London Mathema Lecture Note SeriE
Linear Algebraic Monoids MOHAN S. PUTCHA
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London Mathematical Society Lecture Note Series. 133
Linear Algebraic Monoids Mohan S. Putcha North Carolina State University
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Henry V/// in 1534. The University has printed and published continuously since 1584.
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© Cambridge University Press 1988
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A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data Putcha, Mohan S., 1952Linear algebraic monoids. (London Mathematical Society lecture note series; 133) Bibliography: p. Includes index 1. Monoids. I. Title. II. Series QA169.P87 1988 512'.55 88-6103 ISBN 978-0-521-35809-5 paperback
CONTENTS
Preface
vi
Notation
viii
1. Abstract Semigroups
1
2. Algebraic Geometry
12
3. Linear Algebraic Semigroups
18
4. Linear Algebraic Groups
27
5. Connected Algebraic Semigroups
42
6. Connected Algebraic Monoids
48
7. Reductive Groups and Regular Semigroups
69
8. Diagonal Monoids
80
9. Cross-section Lattices
89
10. c'-Structure
97
11. Renner's Decomposition and Related Finite Semigroups
109
12. Biordered Sets
121
13. Tits Building
129
14. The System of Idempotents
135
15. /-irreducible and/-co-reducible Monoids
146
16. Renner's Extension Principle and Classification
156
References
163
Index
170
PREFACE
The purpose of this book is to present the subject matter of (connected) linear algebraic monoids. This subject has been developed in the last several years,
primarily by Lex Renner and the author. The basic results have been obtained. The subject is now ripe for new developments and applications. It is with the hope of attracting new researchers to the subject that this book is being written.
The theory of linear algebraic monoids represents a rather beautiful blend of ideas from abstract semigroup theory, algebraic geometry and the theory of linear algebraic groups. For example, one of the first results of the author has been to
show that the group of units is solvable if and only if the regular ,-classes of the monoid form a relatively complemented lattice (they always form a finite lattice).
Equivalently the monoid is a semilattice of archimedean semigroups.
These
semigroups were abstractly characterized by the author in his undergraduate days. From the viewpoint of semigroup theory, (von-Neumann) regular semigroups represent
the most important class of semigroups. Group theorists are generally most interested
in reductive algebraic groups. Well, there is a connection. L. Renner and the author have shown that a connected algebraic monoid M with zero is regular if and only if the group of units is reductive. In this situation, the author has shown that the Tits
building of the group of units can be described as the local semilattice of partial -class idempotent cross-sections of the monoid. Going in the converse direction, L. Renner and the author have shown that the biordered set (in the sense of Nambooripad)
E of idempotents of M is completely determined by the Tits building of G and a
vii
type map ? from the finite lattice Yl of /-classes of M into a finite Boolean lattice (the power set of the Dynkin diagram). Another indication of the beauty of the
subject is Renner's generalization to algebraic monoids of the classical Bruhat decomposition for algebraic groups.
Renner obtains his decomposition by simply
replacing the Weyl group in the Bruhat decomposition by a certain finite fundamental inverse semigroup. For the general linear group, the Weyl group is of course the symmetric group. For the full matrix semigroup, Renner's semigroup is the symmetric inverse semigroup.
There are strong connections between algebraic monoids and certain compactifications of semisimple algebraic groups and homogeneous spaces being studied by DeConcini and Procesi [14], [15]. In this regard the classification theorem
of Renner is crucial. Let G be a reductive group with a maximal torus T. Renner establishes a correspondence between connected normal algebraic monoids M with zero having G as the group of units and normal torus embeddings T y T (with zero) on which the Weyl group action on T extends. Since normal torus embeddings have to do with rational polyhedral cones, this yields a discrete geometrical classification of normal connected regular monoids with zero. Renner establishes this classification by
first proving a powerful extension theorem: For such monoids M, a homomorphism
on G, extending to 7, extends to M. For the most part we have included all proofs (in many cases simpler than the original), thereby making the book quite appropriate for reading by graduate students. There are a few exceptions. For example, the recent results of the author on conjugacy classes are stated and explained without proofs. However, enough examples
are given to give the reader a good understanding. The same is done with a part of Renner's classification theorem.
