.

i

Vi

g r')

(i

u.,u*

such that

Vi

Lk.

(Vr,rtOr) ,

v. = rkrk |

€ rt,

i

paths

-

t5

is

determined

and the

indices

As particular,

As indicated

by the

if

by

run

through

no such

notation,

is

iqi

k

induced

I.

Similary

we define

=

T'(V)

by

tVi ,.'orl

and

=

Vi

Vit

I

for

g f' ,

i

€ f '

ua = u,

for

indices

run

through

= O.

Again

L J L

where

the

k

exists,

i

a1l

K

k

such is

.9.

JL

It

that

is

i

€ +k.

lnduced

easy

to

bv

no such

Ui

X.

verify

that

of

I.

of

U = R(X) €t(M,0),

X

If

Furthermore,

both

if

and a quotient

y

is

then

r" (vJ:=

T(y)

and

f(y)

are

an indecomposable

V

subobjects

direct

summand

determines a subobject

-x/ , rt l ] v i ' l ,

where

w

is

T'(V)

a complement

of of

Yinq. fn

the

graphs (with given (1)

-

other

a genrating respect

to

certain

a generating

(v)

of

the

prove,

and iii)

It

is

prove

to

n(t)

from the

s hl

in

to

of

order verify

enaough to by direct

an equation

definition

type

of

Euclidian

r^rith equations

rn

we have

set,

easy

that

every

orientations).

deflnition. are

for

1 I < t

1n(t)

is

conditions

C) we give

set

set

have to ii)

(Appendix

tables

to

sho'

the

prove

(iii),

n(t),

an equation,

a

condj-tions

calculations.

for

!(t) that

all So rrre

i.u.

(n(t)

1B(t,1 >g

can be seen directly).

So let for

X € R(M,e).

each Euclidian

(M',4')

of

(M,n1

Now, our graph

proof

and for

defined

will

each

by the

consist ,,

(t)

non zero

,

of

considering,

a contraction components of

n

(t)

,

- 74 -

and in decornposing

R(X)

into

indecornposable representations

of

( M ', a ' ) . /+I n'-'X

that

Note,

i.e.

R(x) ,

(t)x

n of

restriction

are

the

positive that

composition similar, direct the

notation,

contraction F * F),

-

2*-l*

or

llorv if

n(1)

nt5)

contaj.ns rf

n

(1)*

N o \ , ra s i n

for of

a Dynkingraoh,

must

% is

thus

(3.1), that

occur

in

to

the

and these

show,

isomorphic

isomorphic

t

n(t)x the

nonhomogeneous

E(t).

to

c+E(t)

for

some

for

graph

1 -

R(X)

into

the

types

set

p < n).

if

A2 t

representations.

a generating

is

Ep+1)

g(1),

direct of

consider

(wj.th

2

the

the

realizatj-on

sum of direct

type the

there

(o,j),

indecomposable summands are

('l,o).

of

of

I,

T' (yo) )

must be a direct (2.2.1)

n(1)'

we see,

!

is

summand

additve

( 1 , . 1)

and

T'(y-ol

= g(1).

type

Evidently

(namely

must be a direct

because

surnnands v

there proof

o,

isomorphic

to

summand Vl that

and

n(1)u

E of

a o

Thus

( 1)

o

de-

tables.

the

an equation

= n(1)x,

a submodule

the

to

defined

Dynkin

type

a o,

natural

(1,'l).

dimension

those

is

The dimension

{O,1),

the

of

R(X).

is

and decompose

components

denotes

we will

s x

on1y,

to the

= o

T' (yo)

E(2)

(1,o)

n(1)'v

summand

no regular

is

the

dimension

di.agram.

T" (!t)

of

of

their

=

representations,

of

by

the

and

< o,

are

n

11) t)\ = {E'" LZ, E'-'

:

n(1)

of

we refer

There

by

(t)'

a realization

particular

V1

if

,

determined

n(t)x

course,

First,

for

is

R(x),

summand

determined

(M' , e' ) .

to

roots

of

Ar, (n > 2)

%

(t)

one

if

Ä . , . ,, Ä . , r :

(of

= n

(t)'R(x)

summands are

implies,

For

uniquely

(M'rQ')

Furthermore direct

n

is

x

.

type

(1,O).

-75-

+ im19n+1 = x1

imtQ2 module

Ft

would

(x =

be a direct

be regular

(3.2.2

sunmand of

type

(1,O),

preimage

of

fore

the

From this

it

i).

{E = IZ}

ConsiCer

the

dlmension (o' l),

types

(1'1)

must occur

would by

d.

(2,1).

type

hence

(2,'l) .

type

be surjective,

xt

of

set the

with

82.

summands of n x > o,

rf

n x

or

(2,1) .

cannot

be of

would

negative

I

R(I)

2.

of

%

The

(o,1)

type

has a direct must

1q2

(1,O).

X

*.

(.1,O),

are

isomorphic

and

E(2).

,(1,Z'l

As above,

type

Xp+l.

n = 2*-.1

R(X)

< o,

be not

defect

for

a summand

If

be

g+S(1).

!

equatj_on

diagramm

there_

cannot

an equation

(1,o)

T' (v1 )

be a module

is

n(2)

not

has a direct

tqp+1

T"(yj)

3 n.

Vl

If

under

that

direct

T,(yo)

of

sur]ective,

the

or

sununand v1

cannot

shorrs that

of

and

ipZ f

to

would

R(x)

easily,

contraction

X

hand,

a generating

is

and

I,

projective

the

other

thus i*tqz

othen"rj,se

summand of

on the

follows

The same argument Err

(xi,iaj).

So

Vl

to

C+E,

be

is

wour.d be not

of it

regular

(3.2.2).

and

Ecnt

similar

-

/ tl 8D,.,: {E*" = l,

to case

8..

EQl = Frr";

..Fn-2(l

l r" a s'enerarins "n

set

$tith

equations

The proof For

for

E(2)

= 2*-1*

n(l)

E(1)

is

conslder

the

similar

and to

contraction

=

I(2) gn.

case to

2n*-1*.

11,2)

1<

n.

rf

()l

n'-'X

> O,

The type tive type

a summand (O,1)

defect, (1,1)

would

thus, by

% lead

because

(3.2.2').

of

type

(O,1)

to

a submodule

x

i.s regular,

or of y-

( . 1, t ) X

occurs.

of

posl_

must

be of

- 76 -

Therefore

=t.,

T'(Yo)

o

*o,..+Fp.,.

f.,_Z(

p+2,

ft

^n O +

the modul,e would

o *

...

be a submodule

woul-dn't

of

be regular

_

Fp...".-r(;

by

T,(Vo)

of n hence

,

(3.2.2).

oosirive

of

defecr

and

I,

X

p = 2 and

Therefore

t2l

r'(v^) = E' If

(

n I

O,

( 1 , o ).

of type

T"(Vl)

cDn:

is

to

= -2

n'-' Di--!

F2 ...

r,

p + n-1,

If and

X

is

a direct

then

not

Vl

summand

'o * o ... odo for

"p

the

defect

regular.

of p = n-'l

Hence

BDn

C^ <e

set

*

+n

^i .-t

with

/1\

= -l

nt"'

,

^^-^'der

'i-. r

P4

äö-F: F"FO*.F Inn

a generatj.ng *

have

-

3g+u(2).

f,in t , I' , !n ( 1 ) = 1 r .3 , !F ( 2 ) =

t)\

must

so r" (vl)

negative

r'(v1)

simil_ar

is

R(X)

1 < p < n-1.

some

and

then

tLhr reE

6-0 7 rn_2_ld

...

equations *

F:2* "^^:; id = E ,L "(3)= o,, 3

n(1)

-

. . . . tnr_ 2

O-O /

-1*-2*+3*

*

+n

cu vor nr Ltrraa9cL tr ivol l n

tLoU

tLhr reE udrioaygt dr ranm

the

dimension

orf u

tL-y'lhJ ca

^a3 -

'\

1

/ 2lz In

the

decomposition

direct nF

t.,ri+i-^\.

e! wrrLrrrgl;

rf and

n(l)x

> o,

T,(V^) !

summand vt

1

(using O

^

R(X),

of

summands are

.

O O, ö t,

O

the ^

natural

1

of type

rt I

n ( 1 ) : << o , o

or

!

o

^

1,

i O, j

there mustbe a direct r(l).

JO

I

types

suggestive

1

or

s u m m a n d .V o

1

i

of

the

kind

1

of type

!

r

there must be a direct or

]

r.

But the fi"rst

two

'

cases lead t o a is

of

quotient

lr

type

of

of negative

I

and, as above,

T" (V1)

defect.

Hence

Vl

must be isomorphic

to

c+E(1). For

E

(2)

For the

and

E(3)

renaining

the proof

diagrams

choose the orientatlon j6i

easy (contraction

lre use also

0

in

(x = (xi,j i)) "jMi we wouLd get a subobject

othenrise

the

of

I

o + F

E(1)-o+F2-it*F6-o

o + o + *F6*FU v

+ ',4

r is

(3)

=O+O-F.*F-*F_ tbt

a generating

set with

eguations

o ( 1' '1 nr

=

-2.1

1

* +2

+4 +6

* l

=O

1-2

o l)\

n'-'

/1)

n'-'

- - 1 * + 2* + 7*

A |

-l

,l

*

= -1 +4

=O

o-1

to type

that aII

r.re can

mappings

are monomorphisms: for,

easy to see.

gtt'=O-F2-F1

fact,

such a way, that

- Xj

t xi

j.t is

is

O

of positive

defect,

Ar).

-78-

Da

to the diagram of type

the contraction

consider

First,

9

+ 1

2+ rf

rlr lt'llto,Yo

or

1

hasdimensiontype The first

1

i

positive

+6

defect.

1 o

o

o o

o,1

o'o

o o

1

4 of types determj.ne subobjects of 1 r1l 'l 1 1 ' E" and T'(yo) is of tyPe

three

Thus

%

/1\

< O, then there must be a summand Yr of type n("x ö1oe1 Again'the 1 2 1 o,o 1 1 or 1 1 o,1 1 o,o

If o

four

first vr

cases lead

is of type

)

of

defect'

of negative

I

r" (v1) !

and

1

(all

6+9(1),

would lead to quotients

T" (yl )

for

bilities

1

to quotients

of

other possiof negative

I

defect) . The proof case

ör,

t1\

E\'/

for

and

E(2)

(contraction

= O * Or

A3

to type

F.+

F"

the same Lines as the

follows

E(3)

lo

Az).

resp-

E.

*

o*

F6

*

F7+ o,

o,

t-

E(2) - o + o + o

+

F1 r(

( 1' 4 , q- '

/?f

E(',' = o * F: -F2

is

a generating /1\

n"'= /a\

t\o' /?\

r tt ' '

For - v \

E(1)

=

-2.1

= -1

*

*

*

*

+2 +5 +6 :i

+5 +7

= -2.2

*

*

1 = C

*

1-2

o-1 *

the

O

1

0

0

I

I

U

1

*

+3 +5 +6+8

consider

F7 * FB

equations

with

set

1 ' -or,"' l )

:;(1'1)-2 F i * F;

=O',l

contraction

1 V

-z

of type

D4,

Thus

1

i 2*1*

-19-

for

the contraction

E(zl

of type

5 +

A3,

t+t

and argue For

E(3)

as

in

case

consider

the

E6 . contraction

of

rype

D5,

5 + 3+1+6+8

1

o,

n(3)* t

If

o o o

1, o

1 1

O,

1

of

all

1

mination

phic

,"

-

1

1

1

types

1

or

with

= E(3),

T'(vo)

clude

'l

has dirnension Type % 'tol o o o, o o 1 't,

the

2

rf

1.

a o,

n(3),

thedimensiontypes

O

11 1 o

O

Vl

of

R(x)

tient would

and of

T" (Vl ) : determine

1

2

1

O O

dimension c+E

(3)

a quotient

O

O,

1

1

1,

one,

are

we con-

monomor-

again

a1r but

1

O,O

O

eli-

the

1

1

last O,

canbeelimi,nated.Adirectsummand

type all

,

1

1

last

eEl

'l

0,

by routine

the

then

ooo

oneof

1t

Again,

and

zdl

1

o

1

of

o

o,oo

O

o

exception

because

txi'jei)).

2

o

1

other of

X

;

1

o

determines

possibilities of

negative

the

quo-

T" (v1 )

for defect.

F

r11

E.''=o-)

o

---)o

-

o -J

17 Fz €

Fl -

FB-

r)

'1

Ei2) - o ------'o --J

Fa €

F: *

rr!D+

t' t (1,1) u1 F3€/l-

'o

_2

E(3) = o -----,Fs ----, F4113,"3_-wittr

to= (1'1'O)+(O'1,1)

I' üe

r! €_1)r rf o(o,t,t) ,t (1,1,o\

-80-

with

equations I1l

n\"

n n is

*

(2)

-4.

(3)

E(1)

|

**)t +9

tl

oooo

-z

|

1 u -z

oo10

+9

+ö

z

*+ - 3 . ' l + 3 + 5 + 2 . 7* + 8 +9

a generating

For

1

:t

*

= -2.'l +2 +7 +8

= O

0

1

0-3

1

set.

and

E(2)

to diagrams of

leads

the contraction

type

D4, D5 For

resp., and we argue as in case 87. r1) to the diagram of consider the contraction !'"',

EG,

type

7 .t

5---)3J1(-8(-9 If

r 1l n'''X

< O,

the

R(X).

of

would be positive. zi,

L € {5,3,8,9}

woufd be positive main the

also

1,

2

1

direct

of

positive.

is

,1 ) O,

Then

,7 = 1.

So

the dimension

11 2 1

inde-

If z7 = O' Assume zj = 1. o 1 1 1 1 and the defect of

in these cases.

2

Let

otherwise

1,

1

two of

Then at least

must be one and the defect of

1 v:ltz --€

Hence

cases for

followlng

1

type

must be of dimension

T'(yo)

value

n(3)

would have negative defect.

T'(Vo)

types

dimenslon

be the dimension type of an direct

=9 t1 "5 "3 "g composable summand %

Vo

possible

which the

for

R(x), of z-t

"r,,**-rra.

18

are

there

2., > 2 type

2

2

2

T'(Vo)

and there

re-

of 1

and

to

submodules

2 1.

tz5z

The

first

three

defect.

So

cases

vroul-d lead

a

of

I

and

of r'(v^)

positive :

E(3).

rf

has dimensJ.on type

Vo

n(3)x a o,

indecompcsable summands vl

value

Again,

by simple

2

3

2

1

there are 10 possj-ble dimension

types of ilt..a negative.

1

elimination,

of

R(X) havins

we conclude that

n(3)

- 81 -

I1

must have dinension

F4l,

First Fl

notlce via

Let

that

E2

Et_ -F 1 ' ? 2 )

is

.r

,rr.,) = 1.

such rhat

",

an isomorphisrn of

{r")=o-o-r3,-Fr*Fs,l = L q r t l o - F 3 -F 2- F l - r , set nith

First

- 2 . 1 +* 2 +*5

(21

t

%

of

I -

*

R(X)

E(1)

to

t

rf

n(1)x a o,

and

T"(vl)

Fo!

e(z)

:

'2 "

vector

F1" '1

M F1 F2

spaces.

Then

ft

1 -2

1

= o

1

o

t.

2

1

,r(1,2)

then there is (2,1 ,1) .

So

83.

a di.rect summand dlm(T' (%)

1.

Of course the restrictions

1 -

5

=

of T,(%)

remain indecomposable, hence are

Hence T,(Vo) :

there is c+E(1)

and

to the diagram of type

o,

n(1)x,

isomorphic by (3.1.1). rf

=

a generating

O

o f d i m e n sj . o n t y p e O

2Ml

o , rf, - F, E z I

F1

=O

the contraction

5.

dlm E(1) = O and

*

-2.1 +3 +2.5

2l(1'2)

So

of

equations

(1)

consider

T" (Vt ) 3 g+n (3)

can be considered as subfield

-

tr, " "" 2

(=

and

2M.t, because Ut*1 , . E;

I

type

a summand Vt

E(1). of dimension type

as above.

we argue similar.

Jutt'=o-F2*Fr-"n*rsJ [eQl-o-F2-Fr9ri-")

with

eguations

(2,2,1)

-82-

n(1) -

-2.1*+2.2*+5*=o

2-2

o

1

and

=o

1-2

1

o

isagenerating

n(2)--2.1*+2+4' set,

)-)

(, : F;6nM., - F1

srhere

(3 t.,

The proof

with

type

rr = -2 1* + 2t = O

equation

Fn.,.

E = (Etrrer)

is a generating set,

rt}

tMn

Fn-rnodules.

an isomorphisrn of

goes now as in the case of

{E = o - Fj'

dz,t,

is

F.,-module)

as

(p ! F; + F1€

is such that

1 -2,

the map

while

+ F1 is defined as dual maP to an isomorphism ,e'rz EiOrM1 (:o F1, ). . E 3 z- F 1 o ltz zil ltz! '1 '2 Consider (1,3),

t

If

R(x)

T'(v-;

If

nX

% 3 g.

(3,2) and of course

G2.

and

there is a summand V.

of

3 c+e.

T"(yl)

Sinilar.

l{e have proved now that K-modulation set

(3,1)

of dimension tyge ( O,

type

then there nust be a direct

> o,

nx

of

type

GzZt

1.

summand ru

to the Dynkin graph of

the contraction

in

M

tion

Direct

every Euclidian o

and orientation

there

Thus we can formulate

!([,1,n).

3.2.8 Theorem:

for

(r,il)

Let

the product of

H(M,0)

R(t) , d.""tibed

in

calculation

orientation and

ß.2.7) .

shows also:

h

exists

theorem.

graph with n.

uniserial

with

a generating

the following

be an Euclidian

rtt and admissible

(l,g)

graph

Then

K-modula-

R(M,o)

subcategories

is

- 83 -

3.2.9

Coro1lar: h

Let

n

the nunber of

be the number of vertices elements

representations.Then all .

slmple

glven

regular

by the

L "

a generating

in

OShS3

(f,d)

of

set

and

of regular

andthenumber

non homogeneous representations

I

of is

formula

L = n+h_2 The number

h

is

the graph

of If

0a

is

Ärrtn > l)

an orbit

representatlon

vrhere

Let

1-

independent

of

of

a simple

x € l(M,n).

sion type, if

the

length

or

regular of

of

in

the case

2. nonhomogeneous

C.,

then

0+.

T h e n I € l(M,a)

dim X € N.,

except

h = 1

when

under the action

denotes

O

otherwise

is

of continuous of discrete

dimen-

di.mension

type. 3.2.10 Corollar:

The mapping

between all

indecomposable

mension type and all Proof: Iar Ler

dirn : L(M,n1 - qf

By (3.2.1) representations {s(t) llsr
dirn of

representations

positive

induces L(M,a)

c L(M,0)

induces a bijection

roots

a bijection and all

of

of discrete

di-

(f,d). between all

non regu-

roots of non-zero defect.

be a generating

set of regular

-84-

No!,, it

representations. rooot

of defect

%

E ."*(t) rlss
< h

contained

in the orbit

rn fact, module just type.

type if R

(t)

x(t)

- aim e(t)

for

the number of elernents

o.rd"r the action

the notion

€ R

(t)

,

for

once as composition

of c. From (1.2.7)

of nodules

and of homogeneous representations

E(t) €R(M,n)

some

foll-ows now easily.

in general

that

as

can be ldritten

< !r

ä

r1+r2<1t,

of *(t)

ttre Corollar

Note,

dimension

where

and 11 'r2 €II with

and (3.2.8)

Remark:

,

zero with

every

I that

can be seen lsee

is

of continous

doesn't

an element of a generating

which every factor,

is

of

element of continuous

o

(t)

coincide.

set,

then the

appears

dimension

-

85-

S 3

The homoqeneous representatlons

Let

(r,1)

and let

be again an Euclidian H(M,n)

be the full

honogeneous representations

3.3.1.

Proposltion:

graph with

subcategory of of

H(M,a) is

of all

Proof: Ler in

simple

Let

Et,

Qo

(E1Ezl

Now Hon(81,H) dlnl of

H(i,l,n)

that

(M,n)

of all

subcategory of

Furthermore

subcategory

fi(M,o) =

indexed by the sät " I r US ",

homogeneous representations.

EZ € H(rtt,n) be simple,

= Bf"

(kl

Ir,

dirn (Hr) = ki

(non-symnetric)

!z !1)

lr

,rrrj:;:';:rm

- di*x

Hom(El ,EZ)

dirn* Extl (Et,EZ) .

I O if

and only

Ef

if

are mutually

jrcongruous

(2.1) and the proposition

It

is

easy to see

K

is

infini_te.

that

defined

= o

g

{2,

Horn(H,,Ez) - dim* extl (g1,H2) . Therefore

sence of

Remark:

L(M,n)

an exact abelian

an uniserial

Qr r Qr * g be the "rn (2.1). Then, by (2.1.4) Br"

K-realizatlon

(Jt{rO).

L(ü,0) closed under extensions. irhere UH is

qraphs

öf Euclidtan

and then the simple

components in

objects

L(M,0) in

the

follov/s from (2.1.7t.

we don't

need here the assumption,

-

3.3.2

a) Let x € l(lt{,n)

Corollary:

morphism ring R

an

is

F-bimodule over

b)

is

L(M,a)

a) rf

some

k € f.

If

is

I

by its

b) Suppose there

is

a full

a)

and regrrlar,

"/.ltnl

= F = Kr,

di-mensional over above

embedding

T

Next we will Thus let central over

[(M,n)

and

G, as which

K

(R) and

x ring.

T(R)

of

R

y.

and

Consider

is

indecomporing.

By

have to be one

J(R)/,

= J(R). But by J- (R) cannot exist such an

of tame type.

rings

and 1et

of

a local

there

the notation

be division K

is

Loewy factors

is

uniquely

Kr . Consider

cannot be a division

in particular

introduce

F,G

action

R

= 2, therefore

and

subfield F

and all

K',

(J(R))

di\,

R

fron

length'

R = End*, (xry)

R = End (T(R)), hence

because

easily

folLows

and by its

some f ield

then

hence

for

embedding

wi-th trivial

= K',

T(R) e L(M,o). Then sable

exact

L(tt{,C) for

soc R = J(R) = K' 6 K', *r",*,

socle

regular

sinple

* ! tk

and

rnodule is

because every regular

/$2,yz,xyryx),

Furthermore

the statement

regular,

T : rnod*K' < x,y > <4 R = K'(xry)

".

is prejective

x

then

t

type.

of tame representation

(3.2.'1) and (3.3.1), determined

*r",",

F. In particular

R are one

of

Loewy factors

such that

F

ring

a division

is

and all

x d R(M,o)

endo-

be lndecomposablewith

R. Then there

dimensional

Proof:

85-

of

finite

representations dimenslonal

fMG

be a bimodule,

acts

centrally.

Let

= dim"(r,r")

and

of

a bimodule.

over a cornmon finite

dirnensional

(dn"rd.n.

(I,d)

'a

= 1 .

"rra-.noo".

the

''t

.2

modulation

with M

d., , of

(1,a1

5y

setting

dr''

= din,("M)

F1 =

,82 "

= G,

'

-

=

rM2

rMc

and

=

Zut

87-

I{o*F(FIi{,FF). Choose the orientation

f,MC-

o

(d12'd21) (f,d)

of i.e.

1 .

by

. . 2. Then define

a representation

spaces

Vl

over

F

,p : V1 O,1,1"-Vr. (v1,v2,6;

to

v2 I

and Let

, where

-

6 : vl

(v1rv2ra)

V1

and

? ^12

a

is

v2

erith

cNF

defined

over

cNF

=

as in

tations

are homogeneous. We will

L*(FMc). Notice

for

we will

an arbitrary

S

orientation

is

type,

of wild

following

bimodules. give

So we will

partially

3.3.3

Lemma: Let lation

is

will

of

(r,d)

of

neither

i"

of triples

.r, F-linear

Ln (FMc)

is

with

isomorphic ( V 1, V 2 d ) ,

is

Ä.,.'

of type

regular

or

H(FMG)

First

resp.

we will

modulati-on

show that

in

the next

orientation

H,r-(FMc).

try in

M

to H(M,0)

and

L(M,ny

situati_on

The

of

to the case of

paragraphs

and will

above.

components done in

be an arbitrary

of

represen-

Dynkin nor Euclidian.

the problems

incongruous

map

(V1,V2,a)*

case arl

the more general

components and let

u.H^r1o^*,1t"n&l \

consisting

reduce these questions

M and admissible

rncongruous

this

,(f,d)

treat bimodules

solutions

Remember the notation

in

e , second we witl

categories,

map

homogeneous representations

lemma, which hrorks in

Grothendleck

of two vector

hornogeneous representations

denote them by

graph

(f,d)

if

(f ,d)

tvro guestions.

sinple

Euclidi.an

adnissible

that

treat

of

q

given by

resp.

to be. L(M,n),

G-linear

FMcn. Note that

(2.2) . So, if

resp.

the set

and a

isomorphisrn is

[(F!'lc)

classify

G

are above and

we can speak about regular

rn the forrowi-ng

consists

rMG

Lrt(FMG) be the category

l(FI,tG), where this

where

v2

of

l(FMc)

(2.j).

valued graph with

modu_

e. Let X,y e L(M,0) be

End X = F, End y = G. Let

GNf

-

be a submodule of =

FMc

GNro

following i)

L(M,o)

with

Then there

with

bimodule

dual

a full

is

ernbedding

T(o,GG,o) = y,

= x,

T(Fr,o,o)

exact

and with

properties:

Every

ii)

Extl (x,y)

= lIorF(NF,FF).

T : L*(rM") -

88-

If

image under

GNF

of

all

= Extl (X,y)

modules

Every

T

has an

{X,y}-filtration.

U € L({x.y})

monomorphisrn

T

then the irnage of

X <+ U

consists

the following

with

and every

property:

epimorphisrn

U ->> Y

splits

Proof:

By definition

where I

VF,WG are vector

: VF -

Because

spaces over

of

triples

F

resp.

(Vr, W",e), G, and

g is an F-Iinear mapi (notice that ON,, GNäri cNF). (FF,o,o) is injective L*(FMc), the module in Wn O

(ker q,O,O) is (ker q,O,O)

a direct

is

O = alh

with

L,!(FMc) consists

nothing *ur

T(vF,wc,a) = 3

I

= t

Let di\

,

a r(

s u m m a n do f

(V',W",O).

than a direct

,r"*ands, '/kera,wc,

Of course

sum of copies of

(FF,O,O)

hence ).

so let

e

be iniective.

and assume the extensions

"U O-

Y..*2.

-r

fornr an F-basis o l
y.-y_ -a

-

-r

of

X. +O,

Y , = Y , -rX , = X a n d 1 S i S

-I

NF. Let andlet

-r

g. z=(

be the kernel O zr)Ä1. lsisl-'/Y

of

I

the canonical

nap

89

Because

p

is

lnjective,

ur 4wc

trae have

o cNr = I o*"

an

injective

F-linear

(l = dimc(vc)), i.e.

rnap

we have

a commutative diagram with exact columns:

I

I

I .l

t vr.$

w^ o ^N-

il

I

tl ll o (c^ a ^N-)

.'

3"r+

"T ul g !

I

J

+ 3tr-

o I

I

I

J

o

with

L = J. di\(NF)

A = (VF,Wc,o)

is

- di\,(vF)

the kernel

O (NF,cc,id) * J'-Lr such a map is (f.i..ir) rJ r

given

(1
tttir, But this

by its

13j<J,.1

= J.

r _ di\(vF),

i.e.

of a map

O ( F F , O , O ).

first

component,

which

is

of


' o J o r F 5 .-

o" F".

can be also be considered as a map (ttir)

, oJ @r X ._ol

x,

since

F = End x.

the

form

-90-

Consider

following

the

conunutative

and exact

diagram:

oo

tt ll

,t

O _______>

U -___> ,

O 1
y _______--_____) y* ________--___) O

o 1
!.t

(E 1
-Xr .

tt

JT

o --_____+

o zr _ 'l
________>

o

an epimorphism

L

ll

tl t3t z .......>

Ir

.t

i,

oo

By the usual

arguments there

exists

ttith

Y, embedded in

Z.

kernel <--*

I

by the

usual

---a=it=,

connecting

I,

homomorphism

Z. So we have an epimorphism I\ F _i-il, \

@ -z

-->>sL/vle

J

Define !,4'

T(VF,WG,q)

e

resP.

be defined

(fii,

: 0-

L)

)

be the

[t'(Fltc) Lt , )

a homomorphism. (h1I,

to

t

Ff 9 L.L

Jr

(A'= (vi,wi,o')):

.

J

as kernels O

t,

(o,ß) Ff

Ns, a

induces such

_ I 'I- ä e I -

1
kernel

Lr

Then -

Nn -

o

.

J

that

of

this of

epimorphism. (fiif) -J-

and let

Let

: O N" ._..' J (c,ß) : A -

an F-linear the

L

following

0 Fr. ' L A, be

mapping diagram

conmutes

_ 91 _

-

vr tl" +(r,'l

___* rn.order

vf *

of

the module

notice,

operation dte 1
S ""

that

on

"1,nrr,,

ll.iltJ

?

Z-r J a

O --------+

which

1si
-z' .

o __________)

f,",

T ( A r ) w e n e e d t h e following

Z:

the operation

g,

------+

,,,

to define T(cr,ß) : T(a) -

propertj,es Flrst

*, 54

? "1,,,

leaves robe

g € G

of

on

GNf

6

g, invarianr: Define i in the followj,ng push out diagram,

e Ii '|
so

way, as left i'e'

il

I

OY'
o xi+o 1< i < r

a g?t = , 1< i s r I si
acts on

$

Z, hence on

nultiplication

the folroning

plication

by

g

2j, r

because

Z/t -. soc z_ by

g

acts

N is G-module.

in

fact

,=?=ft' on

NF

in the same

as F_endomorphism,

diagram conmutes, where we denote the left on

Z lle

.tr

-----------+ o

I

lq

{,

and note that

an

U =

0Y*
o -_+

induces

z. ---

NF

by

9 o 1 € End"(Gn I

, Z / u 3 s -xrt f.icr I o t ls

g/l = r, .,=?=.

cNF):

rnulti-

-92-

So we get as J'

a connutative

x J-matrix

O

lrr",or

',

!

T(4') --->

lr

(c,3)

for

map

u , 3

is

T(A')

is

Z J'

easy to see that

ß and

(h11,).

s o c T ( A ) = e -y J

.L (hrl,) L'

of course

X ----+

o

T(c,ß)

I O

t(4)/_^^ ' s o c -r,(^Ar , , = ( E -X S

and

a homomorphism then

T

r

is

exact,

T ( O , G G , O )= I ,

For the proof

of

(orcc,o)

ii)

property,then L(ix,Y])). in

i)

a is and

note that

projecti-ve.

the property in ii). If

so

to the whole of

construction

steps

L !j-:Ly,e,

O o

A = (V",W",ro), where

tations

(in

(f'...-.',

s

induces a c-linear

by resrricrion to soc T(A), rhis induces 3,", and we can go backwards, i.e. T is ful-L.

(the extensj-on of

and

gX__-_-_____)o

J'.I
induced by

-

!

ß : @ _Z J rt

considered

J-L=di,nVF, T(A) -{

c:

is

L

lunt

o Z ->

I O. Obviousl-y

withS=I If

is

(f..,) Xr -r4

O

ü

JtT(a,B)

Wä

J ',lsisr *

"l

I

where

g : Wc -

over G):

----------> gZ------->

O-->T(A)

o ------)

(where

diagram

is

an exact

Lr,(rM"),

i.e.

furl

also

not monomorphic, is

ernbedding;

to represen_

trivial).

By

im T c l(tx,y]). (FF,O,O)

Therefore

u e [({X,y})

T

every

is

injective

erement of

in irn T

is indecomposable satisfying

In(FttO), must have this

a n d V 'ts^o^ c- ,ur i O X forsome J,L€N JiThis means that g c a n b e e m b e d d e di n o z. Now arrsocUIOy

can be reversed

and we get a preimage of

U."

-93-

Renark:

Of course

submodules

modulation f{(l{,a) glven

image of

T

can be deterroined

cNr

be an Euclidian

i{ and admissible

graph not of type

orientation

we can assume of course that in

radical

the tables. of

also

for

proper

or

i'

wlth

of Extl (x,y) .

(f,d)

Let now

the

Let

nf

Q. rn order is

ö

to investigate

one of the orientatlons

be the generatj_ng element of

rrith r resDect respect

Of

i.,.,

to

.Qa. Let r-a+

5

N-,

the

i F L^ the !L^ source -^a fF b. io

E. o

8

(t,d)

if

6

is

of

-7 E^

ö

type

4

-41 F.

'42 'l

G" n

3

G2z

and

,

n.t 1 Notice

that

restricted Ii

= tt,i

otherwise

to

f'.

is

otherwise

short

to

root

the maximal

First,

non negative

for

in

the

of

Now the Lermnafollows

is

by

rirhere di = o

,r^

investigatiän

a root

cases

of

=d

and

shows,

(f',d),

in

fact

Fery

Bn, CDn, Fn1 and GI r

root. I,I

€ Of'

components. This

For the notation

case by case.

f,

(1,,d,)

defined

rhen a direct

restricted

the maximal

a Dynkin graph

= (yi) €or

Let I

(4f = (.t,i)).

3.3.4 Lenrna: I

Proof:

= f

f\{io}

long

say gives

and short

easily

x 2 1, a partial roots

inspecting

if

x _ I

ordering

see for

has on

ef'

example [ 4 | .

the Euclidian

graphs

-

(M',o')

Define the K-realizatlon let

yr = (Yir

y

to

restriction

of

defined by

Yi = Yi

{i,i}cr'g

f

(l',d')

of

!r r where

tyPe

of dimension

Y = (Yrrrar)

(3.1 -1) . Let

l',

and

by restriction

be the up to isomorphism uniquely

e L(M',Q')

i9'r)

rePresentation

deternined

94-

is

L'

the

€ t(M,a)

= O, j(9i = j9r. tf "ro is an inand .t0t= o otherwise. Then of course ! i € f'

if

of

(M,O).

we will

appty

(3.3.3)

= X

injective,

decomposable representation In the

following

Notice

that

F.

-i-

and

I

is

with

because

L(M,n).

F,

, Ye

i

€lisasource

_I

o (2.1.3) .

3.3.5 Lemma: i) ii)

Hom(X,Y)= O = Hom(Y,I). =F

EndX=F. 10P

iii)

Ext'(x,Y)

= ^N-

and

aredi-visionrings-

EndY=G

except if

A..,

is a bimodule of type

( r , g ) l = or" .l n. dö rr.H .t" cNr ," "r .r n" Proof:

i)

factor

of

morphic ii)

Because

Of

!.

End Y 3 End v',

= Ei -o

In particular

image of course

= O, I Yi ^o

it

be a submodule or

an ePi-

Y.

End X = F,^ th"

t""oit

= F

a division

is

for

follows

g Fi for G ancl (2 .2.4) . In fact 'o r | jo a 0' must have the sarne length

iii)

as composition

c a n n o t occur

cannot

{ .,-

!

as

=

; iö*.j

d.,

J'o

y' € of

y;. r

Because

from (3.1.1) '

some j o e f ' ,

By i) and (2.1.4)

d i n ^, , E x r 1- ( x- , y ) = - n Il t x- , v- l

ring.

'.

(2.2.12)

where the root

-9s-

Inspectlng (f ,d)

every

is

= tt^

of

case it

type

iCrr, h"r"

I'

is

of

e Q'

of

gives

this

of

) = 1, therefore EndY=EndYr

Let

This

is

fir.,

T:

type

= ,ro, c'

and But

some O < r € N, therefore

is

the maxirnal

is

the unlque

w €

wn, implies

i.e.

dim,ON) = 1, and

d"=ir"a.

""

tells

us that

there

of course

T

is a furl

is described

(3.3.3

in

we can speak about

exact

is of the same type

rMc

ii).

Because all

modules of

if

A e g*(füC).

and only

if

Then

T(A)

is

A

is

of

continuous

of continuous

continuous

L = dln

ker

(f ,d) of

be not of

continuous

type

4

is

V, - dim ker or) 6örr. fh.r,

dimension

nf = x + I

that

Therefore

pF + (din

type

because

of continuous

if

of

is

of

and only

if

dimension

T

type.

in (3.3.3)

= dirn V".

rüC

(f ,d)

dimension

dimension

Let A = (V,W, q) . Then by the construction dtun ( T A ) = d t u n w c . d i r X rrith ! + L.din

Notice

1

j € fr

l,=w(i),

Proof:

is

I,

(f,d)

type.

type

A

length

except j_f

o

are Eucridian

3.3.6 ternma: Let

Let

if

=

dim(cN)

a rong root.

for

L*(FMG) -!(M,o).

graphs

di.mension

except

the knowredge ot tt"

tro

=F1 . Now f,t=4.fr,

GNF. AIso the funageof

involved

of

hence

11r,4,),

(3.3.3) cNFo. Then

enbedding as

is

Y' = C-rrF,

type

=

#c

as above).

11',g')

Dynkln graph.

simple long root

CNf

NF = 4. Now

now ln every case

6örr. H"."

Is of

= 2

dllr(NF)

,Jo

type

root

that

dln

(jo e f '

uth(Nr),/f*

of

follows

is type

type

dim V, = dim !tO.

not of if

and

4,

type

ftrr.

and onLy if

-

96-

ditn(TA) = dln

W - ( d l n _f + d f u o X ) = d l n W ^ . _nl-- , u

i.e.

lf

contlnuous

(fral)

f4

u.

Is of

iCrr.

fh.r,

type,

if

ls

flC

and only

of

dlmenslon

type

i.,,,

Let

type.

and

of

ls

A

d1n VF = 2 dim WG. But thls

if

and only type

be of

continuous

= dlm WC(+E Y + 2 dtn x) = dim wc.lr,

din(rA)

lf

ls

dimenslon to

equivalent

because here

9r = 2 x + YNotice

that

A€ I-*(E$G)

r(A) ls

€ t(M,o)

lndecomposable,

composable, 1f and only

lf

lndecomposable,

ls

because

resp.

A

End(A) 3 Sna(f(a))

lf

and only

are lnde-

T(A) is

lf

a local

ring.

3 . . 3 . 7L e m m a : T ( f l * ( E J , ! G ) 1c R ( M , n ) .

Proof: Recall in

the

R(M,a)

functor o(t)

there

- up to

+ 1

of

the

in orbits

(3.2.7).

A, direct

(see 2.1.7) extensions.

(1 < t s h)

0(t)

inspection

(o s r(t)

isomorphisms dlrnension

(in

sion ring

non homogeneous representations

regular

is a unigue representatlon

type

R(M,o)),

as composition factor '

slmple

dlvided

n.lt) ) e p(t) r, l

continuous lt

notion

Ct

= (' A l Jt), the

(3.3.6).

Obvious by

H is

and

uniquely with

B(t)

- regular

Then

nxt(e(t) rg(t) ) = i.e. !rHH,

a fu1l

exact

o(t)

subcategory of

closed

be

of

length

occurs just

L11e(t) i) [(M,0)

B(t)

rp.dule

ena(g(t) )

4

orbit

-

= A(t)

and cornposition

4(t)

ß.2.7).

in every

A{ + o. Let ao

determined

socle

E(t)

hrlth

every element of

i.e. of

shows that a+r(t)

< r*) c

under the coxeter

once

is a dlvi= 3(t) under

+

-97-

3.3.8

Theoren: T induces an equivalence. between (1) (h) A *. . .* e x f l ( M , 0 ) = : C ( r r , t , e. )

Proof :

We have to shobr T(H,t(F!Ic))= C(M,a). By (3.3.7), (3.3.6)

and (3'3'3

ii)

it

is

enough to show thatanindecornposabre

ä € R(M,a)

of continuous

if

and only

if

ls

J.somorphic to a dj.rect

dlrnZ ="=

contains

sum of

sum of .

let

copies

"! E fr

sabLe direct

direct

=o

q.

of

itri) ) , hence

hence Z, contains

some simple

I t s t < h),

so every

direct

with

as above,

E

Furthermore every

because jo

restricted

= a"(z) j e r

Z_ is of to

image of regular io

u.

defect W is

and

is

all

Let

zero.

i€r.

submodules of thj-s j-s true

Then

q

is

has negati.ve defect.

-

,o.

öc(l{) *j^

*

w,

c(M,n).

rn particular,

can be seen.

ac(Z/u) - - z. of

regurar,

non homogeneous module

z cannot be contained in q

for

be a indecompo-

E

non homogeneous. But then

regular

Z, but this

such that

19",9'),

ur=i,

forall

In particular

W1 * O, as easily

neishbour

an epimorphic

i.e.

summand of

ö"(u)

by

isomorphictoa

regular,

summands of

nto = O,1w = (W1,

e1a(t)

is

!-

(3.2.2).

defect

s'nmand of

,

. ttoiice that di.m U = J.I o n, = (nr)eqr, because dim z eN.nf.

Z e C(M,n). Because

indecomposabre

bour

z = (zt,iq:1

* = Ei

of

where

Z_ have non positive

ZrU

such that

".a € L(M,n)

,/u"L"

and

C(M,0),

"]

=o,iti

usz

g

of

U = (Ui,iüj)

andU,

isasource,

2, wltnJ=_:,

for

?

-otr^

direct

g

copies

modure

contaj_ned in

g =

a submodule

fQrio+jer

Because io

First

g-

dimension type is

(zil , anddefine

1ti=iei

all

flnhUC)

,nur, is

öc(L). ;r.;, -o

;",

< O. So there

is

O. Note that

!'

of

-o

*j

hence

= o w

is

by (3.2.2),

impossible

always a root

Ler

at

least

neigh_

= dim W

the Dynkin diagrarn

-

98 -

Nou we have to dlstlnguish

11',9').

cases:

several

/

BDrrr Drrr E5' E.7, Eg, F42 and G22.

a) Type

Here y'

dim u = z, - o - - oI

hence

are at most

zi -o

sumnands in

=,,

tro

rmPlies

- Tnis

that

of

g.

into

By maximality

of

I'

all

gher:e

a direct these equals

restrlcted

to

I'

Furthermore

the

first

dim U

Y, because

= 1r

ö"(io)

a decornposltion

modules.

to

must be isomorphic

Also

ac(U) = - z.

and

indecomposabele

sum of

Rr,.

the maximal root of

is

z. . vr. to ör, ana 6örr.

fr,

b) Type Again

is

I'

the maxlmal root

component of every

root

(w' = (wi) € Rf ,),

because

there

just

are

be isomorphic

Here

at most

is

to

and again

U' which

must

Y' € Rf,.

of

by maximality

!

y,

of

summands i.n a decomposition

,j

*.i = t

1, hence

din U - 21

+ O. llow

ri

-

vn-1 3 1

1, and

= 1, because only

ti-r

for

n-1

dim U = 2.,

= 1, hence

JO

JO

roots of

(f',d'),

1. All

first

component

d) Type Again

' I

U. But

*l'=

2. So lt'

all is

of

a decomposition

gSI

root

the maximal short

is

I'

jo = t

in

Rf,

g'n.

c) Type

t.i

in

Rr,.

in

< I'

(f',d').

of

v = (vr)

roots joined

t

are just

= 1, hence

,l_

Therefore

Rr,'

by an edge with

and there tl

Furthermore

")- O = ,1,

1o' Now summands therefore

Jo

which are greater than g'

and therefore

= L',

I'r

have

i.€.

asdesired. io.r. I'

is

the maximal short

.1 = 1, therefore

dim U = zl'

root

of

(f',d').

L. As above

*i

Furthermore = 1, hence there

are

I

-99-

just is

z, only

dlrect one root

has defect e) Type

surulands in greater

zero.

a decomposition

than

U. In

namely the naximal

I, W g y.

As above

of

R, r there root,

which

F4t. that

-Notice

every

root in Rf, greäter than has defect <_2. L, N o w :d: i m U = l and ac(U) = -rL , so there are at most 2 z.t,\ _ 1 0 tto = t"t direct sunmands in a decomposition i of u. Furthermore 23 = 21 but u2 . 2 ior all roots v = (vi) . Rf,. This implies *ä = z' Therefore the nurnber of indecomposable summands occuring in a decomposltion of 1 just g is zio = Z 21 and the defect of every such summand is - 1. Again we conclude W g y.

f) rype drr. (a,b) -

write

ac(w) < o

* bit. Then !' "iZ a n d \ , r 2+ o . B u t J . I '

withOSk.l

,k2et

shows so far

I{ = y

forces

or

s z'

k,=k,=O

Z'e

c f i , l, a ) I

because

t(fin(Fttc) ).

B(1) c c(M,o)

and E s z

for

with

lt

Er = O ^o

(E = (E.. r. iej)). g=?y,rhen

Z'.lf

(3.2.7,

u s z

r,et

so let

2y.l=yZ.This

z rF C(M,o). But

din z = dim 2,.

ugg,

of

Y,, the

i.e. theorem.

in

of course

sirapre non homogeneous modure as above, then

U' < z'

byabovehence R(M,o), so

y G T(fl,r(F,Ftc) )

E < u

because

be the corresponding

J

soc Z, = A(1) g E

of copies of the proof

some regurar

< t < h).' Define

E e ta(tl

of

because

= k1 (2,1) + kz(1,1) + k3(3,2)

llt

A''',

(3,2)

as desired.

we have shown

then there is

= (1,1),(2,1)

u

E<2,.

submodule

Burby

cannot be a direct

uy (:.3.3

sum

ii) . This finishes

-100-

We have reduced

nohr our

case of Euclidian

graphs of

paragraph

investigate

Bimodules

S 4

let

it

of

din

FM

il" = 1

tatj-ons.

Recall

which acts

that

F

centrally

on

dim I G : K] = g < -. geometry,

K

finite

fields

resp.

n = (1,1) root

c

contain

of

rMG.

using

(If

FMG,

rM = 4,

rn general

K,

subfield

: K] = f ( - and

counting

similar

frorn algebraic can be

results Let

arguments.

N

be the

forrn

Q(x) = (x., -2xr)', n = (2,'l).

resp.

dir

a conmon central

but

quadratic

A11, i.e.

or

homogeneous represen-

simple

dirn [F

to be infinite,

the corresponding

an imaginary

AtZ

functors.)

coxeter

all

pl"lc, with

radical

generated by

bimodule"

= 1, dim MO = 4.

rM

the partial

and

for

Q ( x ) = ( x . , '- x r ) -

dim

to classify

found also

)))

Euclidian

Because we wi.1l use here arguments

we suppose

of

in the next

t"

{r.

be a birnodule of type

one of hard

for

fl(M,a)) to the

tvpe

= 2 = ilim MC or

seems to be very

resp.

t (FMc)

fMC

we apply

i., .,

type

Euclidian

paragraph let

In this

din

we w-ill

(to desribe

problem

first

a positive

X=

(x.,,xr) €lR-,

CaIl an element frorn N.n

element

of

Ri

a !{ql.gg

of

M

F..G'

3.4.1

Proposition:

Let

Then

din

X

there

is

just

and for

every

isomorphic type is

is

I.

X -

(xl ,xr,to) € L(f"[G)

a imaginary

or Weyl root.

one indecomposable imagj.nary

representations

y

root i.n

be indecomposable. For every Weyl root

X € L(E.MG) with there

are

infinite

x

dim X = xr many non

H (FMG) = R (FMc) , whose dimension

-101

Proof:

We have only

(Notice if

that

dim X

I

X = n

with

I

Uq

that

G

G

G

has dimension

infinitelVlorbits

in

modules

dim X = n- Let

direct

r

with

summand of

the

smallest

dln

x = n

slmple

inag:.nary root is

on

than

As remarked earlier,

it

of

dln

G:Choose It

is

Let define

dim

,.x O+xo€Fxc

Now xo -

n e M\X. nappings

Urr, so every orbit

= 1 = din

F

Of course from

is

but

there

,, are

j.s _ up to

there

Because

n

is

x srith

a well

defines xo,m

treat

f.XC X"r

n

is

gJ,!", which

be a proper

deflned

by

of

subbirnodule

by

F

Irith

xo.9=go.*o.

isomorphj.sm of skew

an isomorphism

F

case,

forces

and we can identify o:c.*F

51 aII

noe, a speciar

is a basis of

Finto

infinite.

hard to describe

bimodule

anddefine o

of

non isomorphic

type

seems to be very

1

6re

dim G - dim Ur, =e(n) = O.

homogeneous module

= 2 = dlm MG. Let f"t{

easy to see that

fields.

Urr, where the orbits

so the number of non isomorphic

case of a non simple

F"Mc. Then

with

K, together

such modures.

simple honogeneous modures- so we wilr

course

is

be not hornogeneous. Then every

number of

srmpre regular.