NOTATION
Throughout this book,
71, Z+, O2,
Qt+,
Q, Q+ will denote the sets of all
integers, all positive integers, all reals, all positive reals, all rationals, all positive rationale, respectively. If X,Y are sets then X\Y = (x E XIx e Y). If Y c X, then
X\Y will also be denoted by - X. We let I X denote the cardinality of X. K will denote an algebraically closed field, which will remain fixed *
throughout this book. We let K = K\(0).
If xl,...,xn are indeterminates, then
K[xl,...,xn] will denote the commutative polynomial algebra in x1,...,xn
vector space over
K,
then
End(V)
If V is a
will denote the algebra of all linear
transformations from V into V, GL(V) its group of units. We let A (K) denote the
algebra of all n x n matrices over K, Kn = K x ... x K.
If AE '*n (K), then At,
p(A), det A will denote the transpose of A, rank of A and determinant of A, respectively. We further let
GL(n,K) _ (A E An(K) det A 0) SL(n,K) _ (A E An(K) det A = 1) 9n(K) _ (A E An(K) A is upper triangular) .On(K) _ (A E An(K) A is diagonal) 9n*(K) = 9n(K) o GL(n,K)
.0 *(K) = 9ln(K) o GL(n,K)
If A = (aid) E .,lGn(K), B e 4p(K), then A e B= (aiiB) E Anp(K), A e B=
ix
0B]
E
.IGn+p(K).
Let (P, <_) be a partially ordered set. A subset r of P is a chain if for
all a, 0 E r either a 5 0 or 0 5 a. If r is a finite chain, then the length of r is defined to be
jr -1.
If a, (3 E P, then a covers
(3
if a> (3 and there is no
Y E P with a> y > R. Let a, Q E P. If a, R have a greatest lower bound, then this element is denoted by a A (3 and is called the meet of a,(3.
If a,(3 have a least
upper bound, then this element is denoted by a V (3 and is called the join of a,(3. If a A (3
exists for all a, (3 E P, then P is a A-semilattice. If a V (3 exists for all a,
(3 E P, then P is a V-semilattice. If P is both a A-semilattice and a V-semilattice, then it is a lattice. A lattice P is complete if every subset has a least upper bound and
a greatest lower bound in P. A lattice P with a maximum element 1 and a minimum
element 0 is complemented if for all a E P there exists a' E P such that a V a' = 1,
a A a' = 0. A lattice P is relatively complemented if for all a, 0 E P with a < 0, the interval [a, f3] = (y E P I (X 5 7:5 P) is complemented. A lattice, isomorphic to the
lattice of all subsets of a set is called a Boolean lattice.
Definitions 4.21, 9.9 Definition 9.9 Definition 1.1 Definition 10.14
Definitions 1.5, 12.19, Chapter 14 Definition 12.6 Definition 10.22 A
Definition 14.4
2( A
E,EG,E A
Chapters 13, 14
21
%(G), £(M) Ar' -Tr
,aa,Ga,Ua,Ta(a E
Definition 4.17, Chapter 8 Definition 11.11
Definitions 4.43, 4.46
X
µ
Definition 1.20
w(width)
Definition 6.26
ht(height), Vi
Definition 6.21
Ge,Me
Chapter 6
W(Weyl group)
Definition 4.21
N(Normalizer)
Chapters 4, 6 centralizer)
Chapter 6
1 ABSTRACT SEMIGROUPS
As usual a set S with an associative operation is called a semigroup. If 0 ;& X c S, then < X > will denote the subsemigroup of S generated by X and
E(X) = (e E X le2 = e) the set of idempotents in X. of = fe = f.
If e, f E E(S), then e ? f if
An equivalence relation a on S is a congruence if for all a,b,c E S,
a a b implies ac a be, ca a cb.
If S'
is a semigroup, then a map 4): S -4 S' is a
homomorphism if 4)(ab) = 4)(a)4)(b) for all a, b E S. The corresponding congruence is
called the kernel of 0. A bijection *: S -4 S is an involution if (ab)* = b*a*, (a*)* =
a for all a,b E S. A subsemigroup of S which is a group is called a subgroup. S is strongly it-regular (stir) if for each a E S, there exists i E 71+ such that at lies in a subgroup of S.
See [1], [19], [49]. If a,b E S, then b is an inverse of a if
aba = a, bab = b.
An element a E S is regular if axa = a for some x e S, i.e. a
has an inverse in S.