Ä

dirn un. Therefore

{

every

it

the non isomorph!-c modules

honogeneous modules of dimension

nannely the

over

on

must be a WeyI root,

a finite

and only

by (3.3.1)

U,.,, hence infinit{many

I

isomorphi.sm - only

Ug

(2.3.2)

trivlally

smarler

if

of modules

wlth

n . N o e ra s i n acts

type

homogeneous modures of dimensioa

acting

are in bijection

r*1c+61

by (3.2.10).

Furthernore

a variety

Aroup

of dlmension type

under

dimension

to the class

corresponds

algebraic

under

statement

many simpre

(2.3)

there

of course

^^1

infinite

a llnear

of

of continuous

is an imaginary root.)

g. Recall

äir

to prove the last ls

enough to find type

-

FxF-

rFr.

and we may "Mr

-

102

mf=fo*o*ft.*

Obviously,

and

6

both

e

G1

are

*o

f)o

forf€F.

additive,

*

(f1

and

the

m = m(f1

f2)e

equality

=

fr)

=

(mf.,)f,

-_ r' .' 16 f 2 + r ) r ) l * o + ( f i r l l . m shows that vation.

e is

Fxo * Fem

AIso

M. Thus

e is

right

is

F. Conversely,

(e,1)-derivation given

F-structure

of

F. Let

F O O fixes

and

3.4.2

Lemma: The non simple

Euclidian

bimodules of the form

M(e,ö),

6 annihilates

of

Iet

hence equal

to

e be an auto-

aM(erö)

: =

uF

oI|F

a,b, f € F.

K.)

e

so we have shown:

bimodules

are just

e € Aut F, ö an

tile

(e,1)-derivation

F.

Remark:

The representation

choice

-x-- o €

of

o ' G -F. e is

f.

above depends of as M(e,6) fMC ^€ +}.ö ^hoice of the isomorphism

s

fixes

an automorphisrn and

T',

with

ring fi

Lhe

of

Y ^i ' -ö' F"G'

Of course

skew polynomial .i-

\,

an (e,1)-deri-

= 2- AIso e i s a n a u t o r n o r p h i s m d i . r nM ( e , ö ) " = d i m FM(e,ö) is an F. F submodule, so I"1(e,6) is not simple. (Of course

Since

i - ^

is

by

(a,b) . f = (af + bf ö, bf e) for

If

6

F-subspace of

a right

an automorphism of

morphism and 6 an with

F and that

an endomorphism of

K. 6 an

R = F[T;e,61 a F, n €X{,

(e,1)-derivation

of

F, the

has as elements the formal

and the multiplication

is

sums

induced

-103-

by ruultlplication

ln

F

and the

Tf=fer+f6 Let

mod R

denote

finite

dimensional

3.4.3.

Theorem:

rule

for

the

FCE

category

over

of

all

right

R-modules

$rhich are

F.

H(r.M(e,6)r)

j-s an uniserial

is

category

equivalent with

to

objects

nod R x U

of

arbitrary

where

U

length

with

one sirnple object.

Proof:

= (1,O),

Letel

=FOFr

(O,1) €l'i(e,6)

=",|t,

N=Fel

"Z=

-(M/N)F, (trt= M(e,6)), the canocical projection. E-t!, s = (Fp, of "/", n ) is a simple hornogeneous representation

n : Fr, O Then

r

because

,,!t,,

subcategory

of

sition

factors

direct

factor

(3.3.1)

is

surjectlve

and

H(FMF)

of all

isomorphic

to

of

S. Then

show mod R =

theorem. If

(Xr,Y"rc)

A=

{a€Xle(a3er)

s=

{b€Xl3a€x:

F-subspaces (aoe.,

of

Xr:

This

V = (\,Yr,

of

9, )

is

i.e.

U

be the

is

uniserial,

all

of

E!1,,

compo-

and a

So using the notation T,

full

of

in order to prove the

let

and

e(aeel is

+b6e2)=o}. obvlous

for

Note that

A and B are

A. With a g el + b I

e,

also

1u, +bf(t with

b

no\., a simple

+bf ""., also bf

oe2

belonqs to

belongs B.

homogeneous representati_on

= (aF, a(aF O ,,\),c,) q to aF, satisfies a^ V' > O, where

O + a e A, then !' of

=O}

U

[ UH =, H€ S H+S

a representation

+boer)r{"-le)=(a

to the kernel If

is

Let

homogeneous modules with

by (:.3.1).

H(FMF)

we will

dfun S = n.

with a^

and


denotes the defect

- 1 0 4-

with y

respect

to

the

Coxeter

ä. Vr = O

is regular,

slmple.

Obviously

morphic

to

Vr

by surjectivity

and

Xf,

=

XF. If

q(bOel),

g -

c defines A*r

€ L(F\')

where

o(a O er)

= aT

and that

ö

finite

that

all

T

r\,

-bOe.,

defined

into

is

for

is

!Z

onto

D:

F, define

Because of

T

R

V-X*r an e

o e1) = a

tp ls

such that

given

= (\,\,o)

6ta*l

easy to see that

Yr

a hornomorphism,

is given by,p(a

\

and

indeed F-linear Dö :

lmod

R,

some cases,

so

(see Appendix A).

But it

becornes

it

is

to give

classiflcation

C,

hopeless

polynonials

fields

(setting

very

This

usefull

{ t , 1)

0

there

in

a general would lead

in one variable

are simple

also

of

to a classification over arbitrary

F = K, a commutative

homoqeneous representations *02

and

provided

€ kere.

e = 6 = 1 in the theoren above). In particular, sinple

Xp

mod R. Conversely

over

-

of

module operation

r Yt*

from mod R into

K = 1R or

irreducible

comnutative

lso-

= dim Y,

\,

aT = b

p(aO 11 = 99. So

the sj-mple homogeneous modules: of

not

mod n a T.

example if

now clear

diln

an isonorphism

aOe2

The theorem above is

1.)

ls

n X O e1 = O, hence

kero

c = (c,B)

a € A. It

functor

is

a well

of

o

\

for

" Therefore

is

dimenslonal

e,

ir

y

S. So, if

where we set

because

a functor

R-module

Remarks:

is

T , and

!,t,YZ.

c isR-linear,

öo 1 1r.

that

action

then

for

p(? 6 e1)

then

A = A(V) = Or this on

Vr because V is

B = Xr because

becomes an R-module,

ro(aOer)

Because

I .

of

y € T,

if

to

A = O, i.e. thus

fMf

as usual.

hence V'=

lsomorphic

g. we conclude

ker g + X Q e1 = X O

Thus,

(3.2.2),

is

c

transformation

for

have dimension

fj-eld,

and

CC(1,1)C type ('1,1),

honogeneous representations

all whereas of

-105-

dimension 2.1 lf

(n,n)

tytrre FcGcE

are skew fields

dim Itc = illn -andif

CH

there are

then a sfunilar

= dirn

K

the

element of

H.

Appendix

g€H\G

theorem holds factor

flelcl

B we w1ll

treat

ring

of

of tarne representatlon

step ure w111 reduce this

Lernna:

Let

fonn and

(I,d)

(M,A)

on which ,J.1", (dirn,"!t) (dirn t* (E!{c) -

F, which

graphs

K

this

are of

rn the

the free F, where

over

corußute with

every

2., and 22.)

Tn

type.

finite

following

form are of wild

and Eulclidian ne vrilr

type.

show that

In a first

problem to the case of bimodules,

be a valued

graph with

be a K-realization

dimensional

has to be

type

type.

quadratic

z.1,22

K commutes with

of wild

graphs

indeflnite

of

R

FK("1,22),

an ex.rmple of

we have shorrn that. Dynkin

finite

H (FHH) . Eere

(In partlcular

so far

3.5.'t

o2 a F, ß2 e G,

with

for

of elements

Graphs and bimodules

graphs with

= dim cF = 2,

ln two non comnuting varlables

denotes

S 5

with

rG

c€G\Fr

replaced by a certain K-algebra

O
for each

skewfields acts

F

centrally,

Z 5, such that \) [ ( M 1 a ).

(f,d).

of and

G

quadratic

indefinlte

Then there exist

over

K, a blmodule

and which satisfies there

is

a fu1l

exact

embedding

-

Proof:

Iisted

In the

there.

(M,0) to

(1,{)

is

t(tr,{,0),

so *"

i.

(i.{,0) itself

second case we have an obvious

<--+

ltfi,äl

connected. By (1.1.2)

is

graph as submodule or a valued

an Eucli-dian

embedding

(fr,ä)

where

FMG

(dim

hrith

,"t't)

(f,q)

graph

the

ti,ä'l

exact

full

aenotes

(r,Q) = (i,d),

can assume

a bimodule

restriction

but this

means

(dim M") > 5

or

has the fonn (a,b)

There are three

(c,d) 2_3

different

1 ----->2

is

51 : L(M,0t) -

und

F2

t(1M2 O ZM:)

that

the compositions ab = 3 (that

dimensional

as a left

embeddinq

= (X,X &

(x,Y,q)

because

one indecomposable represen-

L(M'Oj)

given by also

and

respectively.

full

an obvious

l(M,ft2)

just

L(M,n3) kill Ft

cd=2or3.

0.1

o

namely

Now there

------> 3

1 _____-___>2<_ 3

-

and

orientations

92

z L(M,ar)

tation,

hrithab=3

1(-2--------+3

Also the functors SZ

(f,d)

We nay assume that

contains

of

106

with

means, vector

lM,

lM'2,y'id,,p), Sl

and 1s not

and it

tZ ti

is

easy to see

are embeddings,

at the same time

space and as a right

vector

onespace) .

-107

Because

abcd > 6, thls (frdr)

so let

we may also (M,O)

proves

be a proper

assume that

restricted

to

-

the

lemma in

(r,A)

subgraph of is

11"4')

of Euclidian

connected. i e f,,

l,et

11',4').

the second case. type.

(M"Q,)

Let

j € I\f,

be

be neighbours.

Conqider

pi € L( l,{', e, ) _c L( /t{,a) . of course (din pr) o, i * "-r so *i = (Qim a-t !i). is arbitrarily large, because = dirn p. -. p r ) dlm C-r *E, ö.(dim nf, where m denotes the order of

c

modulo the radical

By (2.2.4)

End (c-r

= Fi,

so "-r

polnts.

(note that and

Bi

or

di\

depending

= -eltc-t

!i,gj)

Exr(Fj,

c-r

= -"ltgl,

Ei)

on the orientation

respectj-vely.

pi,Fj)

o, and this

We consider

only

so we have to prove (din

,M)

the

dim

acts

"lA centrally K

is

the graph

assume in

(f,4):

to be infinite.

with

are arbitrarily F = Frr

!,Iith

we get a full

exact

representatj.on

a common subfield dj-m and

of

*G

type

of

realization

F and

= g < -.

G, whiihIn parti-

G.

For finite

must be rnodified

than the category

(a,b)

xr

we need some notation.

paragraph

this

now nothing

K

fi

case.

of wird

pM6, and dirn *F = f ( -, contained in the center of F

ments and proofs is

K

is

"i

= dii

first

on

$ 4 we wilr

[ (FMG)

t(rMc)

(dirn Mc) > S. First = b, dim MG = u,

Let

As in

noer' that

fj

dimensions

=extl(c-r pi,Fj) = ll,, and GNF rgc "j, enbedding [,r(FMc) <-+ L(M,a) by (3.3.3).

cular

< o).

are orthogonar

= dij pi)

c-.

O =

if

F.i

pi)

a"(gi!

Hence we get by (2.1.4) dirn* Exr(c-r

large,

pi)

(f',d');

Nr, of

K

the

s1i9ht1y.

the representations F

state-

Fo4G.

of

G. The corres-

-

ponding

Coxeter

108 -

transformation t\ -'1 , f ab

-a

tl

-11

u,

\\ / The nonsyrnmetric

bilinear

given

is

C

by the matrix

\

B = B?

form

given

is

by the matrix

-a'g) (t

B=

sI (x

\o

I

''x2) (:

ef t5,1) = x aIt =

i.e.

= (xl ,Y2), y = Q1,yZ)

r.r

;X)

€ Oz. The corresponding quadratic

form

Qf

is denoted by Q and i5 given also by B of course. As in $ 4 we call :a positiv root in element x = (x1 ,xr)e;2 Q2 weyl root and a positive än irnaginary

root,

In the following T : mod

if

we will

construct

K <21,22>

T consists

< O.

e(x)

the composition

T

wj-th a partial

(2.2)

set

Lefiuna:

Proof: Then

ä=(1,1),if

B(nx_,x_) nru

Of course it B(IorIo)

for

all

let

imaginary

only functor

loss

the image of roots.

simple rnodules, is

still

of generality

a that

full a>b.

5=(1,2).

1 < n e U.

is enough to prove

< -1 . Let

B(5,ä)

b = 1.

= f + 49 - 2ag = ag + 49 - 2ag = 49 - 2ag =

= 2S(2 - a) < -1, Then a >3

S -n

b=1

are

kilf

Coxeter

ernbedding, so ere can assume lrithout

b+|

3.5.2

of

functors

embedding

see that

of modules whose di.mension vectors Coxeter

If

exact

L(FtlG), and we will

Because the partial

exact

a ful-l

because

and, b7 fb=ag,

f = fb = a9 and f >g. Sc

B(ä,Io)

a > 5. Next let =

f +9-fb
b = 2. (2-b) =O.

-

If

b)2,

f >g

and

Thus in every case We vrill

consider

connutative copies ing

of

B(%,ä)=f

B(ä,-o)

field,

and let

a sinple

+9-fbsfe-

b)
< -t.

now a special

proposj.tlon

3.5.3

109-

case.

be a direct

K,MK,

one dimensional

K, = F = G

Assume that

is

a

sum of a number of

bimodule

K,NK,.

Then the

follow-

holds:

Proposition:

There is

an exact

T : m o d * K r1 " 1 r r |

full

ernbedding

(K,MK,).

Proof:LetdimK,It1=m(=a=b).Thenm)3,becauseK.MX,is neither

Dynkin nor Euclidian.

lsomorphic but

fixed

for

any vector

Of course,

space

x

over

Xm = X e ...

isomorphism

X

and

K'.

O X +

X 6

K,NK,

chooslng Xd

'

an arbitrary for

*,\r

ur"

every

m-times vector

space

equivalent

to

j.n a natural over ,1,"2

K',

over

K,

way. Let X is

it

where

[(K'm),

then

is

easy t,o see that

K'm

X be a

is

if

X

a K'-vector

is

a finite

,L*=

X, also

dimensional X, then

Ar(x) e X

K,-vector X

(i = j,2i

€ m o d ' !K ' (

2 1, r 2 ) ,

id)

: xm = X E

and if

c

K'(

z1rz2> -.

x € X), because

T(X) = X = (X,X,,p) e L(K,m), where

as algebra over

with

space and

beconres a

Now define

1,2u,id,...

dimensional

denoted by zr,zr.

is

a | = (2

is

as K'-K'-bimodure

> -module finite

F-'1z1rz2)

free

L(*,\,)

space, where multiplication

are two endomorphisms of

rnodulesetting

considered

K' a ,1,2,

induces two endomorphisms of

Conversely, q1ra2

X

K'.

------) x. rf x,x' K,K,m*, : X ..._Xr is a Rt
e Iinear

l

_ 110 -

g = (crcr)

maP, then (o,ß)

: X -ä

Xr

diagrarn a third

X O

defines

an homomorphisn

be an homornorphlsm. Now m 2 3, so in component exlsts,

forcing

shor/rs that

the proposition

is

is

F

full.

of

course

above we used that

because the corresponding

be a skewfield

Assume

fNf

the Euclidian

(i,,a) ,

T

is

an exact

embedding,

and

fNF

L(K,!,!K,) = !(K'n).

graph is

be a bimodule

a tree.

Consider the K-realizations

+ FFf. graph

Ä2,

,'/

\t

rFr F

\

,tr./ (l/',o), v,/

r,Fr \"

rNr L(M,o) and

L(/l',a)

ar:e not

This

For example,

one dj.mensional

\l

Then

following

and

proved.

Remark: In the proof true,

the

x'

-rIIl

T

Let

I Is

1""' _

Xr.

c = ß.

*,M6r

xt et*,q, This

C : I -

equivalent.

(M,o),

is let

over

F.

(M',0)

of

- 111 -

Let

norr

imply

3.5.4

be again

FM,

Note that

a > b

and

ag = bf

f 2 g.

Proposltion:

i)

components ii)

Let

slonal

of

Proof:

dim

Let

a : FF O

following

o -_+

f,Mc

for

a comrnutatlve fielil. of

L

dj_mension type

just

isomorphism every

Weyl root

are Weyl roots,

e [(fMc)

if

and

one indecomposable qos, al1

x.

fMC

of

I of

------)

surjective. dimension

Note

that

So, taking type

codirnension

1

*

are

1o resp.

I

--------+ o

____>

rMc

. t1u __________+

indecompo_

2. Consider

lnr*", I' ll .r t "-*

o

is

gj.ven just

_____> o

u

let

ker rg , the

yG

rMc

i.e.

%,

b=1,andtet

diagram

kere

4:

< O, so we

0(5)

be of dinension type

be G-li.near. a is

Irr-

o ------+

up to

-yc

modules

G-subspaces of

sqme one dlmen_

< f .

andy=c!,ir

and only

composable

is

of

(3.4.1).

b+1 f.MC

copies

one component

x < xo

X = (Fr,y,e)

=

F = c = K,,

X = x

din

as in

Y=Gorif

if

least

incongruous

ä. sum of

*(End Io)

with

can argue

if

at

with

I

O<4e22

sable

is

type

many mutuarry

a direct

K,NK,,

B y ( 2 . 2 ) there

i)

module

be not

bimodule

wirh

ä

There are infinite

dimension

Ft{C

Then there

ii)

arbltrary.

inde_ by the

the

-112-

Where

f

denotes

nultiplicatj-on (FF,Y,e)

the

vrith (FF,

and

G-linear

he F, and I ls the natural -!,t^ "/u,n) are isomorphic

between tvro such modules that

the

non isomorphic just

correspond sions

1

to

resp.

End X = E

if

29.

din

*End X < f

for

all

in

this

indecomposable

enbedded in

c,

K = {h€

equivalent

is

mod (F

the

F\'

so also

the

of

all

h € F,

of

dim X = xo.

U

is

fMC. has

has K-codimension

FI1G

End X < F

Because

implies

So

and therefore End X = F

End X

can be also

F = c. We can assume of F-F-bimodules

is

rnaximal Fright: of all

subspaces

F-subbinodules, with

of

Flb

therefore

basis

is F\,

V.',r...rv"

= Nr o...O Na = v.,F O...e taF.

(U + 191, because. a > 3). Then

vlF e v2F O U. Now

Of course

of

%

codimen-

kere}.

b + 1. Assume that

f 2 g, this

intersection

h.v,

of

means, =

I

Furthermore kerrp

sum of one dj-mensional bimodules,

h . 11 = r.lho,

Let

X with

U - v3 F O...O vaF =

an F-c-subbinodule

F-subspace

Nosr let

This

gS

nith

of

cannot be F-invariant, case.

F.

, where FoP is the opposite skewfield of " *FoP) the theory of Artin rnodul-es over an Artin algebra.

insection

Cb Let

is

Then

lsornorphism

of

I

F lhm = mh Ym € M}, so the category

to

a direct

modules

f-ag>59.

and because

F, and we can apply

zero,

and every

fMC End X = {h € Fl h'.kerqS

But a proper

ker rp

epimorphism.

of G-subspaces

ker q

if

j_nduced by

fMC

by some element

indecomposable

2. Of course

> 59. Therefore

Of course

induced

1. Thena>5,

K-codimensi.on

that

is

the F::-orbits

and only

Firstletb=

endornorphism of

= vrhr

v1F, and vrF induces

a subbimodule

u € U, then u'

of

are F-bimodules,

two automorphisms

Consider F\. : = hu € U', So

i.e.

o and

t

U O (v1 + vr)F

of F. S

f\..

- 113-

+vr)

h.(u+v,

=u'+

(v.,+vr)f,=u,

forcjlg o= r.

Similar

and consider

U O (v.,+vliro)

forcing

d = ho, hence

copies

of

f o r s c n r af , e r

l s i < a. Now Iet

forall

F < FMF. Then,

for

a certain

ho€

F,

fr e Fr

+ (v., + vrho)fi = u, + v.,ho + vrhoho

nono - hoho.

F = K'

was arbltrary, of

= u'

+ v., + vrho)

h.(u

=v.ho

h.v.

+v.rho +vrht

Therefore,

must be comnutative,

sorne one dlmensionar

because

and

F-F-bimodule.

ho.

is

FltF

a direct

proves

This

F sum

the

propositlon.

3.5-5

corollarv:

rf

ab > 5, then

t(Fl'rc)

is

of wird

representation

tyPe.

Proof:

F = G = K'

Tf

sum of

copies

proved in

is

of some one dimensional

(3.5.3).

In particular

F = G = K, because

if

fy the assumptions of induction

K

enbedding from some wird

4,8 i) ii)

and with

category if

e L(FMG) vrith the following

true,

it

N^ : = Extl(B,A). ro. l . 1

For, by induction, by (:.3.3) L(Fnc).

this

if

is

into

is

a direct

has been pro

f = S = 1, i.e. let

fMG

prove the corollary enough to find

a full

satis_ by exact

L(FMG). so we are finished

we find

two incongruous

components

properties:

L e t E n d o = F . l. E n d B = G . , , t h e n d i r n Let

Flb

on M. So

we will

Of course

(3.3.3),

is

centrally

ii).

and if

F-bimodule,

this

acts

(:.5.4

over max {f,g}.

by induction

a commutative field,

*F.,,

dim

*G.,

< max{f,g}.

T h e n d i n. . _ FrN .' > - 5 - ..

L ( c N ^ ) i s of wild representation type, and '1 "t t h e r e i s a f u l l e x a c t enbedding from t,r.,*n., ) into

114 -

-

L (X) , $rhere X

purpose consider

For this

componentsof dlrnension type infinite.

By changing

a > b, hence < -1,

sion

of

so

E

End E

then

is

X3

by induction serial

composition )(o

An factors

is

L(fl'tc) i.e.

t (X)

and

all

A'

rhar B.,

und

is

a subfield

bounded.

So we can choose

End -n A

(rxt11e-,A. ^'o "o ) )

din

3.5.6

This

proves

Theorem: orientation

o. Let

a K-realization i) ii)

of finite

non isomorphic

with

order < f

*(End L)

an uni-

downwards),

(3.5.4) . Let in

extension

components in

B(Brr,Ar,) = B(2 Io,n.

-2n,

<

ä) large.

becomes arbitrarly

End -n A

such that

ao

> 5. Now set

satisfy

4

=

\, "o

B = B-, "o

the conditions

i)

and

and ii)

the corollary.

(r,d)

Let

n + 1

n

End -o' X , the K-dim'ension of

of

o End An = F.,, then A, B and Fl above.

number

are incongruous

B'

X3€X\ {X1,X2},

L (E!{G) . Proceeding

in

be a non trivial

n > 1. Furthermore

End -n A

(in L(X)),

E

(in this din

of type if

of tame type if

be a connected valued graph utith admissible K

be an infinite

(f,d).

Then

and only and only

if

if

field

and

(M,o)

be

L(M,tl) is (r,q) (f,9)

.

exten-

a non trivial

natural

length

of

ehe K-dimension of nxtl (En,An)

Because is

for

every

is

X

e X. Then

It,Iz

components

for

incongruous

By (3.5.4)

the socle of

In,In_j,...,X1,ä

I"

of

End X2. Furthermore if

of

choosen such

* t {L,...,In} -a bct'(Yn,Xo). Again

is

incongruous

in

Let

there exists

X2. Now X2

are

set

we can assume that

necessary,

max{fr9J = f.

we can construct

object

where

by

a subfield

and

E

if

hence, by (2.1.4)

X1

the

(see (2.1.7) ).

*o

orientation

f > g, L.e.

B(I1,I2)

is

is

a Dynkin graph,

is an Euclidian

graph,

4

- 115 -

and iii)

of wild

type,

Dynkln Finally 1.)

-of

graph

course

it

nor

fleld

and

is

a

graph.

remarks: the

L(M,o)

process in

whlch satisfy

K,NK,

i-s an extension

is

resurts

to the

case,

a K-reari-zation

comes in

(3.5.5)

of

where

same varued

of

----)

(3.5.3).

of

K, and we get

[(M,n).

of course, K,

a full

exact

Here an extension

embedding

field

of

K

play.

some words seem to be in $reen tame and wild

do this

gories

this

cation

of

in

order

categories.

some chance to crassify

all

some speciar

concerning

all

finitely

category

indecomposabre

embeclding.

11,...,rd

it

l_-_->

K-algebras. over

K, let

modL K

M,- e...O

L__j,

"

t{-

(lh

d+2-times 21

acts

as the

(d+2) x

rn fact

For wird

For let C

we

cate-

(d+2)-matrix

by € nod'i R)

R

be

be an

be a full

Define

T : M-

is

wourd suppose the classifi..

S : m o d * K1 . 1 , " 2 > _ _ - _ > C

T : modt .Rc-->

bet-

there

objects.

cases of bimodures.

dimensional

and

the distlnction

For tame categories

seems to be hopeless,

K-algebra generated by abelian

rnay lead to a bimodule

the condltions

fierd

mod* K' 1"1,"2>

where

neither

(f,d).

The induction

will

(f,d)

if

an Euclidian

i-s easy to extend

finlte

graph

3.)

and only

we urant to make some general

K is

2.)

if

exact

-

/"

11 "r=['1

9J":...

['

is

easy to

have in

o

";,

\o

It

fact

see that a full

T

exact

defines

In particular, is

R, so every ring

4.)

of

if

R

an object finite

For simplicity supposed that

isomorphism

dimensionaL

So we

K, then

over

endomorphism ring K-algebra

here only

i-nvolved

birnodules.

ing binodul.

embedding.

occurs

isomorphic

to

as endomorphisrn

in C.

over a common central involved

with

dimensional

we treated all

exact

c.

1s finite on C

sorne object

full

embeddi-nq

ST : mod't R <+

ST(RR)

-

| lo

FMG between

skewfields

subfj-eld

skewfields

Homr, ("I,la,

defj-ne the Coxeter functors

are

K, which

What we really over

K-realizations,

aF)

finite acts

need is, F,G and

(see (2.2)).

dimensional

centrally

that

there

we

i.e.

for is

on all every

a bimodule

Hom" ("Mar G") So, if

occur-

we define

to a

-

(M,n)

realization fields

r'i, (tMr)

ut\,

i

to

0r then

of

of

graph.

Also there

(3-5.5)

is

F = K(Y) over

that

discrete

R-nodule

true,

dimension

we use arguments

(2.3)

and for

true.

rf

R-module

the

for type,

the

and the of

statements

in

reduc-

course we

K

with

aeopara-

this

(2.1.4).

we take the function field

example

from algebraic

proposition

for

closed

n = F[T,61,

is

infinite

A and B, [ 21 ].

representation and (3-4.3)

type

There_

fierd

usual

deriva-

then Robson and l{acconnell

has

K

as endomorphism

there

is

dimensional This

inplies

(see 13.5.3)).

over K, for

that

ring

have

and

U

is

mod R

any two simpre is

of wild

On the other hand, as in

a non simple Euclidian

,{(FMF) = mod R x U, where

bimodule

an uniserial

such that

Flb

category

(3.4.2t

with

one

object.

Now ret

F

be a di-fferentiarly

closed

field

Then, as Cozzens [ 6 ] has shohm R = FtT,6l module, Euclidian

namely

RF,

bimodule

the number of is

remain

vrlth respect

1

Ext'(A,B)

simple

lecture

(t,q)

in

K, if

no longer

any simple

j

=

(iMj,ti)

Ho\,.

to the case of birnodules.

nor substitute

we set

such that

skew-

r^rith

iMj

non homogeneous modules

an algebraically

6, and if that

for is

this

regular

subfield

in particular

shown,

in

to be a set of

F.-F.-binodule

1.-.

the modules of

the

netry,

tion

arrows

the homogeneous part

need a central

fore

all

of

an

,rt, I = dii

ut\, for

with

many statements

classification tlon

together

= dty

the crassifiction

(f,d)

of a valued graph

e l,

= Hotnn'. (rM.i,Fi)

-

llI

equal to

and this p\,

module is

such that,

indecomposable

for

has just

injective.

L(FMF)

of is

Fl"tF of

6.

one sinple

This

every positive

representations

1 or 2, in particul_ar

vrith derivation

leads

to an

integer of

length

tame type.

n

With other that

is

t(FMc)

dimension of

M

ments

(orß)

in

is

too

not

(3.5.6)

ii) If

of

over

large,

type,

of wild

type,

is

(M,a), if

if

the

involved

if

if

1,1") )

the

the set of ele-

first

bimodules

4,

for

m e M),

all

inaccessible

strongly satisfy

this

condi-

by the following:

if

(t,d)

an Euclidian

References:

(3.1.1)

quivers,

e.

for

The generalj-zation

j-f

and only is

(r,d)

neither

graph

L(M,0) is of wild

i.

(din

cm = mg

than the

snaller

t 25 I

"U) of M (i.e.

F x G, which satisfies namely

has shown in

(din

if

Z, the center

an be replaced

finite

(f,d)

type,

Ringel

r:

l(M,n) i)

of wild

number.'So,

cardinal tion,

than used here,

arguments

it

is

a Dynkin

Euclidian

depends on the

graphs of

to valued

nor Dynkin. reallzation

or tame type.

has been proved by P. Gabriel valued

graph

type

in

$ 1 and $ 2 can be found in

in

[25].

I 11 l,

of

for

Arr, Dn, 86, E7 and Eg.

graphs has been done in

of

I 18 I

[1O].

the remaining

l,tost

paragraphs

.

1.

VORLESUNGEN AUS

DEM FACHBEREICH MATHET'IATIK

DER

ESSEN -

In

I,'NIVERSITAT

this

series

H. Stichtenoth,

already

2.

H. Brückner,

appeared:

Alqebraische einer

GESA"UTHOCHSCHULE

Funktj-onenkörper

Variablen,

Explizites

1918

Reziprozitätsgesetz

und Anwendungen, 3.

M. Chamarie,

Sur les ordres d 'Asano ,

4.

J.C.

Robson,

Some results

5.

L.w.

Small,

Rings satisfying

1979

maximaux au sens 1q7q

on ring

extensions,

a polynomial 19 8 0

identity, 5.

H. Jacobinski,

Maximalordnungen Ordnungen,

1979

und erbliche 19 8 1

118 -

Appendix A:

s1

Applications

Alqebras

Let

(f,d)

the

tensor

Let

F.

be

€ f),

(M,0).

of

bimodules

resp.

and make M into

a

=

write

is

decomposition

form

the

n-fold

in

r

^Ml

the

, wrt A =

via

the

skewfields

defining

O) the

M =

Fi, -

.o_ i€f

construct way:

following

O i.*.j inf

r

1Mi

A + F.

Drojections

because O iM.i, r.+.J of A-A-bimodules.

(i

€ l).

decomposition

this

n > O,

for

llow,

Mn = M O^MA^...8^M,

grroduct

tensor

in

Then set

A-A-bimodule

I{e may also a direct

'j

(M,0).

K-realization

(M,Q)

of

(i't

rMj

IIJ-

wi.th

T(M,A)

algebra

(i

graph

a valued

we

where

@

=

Mo = A, ltl = M, and putT=r(M,O) T through

a multiplication i i i+i Mr * M'-r M'e.

and extending

becomes in this

way a ring,

K-algebra

e Mn. Nowwe define n=O isornorphism the canonical

of

R a tensor

it

(M,Q) of a valued graph (f,d). dimensional oriented

R = f(M,n)

if It

over K, because Q is in

circuits

f.

same K-realization

for

is easy to see that i.e.

admissible,

Furthermore

J(T),

T is

there

i2+...+i*

for

particular

the Loewy-length

one of alL

occurrj-ng

A 1.1

Proof . as

Proposition:

Define follows:

€ f in

some i., ,...,ik of

T is

the Jacobson radical

I with

of

and

respect to 0. In

the maxirnal length

nlus

pathes

oriented

rnod* T and

R : rnod*(T)

just

finite

are no

is given by e ut, more generat .rk(T) is given by o Mi, i=k i=1 . uK + (O) , if and only if there exists an oriented path i1{

a

and we call

or even a K-algebra,

algebra,

T

obviously

by distributivity.

-

L(M,ff)

l(['{,Q)

equivalent

are

and P

:

L(M,Q)

-

categories.

rno
T

-

Flrst,

if

X

ll-module. x^

a rlght

T-module,

n =

Because

is

,-?at decomposes uniguely into

right

F.-vector

Vte define scalar

space, /l-Iinear

the

multiplications

Xr O .M.

tj

is

119 -

into

linear

lnduced by

rl

R(Y) = (Yt,

R(q) =

deflne implies fact

I

:

X8A"^**

m € M (p

is

on

Xn

=

The fact jri

,

that

Fi

,8, The scalar

q(l ) -

o i,j€f

=

(using with If

the

fact

(q.)

P (q) =q= I-1inear.

Ji 9 .

that,

=,p(,p(t) X O oj i€f

Xi

that

is

y

-

Xj

Lw,n) joi),

of

q

is

f-Iinear

R(o)

to

is

Xi

and

in

Mn

define

Xi, h,here r?, the projection 1 -

via

by

on

X

K..

is defined

: X6Mn+Xwith

: x E^ M =

(Xr8o ^ 'i

e i, j€t

q(n+1) o =

on

multiplication ,(n)

right

L(M,n)

operates

inductivelyby

the

maps

because

o

Conversly, given X = (Xr, jef)€ t(M,0), X = P(X) € mod* T addirively by * = n =

by

yrr

a

€ f

g

Then

R(X) = (Xi,

q (Xi)

is

bythe

be the restriction

a h o m o m o r p h i s mi n

+ j

ox, iMj iMi = = R(x) (xl, x j,pi)€

that

q = (ci).

Xi

= xi

in mod* f and

oi

as right

decomposition,

i

determined

x:. on

(oi o 1) = oj

jüi

I

X

where

map

jqi,

Let

Xr,

= O for

j r . l 1 r) n o t i c e

^-1inear.

a is

x = oigt F.

we define

Also given c : X + y

direct

X.

thus

mapping

a ring

and

of

X.,

consider

for o t) rs

,M.)

i :

+ k,

o i,j€f +

e i€f

Xi

i

P(Y)

X.=X )

En r.M.i =

"'-'cool

(X An M..)

a morphism in

: P(X)*

x, o. .M. r- A I l

E.l M L (M,Q),

can easily

-

(O))

and

(9 X 6_r. M + X then

be seen to

be

_120_

Now it

is

between Note,

easy to mod*I

that

1s called (Ieft-)

is

right

exact

semiprimary is

all

are of

equival-ently,

if

any (left-)

right

(i.e.

R

of

. Our aim in

the

J(R)

right

K-algebra,

finite

tame representation

finite

resp.

K

*/"(*)

and

and left to give

type,

hereditary a classificatj.on

(i.e.

in

R,

rings

a commutative fie1d,

tame representation

R

For

Artin

type,

of

Extt (",-)

nilpotent for

called

ideal

functor

in

hereditary or

right

hereditary

paragraph

this

F. is

R-module x. is

is

property

this

a ring

, in particular

being

T-module

each (left-)

if

Artinian)

prcPerties

coincide

of

for

rings

semisimple

the

of

or,

an equj.valence

algebra with

More grenerally,

right-hereditary,

projective

define

a projective

An Artin

hereditary.

is

p

and

submodule of

(2.2.8).

by

R

L (M,0)

and

every

projective

see that

which

mod*

R

is

the sence of

Chapt. 3). Of course to do this K-algebras

R.

basic

B (R)

Pp

rinqr

is

If

o n l yof

course 1f

Morita

equivalent

with

is

an aribitrary is

is

is

Artin

nonisomorphic

a direct

Morita

ring

the

projective

R = B(R)

if

and

sum of skewfields,

equivalent

dimensional

to mod* B(R)

basic

as End (pR) , where

defined

all

a finite

above, is

K-algebras.In

enough to consider

R-module. Of course

resp.,

Of course it

is

sum of

/, / t"ß) r i l o dR is R

is

R

J(Rl

if

defined

R to

the direct

indeconposable

it

and

to mod B(R),

K-algebra,

. Note that

mod* R is

f(M,Q)

,

basic. also

enough to consider

particular,

K-realization

(l{,Q),

if

(f,d) then

is

indecomposable a connected valued graph

T(M,0) is

indecomnosable.

-121-

Note that

an lndecomposable

finite

resp.

valued

graph is

tame type,

heredj-tary

type.

There ls

hereditary

graph

A'

where

F

central

is

of

by

finite

.on"id.t

the following

K, and

way all

type but

tame indecomposable

a skewfield,

subfleld

Euclidian

of

of

the corresponding

which are no tensor

algebr""

trith

if

is

fact we get in this

a class

this

algebra

and only

K-algebras

K-algebras

To deflne

if

Dynkin resp.

(3.5.5) and (A 1.1).In basic

tensor

first

basic

hhe Euclidian (11,0) ,

dimensional is

fNf

tame

alqebras.

K-realization

finite

not of

over

the

a one dimensional

F-F-bimodule:

F'F

rFr

rFr /

\ rNr

We.can identify F

acts

It

is

on

the left

FF

easy to

vector

from right

see that

(n + 1) x (n + 1)-matrix

space

FN

with

by an automorphism -- T (M,0) is

T

isomorphic

ring FF...

FM

; I E i F F

[' Än(s,o):=

F

o

\ \

.FF

I

l I

.:;; .F

\

F

,F, e of

where F.

to the

-122-

M =

with

O

as

"F" "N" be an automorphism and (where

e

fixes of

definition Ärr(e,o) by

€ F)

(3.4). a basic M(e,6 )

is

a tensor

algebra

holds

K).

RecaII

hereditary

indecomposable

= M(e',o)

if

i.e.

We will

graph

if

= T(rM(r,o)r),

see that

we still-

D > 1 ,

argebra

n + 1

t141.

< n ; then we have also (r1,

for

permutation

(1 2 .

(ra, rr,...,

td)

and (rr,...,.d) called .,

Extl

(snrso)

satisfying

a1r

t,

where

n-1 o n).

n

is reqular.

td-2

+ o

for

+ o. A o < rt

:

n

is

the cyclic

Furthermore,

= n -

if

(rj,

is regular. Finally, if

which

11 + n, rd + o

if

is called preprojective

preinjective rd-l )

(Sa,Sr-.,)

regular

is called

ra*1 = n(ra)

Än(e,ö )

Extl

integers

.,rU) of

(1 < t < d €N)

Evidently,

sj.mple modul-es So,...,Srr,

are ordered in such a way that

sequence

For the

is of tame type.

Är, {6,c)

the

in what follows.

In particul-ar

need some notation.

has precisely

(r1,.

of

l,

11 = o, t2 = 1 .,rd-1,

td = n

is

i.nner

tensor

the

n > 1

Ä,,r. so always

proof of the statements see

is

is

6

6

cf

for

. consequently,

5]

describe now mod* Än{6,e).

we will

and

. c -

= et

d

lf

and only

"M(e,ö)"

1 < i

the

K-realization

Euclidian

First

F

of

M (= M(e,O) ) in

and only

if

Är, {e,6)

e

Replacing

F,

of

c € F,[

a suitable

for

n = t the

in

(e,'l)-derivation

Är.r{ö,e)

to

annihilates

Now

t.,O ).

\

an inner

For

6

M(e,d )

no\^r

(e,1)-derivation

arl

ö

M(e,ö ) we get

K-algebra

(f

and

Let

F-F-bimodule.

and

rd)

-

A 1.2 are of i)

Theorem:

The indecomposable objects

the following

in mod* tÄrrte,O;1

types.

For every regular,

preprojective,

(r.,, ...,rU)

is a unique indecomposable module

, there

X € mod* (Ä-(r,6)) n'

xt/"

such that

"t- 1 is an uniserial The'direct

or nrej-njective

sequence

hrith composition series

O = Xo.

ii)

123 -

X., -

c Xd = X,

...

- s't

for

all

t.

Furthermore

End x

ring. j-ndecomposable

sums of the renaininq

m o d u l e s i n m o d * ( A r .(,e , 6 ) ) subcategory which is equivalent (For the definj.tion

of

Flt;

The equivalence

ii)

is

gj_ven in the followinq

is

assoziated with

i_n

The

FIT; s,6 ]-module

W

the

Ä-(e,ö)-module n-

I . Je . . . O W

ärr{e,6)

(arb)

A 1.3

Corollary:

€ M

on which

matrix

operation

with

the additional

and

w € W, we def j_ne r^r(a,b) = hra + rrbT.

condj-tion

that,

Är.,(e,ö) is of tame type

Proof :

B y ( 3 . 4 . 3 ) m o d +( F ( T ; e , 6 ) ) i s e q u i _ v a l e n t r o

a direct

factor

every

of

indecomposable

an uni.serial is

wav.

=

operates. by the ordinary

from the right, for

see (3.4 ) ) .

e,öI

; '

to mod* (FIT;s,ö]).

H(M(r,ö) ),

module in mod* (F (T; e, ö) ) has

endomorphism ring

tame. So End

X

in partj-cular

is

uniserial

m o d u l e i n m o d *{ Ä r . , { c , 6 )) a n d

and mod+ (F (T; e ,6 ) ) for

every lndecomposable

Än(e,6) is tarne (see also

(3.2.)))

-

Rernarks: Coxeter First

1 ) In functors

of

-

the representation play

we constructed

that

tz+

the basic

theory

an importent the partial

role,

but

the

also

situati,on

endofunctors, usually

case,

the elementary

j-n

(A 1.2.i),

that

rnodules

one projective injective or

It

the modules

preinjective

So

f*

the

out

and

f-

are

same way as one

is

6

if

is

a tensor algebra,

that i.e.

the indecomposable (r.,r... rr.) O i

,

r € N,

modules are

End X = F

for

X. Also

is and

just

the

has preci-sely

Är., {e,6)

So, and precisely

modules

1-(n+1)

"116 the preprojective

f-tto

(observe that

indecomposable

Än (€,6),

(n+1)-powers f+(n+1)

indecomposable

in particular

imitates

possible

(In particular,

the

turns

is

for

Är.r{e,6)

sirnple moduLe, namely

Sr,) .

f-

whose corresponding sequence

, o < r €N.

t*ttr,

if

are just

the preinjective

sinilar

them in

X € mod* Ärr(e,6),

are just

preprojective,

i.e.

functors).

indecomposable modules X

this

the Coxeter functors.

then the Coxeter functors of

in

the

and then sre

Now it

is

(e, 1)-derivation,

an inner

(2.2). f+,

and one may work with

does with

group,

"elementary functors" much easier

algebras,

whose behaviour

i.n the rteyl

reflections

tensor

as we have seen before.

functors

composed them to the Coxeter functors to construct

of

one simple preprojective

all

a dimension

vector

h+1

(€ f") for

and the defect

K-realizatj.on

descri.bed regular of

2)

of

in teuns of

the defect.

the

regular

fn particular

Sinilar

modules can be

one get

endomorphism ring.

as

that

all

For details

s e e [ ' 1 4l .

There iE another'way understanding

Consider

graphs,

Euclidian

modules have uniserial

the proof

better

of a module can be defined.

the following

of

to descibe mod* Err(e,6), the

internal

valued

structure

graph lvith

of

which gives

a

the modules.

the K-realization

(t{,4)

-125-

as foIlow:

f,r

FEF

f,F

F-_--> 'n-1 4

----->F

F '2

FEF

M ( e , 6)

Thus X = (Xi,jpi) n linear

naps

€[(M,n)

consists

: Xi - Xi_j

i_.lei

(n+1) F-vector

of

(1 < i

spaces Xrr

< n) and a linear

map

t X' O M(e,6 ) - Xo. o9r, Let

Lc(M,Q) be the full

X = (Xi,rer)

subcategory of

such that

the following

Xr, 6 M(e,6)

where r ,

S

FFf Then Lc (M,Q) is

For the next

*_._

n-19n

M(e,5) is

the natural r^

equivalent

h^^*

lemrna 1et R be an

Suppose that

€ R be orthogonal

F = fRf

R is

basic

primitive

above is

allsays satisfied,

iA n \ rc-r u r r /.

finite

of

(It

is

fR resp.

R-modures. observe if

R is

homomorphism between tsro projective

dlmensional

and indecomposable.

idempotens,

and G = gRg are skew fields.

indecomposabre

embedding (see 3.4).

arbitrary

and G are the endomorphism rings projective

representatj.ons

xn : xnolFF o91'192'

f,9

of all

diagram corunutes:

xo

K-algebra.

L([i,0)

hereditary,

Ler

and. assune thar well

known that

F

gR, the corresponcling that

the assumption

because then every

rnodules must be iniective).

Let

fJ(R)g,

FNc:-

Notice

that

as

considered o fR9

FNc

a full

exact

--+

mod*R

T : t (-N-) rb

Leth=1-f-9,

idempotents. we get

J

R!

form

R with

End(*R)

fJh\

c

q;h

(Note

\

Let Then Set

x

=

M = Mf

(XF, Yo,to) €

define

'r 1

R3r = int-rt li.

'

nJ5r'

to

I(FNG),IP

.i

as

:

XF Q

1n the

f,

because

R is

a direct matrices

sPace.

operate.

Let

\

withrl€Frrr€G,rr€R'

nlag'

nr^n 'l I

)1,.-.,j5€

element of M. Then define

sum of

way:

9Jon 1

@ fjh)'

basic).

+ Y".

r)

m = ( x 1, y , x ,

idempotent

an

those

"N" following

K-vector

= fRh,

contain

into

on which

fJh

exarnple

h cannot

decomposed

O Mg O Mh,

f i h \ -z

' J 1 aa

be

can

(xOF fJh)

€4

for

isomorphic

= M €mod*R

T(X)

M = x 0 Y o

'

ret

M € mod*R

spaces

orthogonal

= Rf O Rg O Rh

and using

that

because

l;"t hrs "* ,}

K-vector

way.

"R

fJg

Any module

of

set

a complete

= J(R)

E I / o.rf

natural

embedding

so f,gfrh

Identifying

srith

the

o Horn*(9R,fR).

A 1.4 Lemma: There is

Proof:

in

F-G-bimodule

(x,,xr€X,

Y€Y,

J(R),

and

j €J(R))'be a generatj-ng

-127-

m.r

= *1

t1

= *1

r

the

onto

m2 = yr2+9(x1

11 € x,

and.m,

I

(m1,mr,mr) €M by setting

:=

f)rh+

whole

x,

of

of jls)+o(xrotihjrg)

O fjhrrh€X

M by

AFfJh.

linearity.

= O and gJh.h.Jg

= O, because

fields.

usi-ng

easy

in

this

way,

If

(c,B)

shovr that

Of T is

Xi€L(FNG),

i

So let

c

Let

x€Xr

, 11 -

course

A 1.5

Lemma:

K-algebra,

is

f

is

=

r

out

A.T-A.T R:=^/t Then R

So r

A be an

an ideal

J.I

of

=oO

gFfJh)

C (Xi

remains

F-,

to

map. Now

and 3

x2 @FfJh is

= o(x).fjh,

T(q,B)

= T(Ii),

and sinilar

c is

be

c y2.

o(yt) is

G-linear.

hRh-linear.

because o is

R-linear

1 and it

is

easy to

see that

L(FNG), i.e.

o=

T(o,3).

So t

finite

arbitrarv

A contai-ned

in

is

dimensional

Jz(A),

If

A/_

is

(O).

_
we can assume first

-/tL>^/I=R,

hereditary.

e y.

-

to

shows that It

: X1 8"fJh

Then o(x-fjh)

Let

then

Factoring

t

skew

a R-module

T(q,B)

embedding.

. Obviously

a morphism in

[17]

define

m e a n s o ( X . ,) c X ,

As above

=(O),

fJf

becomes

M1 * M2 be an R-linear

B := olr,

c

of

= X OF fJh.

verification

Mi = Xi

action

f.Jg.gJf

M = T(x)

L(FN.),

an exact

So let

= c(x)Ofjh.

hereditary,

because

T is

this

= F and gRg = G are

that

in

A direct

c M2f that

X,

see

a morphism

and j €J(R).

(q,ß)

to

that

fRf

M9 = y and llh

T(Yr).

ol*r""f"n.

t(xefjh)

Let

-

:= ol*_ and "1

hence

Proof :

is

full.

= o(Mt).f

:=

is

= 1,2 and leto:

s(M1f)

t

x,

: T(x.,)

R-Iinear.

Let

i.t

and Mf = X,

: x,' r

(o,B,oo1 ) is

thls

Extend

Notice

fJh.hJf

€ Y,

Notice

hence that

I,J

"/, and

J.I

=

(O).

isprojective, 12 = I

are

ful1.