S
is regular if each element of S is regular. An(K) is a
regular semigroup, and by the Fitting decomposition it is also an sitr-semigroup. A semigroup with an identity element is called a m not S
1
=S
.
If S
is a semigroup then
if S is a monoid, S 1= S u(I I with obvious multiplication if S is not a
monoid. Let M be a monoid. An invertible element of M is called a Unit. Let G denote the group of units of M.
Then M is unit regular if for each a E M, there
exists x E G such that a = axa. Equivalently M = E(M)G. If M is unit regular, then any submonoid of M containing G is also unit regular.
2
Definition 1.1. Let S be a semigroup, a,b E S. Then
(i) a , b if ax = b, by = a for some x,y E S 1.
(ii) a .`b if xa=b,yb=a for some x,ye S (iii)
52 0
052, W= A n
(iv) a b (a divides b) if xay = b for some x,y E S 1.
(v) a$b if albla;Ja={xESIa sx}. (vi) Ja >_ Jb if alb.
Remark 1.2.
For S = 4n(K), I, 5E are row equivalence and column equivalence,
respectively. If a,b a S, then Ja ? Jb if and only if p(a) >_ p(b).
Remark 1.3. Let S be a semigroup. Then
$, 52, .91, ad , .0 are equivalence relations called Green's
(i)
relations. See [11], [24], [33] for details. (ii)
If a e S, then a lies in a subgroup of S if and only if a M e
for some e E E(S). In such a case, the M -class of a is the group of units of eSe. (iii) If
S' is an snr-subsemigroup of S, a E S', e e E(S) and if
a M e in S, then e e E(S') and a M e in S'. (iv) Let a,b,c E S.
Then a 5l b implies ca 52 cb and a Y b
implies ac -V bc.
(v) Let a E S, e E E(S), a 52 e, H the a6 -class of e. Then Ha is the 26-class of a. (vi)
Let e,f a E(S).
Then e
.
f if and only if of = f, fe = e.
Similarly e ,f f if and only if of = e, fe = f. (vii)
Let a E S be regular. Then a = axa for some x E S.
So
e = ax, f = xa E E(S), e A a ,' f. Thus a E S is regular if and only if a 52 e for some e E E(S) if and only if a I f for some f E E(S).
3
(viii) Let D be a .-class of S. Then an element of D is regular if and only if each element of D is regular. Let a E D be regular, x an inverse of a.
Then a , ax ,V x. Hence X E D. The following well-known result is derived from Green [24], Miller and Clifford [48] and Munn [49].
Theorem 1.4. Let S be an s7cr-semigroup, a,b,c a S. Then
(i) a $ ab implies a A ab; a $ ba implies a
ba.
(ii) ab $ b s be implies b $ abc. (iii) If e E E(S), J, H the $-class, M- class of e, respectively, then
JoeSe=H. (iv) / = .0 on S. (v) a / a2 implies that the M -class of a is a group. (vi)
a s ab s b
if and only if a -V e SE b for some e E E(S);
a o ba s b if and only if a .5 e. b for some e E E(S). (vii) Any regular subsemigroup of S is an snr-semigroup.
Proof.
(i) Suppose a f ab. Then xaby = a for some x,y e S 1. Then xl a(by)1 = a
for all i E 11+. There exists j E lL+ such that (by) I X e for some e E E(S).
Then
a = ae E a(by) jS c abS. Hence a 5B ab.
(ii) By (i), ab f b. So abc (iii)
be $ b.
If a E eSe n J, then by (i), e 5¢ ea = a = ae . e. So a X96 e.
(iv) Let a, b E S such that a s b. Then there exist x,y E S 1
such
that xay = b. So a s xa s xay = b. By (i), a .. xa A b. Hence a .0 b.
(v) Let H denote the a-class of a. By (i), a2 M a. So a2x = a for some x E S1.
Then ai+1x1 = a for all
(i), al E H for all i E 71+.
e E H and H is a group.
i E 7L+.
So al 5E a for all i E Z. By
There exists j E B+, e E E(S) such that al N e. Then
4
(vi) Suppose a / ab 0 b. Then by (i), a A ab I b. There exist x, y e S 1 such that abx = a, yab = b.
So ya = yabx = bx.
Then aya = a, bxb = b.