-123-

= J/,

.l(n)

Also

pJ91.r -fo

be a projectlve rn

rl

.1

.r -\"7,

eJf

sider

ker g c ,.J/I -1 q- | (r)

, .t

because

= P." /t tr,

is

hence

ker(rlo)

projective.

R

cause

is

with

R-module.

Let

J

as

R-module,

and con-

F{

+ (o)

rf

hence

Qü,

means

this

an

is t/".,

= {o),

lndecomposable

basic

be a complete set of = erRer(=

Fi

Set (f,d)

a valued graph

define

Be-

End etR)'

a skew field.

to be the graph with

n

Ütp splits'

by the Nakayama lemma.

is

,n."

.

Also

PBut

.

Then

epimorSrhism'

in

small

J'I

idempotents,

heredj-tary,

orienlation

^=

€1r...r€r,

Let

r= ei(J(R)/"21n))"j

iMj

/,

dimensional

a finite

primitJ-ve

of

f

I = (O)

K-algebra.

hereditary

the radical

E P'" /t

If

the

is

the natural

is

= (o))'

(t2 =- l'r

'

/,

consequently

1 = (9). hence

r

of

cover

ker g

be again

R

orthogonal

I c J.

o

because ,

I = J.I

Nosr let

t

where , r,fr is

,

isomorphlsrn, i.€. 1.e.

12 = t.

In particular

J.

R-module

hence

,

r

by

are annihilated

because they

R-modules,

1, " '.rr

vertices

'

wherei,j€farejoinedbyanarrowio-)ojrtithresPectto

i

anil on1-y lf

lf

0

M, = (O) ,

d..

Lf

the

iMj

R

because

(note

+ (o); is

hereditary).

Define

g

€ l.

Let

= d i r n o ( . M a ) , d + r = d i m o . ( . , l i t . ' , ), L , j .i ^ :

inplies by (M,0)

be

€ f)'

Then

JL

J

"j

+ (O)

. uj

that

K-realization

of

(l,g)

given by

Fi

,

iMj(i'j

(f,d)togetherwith(M,n)iscalledtheK-realj-zationassociated with tensor

R (or the algebra

K-realization (M,0)

is

a

K-species t(En)

is

associated K-realization

to

R),

denoted by with of

and is I(R) '

the tensor

denoted by obviously

algebra

some valued graph,

ER the

T(M'0)' is

The

again

r"here (l'{'fl) '

- 129 -

n o T(fn)

so

A 1.6 Lenma:

if

and only

R

ls

if

R

a tensor

ls

a tensor

algebra

)

if

and only

--t

O + e, J- e, + ei J e. - er(-/"2)". Fl-Fj-btuoodule sequence for

Proof:

Obvlously

a tensor

algebra.

all

the condition Furthermore

* O

i;j

if

splits

as

€ {1,...,n}

above is

the

algebra.

, nhere

satisfied

condition

is

if

R

equivalent

J = J(R).

is to

the

(J = J(R)):

follottlng,

).a -/r2 O + J- + J +

+ Q

splits

as

Ä-bimodule

sequence, rjrhere

nn

A = .O- Ft = .O- e.,. R e. i=1 i=l

.

So we can assume that

= nRA rrArrsA e /Jiln o ^J2n Conslder

Of course

T(R).

pl

is given by

h

, iüj

, i,j

€ {1,...,n}.

In partlcular r(R) =

^NA r

t

=.o^

Mk , M =

-"

k>2

nÄl ä

ri i { iJ

i,j=l

by

^NA-AJ-^

Novr rre extend

r

e,L (/Jül =

where

nN^,

in1l J / .ur i l ^

= R.

Define

=m1 .-.mk,

t(mr8...om*) to a bimodule

o

homomorphism

(mreMcR)-

- ,Rn. Af(R)n Of course r is onto and rnultiplicative. Furthermore ker t c nNn ) = and = (o) and J- (T(R) ). B e c a u s e R i s h e r e d i t a r y , k e r r ANn r

ls

an lsomorphisrn by

The next fact

it

Leruna makes sho!,ts that

from tensor

algebras.

(A 1.5).

(A 1.6)

basic

r ,

a little

hereditary

Artin

bit

more reasonable. alqebras

In

are not so far

-

A 1.7 Lemma: i)

e.lr...r€r,

3,

Let

There is

Let

onto

(o) e, lJ / ,2) er 'v+1 + ^v "

.,ia}

{i1,.

(1:vSt-1) :=e.

I 1

T =

Proof :

Je.

)2

.

Assume that

Let

12 e. J e. e. ... ru_1 rv+1 .u

e. tt_1

J...ei

Ju-1

...e. J2a-, Je. rr-1 lu lu+1

r-1 S . , < e -i 1 a l e . I v=1

...

r-'l ... E T . , -< e . J e . 11 r2 U=1 Y S+Ul

First

note that

Ut\i(en

r

such that

the direct a submodule. P

of

.kR

sum of

Ja.i tr

,l er)

,

=: Ur Je. e+ rr-1 Jt

qnd

,I+tJ2

=: Ur

Je. e. 't-1 tt

U1 nU2:SnT:.tJ.j

e k J e j s H o m ( e r R, e * R ) ( j , k € = rjk

is

iust

the

largest

a submodule isomorphic contains "kR To see this let .jR. Q copies of

Obviously

it

isomorphic

to

j-s enough to "j*

is

show that

contained

in

every O

{'l,.'.n}).

natural

J

number

J e, tt

Let

Then

Furthermor"

such that

{ 1 , . . . ,n}

and

(1.u:r-1). S =

by

e* (J/ ,2 )e* + (o) 'u*1 'u

and

+ {j1,...,jr}

S := e. iI e. J ... 11 ,2 v

u

frorn

, 1 a U 1 r-1.

1 I v S t-1

for

given

'-j1,)2,...,ir=)

,

indices

J = J(R).

isomorphisrn from

J ek J ej

ei

i=it,L2,...,ia=i

be sequences of

T

as above and let

Fi-Fj-bimodule

a canonical

J o e* J e, "i "k multiplication. ii)

130 -

to be such

subrnodule

(that

is

-

tJt

-

is "ejR-homogeneous',).l,et M=e+plekR. Then M "kR projective and the canonical epimorphism e O p - M splits. maximality

of

r

novr e : p.

implies

ejR-homogeneous component of rnodute

P

let

the sum of

all

an"

component of

e{

using

the

component of

(e, J eO .r er) = rik multiplication e.

e. (R)(k, j),

J e*

the

l.e.

e*F.-homogeneous

rki

i

map from

J e.

and

comparing

dimensions

ii),

notice

u t * , , . n l J/ , z l . 1 "

that

is

just

the number

summands of

.kJ

be

"iR).

the rank of

dfut

-j

p,

i).

To prove of

just

iruplies

onto

AJ

to

This

canonj.cal

J eL O er- J e.

we get

is

e.R-homogeneous

"iR.

a projective

for

O.

J e* J er)

rank i,

isoroorphic

e

The

the

erR-homogeneous component of

the rank of

Noh, U\,.,.,

thus,

the

(more general,

.kR

submodules of

r = rik

Call

the

P(j),

e = ekR(j)

Call

is

into

isomorphic to in a decomposition "kJ "jR indecomposable modules. Now ii) follows easily.

A 1.8 Theorem:

Let

R

be an hereditary

flnite

of

dirnensional

K-aIgebra. i)

ii)

R

is

of

finite

representation

type if

B(R) o f(M, fr,

where

finite

union of Dynkin graphs.

R

ls

disjoint of

(M,n)

and only

tame representation

type

, ) e A 8 ( R ) g f ( , 1 {Q

(ring

is

of a finite

a K-realizatj.on

Euclldian

graphs,

and

A

is a K-realization

direct

is

if

and only

s u m ), r v h e r e disjoint

a finite

if of a

if (,t{,0y

union of ring

direct

sum

-

of algebras of type defined

Proof:

So let

(A 1 .

\{ith R

R

is

a complete

F. = e. Re.

in

set

of

1 < i

N let

2 .JI €.J*

Thus,

if

there

III

Chapt.

be the

dim- (,N.)

exact

mod* R

of

Because

R,

we can be

€1r...r€r,

idempotents,

< n.

iNj

a full

> 4,

type

primitive

- - ) , (, J- /, , 2^),* €

is

drr(iI)

is

K-algebra.

hereditary

orthogonal

= End(e.R),

LJrJ!uJ

(A 1.4)

in

and indecomposable. Let

+J.i.l.j*,

o +

proved

theorern is

the representation

d.. (N) = dim- (,N-)

*. By

the

dimensional

basic

For each R- bimodule and

Ärr{e,ö)

) and (A 1 .3 ) .

interested

assume that again

of

'l

be a finite

we are only

where

Ärr(e,ö),

as above.

The "if"-part

together

-

tJz

('l < i, j

< n).

r

Consider

O.

of

type

wild

c9

L(iJj)

embedding is

erNei

Fr-Fr-binodule

mod"

(3.5.5),

by

R. and

L )

mod

R

cannot

be of

i)

Let

i,j

€ {1,....,n},

R

be of

In particular, for

i, j

all

i.e.

R o

graph

by

ii) then

Let R e

finite

finite

type.

hence

dimo (iJ*) 'i-r'j'"

= (O)

.1 Ja l €

type,

and

5

dij

(J)

1

or

,tJ / rzl:

By (A 1.5)

{ 1,. ..,n}.

T(R),

Then

or

> 3.

dij(J)

if

I R

1

dim,

tot

aII

(.Ji)

*

and

1

1.

splits

is a tensor algebra,

must be a K-realization

ER

for

3

of a Dynkin

(3. 5.5 ) .

R

be of

T(R)

by

tame

type.

(A 1.6)

splits

If and

K

is

for all-

a K-realization

i,j

€ {1,...,D}, of an

- 133-

Euclidian

graph by (3.5.6 ) .

So let

I,j

drr(J)

= a

in this

jarticufar

r.,

this

inplies

IJet

annihllates i

= i

Because k

1 S t 1 k. ,' sl J / " 2 \ i "

tt

and

such that

using

t

a

....,iki

and

.K-

i '.r - 1- { J / r"Z ) i .' t

OR of

*Jr_

than one. So, changing

t*

to

and restricting {O,1,...,k} 1 ------)

= (O)

+ (o)

implies

o I t:

2

,

Z. k

for

circle

such that

in

or agaj-n

the graph

s=t-1

fR

there

is

. --t

way

the graph of a,. (,f2)

rrith

i=

ER

would be ia

by

orientation

k_l

from

+ (o).

s € {O,...,k}.

(replaclng fp

i.J*

Ire get

no additional

,

Let

for al1

the indices

---)

be

for

,/\

\"there in

F,

we see that

ar.r{,f2) :

could not be nilpotent.

, O s t S k),

for

v 1 k)

+ (O)

L-)

greater

of

6

O!

J, -.,t - 1 - t

(A 1.7 ii)

we would get an orrlrra.a

J(R)

.J, r M(e,ö)

for

,1st5k)

othen"rise

Then again by (e 1.7 li) otherwlse

and

(iv € {i,...,n}

(osssk-l

+(o)

m € {1,...,n}:{i

dij(J2)=arrtJ/"2)=t.

(e,1)-derivatlon

is maximal,

l'urthermore,

s = O , t = k;

or

Of course

K.

a sequence of maxlmaL length k.

F. =; p

F. r

and some

6

= 1o,i1,...,in

1 S t:

does not split.

case. Necessarity we have

some K-automorphisn lrhlch

*

such that

€ {1,...,n}

t'

-134-

O:

"

, t.

By (A 1.7 i)

k.

J

"o

e1 J O ...

I

"1

J.k

e.k-1

eo J2 en . . . e n - . , J e * e r n b e d d sc a n o n i c a l J . y i n t o J J "1 "2 "o = so dt-1,t(J) (note that k > 1, because eo J2 en + (o)). for

In particular,

1 s t S k.

n > k+1 , that

Assrme that complete

set of primitive

an index,

say

(the case

+ (O)

**rJa

be maximal with o: and, to

"

can be treated

and

OR ,

$re qet

1 t, J

and considering

and at most one additional if

oJk

.----_-l

Tf

X =

setting

=4,

OK

T

:

in

k+f

to

kJk+1

(Xo,\,XL41,peor11191) X = Xeo O Xe1 O ...

*

lrtith respect

If

there

ls

no

t = k+1 '

if s r k-1 t = k,

(A 1.7 ii)

to

k+1,

consider

. Fk*1

[ (,u,0)

L(M,Q)c-1mod

by (A 1.7 ii),

fR

see that

all

for

using again

Also'

K

d,(J)

Now define

J e" = (o)

. ------------J

o

Because

O s t : k

= tO) ' *.rf,*, (M,Q)

K-realization

the following

Let

way from some O:

Always holds

t -1 k-1.

s : k-1

O :

similarly).

easy to

is

it

O 1 t 1 k,

circle

cannot be nilpotent. oJk+1 ,

some

dok(J2) > 1

t+1 S s'

is

Then there

for

.k*j

Then

an orj.ented

way from some

additionaf

idempotents.

(O) aJn*., +

property.

this

does not form a

€o,€1,...,€k

orthogonal

otherwlse, if

s S k. if

is

such that

k+1,

F1 .

Fo = Fj *...e

j-s of wild

R

type.

as follows:

€ L(M,a) O Xarr_1 as

define

X = T(I)

K-vector

sPace,

6 mod* R

-

rrith

Xes=Xo

135 -

for

O!sSk-1

for

k+2 a s :

Xen = X* xtk*'r

= xk*l

Xe" =

(O)

necäll

that

O 5 s :

k.

P" e F,

= FF

s_1Js

Now define

n-1. (as

an R-operation

F-vector on

X

space)

using

the

for follor,ring

mappings: id X""

E

(O 5 s :

Xe. "J.----f id

X."

6

Xeo S

oJ;

"Jk+ for

1 :

s S k-1,

r

1 k-l)

181 " C-----+Xeo

where

r ,

o.ff

kgo E

oJ*-----.tXe*

*o"n

is

the

canonical

embedding,

k+19k X.k

a

kJk*1€Xek+1 k9o

Xeo I

Tf

,

oJk---+Xek

all

other

: X + Y

is

a morphism in

can be defined

as

in

is

o

a full

Next

exact

So

R

O S t 5 k-1,

spli.t. by

This

is

cannot

zero.

T(o)

: T(X) -

easy

to

be

tame

see that in

thls

T(y) T case.

and that

)r

not

L(M,Q),

and it

o+tJi*rltJk*r

{O,k,k+l} this

embedding.

assume that

** does

(A 1.4),

maps being

1{"/J2)k*.tlo

reduces

to

{t,k+1,t+l}

,

and if

**

the

and

case R

above,

cannot

i.f. we replace

be tame aLso in

case.

If

O S t

for

alL

I

k-1,

O < s < k

the

splits,

secrrFn.F

it

is

easy to

see thar

136 -

-

4"Jk*1 --+,{J/ ,z)k*1 {

o -""i*., In particular

wild

of

spl its .

s = o.

for

(M,0),

K-reaLj-zation

following

the

Consider

o

which

is

of

to

be

course

type:

FA

o

K

Notice

that

tame.

Define x =

For

setting xes

= xo

X"k

= Xk

do,k+1(J) :

T

4,

1

)mod*

L(M,A)t

€

X = Xeo O ...

as

,

o < s :

as

X

on

=

space,

with

J

+

"k*1

""

(O)

€ mod* R

(o), the

Define

otherlvise. mapPings

following

the

by

above

J

Xe"

and

K-vector

"o

x = T(x)

def ine

L(,U,4)

if

or

k-1,

= Xk*l

X"k*l

R-operatj-on

e Xen-1

was assumed

follows:

as

R

(xorXn,Xk+1,keo,k+leo)

for

R

because

id Xe"

(O S s < r

I "Jr---------äxer id

xe"

k-1 ) k9o

181

^ e

a

:

I

xeo

oJ;

,

oJp------JXe*

(1 s s 1k-1)

"J;:--)xeo kQo Xeo

@

X""

I

oJn--------)Xen k+1Qo

16'r 6 eo J es J

Xeo @ eo J

.t*f

Xek+1'

et+t+

"Jk*f------+Xeo if

the Note fR

J.k*.1

J "o

canonical that with

+

(o),

where

r

eo J es J

:

"= all

embedding,

either respect

oJk*1 to

=

QR ,

other

joining

zero.

maps being ,

olJ/rZ)k+1 o

(there and

is "t*l+oJ.k*.1

is

k+1,

only

one way in

namely

o + k+1 ) '

-

or

=

oJk*1

respect

.1

to

narnely

h,

at least

one

-2) = odk+1

o

=

oJk+1

"o

)

J t"

J tk*1'

J

J .k*1

r,s

€ {O,...,n-1}

fn

with

-

respect

O + k+1 ,

s).

the

R-operatlon

this

way a full

the assumption that

is

o eo J .1

Remarks:

x

is

exact that

fn

K-argebras.

basic

R

easy to see that

and that

Thus we get n > k+.1.

implies

paragraph

r

defines

in

R = A*(e,6).

we only

treated

course we can generalize or

and left

semi primary J(R)

that

realization

dimen_

the results

to

rings

coincide.

f(M,n) of

Artin

finite

is nilpotent).

hereditary

seen j-n the case of bimodules statenent

is

splits

orthogonal

this

true

,r*

that

one

forms a complete set of primj.tive

semi simple and

j-s an arbitrary

fact

the other

n = k*.1 r

rings,

remains no longer

r,

In al1 cases it

tame and

Artin

hereditary

( t h e r e a r e t \ , r o r . r a y si n

sorne

Therefore

is

R, /.f(n)

is

or

to

R

of

and

a contradiction

Of course this

1.)

one contaj.ning

r + s,

from the

embedding.

with

for

werr-d.efined,

hereditary

Let

,

In

J .k*1

one containing

follows

€or€1,...r€k

idempotents.

sional

on

{o,k,k+l } ,

do,k+1(J-) = 11,

eR,

This

are two ways in

and as additional

, s I

(A 1.4) and (A 1.7 ii).

using

right

j

{Orkrk+1}

to

(there

,

oJi*t

s € io....,n-.l

""

containi.ng

O

o(-/J2)k*1

137 -

is

Of course

always tarne, j.f

remark 4).

(i.e.

For such rings

an Eucli-di.an graph, (3,5.6

R

it (M,e)

as r.re have

gut the follorvln
holds: be an hereditariz

semi primary

and indecomposabre (as ring).

ring.

Then

Assume that R

is

of

finite

R

is

tvoe

138 -

-

and only

if

o f tane

fhere

2.t

then

rr7

K-realizations

of

,r21n1 = q.

B(R)/

again

,J(B(R))

=

(o) iMj +

finite

is

and all

Now define and joining .Mr + (o).

L

is

(lR,dR) -t i i-

.

l<--/L.)JL

reflects hereditary.

j

(if

R.

Then

j

over

by an arrow '

LJ

(rM.) j

-

t

many oriented

But of

course on

K

acts centrally

K,

(comp. (2.1.1 ) ). as vertices, j,

to

if

f rom

i

, dii

= dimF (iMi) -

-i

(Ki,i!4j). just

circles,

of

in

about

of this

R,

if

R

this

'

of course the form

and we cannot exPect that

behaviour

information

in general,

n),

!

{1,...,n}

(MR,QR) we take

the representation

sum of

Note that

sides.

(1 a i,j

QR taking

dlr. = din,

K.

subfield

i.s a direct

on both

because

,

i M i + ( o ) + - ' M ,)

To get

to be the

rp

sum of skew fields,

ä ,tl

= o

jMi

and

K-realization may contain

with

Lrj=1

dimensional

r.rith

Define

o.f

€heory R

define

the central

dimensional

finite

(fR,gR) i

graph-

assumed to be hereditary.

not

)

case. As

t

e Homo (itlirFi)

Homo (iMi'Fi) ^i-r-'j-,

an Euclidian

R

of

over

1e r z ( R { R ) ) =

does not imply R

because

cholce

K-algebras

is a direct

e Fn

\0\Ar

Fi-Fj-bimodules,

algebra

dimensional

J(ß ß))

Furthermore

a suitable

is

:

...

them finite

each of

iMj

II

R

If

the representation

K-algebra

be the basic

F -t

of

dimensional

finite

K-realization

I (R)

Let

to

For an arbitrary

following

of

an applica€ion

is

for

a rea]-lzati-on

is

:R

a Dynkin graph.

of

R e Arr(e,ö)

eithef,

or

also

a realization

is

:R

type,

ö and.

e,

if

is

rR

not

we have to consider

the

-

so-caIled is

separated

defined

There is if

there

we define

=

with

this

notation

= (O)

;2(n)

R

K-realization of

to Artin

MüIler

I 24).

Normal

of

finite

finite

a finite

disjoint

rings

R

(1 s i,j

and no

{f*s,gos)

and

as follows:

;v € {o,1} )

and

s n).

K-algebras

and only

R

\,rith

type can be

if

:i

is

a

Dynkin graphs.

A generalization

,f2 (n) = (O)

with

(j,O),

given

union of

see [ 10].

= u*rr(iMj)),

and

representatj_on

type if

(dii,dii)

, dii

defines

dimensi.onal

be again a valued of

x = (xi)

the skew fields € Qf , o :

how we can interpret x € L(M,o)

finite

the set of vertices.

(i,O)

is

This

of

this

was done bv IaI.

forrn problems

d)

consisting

of value

(o) , iMj +

the proof

theorem

(f,

the

is

of

For details

(j,1)

(i € { 1,...,r}

which are of

classified:

is

This

I*.

resp.

{O,1}

{M*s,ft*s)

if

.

iMi

R,

= Ut*"r,r"r)

(i, 1) .

and

= Fi

F(i,r)

(i,o)M(j,'l)

Let

(dij

Now the K-realization

n*".

Let

to

of

are no edges between (i, 1)

ones between

(i,O) (o)

iMj *

I;

:= f*x

fR

an arrow from

Note that

S 2

K-realization

as follows.

and only if

139 -

of

the

dimension

graph with F.

€ i)

isomorphlsrn classes type

x

(M,s?),

and bimodules (i

x. € z

K-realization

as orbits

(i,j € f). iMj , (2.3) we have seen

rn

of of

representations a certain

varretv

x

under a certain

algebraic

V* = lt : i.i.j

n

group

linear

H o m F .( " r * t -j

I

, Fjxj)

iMj

(i,j

cL(xi,Fi)

G:

f).

€

i€f

fixed

choosing

(j € f) ".,*j (x..d..,1*)-^.trices

as collection

of

L

are called

collections action

c,

of

classes

of

dimension

oriented

similar

of

IJ

and without

normal

possible

forms of

to oriented

graphs without

multiple

edges does not

seem to be

problem

of

the normal

leads to the following

K

all

€ f).

(i,j

F+

similarity

restriction

classical

over a field

matrices

over

for

"rr"n under the

J

circles

the

runs through

J

of vi.ew the

Just

x

the

finding

in

(x,.d.rrx;)-matrices !

are conjugate

words we look

point

desirabLe.

they

if

where

v'Iith other

where ttl

F., J

and we are interested

types.

over

J

such collections,

collections

From this

LJ

U*

we can interpret

bases of

form of

square

K-realization:

t\

K.J is

easy to reduce

But

it

and

multiple

lne

representations

K-realizations

with

oriented

edges to the usual- K-realizations: of

1"2 Fz

'

----------------

-

!'

n-1

o",,,/"1 M

can be identified

with

representations

(Xi,toi)

of

circles

-

t9t

-

M

1"2 F 1* F 2 -

'€Fn-r

n+1Mo

n+1

-_______+

ll F

\M

\F'

n"o

such that (in

particular

Also,'

, Xn+1 O

nen+1

F n

n+lMn

n+lMn

F

= Xn+l -------lXn

is

the

-n

F

identity

di.Frr(x.,*1 ) = dim",r(xn) ) .

the category

of

a1l

representations

of

the K-realization

" 4',) '1'2

F1 '

is

precisely

the

F

category m

of all

e-

Fr -1

L - t

(we already used this, As fj.rst given by

trivial

representations

of

l+\

'2 I

for

-2 example in

(3.5.5) ).

exarnple 1et us consider

the K-realization

(^/,A)

K

K.-b.K.

Then

modules of dimension type

t(iq,A) (1,O) ,

has three (O,1)

and

indecomposable (f,1).

Given a

142 -

-

representation

p : X1 + X2

e1 o ,pz o p3 ,

where

to a direct

corresponds

+O

92rX1

vre can decompose rp into ' - xr' is an isomorphism, 91 : x, sum of

copies

of

id

isadirectsumofcopiesof

and

: K + K, O:K+O'

and

O : O + K.