So ya a E(S), a I ya = bx A b. Conversely assume that there exists e e E(S) such that
a I e 5E b.
So xa = by = e for some x, y e S.
Hence ab lxaby = e a ab.
Thus a s ab. (vii) Let a E S'. There exists i e 71+, e e E(S) such that b = al M e in S.
There exists x e S' such that b2xb2 = b2. Then bxb = e. So e e E(S')
and b M e in S'. Let S be an snr-semigroup. A
Definition 1.5. E(J) ;& 0.
$-class J
of S is reg l if
Equivalently some (hence every) element of J is regular. Let 2l = W(S)
denote the partially ordered set of all regular $-classes of S.
If J e ?1(S), then let
J° = J u (0) with
rab if a,b,
aob=j
0 otherwise
Let S be a semigroup, 0 # I c S. Then I is a right ideal of S if IS c I ; I is a left ideal of S if SI c I; I is an ideal of S
if S 1IS 1 c I.
The
minimum ideal of S, if it exists, is called the kernel of S.
Definition 1.6. (i) A completely simple semigroup S is an snr-semigroup with no
ideals other than S.
(ii) A completely 0-simple semigroup S is an snr-semigroup with 0, having no ideals other than (0) and S, and having a non-zero idempotent.
Remark 1.7. (i) This is not the standard definition of completely simple or completely
0-simple semigroups. However this definition is equivalent to the standard one by Munn [49].
5
(ii) Let S be an s tr-semigroup, J e te(S).
If a,b a J, then there
exist x,s,t a S1 such that sat = b, axa = a. Then b = (sax)a(xat) a JaJ. Thus Jo is a completely 0-simple semigroup. (iii) Let S be an s3tr-semigroup, J E te(S).
If E(J) 2 c J, then by
Theorem 1.4 (ii), J2 = J and hence J is completely simple.
(iv) A completely simple semigroup has only one / -class while a completely 0-simple semigroup has two O-classes.
Definition 1.8. Let G be a group, t, A non-empty sets. (i) Let
(i,gj)(k,h,l) = (i,gP(j,k)h,1).
P: A x t -+ G be any map. Let S= t x G x A with Then S is a completely simple semigroup called a Rees
matrix semigroup without zero over G (and sandwich man P).
(ii) Let P: A x t -+ G u (0) be any map such that for all i s r, there
exists j E A such that P(j,i) * 0, for all j e A there exists i E r such that P(j,i) * 0.
Let S= (r x G x A) v (0) with r(i,gP(j,k)h,l) if P(j,k) # 0 {
l
0
if P(j,k) = 0
Then S is a completely 0-simple semigroup, called a regular Rees matrix semigroup with zero over G (and sandwich map P). The following result is due to D. Rees (see [11] or [33]).
Theorem 1.9. (i) Any completely simple semigroup is isomorphic to a Rees matrix semigroup without zero over a group.
(ii) Any completely 0-simple semigroup is isomorphic to a regular Rees matrix semigroup with zero over a group.
6
Proof. We prove (ii), since (i) follows from it. Let
S
be a completely O-simple
semigroup. Then te(S) = (J,0) where J = S\(0). Let e E E(J), H,R,L the a-class, A-class,
-class of e, respectively. Let F = LI A = UM, A = Rl e = RIM. For
), a A, choose rA a A
,
for y e IF choose lY E y. Let A E A, y e I'.
then by Theorem 1.4 (i), rAly e H. by P(A,y) = rAlY
Thus we have a map P: A x F -+ H u (0) given
Then since rA is regular, there exists f e E(J) such
Let A, E A.
that rA t f. Since f_ .
If rAly e J,
there exists y e IF such that IY 5l f. By Theorem 1.4
(vi), rAly * 0. Similarly for each y E t, there exists A e A such that rAly * 0.
S' = (I' x H x A) u (0) be the Rees matrix semigroup with sandwich map P.
lyhrA = lP,r?,.
Define
Since ehe = h for h E H, we see by
yr: S' -4 S as yr(0) = 0, W(yh,A) = IYhrA.
Theorem 1.4 that lY 52 IYhrA I rr.
Let
Let h, h' E H, Y E IF, A e A such that
There exist y,z E S such that rAz = e= y1Y
It follows that h =
ehe = eh'e = h'. Thus yr is injective. That yr is a homomorphism is immediate. So we need to show that yr is surjective. Let a e J. There exist y E F, A E A such that IY 5E a e rA. There exist y,z e S such that r;(z = e = yl). Then ya A ylY = e,
az l rAz = e.
By Theorem 1.4, ya 52 yaz I az.
Hence h = yaz a H.
IY I e SE y and so f = IYy E E(J), f Sl IY 52 a. So IYya = a.
Now
Similarly azr7 = a.
Thus l,,hrA = a. This proves the theorem.
Definition 1.10.
Let
S
be a semigroup,
S=uS
a partition of
S
into
aEQ
subsemigroups. Then S is a semilattice
ni n of Sa((X E fI) if for all ay E S2
there exists 8 E S2 such that SaS U STSa c SS. Y
Definition 1.11. A semigroup S is completely regular if it is a union of its subgroups. The following result is due to Clifford [10].
Theorem 1.12. A semigroup S is completely regular if and only if it is a semilattice of completely simple semigroups.
7
Definition 1.13. A semigroup S is archimedean if for all a,b E S, a b1 for some 71+.
1E
The following result is due to Tamura and Kimura [114].
Theorem 1.14.
Any commutative semigroup is a semilattice of archimedean
semigroups.
The following result is due to the author [62]. The proof given here is due to Tamura [112].
Theorem 1.15. A semigroup S is a semilattice of archimedean semigroups if and only
if for all a,b E S, a b implies a2 b1 for some i c- ]f+.
Proof. The necessity of the condition being obvious, assume that for all a,b E S, a I b
implies a2 b1 for some i E 11+. Then for all a,b e S, j E 7+, there exists i E ll+ such that aI 1b1. Define a relation rl on S as follows: a rl b if a 1b1, b I al for some
By the above, rl is an equivalence relation on S.
i,j E Z+.
So a2b I (aba)2 I (ab)1 for some i E Z+.
aba ( (ab)2.
there exists k E
j r= 71+,
Let c E S. x,y E S1. 71+.
Then
Continuing, we see that for all
Now let a,b E S such that a 1I b.
such that alb I (ab)k.
There exists i E 71+ such that a
I
b1.
Then xay = b1 for some
So cxa I cb1 (cb)l for some j e Z+. Hence ac I (cxa)2 I (cb)k for some k E
So ac I (bc)k+1
Similarly ca 11 cb.
a,b E S.
g+
Let a,b E S.
Similarly
be I (ac)l
for some l E 71+.
Thus ac rl bc.
Hence it is a congruence. Clearly a rl a2 for all a E S.
Then ab I (ba)2, ba I (ab)2.
Hence ab 11 ba.
Let
It follows that S is a
semilattice of its 11-classes. Let T be a il-class, a,b E T. Then there exist x,y E S such that xay = b1.
Then bxayb =
b1+3
and bx rl xay 11 b rl yxay rl yb.
So
bx,yb E T and a, bi+3 in T. Thus T is an archimedean semigroup. This proves the theorem.
Let S be a semigroup, I an ideal of S. The Rees factor semigroun
8
S/I = (S\I) u (0) with
ab if ab a S\I aob =
0 otherwise
If S/I is a nil semigroup, then S is a nil extension of I.
Corollary 1.16. Let
be an sitr-semigroup.
S
Then the following conditions are
equivalent.
(i) E(J) 2 e J for all J e Yl (S).
(ii) For all a e S, e e E(S), a I e implies a2 e. (iii) S is a semilattice of archimedean semigroups. (iv)
S
is a semilattice of nil extensions of completely simple
semigroups.
(ii). Let a e S, e e E(S), a e. Then xay = e for some x,y a S. So
Proof. (i)
ayex,yexa a E(Je). (ii)
Thus (yexa)(ayex) a Je and a2 Je. (iii). Let a,b e S
such that
a I b.
Then b' M e for some
e E(S), i e 71+. So a l e. Hence a2 a bt. (iii)
a E Sa.
(iv).
Let Sa be an archimedean component of S.
There exists e e E(S), n e
Z+
such that an ode in
S.
Let
So there exists
x e S such that anx = xan = e, ex = xe = x, can = ane = an. It follows that e,x a SW
Hence Sa is an s7tr-archimedean semigroup. It is obvious that an sitr-archimedean semigroup is a nil extension of a completely simple semigroup. (iv)
(i).