This

tt

e3 : O + X,

is

sum of

(nrm)-matrix

means every of

a direct

g

rhe form:

there

For the

following

are matrices

we need still

over Droduct

and

A

is

the

zero matrix,

So we can express over

K

is

product of

a)

of

we call

statement

1 x 1-matrices

(1).

KOK K.(-.K

,

indecomposable

Preprojectj-ves: the dirnension sentations

we shall

A

and

B by

understand

the product

a zero-augmentation.

above by saying

(type

every

n x m-matrix direct

to a zero-augmented

AC ).

The possible

dimension

modules are

( k + 1, k )

types

If

the matrix

B

(by two matrices)

similar

Next consider types

the

F,

AtpB = M.

such that

"l Bj

IA

B

A,B

some notation.

some skew field of

to a matrix

similar \

matrices

I lo If

is

/ r..

are two invertible

the direct

K

over

l\o 'll \ \ o'.. 1 \o/ \

M=l

that

of

copies

of

o!k€Ei

the projective

( ' l, o )

,

(2,1)

indecomposable

are

repre-

-

b)

143 -

Preinjectives:

(k,k+1) , O : k € z;

dimenslon

of

types

(k,k)

Homogeneous:

We can interpret

the

injective

(O,l )

( ' l, Z l

lndecomposable

are the

modules

1 < k e Z,

the classification

as normal form problem of

of

the representatlon

(41,A2)

pairs

of

above

1lcx6)matrices

over

l l t l

K,

(41 ,A2 )

where

and. only P

and

if

there over

e

we identify sentations

is

are K

calIed

two invertible

suchthat

K.-.K)

of

indecomposable

to pairs

(k+'l,k)-matrices

pair

is

similar

of

to the pair

is

with

if

size) (where

the repre-

easy to see by induction

over

size satisfies

correspond

and each such indecomoosable (k i O).

(') = nn

.t

t'-'.t,

matrices of correct

(L=1,2),

representations

\ " ":?/

(It

(of correct

K. K.+l-.K

P1,PZ ,

ll ."\

nntt'=

(A1 ,A2 )

to

.

So the preprojective Al,A2

matrices

Ar'p=eAi,

the representat.ions of

to be simllar

(':

r

\1

that a pair (i) (i)o, = ^p k, Pp ' k, k

(P,Q) i

of

= 1,2 ,

and only j.f P ano Q are scalar matrices; therefore the 5rair /'r\ tt\ (P,. is indecomposable and all indecomposable gairs ,t*'-')

if

(A1,Az)

of the same size are similar

(3.2.1O)). Dually to this

every pair

to

,

is

simirar

matrix of

e*(i)

,-o(t',rot'', (i

= 1,2)

where

to

,nott',en(2) )

(A1,A2) trtt'

(The preinjective

of

by

(k,k+1)-matrices

is the transgosed gairs).

-

remains to describe

So lt and to

translate

it

into

to

equivalent

where

(1,1),

in

fact

') n : K O K' is

by (f.4.3).

q

= K O K + O O K = K

indecomposable

with

= (K,K,n) € L(Kt),

!

the corresponding

(k :

R. = K[t],

category

matrlx

elements

is

pair U

in

is

^ \ '.'1 o I

t..l

I |o/

g(x2) over K),

ring

length

objects

of

arbitrary

Now g

is

of dimension

where projection,

the natural just

( (O), (1) ).

that

So the

k x k-matrix

to

correspond

pairs

/ 1. . I t\t

X

'

\ol

be a simple

Then

R-module.

polynomial.

the following

x=(X"rX*,q),where is

d.efined

be

the

by

degree

-/!\ = R '/t, p (E,) \

of

p(t),

and

so

to

in

,

where

/

given

€ R,

P(t)

(3.4.3)

in

L(x2):

e:\gx2=XKOXK+XK = X r Q(o,x)

e(x,O)

/

R//-/+\\ X =

Using the functor

representation

,-"k/

'1

\

an irreducible

we get

v^ \ \

t\t

\P\L'

x

K2

Of course

(the polynornial

/o.1.

is

L(K21,

1).

l \ \ v l

Let

of

(K2 = t"t(1,O)). So

K-K-bimodule

mod* R* U,

one simple object

with

language of matrices.

the

a uniserial

is

U

t h e h o m o g e n e o u sp a r t

H(K2),

is a non-simple Euclidian is

-

t{9

then -x

= 1'g

1,t, -..,tk-1

corresponds

the

for

x € forms

X* a

Let

.

K-basis pair

following

k

of

k x k-matrices:

un= "111,=

('::,)

..(2) p(E,

/olt i : ' o- ,

\

\

o

-a^ -?; \

\

-ux-zl 9' - t t - t /

-

= -(ao + alt

p(t)

whele

To a m-ford

+...*

indecornposable

mkxmk-matrix

pair,

(1)

145 -

where

uk_1tk-])*tk

extension Ek

of

.

x

denotes

corresponds

the

(k,k)_matrix:

the unit

I

",1'J1.,=E,k={o...o ) ""1'31.,={;l:l--" tr ) -"o-\

of

course,

in

ej

= id

* .J

.

matrices

over

with of

we get

the

that

So we also K-

u(*2)

Jordan

Theorem:

get

the

form of

the

they

following

of

1)

if

algebraically

dimension

of matrices

2)

Of course,

for

the

as ring,

if

X

irreducible in

fact

is

is

square

closed

(Kronecker

is

similar

K = €):

for

to a uniquely

of pairs of (1t (?l t (Ul.',rur.-,) and

€N

€ KIt]

r P(t)

corresponds

the

form

irreducible).

to direct

summands

( o , . l) .

resp. a simple

polynomial it

is

form of

product

The zero augimentation type

K

theorem

(k,m,l,r

( 1,O)

representations

rational

zero augmented dlrect ( 1 ) ( 2 (1) , (2), ')), /t D rk , tok , r( r m rta ) r

Remark:

the

canonical

determined

,

xJ.x

just

are

contained

square matrices.

Each pair

1 () a ) , t . ,. (p2( )t ) ) tf tHt (. p

/ "J?1,

which are not

(xK,xK,e)

is

rn particular,

So we have proved

A 2.1

of

the representations : X* + X* r

o-

\

the representations

are just

u

/

p(t),

an extension

R = K[t],

R-module, then fleld

x K'

R/(p(t) X =

can be considered of

K.

So

Kr

is

)

-146-

the endomorphisn corresponding

field

to

of

X; or

arr matrices (p,e) (L = 1,2) ticular,

K = lRr

(8,

and

if

then

g

corresponds

H

(i.e.

""Jil,= "lil,O the

of

simple

H = (1,1)

din

if

of

ring

polynomial),

to a linear

the corresponding

polynomial

quadratic).

is

C,Il

denote

this

the

field

we want to describe type

Ä.',,

Note that

!a

on is

is

namely

two copies

of

acts

of

A,

on

is

ACE

M,

algebra,

a simple

of

lR-realizations

aII

lrh

are

so there

CCC

and

%

lR

is

the center

on which A

which

IR

a product

is

and

where the right

Ä.,,,

lR

Of course

of

on which

G-binodules,

CCd ,

of

acts

t'$to simple

type

lH-----J

K,

F E* Gop-nodule.

as a left

A

given by conjugation.

of

are skew fields, subfield

because

Furthermore,

.

T{ ]IflR

H

F,G

ll-birnodule,

one simple

lR-realizations

rR----)

where

can be considered

dtl 1. )

FMG,

(comp. (3.5.3 remark)).

L(CCEO nA6)

different

Let'

field.

numbers respectively.

some conmon central

namelY

centrally,

O

of

the representations

only

centrally,

R

quaternion

complex'

F-G-bimodule

UoP

so there

the real

;.a

centrally

Novr, n Il,

an

of

dimensiona.L over

finite acts

and

K =IRr

paragraph let

of

For the rest

*

lR

is

X,

End U = K. In par-

Of course

the endomorphism rlng

H = (2,2)

din

pairs;the

to matrix

K'.

E = U, or

either

homogeneous representation

(of correct size) suchthar

homogeneous representations (i.e.

simple

translated

isomorphic to

is if

the

action

L(CCC O CCC) o

So, there are only six and

Ä,,,

:

- 147 rFr o FFr

2.)

F ----_JF

3. )

cac o cacc -----------)c

Problem

2)

F = ü. glven

has been treated

Of course, in

F = tR,

with

the preprojectlve

homogeneous representatl.ons case.

It

renains

the polynomj.al

sequently,

each right

of the form finite R. /p

dlmenslonal

P

just

of

R.

leading

coefficlent

q(t)

of

ideal

€ R.

there

Let

O I k I n-1

b € II

if

deg q(t) is

R.

if

is

1,

z n

of

p(t)

p(t)

€ Rttl

generated. is

ideal

R =It[t], It

R.

Con-

i.e. and

for

f

is

*/,

is

every primitive

of

R

corresponds

narnely just

the

1 = p(x).R

Let

vre can assume that

d.egg(t)

Let

the

O + q(t)

be the degree e f.

Assune

which is not contained a* ( lR.

I ) bp(t)

Then there

- p(r)b

= center of

Of course,

irreducible

the

rn particilar,

H.

p

of

t.

in

€ R,

ring

for all

p(t)

But

U

so assume that

t + ao.

deg q(t)

so

p(t)

are

the comnutatl,ve

is princlpal,

as annihilator.

Of.course

a*b + ba* .

centrally

and only

p

as in

R-modules,

be maximal such that

< n.

factor

space over

simple

+...*.1

Is a coeffi-clent

such that

R

of

parts

algorithrn

hence simple

p(t)

of

Then

that

and

of

= tn + arr-, tn-l

P(t)

vector

R-modules, which have

be a two-slded

R

So, to each prlmitive

one isomorphism class

simple

of

So let

indeterminate

some polynomial

Artinlan,

C.

R-modules with

a central

l

as right

1s a prl-mitive

ideal

of

for

the direct

un.Euclidian

ideaL

p(t).R

Also

is

T there j*

and

and preinjective

the simple

over

1s easy to see that

II

can be determined

to determine

ring

or

F = lR

above for

same tday as above.

the

C

R

1 = p(t)R

as real

in

lR.

exists

=: q(t)

+ (O)

and each ideal is primitive

polynomial.

So there

148 -

-

are

two

classes

p(t)

1. )

-

= t

ideal

of

primi_tive

ä,

a €lR. Here p(t)

S- = R/ u /I

and

The R-operation End-(S Ra

Then

R/ ,I

over

C,

is

and S. -

the

of

ring

to by

m.axirilal as right as

I!, ll

h.t

ig

1$-ulR

R-module

End_ (S ) t{a

= ha

}l-vector for

space.

h € Il,

and

primitive

all

with

the

to

-

the

uf,to

2x2-matrix

isomorphism

corresponding

to

ring uniquely

p(t)R.

= C.

the

ideals (not

R decomooses

now Il

isomorphic

= ^t

simple

As a consequence,

I{e identify

also

R.

f o r s o m ea € C - l R .

a simple

Furthermore

element

given

of

= II.

)

determined

each

.R is

l s o m o r p'h i c

on S^ is

rl p(t) = tt-al tt-äl

2.1

of

is

p(t).p.

ideals

ring

in

R are

uniquely)

of

naxinal

into

linear

2x2-matrices

aII

and factors.

over

C

form

a

b\

\-b

;l

i I

with a,b€C

I

So 1et h

"

=i

a'+b 'i,

/ a+ui,

c+di l

i \-c+di,

a-by t,

I

\-c

r ^ r i t ha r b r c r d r a ' , b ' r c ' , d '

c'+d'il i-

/

'+d' i,

€ lR,

(i2

hr

,

(h,h' € rt)

a'-b'i/ =

-

,?\

Consi-der -c-di ("-ar, c-di, \

h + E, hF e n, so "/(t_h)R i.e.

if

i.e.

a+bt

'l nen

)

(t-rr) (r-h') € R trl.

= ^/(r_h,)*

and only if

)

if

a = a'

and only if

(t-h) (r-f)=(t-h')(t-fr''),

u , - , 65 2 + . 2 + d 2 = b ' 2 * " ' 2 * 6 ' 2 ,

't49 -

-

and then *r,a-nrR

= s* with

With other

if

words,

x=

x €C:lR

there

are

infi.nj-te

ma$/ h € It

R/(t_h)R s* =

such that

Translating

this

to matrix

pairs

A 2.2 Theorem: For each pair there

a+(b2+c2+a2)ieC.

exist

p B e-1)

(pA e-1,

of matrices

of

the

(A,B) over II

p and an invertible

(n,n)-matrix

e such that

product

direct

(n,m) matrices

of

an invertible

(m,m)-matrix

we get

is

a zero ausmented

form

(. ^pk. ( 1 )' ^. p . -r Ä 1 ) , n. (t a2 ) r l , k . ( 2 ,),t . \rr rm( 1 ) .r r. (m2 ) r ., , rr ur rl ! 1 )' 1r 1r ( 2 ) ,1 a r u /t npS( a uniquely

determinded

up to the order,

a = q+iß € O, (i2=-t,o,ßgR), li

and Ui*'

\

(i=1,2)

ß > o , , " t r e r ee { i ) , , (A 2.1),

are defined as in

€ N,

(il and

''.\ o

l 8l \

I

'Jl' =", =f "r.. lo

I

()\

I Io

,)

\

In particular,

1)

k,n,l,r

t

I1

Remarks:

with

\

for

square matrices

over II,

Normalform,

where all

,'glgsnvalues"

are complex numbers !,rith non-negative

imaginary

part,

there

exists

(taking 2)

A = E in the theorem above).

Of course it

to arbj-trary In particular, for

a Jordansche

is

easy to generalize

skew fields, there

square matrices

exists

finite

the statements

dimensional

a canonical

over its

rational

over such skew fields.

above center.

norrnal forrn

-

Next we will

M =

so let

3.)

CGC

O

CA6

t (M). As above we know the PreProjective

consider

preinjective

a uniserial

we can determine over

category

and R = O[t,-],

also)

t vtith

C in one variable

and

object

one simple

with

and

fl(l't) = U x mod*R,

By (3.4.3)

representations.

where U is

ring

problem

treat

150 -

U (which

the skew polynomial to complex conjugation,

resPect

n3

R = { E cutJlcu€O , n€N} with componentltise addition 'r J j=o i.nduced by the rule tc = Et for c € O. Again and multiplication

that

is

each right

that

Let I = p(t).R

be a two sided ideal

of R," ) = r1(t-)+rr(t-)'t D(t)

= tn+cn...tn-1*...to'ltrite

p(t)

(picking

the odd and the

deg q(t)

> deg p(t)

such that

cn (R.

even Posters of all

for

a polynomial rf

n is

degree

€ I has degree k < n,

Let

I

of degr..

= rr 1t2;+irr(t2).t,p(t)i=r1

than n in

smaller

all

prinitive

ideal

= t2-a,

ft)2-ft2(t2)t,

is

i

hence zero.

a polynomial of

is

hence zero-

I,

over

right C so

the annihilator

ideals

as polynomial

of

ideal R,z,

is

only

Thus either

even or

o + a€lR or p(t)

*iah

Artinian

= t or p(t)

a maximal is

€Rlt2l

Therefore

= (t2-a)(t2-ä)

ring,

r.rords,

other

Let p = p(t)R

of R [t2].

it

R. Then "/,

a simple */-.

of

are maximal.

of R. By the above p(t)

irreducible p(t)

- p(t)i

I,

eBtt2i=

t have non zero coefficients.

< R be a maximal

where P denotes

n in

than

"^"tf.t = ip(t)

even 2irr(t2)

dimensional

finite

consequentlY r., (t2),r2(t2)

n is odd, then 2rn (t2)=ip(t)+p(t)

odd poqters of

only

Let O < k < n-1 be maximal

- p(t)'t

center of R. consider ip(t) (i2=-1 ) . rf

Of course

t).

€I.

o < k < n-l.

arl

'

o + g(t)

Then tp(t)

hence c* € lR for

principal-

R is

I of

ideal

hre see easily,

ej-ther 9(t)

for

= t,

somea€c\lR'

or

-

t2 - a€R.

Let a € G and conslder tz-a

= (t-b) (t-c)

for

U+E = o and bc = -a. of

this

equations

same b,c

we see that

p(t)

= t.

and So = "/p

2.1

is

End* (So) = O.

p(t)

= t--a,

right

ideal

)

O < a €R.

tS")

p(t)

= t2-a,

o > a € R. Then (t2-a)

ideal

and

*rn

= (t2-a) (r2-ä),

a maximal'right simple

=

the -

R/o module.

overJR, and

by the

"/p

the

complex

R is

also rnaximal-

R/"-rnodule

The sinple ". Il.

a € 0\lR.

ideal

"P /Dn - m o d u l e .

So we can paranetrlze R-modules

- simple

determined

is

= n.

ena*

p(t)

over O

\,iä).Ris amaxi.mal

R/f and S. =

ring

ideal

dimensional

Then I = (t -

(2,2)-matrix

Sa has endomorphisrn ring 4.)

also maximal as right

R-module,.one

uniquely

'/o = Y!(2,F.), the

as right

is

over p = (t2-a)RcR,

D

part

cases:

a simple

with

and imaginary

O < a € n.

four

up to lsomorphisn

3.)

the real

Then P = p(t).R D

Assume

€ 0. Thi.s forces

Separating

Sd we have to distinguish 1.)

-

Ilt

Then r=(r2-a)R is

over p = p(t)R

R7, and S. = i"

. M(2,C) and End* (S") = C.

isomorphlsm

classes

of

the

simple

numbers a € iD, whose imaginary

j.s non-negative. Again with

(3.4.3)

we can describe

t-lul

(M =

ncn o na6)

Of course

this

can be interpreted

normal form problern of

(n,m)-matrj.x

now fl(M) and therefore

as solution palrs

of

the

(A,B) over {D

part

th"

- 152-

!,/ith

the

definition

followinq

and (A',

Br)

if

similar, (of

comnlex matrices

of

there

correct

But the

(A.B)

Call (P,Q) of

a pair

is

invertible

such that

size)

PAQ-I = Ai and pEO-1 = E', conjugated

sirnilarity:

d e n o t e s t h e complex

where E,E'

to B,Br resoectively.

matrix

interpretation

matrix

following

seems to b e m o r e

interesting. Note that There

CAC

is

all

= COE Ch

a natural

Hon6 (vn 6 for

O

vector

Now L(l,l) is

0-C-bimodule.

"t

Homn (r*Gnrw6) ) " Honn (vAoAh%.CA,

spaces VC, WC over

the categary

of

where g,

tp' : Vn O

if

are automorphisms

there

Ro'

isomorphism.

C*,

A

O

Ah.

O

(VC, .ryC,o) ,

- WC äre called

nAC

,

O .

triples

all

I{A)

equivalent

q of Vn and p of WC

such that

wc

vcoc%encc

I

I s8 1 0I1

lp I

J

vcoch

isomorphism

;"äT" notation Two

of

similarity

lR-Iinear

are calLed

q:VC'VCandP:Wn

(pt

if

wc

onoc

above this in

is

Horn* (Vn I

transformations

similar

.t

there -

exist

ü,ü'.

equivalent Co*, vc o

regular

WC suchthat

to

the

following

Homn(a0n' I{n)) C%

{

C-linear

:

HomC (ncc, wc) transformations

- 153 -

vcoah

"otn,(11cc,

I

I

I

I

I n"'' I

J

(ncc'P) Homc

J

ut

vc e ch.

wc)

Honn (*0n, wn)

conunutes. Identifylns lrith

the

vC o

resbdctLons

the

following

CalI

a real

of

normal

2)-blocks

has the

if

exlst

there

((2n,

2nl

and

real

nurnbers we get

forrnally

1 < I

cornplex if

< n) of

its

every

partition

into

form

tkt'

^*,

\-on. (2n,

(naturally)

on'\

( "*'

Ttro real

the

!{C)

problem:

form

(1 < k < h,

h0a,

Vn, WC to

(2m, 2n)-matrix

(k,l)-block (2,

and Homn

Cf*

2rn)-matrices formatly

bkl

€ R

)

A,Ar

complex

( 2 m , 2 m )) r n a t r i c e s

are

said

regular p,e

to be O-similar square

matrices

such that

PAQ-1 = g' With

other

words we consider

transforrnations

normal

forms

betereen complex vector

of

real

spaces \,ri.th respect

to complex similarity. By the

above it

i.nto matrix

remains

terms.

to

translate

the

results

for

t (M)

154 -

-

by (:.3.1)

Flrst, only

an (2n, 2m) matrix

n = m+l (preprojective

if

matrix),-

- or if

n = m+1, and dually

So, if

matrix)

one indeconposable

(2n,

if

or m = n+1 (preinjective just

is up to sinilarity

in these cases there

indecomposable matrix

can be indecomposable

one

m - n (honogeneous matrix).

m = n*11 we have to find

just

2m)-matrix.

Let

Eo=

(Ä:)

c,ß €R, and consider the

for

?)

"-=(S

(2(m+1), m)-matrix

IE

'

p-

=

\

'-.'.

i

P*

with

space

In that

order there

to

transformation

a

(n+1)-dimensional { v . ' | , v . ,i , v , v r i ,

. . rw6*1 , w,o.,1i} (i2

{w1,.

prove is

I

an lR-linear

V into

{ w . ,r w 1 i , w r t w r i , . .rv*},

I

I

.--

to lR-bases

respect

{v1,.

II

I\ o ".". '.''o

|

describes

C-vector

\

..

\\ ' " o l

Then

\

^u

'.'.

|

(over lR)

\

I rE ' , . ' . /

andEsB=(ä-l)

.rw6*1}

are

no non-trj-vial

c-vector '..,v*,v.i}

= -

of

an m-dimensional sPace W, and

1) , where

C-bases of

indecomposability

g of

P*,

decomposltion

v and w,

we have

respectively'

to

show

VC = Vö e Vö

,

- 155-

WC = Wö O Wf such that

tp decomposes into

gt.

W".

-

Vt

This

is

gt'

Wt,

Vtt +

:

trivial

for

m = 1r for

without

R-linear

loss

maps

of generality

V'

= V,

hence tp" = O, Wn =,OT'!VI O W"C = rp(-V)-O Wf,, where denotes

aWI

subspace of

'l For m > let

generated

,

the

least

C-vecror

indecomposable.

V be the O-subspace (ü = (O), if

m = 2),

of V generated

by

and fr be the O subspace of W

fhen 6 = el?

by w2,...,r^/m.

(by induction,

i.e.

tp(V) . But 6-[VJ- = WC, hence

W contalning

Wö = tOl and g is

u2,...,vm-.'

of p(v),

the O-closure

: V - fr is

indecomposable

m > 2 , a n d b y + _ h eO - d i r i r e n s i o n s , 1 f r n = 2 ) .

if

Now e(V) = e(V')+e(V"),e(V')5W',9(V',)etv" i.mpli-es O(V)= (p(v) n W') O [a(V) n W,']. Nore that lalgest

p(V).

C-subspace of

Let o = o1 + u,

ut € 9(V) nW', 12 €to(V)nW". Then ui and ui

= *1 * x'

with

€p(v) nwr,

t

fr is

rhe € fi c

A(V)

,

= u,,i + uri €ff , x2 €a(v) flwr'

Now'

W = Wr O lrl" and W', W" are O-subspaces of w, hence = x1 , uri

rli

= x, and u.,i,

Thls irnplies fr c the other that

(ffnw')

inclusion

is rl is

injectlve.

(finW"),

e

being

trivial.

in

fact

,p-1 tfi)

is

g-1

(ff) c v,.

p,

Ofcourse

(Vnv"). i.e.

the whole of v,

O (fifl!{,'),

has full e

By induction

rank,

(o-1 (fi)n v.), we nnay

fr c W'. Again by i-njectivity

Now e(v1),

e ff,

a(v*i)

the JR-subspace of V generated

{v,,vrr92ir...,vm-1rv*-1i,v*} ,p-1rfi1 is

o

(say) fr = finW',

of e we see that

so fi = (ffnw')

S o t p - l t f i l = ( , p - 1( f i ) n V ' )

and, as above, ? = tVnV') assume that

u r i € r p ( V ), h e n c e u 1 , u 2 € f r .

by

, so the G-closure q=lfil so y = e-1fi1

w = e T V fS W ' , i . e . v = v ' , w = w ' .

.

v,

,

of

- 156 -

t

Let

I.

= Pfr

I^

is

indecomposable.