Let
e,f e E(S), e , f.
Then e,f
lie in the same
archimedean component. Therefore e s ef.
Corollary 1.17. Let S be an sltr-semigroup which is a semilattice of archimedean semigroups,
S'
an sar-subsemigroup of
S.
Then
S'
is a semilattice of
9
archimedean semigroups.
Proof. Let J E Ye(S'), e,f a E(J).
Now (efe) i M h for some i e
71+,
h e E(S').
Then e >_ h, e / h in S. So e = h by Theorem 1.4 (i). Hence of I e and of a J.
Definition 1.18. Let S, S' be semigroups, 4): S -4 S' a homomorphism. Then 0 is idempotent separating if 0 is 1 -1 on E(S). A congruence n on S is i m otent separating if for all e,f a E(S), e tt f implies e = f. The following result is due to Lallement [40].
Proposition 1.19.
Let
S, S' be regular semigroups, 0: S -4 S'
a surjective
homomorphism which is idempotent separating. Then (i) 4>(E(S)) = E(S') (ii) If e,f a E(S), then 4)(e) 5B 4)(f)
implies e
.
f; 4)(e) .1 4)(f)
implies e .2 f; 4)(e) >_ 4)(f) implies e >_ f. (iii)
If a,b a S, then 4)(a) = 4)(b) implies a a6 b; 4)(a) , 4)(b)
implies a s b.
Pr f.
(i) Let e' a E(S'). There exists a e S such that 4)(a) = e'. There exists
x e S such that a2xa2 = a2, xa2x = x. Then e = axa a E(S), 4)(e) = e'.
(ii) Let e,f a E(S), e' = 4)(e), f' = 4(f).
Suppose e'f' = f'. Then
there exists x e S such that (ef)2x(ef)2 = (ef)2, x(ef)2x = x.
Let fl = efx a E(S).
Then 4)(f1) = f' = 4)(f). So f1 = f and of = f. Similarly f'e' = f' implies fe = f.
(iii) Let a,b a S, 4)(a) = 4)(b).
byb = b.
Let e = ax, f = by a E(S).
4)(b) A 4(f).
There exist x,y e S such that axa = a,
Then a ,R e, f 51 b.
So 4)(e) .5E 4)(a) =
By (ii), e A f. So a .A b. Similarly a . b. Hence a A b. The
second statement is now immediate.
10
Definition 1.20. Let S be a regular semigroup. The congruence µ on S defined by:
a.t b if and only if xay X xby for all x,y a
S1
is called the fundamental
congruence on S. S is said to be fundamental if µ is the equality on S.
Remark 1.21. Let S be a regular semigroup. Then By Proposition 1.19,
(i)
g
is the largest idempotent separating
congruence on S and S/µ is fundamental. (ii)
Let a E S.
If e e E(S), then µ eSe is the fundamental congruence on eSe.
Then ate if and only if a X e and of = fa for all f E E(eSe).
See
Hall [31; Theorem 5]. (iii)
If S = 4n(K), then µ is given by: a µ b if and only if a = ab
for some a E K*.
Definition 1.22. A semigroup S is an inverse semigroup if each a E S has a unique inverse, denoted by a 1.
Remark 1.23. (i) A semigroup S is an inverse semigroup if and only if S is regular
and of = fe for all e,f E E(S). In such a case a -a a 1 is an involution of S. See [11], [33], [61].
(ii) Any commutative idempotent semigroup (called a semilattice) is an inverse semigroup. (iii) If X is a set, then the semigroup J(X) of all partial one to one
transformations on X is an inverse semigroup, called the symmetric inverse semi group
on X. (iv) Let E be a semilattice and let TE denote the subsemigroup of J(E) consisting of all isomorphisms a: eE - f E where e,f E E. Munn semi group of E.
TE is called the
11
(v) Let S be an inverse semigroup, E = E(S). Let a E S, e = as 1,
f = a1a. Then a a E TE where as eE - f E is given by has = al ha. Then e: S -+ TE given by e(a) = as is an idempotent separating homomorphism with kernel .t. Moreover µ is also given by: a µ b if and only if alea = bleb for all e E E(S). See [33; Section V] for details.
We therefore have the following result of Munn [50].
Theorem 1.24. Let E be a semilattice. Then TE is a fundamental inverse semigroup with idempotent semilattice E.
Moreover every fundamental inverse semigroup S
with idempotent semilattice E is isomorphic to a subsemigroup of TE containing E.
Remark 1.25. Let S be a regular semigroup, E = E(S). (i) Fitz-Gerald [23] (see also [21]) has shown that < E > is a regular semigroup. Hall [31] constructs a fundamental regular semigroup TE and obtains an idempotent separating homomorphism e: S -4 T< E > with kernel µ.
(ii) The complete generalizations of the Munn representation to regular
semigroups have been obtained by Grillet [27], [28] and Nambooripad [51], [52]. Grillet's approach has been to axiomatize the structures of the partially ordered sets S/A and S/-s," and the connections between them. Nambooripad's approach has been
to introduce a biordered structure on E and to axiomatize it. It is this approach that will be most relevant to us. See Chapters 12, 13, 14. (iii)
For a class of regular semigroups called strongly regular Baer
semigroups, Janowitz [36] obtains an equivalent of the fundamental representation. His paper precedes that of Munn [50]. See [57] for details.
12
2 ALGEBRAIC GEOMETRY
The algebraic geometry needed in this book is of a relatively elementary
nature. We list in this chapter the needed results. In Chapter 16 we will need a few additional results. We refer to [32; Chapters IX, X], [108; Chapter 1] for details.
Definition 2.1. (i) X c Kn is closed if it is the zero set of a collection of polynomials in
K[xII...,xn
In such a case
let
K[X] = K[xl,...,xn]/I
I=
where
{f e K[xl,...,xn] f(X) = 0).
(ii) A closed subset X of Kn is irreducible if it is not a union of two proper closed subsets. Equivalently K[X] is an integral domain. (iii)
Let X c Kn, Y c km be closed sets. Then a map
gym)' X -4 Y is a morphism (or a polynomial map) if each 4i e K[X]. *
a case, there is a natural K-homomorphism 0 4(X) = Y.
K[Y]
K[X].
In such
0 is dominant if
In such a case 4* is injective.
Remark 2.2. (i) Hilbert's Nullstellensatz establishes a 1-1 correspondence between the
closed subsets of Kn and the radical ideals of K[xl,.;.,xn]. (ii) Hilbert's basis theorem states that every ideal of K[xl,...,xn]
finitely generated.
is
Thus the closed subsets of Kn satisfy the descending chain
condition. (iii) If X c Km, Y c Kn are closed sets, then X x Y is a closed subset
of Km+n and K[X x Y] __ K[X] ®K K[Y]. Note that the topology on X x Y is not
13
the product topology. (iv) The topology on Kn
is called the Zariski topology. It is not
Hausdorff. However points are closed and every open cover has a finite subcover.
Definition 2.3. Let X be a topological space. Then (i) X is irreducible if X is not a union of two proper closed subsets.
(ii) X is Noetherian if it satisfies the ascending chain condition on open sets (equivalently the descending chain condition on closed sets).
(iii) A subset Y of X is locally closed if it is open in its closure, i.e. Y is the intersection of an open subset and a closed subset of X. (iv) A finite union of locally closed subsets of X is constructible.
Remark 2.4.
(i)
Any locally closed subspace of a Noetherian space is again
Noetherian. In particular any locally closed subspace of Kn is Noetherian. (ii)
A Noetherian space X is uniquely expressible as a finite union of
irreducible closed subsets, called the irreducible components of X. (iii)
If X is irreducible, U a non-empty open subset, then T = X
and U is irreducible. In particular the intersection of two non-empty open subsets of
X is again non-empty.
Definition 2.5. Let X be a topological space. Suppose that for each non-empty open
set U of X, there is associated a K-algebra 0(U) of K-valued functions of U such that with
(0), we have, (i) If 0 * U c V are open sets, f E 0(V), then f IU E 0(U).
(ii) Let U be a non-empty open set with an open covering Ua(a E I').
Let f: U -+ K such that f IUa E 0(U.) for all a e r. Then f E 0(U). Then 0= 0X is a sheaf of functions on X and X = (X, 0) is a ringed Space.
14
Remark 2.6. (i) Let X c Kn be a closed set. If U is a non-empty open subset of
X, then let 0(U) = OX(U) = (4>: U -4 K I for all x E U, there is an open subset V of
U containing x and f,g E K[X] such that g is non-zero on V and 4> = f/g on V}.