(the

transposed

Next vre want to describe H (M), M = that

O

nOC

the elements

generate using simple 1.)

C%.

we get

=

It

C O C - OC C 6 = 1 @ 1 - i

U = (O,O,p) , tp : lI I M -

Corresponding we get

the

9.

a)

t S" O

+ S.

q
c € O

given by =-

(c €c)

R-module S",

given

ä = q+9i €0,ß

> O

homogeneous representations:

by the

mapping

following:

over

C with

basis

xu say:

(x" E e., ) = x.

Q" (x" 6 er) = {ut

b)

= Eer,

of M: C

irreducible

is

O i

the non-isornorphic

H(M), where the O-linear

nM

e.

.2..

of

O < a €JR : S. is one dirnensional Then

= 1 O 1 + i

e,

list

to the simple

(sa,sa,ea) €

in

easy to verify,

particular

e(c6er)

following

. Then an above

representations is

O ir

the following

=o,

P*)

irreducibel

homogeneous ieDresentations

p(c6e1)

2.')

the

"1 the direct summends, in

(3.4.3)

of

matrix

B >O:

Sais

= x"'a

tttodimenslonalover0,

c =. R=, / -) 1 ' )a 2 _ . ) * u t d S

X" = 1+(t--a) R

form a

S".

C-basis

of

Then

oa(XaOej)

=xa

ea(xaEe2)

=ya

ea(yaOe1)

=yu

aa(Ya I

e2) = xa'a

Y"

infact

t+(t--a)R

-

Next we will our

calculate

normal

-

/

the corresponding

real

matrices

for

problem.

form

Let VC, WC be {wr r...,wrr}

lf

C-vector

spaces with

respectively.

Honrn(vn o M, wC)

-

bases {v.,, .. . rvm} ,

Then the isomorphism

Horn' (vC o af',

Hornn(O0n,w.)) I Hon6 (Vo,

\)

i s g l v e n b y g - ö t . pe H o r n n ( v no M , w c ) , w h e r e t h e l R - I i n e a r m a p is defined by ö : v* - 5

ü trrl

=e (v 6 1 e 1)

t9 (vi)

=tp (v O i

So,

the

VC

resDective

matrix

of

O 1)

ö with

resDect

{ w . , r w . ,j ' , . . . , w n , r t n i }

= e(vj I 1 o 1) = ] ö ( v r i l = e ( v , o i o ' lI = j öt"rt

Using

thls

lre get

the

a = c+B i

€ CrB > O,

O-base

Sa is

a)

of

O < a €lR

to

real

choosen

as

of

Mu r correspondinq as

C-sDaces

to

(p.,

and

the

above):

( a = o):

"\ t -

\o < O or

by

JI.J1

vC = wC = S"

( ""=+ s

be calculated

t , o ( v ,o e . , ) - r o ( v , 6 e r ) ) . i

l*"

b)

}ln can

tv., ,v.,i.....o*,.r*i)

t o l , r . ,e e r ) + o ( v . , & e " ) )

matrices

(where

the lR-bases

"/

l

ß > O :

/.

1ln

I'

I'r--l

az1

\o

t

o

q

1

ß-o

1

I

O,

?

o

1/

ß\

I

/Er I

E\

oul

E1

I

of

158 -

g

To

c)

corresponds

o\ -,)

lt \" that

Notlce

from the all

of a)

left

and the

) := "Jt For

€ r ß : o,

€

by

)

(Note that

't

rf

Q : vc + wc

VC

and

respect

is

a

tp

to

mean b7 irreducible series

etc.

Using

(3.3.1)

uniserial and it.s

(Vö,wü)

pair

are

(9(vö) I !'fö

a subtransformation

of

.

I

matrices

two

map betlteen

Vö , wö

if

and if

respectively,

u € H(M)).

to

lR-linear

\.re say the

WC,

or

/

corresponds

"il

B > o

where either

u"u)

("''

_

\"-l

'.c

given

o:

d,<

and

H(2) a

grrr

ls

square matrices

(Normalize r4u by 1fur

-d,

a=ct

ß=O

with

So a llst

frorn the right).

matrix

irreducl.ble

matrices

scalar

c € JR, lol

For

b)

identj-t:'

non-isomorphic

(applylng

matlices

hre can normalize

is

c-vector

a subspace of

C-subspaces In

(This

thls

clear

factors,

(VC'WC)

VC , Wc

case lve also

makes it

and composition

of

spaces

call

what we comPosition

). we know that

uniquely length.

For

an indecomposable

determined

up to similarity

1 : m € 7'

define

square matrix by its

is

simple

socle

-

f ,ttl;

. .( 1 )

/ "a l.'\ t.t

ma

tol

n=

159 -

-o

the

= E_

and'E

t{e will

diagonal

and

l- = I

for

show that

ais

of

factors

slmllar

into

to

f-) r".

are all

"t1) can decompose only

L=2.

.

sirnllar

unj-serial uslng

to

matrices

inductlon

/i

positlon out

I

H;;'

the

an indecomposable

(n-1 )

length uniquely

:

"[i]',"

contains

sirnitar

determlned

with

over

m

(il

)= za

"="jt),a=gi-I C-vector Then

space

VC

ls

an irreduclble

is

decomposable.

WC , i{ö this

on which

VC=VöOvö,

ment with

(2'2r,

if

lY

\o

to

A,

Vö O O

means that i

ln

there = 1,

x\

")

lu

E\

Hl

VC

exist

\"

of

lR-llnear

A.

is

of

com-

factoring of

transformation. ."nd

there

exist

(4,4),

X if

E\

la

v\

")

\"

")

A

has a cornple-

two complements

A}IA =c vIö

C-matrices

vöOO

Nohr, assume that

(Vö O O , Vö e O)

such that

and size

lu

as

A(x,y)=(Hx+y,Hy)

that

rre can assume

anaconsiderthenarural

I acts

pair

composition

m = 2.

subtransfornatlon Then the

and,

subtransformatj_on

t" \o .\

where

respect of

maxlmal

So

subtransformation

" [*]., ,.

we can assumethar

"Jä)

vtrite

ro

.

"jt)

"jt' that

(L = 1,2) ,

as above

are j-ndecomposable. of course the

"j:' 1r\ H;;'

ri \ H'-'occurs a

where

'

defined

=(:

, E

all.

composition factors

/

a/

\

in

I

.H(i)

\

m times

\

'F

.

|

\

o

and L = 2,

In matrix

y ,

of

terms

sj.ze

such that

= ln lo

o\

"/

,

160 -

-

(r[-rnatrix For

= 1,

i

implies

this

that

01 + o3 =O,

which

is

equations

o1,...'cr4

* u2,

O=cna

in

leads

the

entries

to a non-homogeneous linear of

X

some

to be unsolvable.

there

exists

(3.4.3)

(2n,2m)) matrices il

For every

(non-zero)

(A,B)

a pair

the

(A of is

following

"j:' and (3.3.1) the

to one of

real

(2m,2m)matrix

"j:)

size

(2 (p+t ) ,2p) -matrices

(2n,2n) ' B of

a zero-augmented

(P = t t2,..-)

l Eu-o . " - .

\

o

\

\o'nl

"": I

\o

(p = t,2,---)

(2p,2(p+1))-matrices

/\ I E-Eo

\

le'nl l.i l.'t lr \n.tl \FFI \

\/

a.l "a-o/

D

complex matrices

tyPes:

{l . r l

ii)

So

similar

fornally

of

A D B

such that of

i-n 16

So we can state:

both invertible,

B'

and

is

square matrix

m e E' .

1 t

Theorem:

A

= -1,

system of

( 1 5 equations

Y

and

can be proved

which

indecomposable

A 2.3

such

€ lR

o1a+o3a

indecomposable in every case and again by

every for

crra *o4

this

indeterminants), is

there are

that

Ex + HY = -E.

Consequently

impossible.

! = 2,

For

complex).

means fornally

size

Product

of

-

151 -

square (2P,2P)-matrices

1il )

lr

o

t.l l..l tDl

\

\

o

l"a"-

I.'\ t.'\

(p = t ,2,...)

lat < 1 a €lR

with

\

'."'/

o \ \. e i

ot/

and

(4p,4p)-matrices

square

iv)

and E.

(p=t,2,...)

0
I

O

O

O O O E r. E ^ O 0E E_, E. O

E

O

ctb

E_1Et o o o o

with

I

either

or

ß>O

ß=O

o,߀lR

E O

o

L4 'E

E

I

-t

o o

. ClE

F

U

L

A

A

I

o o

E r. E ^ üb E.E" I

These matrices and,

ln

order)

following:

iii):

of

D,

with

respect

they

are determlned

to

c-sirnirarity (up to their

unlquely.

matrices

il)

lndecomposable

the decomposition

Furthermore,

i)

are

the rlng

which

fix

the

Z

of

all

pairs

indecomposable

of

regular

matrices

is

ztnjl)l

= c

formally given

complex

by the

For case z=C for

0 = 1 ,

0=Ora=g

otherwise

z (Hjl ) ) .o rR. For indecomposable matrices "J:'

of composition tensth

-162-

pairs ls isomorphlc to m the rlng of centralizing *ttl,a^, R =lR resPectivelY c' , with iv) :

respectively

C.

Consj-dering

the

bimodule

of

(2rn,2n) complex matrices

forms

as above normal

Now it

to H-sj.mi1arity. so this Every

is

(non-zero)

(2m+2,2m)

matrices

of

see that

the

of

ls

respect

with

8C I{'

rF

pairs

Kronecker

(2rn,2n)-matrix

complex

hte can determine

SC tff

rfl

to

easy the

can be done using

augmented product i)

. otherwj.se *[t]/.o., rntith R =rI As above z(Hj3)) =

= c.

zß:2))

Remark:

z = z jH!z)) =H

a = g < o:

for

@ 2'2)'

-

O

fh

So we get: to a zero

ll-slmilar tyPes:

following

matrices

-o t.

--,| F

"1 F

transPosed

the

iii)

(2m,2n)-matrices,

ln^ I l-llFl \ \

\ where

""

of

matrices

ii)

'...

o =(tr'

((2rn,2n+2)-matrices)

c = a + bi

etith

E 1' . o

i)

\

l"

\

.Er '. | l '""/ /

I

".

and

"

"

o

\ ,?.)

Er. "

\ \ =ß

a : O

with

€ C,

\

\

Ej I I

"l

?)

These m a t r i c e s a r e t r l - i n d e c o m p o s a b l e a n d , i n t h e d e c o m p o s i t i o n ofa

complex

(2m,2n)-matri-x,

they

are determined

#h

(up to

their

-

order)

It

uniquely.

remaj-ns to examine

uslng

163 -

partial

Coxeter

these bimodules. treat

that

is

M =

is

ue restrict

ourselves

is

we consider

only

to

with

no direct

lR-subspace of (u',V') : V + V'

f

rte get a classlficati.on (Dually

spaces.

subspaces of (U,V)

natural

nunbers

Set

(this

M

is

lH-vector

real

the first

of real

be the pair fact

an

vector

of ar1 quaternion spaces.)

For

(
91 = (4t1r,

vector

of

of

the

c

9r2= (1,O).

i.ndecomposable representation p2 = (2m-1,m) "-m indecomposable pairs in or

j_n the second case by ü

for

Then

denotes the corresponding

C-m p1 = (4m-4,2m-l)

is

is

In other words

vector

the dimension

(where

A(2m-1),

the endomorphism ring

and two

t* (:{) ) .

in

p2 = (111),

= U'.

a crassi.fication

Denote the corresponding

case by

V,

subspaces of guaternion

element) : Every preprojective

( 1 g n e Z, ) .

space

f(U)

representation

has dimension type

(U,V),

palrs

such that

gives

imply

we can

i-somorphic i-f there

arr

in

summand iso-

are calred

dim(u,V)

and (3.4.1)

ooxeter

that

let

P1 = (O,1),

(1.2.8)

of

t*Srh)

the

the quaternionification

a palr

corresponding

of

Deterr,line all

.

(that

A

simple representatlon),

is

t-j.somorphism

(of courseottl-n.]rh)

representations

where

(u,v)

we rvill ({h,E',e),

triples

monomorphic

our problem as follows:

such pairs

one of

L(M)

with

formulate

an

R-linear

of

triples

morphic to the unique injective

U

Instead

.

n*h

the category of all

ff

Of course,

, nILr

#h

i.t 1s enough to treat

rh-\r%Ih=h

e:

!1 =

for

functors,

So let

1t(It),

where

t(M)

A ( 2m-.1)

and

B(n). tR

for

Note B (m) .

-164'

we have two series

Similar

denoted by

lR

(4m,2rn-1) = cm(gt) find

for

V

over

ove!

II.

forms an

we denote by

]R-basis

f

. lR

/

of

NNNNN\N

In

as the the

corresPonding

bottom

row'

and,

finally,

e1in,...,"2ro-1frt in

the

toP rovt

subspaces corresponding

f

as the elements

the

nal which

ll-subspace.

Now, the

are contaj_ned in

in the third

u,

row

to

left

the

indicates

right

; in the

e.,JlR'... 'e^-.tF;

column generate

and the diagonal

the

In Particular,

e1lt,...,e2r-1kR.

shaded area

of

JR-subspace

one-dimensional

sre have from

to a single

t2*-1

t2tn-3 t2.-2

em+1

be considered

should

'a

\

O

,,

I

or

V:

a\

\

er1R,erB,.fl,...,..-fl, e*R,e*+fl,...,"2.-fl,e2m-fl,"2n-fl second row

Il

of

way (n = 2m-1)

\

em-1 en

v,

following

t NN N N

squares

of

basis

the standard

the

/

e2 e3

R-basis

in

/

I

The small

enouqh

be a basis of

{e,t....,err}

{1,iri,k}

is

gair.

above one indeconposable

and let

e(2m-1)

NN\}N N "1

lt

V.

of

I I {

Again

II.

{ e . , , e . ,i , e 1 j , e 1 k , e 2 t e 2 J - ,e 2 j , e r k , . . . r e r r r e r r i , e , , j r e r r k }

We may illustrate

{

type

be a pair,

Then

IR.

endomorghism ring

wlth

with

"*+1(g2) dirnenslon type

of

E(2m-1)

each dimension

(h,h)

So let

and

objects'

indecomposable

(2m+1,m) =

type

of dimension

o(m)

endomorphism ring

to

prelnjectlve

of

the

a one-dimensio-

those

connecting

subspaces the

flR

squares

-

flR

and

neither

f

So for over

f'lR

lndlcates

nor

that

belongs

= t$,ft)

A(2n-1) n'

fr

by the

165 -

to

rhe

1:uSm-1

eul'+ .u*1j ,

1 < u < m-l

em + em+li

and

ep-lj + eul '

B(m)

belongs

to

U;

of

V

but

R-subspace

t6

is

and

n+1Sy!2m-1

m-2ap:2n-1

may be lllustrated

(as above)

by:

k j

1

-1

"2

"3

-m-2 -m-1 -m

AA

o(m):

k j

1 -1

"2

J

that

U.

elements:

tu'

'

f + f'

em-2 em-1

".

generated

-

155 -

E ( 2 m - 1) :

a

1 \

\

N \

N

\

\

\a

\

NN

-

:1.'.rS

\. ).:..

:r\ \

e1 e2 e3

\

/__ /

N

/

em-1 en

\

em+1

e 2 r n - 2e 2 m - 1

"2*-3

that

(A 2.3)

as for

Slmilar

pairs

these

preprojective

are

, hence they

indecomposable preinjective

respectively the proof

detaj.ls of

fact

in

m

over

can be shown by induction

it

N

/

N NN

i.'\

\ t.....

,2

,/

/

__

are

just

pairs.

indecomposable

For

see [13J.

the

Next we want to describe

(homogeneous) pairs

regular

sirnple

(h.,'h). First

consider

the

corresponding

tensor

A = Tffh).

algebra

Obviously

By (A 1.1) where the Let

D

every pair additive

I H/

(Ih'VH)

structure

be the basis

H\

/n I \o

A'

algebra

can be considered

of of

this

is

A-module

B = A @*C.

as

A-module'

given

by

U O V.

Then

/\ a o a \ la. De I I \oa./ is

the

category gory of

tensor of

algebra finite

of

the bimodule

generated

c0c

D-modules is

Kronecker modules described

in

O

CCc ,

equivalent

(A 2.1)

that to

is,

the

the cate-

again by (A 1.1).

_167_

(U,V)

Now let lhen

be a slmple u = 2v,

we knorü that

morPhism ririg

dlmensionar

Let

M

thit

the

so it

(U,V)

of

nlte

regular that

either

u + 4v = 6v

of

M

l,t 9n'C

correspond

Pc--1

ac

(given

t\do vector

pC

and

or

either It

M(2,C).

the

and

of

which

c-rinear

QcoQc=vo*c

ts

or

a fi_

II.

Let

well_known

ring

The

End(MO) OnC

,

R-dimension M gRC

of

and so to Kronecker

over

ee

The endo_

rR, c

is

€-dlmension

D-module,

by a pair

spaces

Pn=uo*c

also

to a

is

ring

(u,v).

type

= (2v,y1.

has endomorphisn

C ORC is

it

A-module.

M gRC A,

dlmenslon

(u,v)

ls

thus

be the correspondlng

is

of

has to be a division

]R-argebra,

B-module

palr

.

Now

module

transformations

A)

satlsfying

so

dlrnPn=u=2v

between

and

dlmQC=2v. Ite have now obtained

a Kronecker

dirn Pc = dlm Qn = 2v , or

M(2,c).

only

with

v = 1-

for

(A 2.1)

it

rndeed,

the

sirnple modules facing

and our

Kronecker

ls

(Pf-1

the

direct

Ql) o

module sum of

(P2-

are lsomorphic

Now' applying always type

exi-st (k+l,k)

the

easy to see that

onry

(then its

its

it

(k,k+1)

(k € N).

c

are

rt

is

modules

the

cannot

happen,

forlows

no non-zero

that

horno-

endomorphism ring is

is

modules

G @* c)

M(2,C) ) .

easy to see that

homomorphisms betvreen Kronecker and

is possible

Kronecker

So thls

endomorphism ring

functors

c aRc

modules

which have either

(and then

is

be indecomposable.

two indecornposable

02)

coxeter

ring

a,

thls

indecomposabre

dirn en = 1.

cannot

morphisms between each other or

is

whose endomorphism

regular

lt

satisfying

eC

whose endomorphism i.s either

dim Pc = dim Qn

with

pC -J

modul"

of

So in both cases

there

dimension

168 -

-

din Pa = din Qt ft This

shows that

type

(2,1).

is

there

(1,o)).

dimenslon type element

h € I{

(h Ih,II)

r

(th,II)

every pair

yields

forsome of

]R' = 1'

h€IIUR.

for

S.

ClR+hR)=a'+hrR

so

End(E + h IR,II) = lR + h lR s C.

morphic given

@ + hR,It)

to

by the

left

Inpartlcula!

if

Next

Then i)

= m is

Uß

,

of

the

follows.

Let

r€4,\{o},

=

t€8,+hß,.

(Ih,Vn)

{e.,,...,e*}

regular

(hrh'

Conpairs

- up

E{ + h 1R,II),

forn

6111

if

and only

s€lR.

(l

s m € Z,h be an

€ II\IR)

ll-basis

of

b:f

and (25u51),

if

hcn+i.R(=a),

if

h€lR+iR(=a).

(1 < u < m)

e..

"1h

c(m,h)

generated

eu_1i+euh ii)

g-1

is

(1 < u < rn)

e., ".th

as

pair

be lso-

g € II.

by an element

@ + h lR,rI) e Et + h' IR,IH)

the

9€]It*hlRr

(R + h lR,fi)

So the simple

the pairs

forsome

Furthermore

Such an isomorphism

Il

and

lsa

lR+hlR

andonlylf

Now let

on

s-'l€n'+h'R',

and

define

if

and

and we get

]R' = Lh

C.

€ Il\ß..

multiplication

h'=h.r+s

dim V,

h,hr

,

to isomorphism - are just h € rl\n

to

9€U

lR + hIR =]R + hr lR.

sequently

b:f a non-zero

Observethat

isornorphic

(2'1)

summands of

an isomorphism betsreen (URÄI)

which is

H

of dimenslon

of dimension type

multiplication

so w€ can assume that

U=lR+hlR

pairs

have direct

we don't

Now left

v = 1.

out that

regular

slmple

(Note that

simple.

turns

and it

are only

Evidently

regular

subfield

= 1,2),

,

and

eu_lj+.uh

(2 1u:m)

,

and

with v.

-

Hele vre can obvLously C(n,h)

assuine

169 h = i

(because

i

€lR + hR).

(as above) by:

can be illustrated

1)

N /

/

J

/

N

{

N

ii)

The dinension = ß

C(l,h)

type

is

+ hR,II).

F u r t h e r m o r e there

C(n-2,h)
cln-'t,h)

obvious

m = 1r

$re get

embeddings

and

= c(1,h)

enough to prove easily

unj.serial, Using

are

For

/ c(m-z,tr) o c(2,h) is

it

= (2n,m).

di:n C(m,h)

follows

where all

(3.3.'l)

we get

that

that

C(2,h) C(n,h)

ls is

indecomposable.

indeconrposable,

in

fact

are j-somorphic to

composition

factors

all

indecomposable pairs

regular

From

in

this

C(1,h). way.

-170-

Now, if

= (U,V)

C(2,h) g C(1,h)

C(1,h)

a subring

is

then

to

of

ejj

hEE+i-n',

then

R + h lR = lR + hIR

(h,h'

€ U).

can be parametrized P

this

is

not

andif

h=1,

simple

regular

+ e2i € U\Ui C(m,h) e C(m,hi)

by

lR + hn.

because

e,|i+erh€U\Uh

As vre have shown,

pairs

(u'v)

of

easy to check that

is

to

isomorphic

U = Uhr

implies

it

However,

is

cornposition factors

This

C(1,h). lH.

If

thecase:

because all

,

are isomorphic

decomposes' then it

a f ixed

set

of

if

words the

In other

by the projective in

representatives

lf

and only

II

of

Denote

P F/R).

space

e FZ*t

for

,

example vre may take p = {ia + jb+k

I a,betR}

U

{ia+

j

u

I a€rR}

{i}

So we have shown

^ 2.4 to

Theorem:

Every pair

the

- uniquelY

orders

(1 1 m € z

pararnetrized Novt P2m)

^

by the is

lR[x,Y,z]/

-IR[xrY]/

'

be endowed with of

i

and

j

direct

sum of and

, D(m) , E(2m-1)

the

points

of

the

projective

)

'

{X-+U2+221

a 1 , r 11.+y-+1 )

one point

with

t o a - up

isomorPhic

pairs

of

C(n,h)

, h € P).

We have seen that

Remark:

is

deterrnined

A(2m-1) , B(n)

the form

rina

(h,h)

omitted. the in

i+a its

the

of

real

variety

paj-rs can be

regular

corresponding

h-ei'5r e hÄ^+rrrm maxj-mal sPectrrm

of

A-module an

plane

projective

Then' consideri-ng

Given an

structure

terms

simple

the

P2B).

to the graded function

narracnnnrr corresPonds

lla,

then

JH-space by defining

to

ring P2 G)

lit O I'1 can the

action

-

/o \ ll,and \-1

induces

Remark:

mod*A

(M O t{h) .

mod*A

and the

(1:m6Z)

c(m,h)

is

of

paragraph

further

applicatlons

we tant

of

treat

real

subfields F;

of

subcategory

h€IR+IRi

subspaces of

is

rather

trivial:

(AAJh)

there (O,C) ,

the pairs

complex

spaces over

Conseguently

nanely

we will

and

filtered F.

containing

F

thus if

the

The corresponare only

three and

%,AA)

@C,%)

some indj_cations

in

its

theory

spaces or,

K

K

there

is is

of

center

more generally,

S called

such that

ordered into a

the

for

graphs.

be a commutative

(partlally)

mapping of

containing i € S,

vector

So let K

A finite

preserving

give

still

the representation

a field

dimensional. an order

of

to define

a skew field

finite

functor

applications

last

K-structures

82.

(O,$) ,

respectively

In this

Flrst

type

objects,

Further

S 3

ful1

category

(3.4.3)).

But here the result

binodule

to the This

with

spaces or complex subspaces of vector

(AR,AA)

of

between

pairs

indecomposable

wlth

from

f

Of course we can also

quaternions.

F

the functor

an equivalence

(see remark 2 in

ding

-x/

r (Mo) = ( (M O o)R ,

by

regular

vector

-)

"/

pairs

of iff

(r t

iJ-

Thus, we can define of all

171

set

field FX

S

lattice

F.

i_s

together of all

K-structure

given a subfield

and

SK(F) of

F

172 -

-

and, moreover, An

(w,wi)

S-space

ove!

F

together

such that is

K g Fi =c Fj

an

gory.

with

an

if

i

The category

= F

a right

Fi-subspace j

:

in

t{i

S.

of

all

i € S,

to be a filtered the

(I{,wi)

(1,dim o li F

i

< j

F)

for

all

k

the

follorving

Ä

(We denote

in

If

a (valued)

fS

full and is

q(w,w1) * for

if

for is

(Vt',Wi) all cateif

and only i

all

€ S.

an

S-space, we

K.

To

SK(I')

-i

Fr)

-k

J- j

_

(f,d),fl

graph

(i,j

€ s)

from

and from

given

i

+^

k

by i

to

nF

o

rr:lrra

of

value

J

.

(1,dim

€ S,

an addltive

space over

where we drarv an arrow if

F*)

w

K-realization

valued oriented

f = {S} ü {O} ,

i

S Wi

e(Wi)

O (wi n wi' ) and

space

each

S-spaces is

vector

f

associated

vector

A morphism

E(SK(F)) = (F,Fi,p.FiF,,r. with

S.

lor

such that

!{, = (w, n wi)

for

vte can associate

is

of

(W,Wi) = (W'',i{i) O (W" ,Vti' )

and

(Vt,Wi)

say

sK(F) )

t(S*(F))

In particular

K = Fi

i 1 j

each

map tpr : Inl + Wr

w = W' O Wir If

(over

wi E wj

F-linear

i € S.

for

is

embedding

€ S

tree,

which

are

in the given

maximal

oriented

then

1t can be shoern that

t(SK(F) ) qt(r(SK(F)

valued

q r a p h b ..Jz

this

there

For example, if

).

order.

f^

.)

l-s a

S = {1 }

lR of I{, then L(SRCI)) is given by the subfield hGI) just the category of lR-subspaces of vector spaces over lH

and is

equivalent

tatidns

of

example also So in

dh

to

the

without

shows that

full

subcategory

t (d[fu) of

subobject

of dimensj.on type

L(SK(F))

is

some cases the knowledge of

classi-fy

of

the indecomposable objects

not abelian

L(I(SK(F)) in

represen-

(1,O).

This

in general-.

can help us to

L ( S n . ( F )) .

Conversely,

_173_

before

the Coxeter

to classify for

t(SK(F))

ls

nunber of

finite

glven

in

be of

finite

filtered

of all

[ 1O].

It

oriented

graph

K

So let the

there

is only a finite

indecomposable

representatlon

that

again

type is.

the property

depends on

fields,

S-spaces.

that

S is

to

and the on the valued

theory of

of bimodules

torsion

we refer

to

free

can be used in

abelian

groups of

fi-

[ 7]

Examples

is is

in a

(3.1.4)

flnite.

graph

the results

K-realizat.ion

K = cF(2)

Dlmkin

if

of finlte

the involved

theory

For details

As indicated (M,fl)

type,

type only

the representation

B

spaces see [ .lg] . ) Of course

can be seen there

of

S-spaces

fS .

the classification

fleld

SK(F)

people used

(For example Gabriel

type.

non-isomorphLc

representation

dimensions

Appendix

finlte

vector

dimensional

reratlve

if

of

sald to be of finite

A classlfication

ni-te rank.

have been founded,

K-reallzations

the case of

Finally

functors

First

type

(2.3) do not hord in general,

some valued give

we will

and consider of

of

of

the

graph,

erhere the

here a counterexample.

following

K-realization

of

D4 ,

1

oK

,2f\ ,/l\ Ko2 Ko3

Ko4

,

(where

all

birnodul-es

are

Just'

xKx)'

-

| Iq

(hereditary)

The correspondlng

/

t .ensor algebr a i s

o

l"

o o K o

K lo o \o \oo First

we will

list

corresponding

c-

i'2 -)

where

C - P .-t'

consider

c

the orblts

dimension

-

of

*\ K] KI Kl

the

'Coxeter

and the

types:

P.

-Pz^

1

,l

:3

=4

1

1

ooo

100

o10

oo1

2 111

1

o11

1 101

1 110

1 111

o 100

010

o oo1

= r -E .

(t = 1,...,4)

P, = x .

functors

Then

I

can be described

as follows:

z*R*\ K@1

Now consider then

dim

= -c - 2

-I r.

(X)=

(r.lrx )tr^

t

O + Fo -

for

it

is

Yf

but

,

X + -fl n + Q

easy to see that

three submodules

(corresponding

:P' l.

+ dfun L-

(1,1 )K denotes the where diagonal enbedding k + (k,k)€ROK for k€K.

lf

(t = 1 , 2 , 3 )

there

to the Iast

exists

row of

no exact

A),

sequence

,

is not a quotient isomorDhic to

of lt

I. ,

I

narnely

has

- 175 -

,t = of f "t", Now lt

is

(1 S t

!

l

K

€

easy to

!2 =or"Et'\o, andy, = x/yt

see that

Of course,

J.f

erlth

O + ), + 1

and we have then

K + cF(2)

is

a field,

4/-rz-

a submodule

g r. -l

The complete

Diagram

(2.3),

(3.1)

Given a Dynkln graph

(f,d)

of

lnfinite

fleld,

consider

to

r

(up to

each

tatlon ls

I.

a sünple

Ir^ -2

,

and

the ,

isomorphic

and for root, X.-)Xr!2

with set

Xr)--lXr-1 lf

such

11,

that

two respectlve

cokernels

!2

graphically.

(Mrfl),

positive

uniguely roots

roots

determined)

r is

an epimorphic

kernels

ls

Ä:1

K r,

represenlZ-Ll

a submodule image of

one may indicate are

an

attach

such that

!1,\2

if

l4oreover, or

as follows:

be illustra.t.ed

all

of

I"_ 11

each other).

Examples:

to

page

of

pair

every

cases excrude

arrows

looks

K-reallzation

j.somorphism

,

X

can easlly

the

draw

:1

of

see next

(the

the

(1,tr)K+KOKr

.

submodule structure

The results

w e can choose

an embedding

y" = - z o r ö . o,/t,ir\

that

/" i

ra e c*1a = C-pa o ^ t 5- 2 t '^

3).

resPectivelv

-4"

isomorphic

the

of -I-! 2 fact

by drawing

parallel. (f,d)

= 1?-2F-3

( 2, 1 )

(type Br),

r.vith K-realization

,

-

| /o

-

A,

(A1A2A3)+

A4 (A1A2o)

A4

jor oA:)

rc-lzolr

.) P,OP.

A4 ( A 1OO ) dP^|EP. -a -L

a1 ( A 1Oo )

A1

(oo-o) e P-t .

where (suggestivelywritten) and

A4=KeX

At =KOO,

A2=

(1,1)K,

A3=OOK

-177t- F u^ FF-F(-

nhere

cEF

such that

G

a r e skew flelds, G.. Ä

and

Ff

are finite

tensor

/\ lc

F

F\

1o

F

\o \/

o

F I F I

\t

of

the Coxeter -p1

functors

'2

^3

100

110

1't1

010

122

0 11

112

o12

001

DD

-) where c-!t=It

(1Sra3).

The correspondence

between the root

is

described

dimensional

(heredj.tary)

t\

-) c-

K

in

their

and

center

dlm3 b

The corresponding

The orbits

containing

by

algebra

= 2.

is

are

system and the module structure

178 -

-

If

we change

and

orientation ^ r

^F"j

the

consider

K-realization

)n

t'

-

tre qet

as

tensor

algebra

(" \"

\o of orbits

and the list

P1

Asain

of

ol 1

c

2^ 1"

o

^ c-z

2, ' 1

c-2qt

= It

The correspondence

F

o

'l

]

functors

D -J

11

o ''t

-1

I I

o^

1

öo

1-

(1 a t

F

the Coxeter

-P2^

o1 o

'\

"

1 3).

between the root

and the module structure

systen a-D

is

described

by

I

II

I

u

-/, ./\

./

t\ lz

,r/'-" ,)l

,/l ,rt t'/

.(" DI :1

I |

Y -?

_179_

References:

Appendix A, $1:

Appendlx B

f'or further includlng

applications in particular

[1O] ,

t14l

52:

l12l

,

[13] ,

tt6l

$3'

[18] ,

[10] ,

[7]

:

t15l

and exanples

vre refer

a famous bibliography.

to

[g]

and [9],

180 -

-

Appendix

C

Tables

The following

tabl-es

following

the

of

on the

information

3 on the

chapter

non-homogeneous representations.

indecomposable provides

summarize the results

Each of

tables

the

valued

resPective

reqular

Euclidian

graph:

1. )

of

The type

the

graph

valued

for

the notation

including

the

vertices. 2.)

The quadratic

3.)

The vector

vector

The defect indicated

5.)

under the action

stable

nt

of

and the tier

Wl ,

g.

number 4. )

the graPh.

form of

is

here. M = (F.'illj)..

A modulation

on both sides,

sional

which

to an orientation

respect

with

ä.

As a rul-e, if = Fj

fi

we can assune

is one-dimen-

illj and

'pi .' Ij 1 r .

=

F - i. F i^F- .i ,

withoneexcePtion'namelyforthediagrami,,,becauseitis< (compare (3.5.3

not a tree F.

= F

for

arJrr

AIso,

if

is

iMi

.M. = oFo

€ f and that

i

all

here ll'e can assume that

remark));

but

all

Fr-moduler

as left

one-dirnensional

for

one edge.

r'te may

L )

assune that slmilarly Fr-module.

tj

for

is the

In the

a subfield

case when tables

Fl

of

is

iMj

we omit

=

and, F.FiF*t rl as right one-dimensional and

ilti

these

bimodules

representations,

divided

to display

explicitly. 6.)

The simple

non-homogeneous regular

the orbits

of

equations

I

C*, r

with

(written

their

dimension

as vector).

types

into

and their

l

-

7.\

The source graph root +^

8.)'

For

fr of

f

io

€ f

= f\iio] (f"g')

such

181 -

that

is

the

where

d'

restricted

nf maximal

denotes

respectively the

to

the

maximal

restriction

l

I : t:

h

the module A(tl

deftned in

Dvnl
(3.3).

of

short d

-t82-

T '^11

(4,1) I-2 =

(x^ -

!r

=

t-2'

-c3

=_2+-L

A - r - (' x - )

2x,)2 L

z

) -l

rU all

=

,--1

/

l_e__?\,,

<-cw:.th[F:G]=4;io=2

f

regular

representations

are homogeneous.

4,. L '

(2'21 L-2 Q, (x) =

{x, - xr)2

1-cA

I'

t1 = ,t IY

=-2<-2 M

itf

all

=

tr4

regular

F, wlth z representations

di-m

', . M I

are

= dim Mr^2

homogeneous.

= 2;

io -

= 2

183 -

n+1.

where

(n > 2)

ä.,

,/2-.....-p\ T. \ / '- p + 1 _ . . . . . . . . - n Q. (x) =

-

t ( x , - x r ) " , where

the

edges

-

i

< p < n+l

I

sumnation

runs

all

through

j

'" '''-t\r, 6---=' 1 ) , lfY

,/'

n -l

\1-1

""'-r'/''

A

"tI"..._,>" =

where riqht

given

N is

of

of

K=center

fixes f

lrl

f

E'-' F O -- . . . O

oo oo-....o

E'-'

to. -.. -o oo o o .- - . . o

o. -. -..o

o-. -. .oF oo o"':

"o

oo o... - -.o

10-....o 1

o o -- - . . o

o 11" "'1

o--.. -.o

o" ".ol oo

o...o-11

u...

- r.9

: oFo" "o

(1)

v f

o. " "'o 1t

FF'.."F

r

{1)

dim C-

o 1 0 .- . ' o oo o......o Z=

ni.,

the

F on

by a field

automorphism

c'

and

UF

"N action

o" ' :"o

F of

(vrhich F).

-184-

if

p


o'.... --o

o---.---o o

FO--....o

10.-....o

10- -

1-- -....1

-rlu

o.---...o

o. - -....o

o-.-

o-----..o

o. -. - -..o

o.-- ...o

o-. - -. -oF

o--....01

o-..

: o--..---o

: o-.-----o

o... ---o

o10-....o

-1 10

F--.....F 1

o

n-p

o

o oFo.... -o f

to

V-II

= n+1

^ ( r ) -- L ^ + -D ( r )

11

A(2) _ c+ E(2)

r

4 n

-. -o

- 185-

(n > 2)

Er,

1

(1'2)

- n (2,1) n+l

3 .....

2-

Qa(x) = (x]. - xr)'* r.alrr_r. tr

=1-1-.

-ac

= - 2+-O+-""'+O--2

,t{

=Fl

-

1-lrS=@

F-F.--..

crE

F-Fn+lwj.th

[F:r, l=

dim cr E

o

oFo.....o

o l o .- . . . o

I

FFF.....FF

2rr.....rL2

oo.....-11

o. - - - -oFo

o.....o10

o.....o-110

o o F o .- . . . o

oo10... -.o

o-110-....o

io=r+l A

(xa - xa*r)2 a (xn - *rr*t)2

=c+E

-110--...o

+ I

f gocr =Osn

lf:rrr*tJ=2.

^r^

r

dim cr E

r

o

o F o .- - - - o

o10-. -. -o

-210--. - -o

1

'

11""'11

o-.. - -o-l2

o. -.. -o10

o. -. - -o-110

o o 1 0 .- . - - o

o-1lo-.. -.o

1-

'^n+1

: ooFo--.-.o

n-l

10 = n+l A

=C

+

L E

=

11p

-t

a -c Ä{

.....

-2+O+-

=

= Fl with

F -

'

F -

o

olo

' 1 "F F . . . . . F

ar.

F

z

o- - -..oFo : o o F o -- - . - o o

n-1

1o = n+l =

dim cr

oFo.....o

I

:

+ C'

oo10

E

= [r:rn*rJ

= 2

r v a

-2IO.

.3t 2 \?

o....

o10

o---- o-110

. -.o

o-110

L __!arE

Fn+l;

Fn+l c F c F, and [rr:fl

crE

f

+-O+--1;

.o

r

-(1)

ct

r

l(r)

oro...-.o!

olo.

. v^o o

-IIU.

"vo

I

l . l ' . . . . . 1 , F. F

2t- .

rl

oo---

^ 11 "-' 1

2

oo --.-or!

oo. -

^- *t oo

oo...

^

oo"ol....oB

oo10

..o!

L

i-o -

L

A(2) _ "+ "(2)

:

-0.

tz t

-ro.---.of

= n+1 -+ - (1)

, ro

-ro.....o!

-o

-1-

=U

EQI

r

Z ^ (1)

ct

din

a)

' 1 'F F

^o

t

c' ^ ( 2 )

5

ct E ( 1 )

dim

E

=

d\_e -'

_189_

(n > 2)

a;"

n-r/n*u

$x)

= 2(xr-2xr)'*

ta.alrr_r(*t-*t-tr2

sr =1-2-2

+ ( 2 x n - * r , - 1 ) 2+ ( 2 * r r * 1 - * n _ 1 ) 2

,
-ac

=

-2 +O1

=Fl -F-'

ff

r

{rl

c- E'^'

t

o

oFo-...-03

I

r1!!""'!F

2

o.--..oF3

o.'

ooFo.....03

oo10.

n-2

-F

.r

cr e(1)

ü

r)

o10

"

(1)

- 2 1 o .--. . 0 3

"o 1

o--.-.o-11

la

o-..-.o-113

.'l^

r1'

=2i

withlFrFl

"<:

-1

at

-o

o-110.....03

L _

-

dim c'

"r "(2) o

-- 1

_o

(2)

,tl

E\''

at

^o

-10.....o?

^2

-1o.....oä

]I

I

1

-f t1

.....F'I

--o

Z = 9*n io = n+1, A(1) = c+ E(1),

A(2) -

t(2) "+

(n z 4)

ö.,

"-'(1.

;>'$a)

2+ ( x 2* = ( 2 x r - x . ) 2+ ( 2 x * r - x3 l ' . r : , : ( x . - x . * , , ) n - r - 2 x .) { * , r - , 2 x n * 1 1 2

ä.=_l>".-o.....*-"
r

ar

E\"

c'

"(1)

o 1

ÖA "Fö.....arv

-o

o--

-F.....FII

2 n-3

n- nr.). . . . ö F "'n

: o^"^.....^o "o o"^" r

-1t.,.....^o

1r.....r1 1^

o^ ^ .I oe" "'e-r1

o^ ^.o ou.....uro

3o--:--o-rr!

ö(.} öolo..-..oö

aro ö-110.....oö

I )\

r

dim c'

1-

-o

oF

c'

o I

-F.

ooo. I

11.

o-

/?\

( )\

r'-'

cF

-!.

\z)

. 1- 1o

-1^.....^o -1 o"

' r o1

o ^ . . . - - -^n 1

d:-^ cr e

Et-'

.-o

1^-

:

o"-..--oo

I

.0.

.\^ -.oö ö1o-.-

c- E'-'

I I

(1)

/1\

dim c'

(3)

cf

(3) -!,

'l^

ot

tl

o r . . . . . r 1' o 1'

(1), A(2) = c+ r(2), A(3) - c+ E(3) io = ,+t, o(1) = c+ e

7=4r.

-

191 -

-6 5 I I 4

II 3-2-1-6-7 $l)

= (6x, -

3xr)2 + (6xz -

+ 3[(3x,

-

3xU)2 + (6x5 -

2xr)2 a (3x6 -

3xn)2 +

2xr)2 + (3xn -

zxrl2);

11 ll 21

| =1-2-3-2-L;ln=.!,

nf

.t !.=1-+1--r-3<-1+1

,-1

F

I F -

rt{

I F-F-F-F-F;

(1)

r

r ^ i h

11\

a _

F \ _ ,

"r "

"r

n

o(1)

U

I

o

I

OFFFO l1

o1110 1lF

( 1 , 1) r I

FOO FOO FOF OOF OOF

(2) "r "

o o

1

^111^

I

I

'I

1 1)1r

o 1 0 -l o 1

dim cr p(2)

"r

n

o o

OFFFF

o1111

F r OOFFO

1 1 oo110

(2)

o o o

I

-1

0

1

oo-1

o

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1

1

rltoo

I

lo-1

00

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r

o

1

2

r

c'

c'

/?\

E'"'

dim

F OOFFF

1 oo111

o o

o o

FFFFO

F OFFOO

'o

A(1) _ "+ "(1) e(2) = "+ "(2) A(3) _ "+2 "(3)

1_l

/r\ E'-'

1 oo-1

00

o o-1

110

,l ,l o1100

(3)

r cn

to

1

o

o

I

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0 0

_ 193 _

El

4_3_2_1_6_7_8

9(o)

? I

= o[ (2x+-x3)2 + (2xr-*r)21 + 2[(3xr_2xr12 + {zxr_zxu)2) + + ( 4 x 2 - 3 x 1 1 2+ ( a x u - 3 x r ) 2 + ( n x ' - 3 x r ) 2

= 1-

tr

2-

3-

2 I

4 _ 3 _ 2 _ 1,8-_:-.il ,.^_*<

2 -a.c

=

1 +

1 ->

t

1 +-4

= F-F-F-F_F_F_F

ll

r

<-

1 +

L +-

I

F I

,(1)

aim cr e(1) l

"r

^r ,(1)

F

o

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oo11100

1 oo1-2

100

(1,1)F 1

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1t rrtzJ..LI

F OFFFFFO

1

1 100-1

1 nl1111^

001

o o10-1

010

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( )\

r

c- E '-'

r

dim cr s(2)

.t

o(')

o

F OOOFFFO

ooo1110

ooo-lolo

1

TFFFFOO

1 1111100

100-1100

F OFFFOOO

1 o111000

1 o10-1000

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I

o 3

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r

|

(r.1)F FOO FOO FOF FEF OOF OEF (1,1)F I

io

FOO FOO FOF FOF OEF OEF OOF

=$

A(1) _ c+ E(1) A(2) _ "+3 "(2) (3) A(3) _ E

o

r


c- E'-'

O

o

Ä . E r

A I

I o1122tL

1 LI221Lo

F \ J '

,(3)

I o10-2LOL

L

V

L

t I - a a ^ l ^V

I

V

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=

ll

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I'L

o

txl

ilil;; EJtsu'l

rrol llCa EIN#I

oooBo o.

o

hd

^ä

I

lt.i;

o

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oirrhtäo

NF

n.1sOl llXIJJ

o n

qEhnäo A

ltd

|

Ero

ErEr

IIH,E rt@ -rc

ts--J

X

xxl N

o\-tl

N

ä lFt s E

!rx

lr lt X a CDl

Erhr

r. .J N N X

r{

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r4orlBqr

I

u

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lroo8o

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lp, Its.

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t

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n ino

orl N.

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trtErqäEl

|

A I N

+ N

x +

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l>

l>

l>

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ö

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r9

e

oooo\ +++ lr'!

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e

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e o

e o

rlt

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o

e

nl

4

e

e

hc

rd

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hl

o e o

e o

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e

fd

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e rd

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lo

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9^

lkl

lI Ee tl ^le It vY IE l'

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rd

(E

o

B

IE

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El

o

e. o l-. hi9 Ox

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E

e |{

rd 6

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o

El r. r9 :-1 rl

UP -14

t1

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l(E

e

o n lr'l

r{

o

I |

El

"l

e El

le IE

U

I I lo

EJ

lh t' I

o

I

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lo lH'

lP.

HP NP

IF (rts

H

N

FI

NN

IE

N

NP

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n

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lr

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otsH

P

POP

NH

NP

-L97-

-41

(L'2')

a -

"

1 -

c

$l)

= 3(2xr- *r)2 + (3xn- 2xr)2 + 2(2xr- 3xr)2 + 6(2*r- *1)2

llr

=1-

2_ 3-2-

trn=o

-a.c =1->1+1-+-4+-2 = F-

M

F -

F-

Fl -

F, wirh [Fl:F] = 2;

dim

c'Et"

o I

ooF1FlFl

FOF FeF FeP FloFl

-1

-1

.r

o 1

I

'1

FA

100-1

1

o 2;\t o T j. 2\1-

o10-1

0

r

C-

lll

E'-'

"+,(1) = A(2) c+ E(2)

.r

u

n

r

v

11210

101-2

t'

L

-z

r

(2)

o1111

=t^

a(1) =

(1)

2222r

I

^4 -oA

n

u

"(2)

'1

r

L

oo211

dim

'1

OFF

(1'1)Fl

( 1)

cr

w

- 1a a

a

0

-

-

lvö

F nr. (2,r)

3 _

4 _

1

olx)

= o(2xT

.r

=1-2-i-4-z;s=(il;

*r)2

+ l(2xs-

*12

5;

+ 2(3x2-

2xr)2 + (3xn- 4xr)2,.

\-/ ä"

=1-+1->-3<-1<-1;

M

=Fl

-

-

Ft

-

Fl

F_

Fwith

^r -(1)

r

LE

= 2;

lFl:Fl

oFlFlFF

ol

1

ooFlFlF

ool21

2

PtFlFlFO

oF1F1FlO

1

(1,f)F1 with

(r'f)Fr

l

t

1

11110 )

c. E..,

FroFl FoF FoF

r

ai,n ct n(1)

=

v

-t

(1)

-z

a

u

L

oo-2

11

20-2

Io

t4-

dllt cr r(2)

(2)

a

-!.

o1120

o1-2

TL222

I O -1 0 1

f € rr:F

t=

,
lo

-199-

G,,'

2

(1,3)

1

O-{x) = ( 2 x a - * 2 ) 2 + 3 ( x , l-

-

2xr)2;

=1-2-L;g

-l

=1+1+-3

A :c

=F-F-Flwithlr:Fll=3;

M

o31

FeF+(e,f)r

FoF+(e,f)F

FloFl

where {1, f ,e} form a basis

ro

=l

=

+ C'

E

of

JJZ

F.,

-2oo-

-22

(3.1)

I

2

9(:i)

= 3(2x3 - tr)2 + (zx, - 3xr)2;

tf

=1-2-3;g=1

jc

= 1 -+ -2

= Fl -

M

<-

F1 -

r;

F with

crE

r

oFl with

I

-1 I

-i o

=1

A

=c'E

= 3

dlm cr

t+r-r'

f

[Fr:rJ

E

wLz

w

€ I'- 1 \F '

F

I.LI

t=

ctl]-

an

-5

z

- 20't -

REFERENCES

t ]l

I.N. functors

Bernstein,

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t 3l

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t 4l

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