Then X = (X,0) is a ringed space with 0(X) = K[X]. (ii)
Let (X,0) be a ringed space, Y c X. Then Y = (Y,0') is the
induced ringed space. Here Y is considered with the induced topology and 0' is defined as follows.
If 0 * U is open in Y, then 0 '(U) consists of all functions
f: U -4 K such that there is an open covering U c u Ua by open sets of X such aE r that for each a E r, f I U n Ua = fa lU n Ua for some fa E O(Ua). If U is open in
X, then 0'(U) = 0(U). See [108; Section 1.4] for details. (iii) Let Y c X be closed sets in Kn. Then the sheaf of functions on Y, given by (i), is that induced (as in (ii)) from the sheaf of functions on X.
Definition 2.7. An affine variety is a ringed space X isomorphic to a closed subset of
some Kn. In such a case 0(X) is denoted by K[X] and called the affine algebra of X.
Remark 2.8. (i) A closed subset of an affine variety is again an affine variety. (ii)
Let
X
be an affine variety,
f E K[X].
Then
Xf =
(x e X lf(x) *0) __ { (x,a) x e X, a e K, f(x)a = 1 } isaffine. (iii) GL(n,K) = (A E %n(K) Idet A * 0} is an affine variety.
(iv) K2\{0} is not an affine variety. (v) An open subset of an affine variety is a finite union of affine open subsets.
Definition 2.9. (i) Let X be an irreducible affine variety. Then the field of rational
functions on X, K(X) is defined to be the field of quotients of the integral domain K[X].
The dimension of X, dim X is defined to be the transcendental degree of
15
K(X) over K.
(ii)
If X is an affine variety, then the dimension of X, dim X is
defined to be the maximum of the dimensions of the irreducible components of X.
Remark 2.10. Let X be an irreducible affine variety and U a non-empty affine open
subset of X. Then K(X) = K(U) and hence dim X = dim U.
Definition 2.11. (i) A prevariety is a Noetherian ringed space X which is covered by affine open subsets. (ii) If XI, X2 are prevarieties, then X = XI x X2 is a prevariety with
the topological and sheaf structure determined by the affine open sets UI X U2 where Ui is an affine open subset of Xi (see Remark 2.2 (iii)).
(iii) A prevariety X is a variety if the diagonal A = ((x,x) I x e X) is closed in X x X.
Remark 2.12. (i) Any locally closed subset of a variety is again a variety. (ii) If a prevariety X has the property that any two points lie in an
affine open subset of X, then it is a variety (see [34; Lemma 2.5]).
Definition 2.13. (i) Let X be an irreducible variety. Then the dimension of X,
dim X is defined to be dim U where U is any non-empty affine open subset of X. (ii) If X is any variety, then the dimension of X, dim X is defined to
be the maximum of the dimensions of the irreducible components of X. See [32], [34] or [108] for the following.
Proposition 2.14. Let X, Y be irreducible varieties. Then (i)
dim X'
If
X' * X is a non-empty closed subset of
X,
then
16
If U is a non-empty open subset of X, then U = X and
(ii)
dim U = dim X.
(iii) X x Y is an irreducible variety and dim X x Y = dim X + dim Y.
The affine variety
Kn
is called the affine n-smace and is usually
The projective n-anace
is defined as follows:
Pn =
denoted by
An.
(Kn+1\{O})/-
where for a,b E Kn+ \(O), a - b if b = a a for some a E K*.
Pn
A
subset X of Pn is closed if it is the zero set of a collection of homogeneous polynomials in K[xo,...,xn1.
function 0: U -4 K is regul
If U is a non-empty open subset of
Pn,
then a
if for all a e U, there exists an open subset V of U
containing a and homogeneous polynomials f,g E K[xo,...,xn] of the same degree
such that g is non-zero on V and 4) = f/g on V. Let 0(U) denote the K-algebra of all K-valued regular functions on U. Thus Pn = (Pn, 0) is a ringed space. Let a = (ao.... ,an) E Kn+1\{0).
Ua/- . Clearly Ua
Let Ua = {(ao,...,an) E Kn+1\{0} IEaiai # 0), Ua =
{ (bo,...,bn) I bo,...,bn E K, Eaibi = 1 } is affine. Any two points
in Pn lie in some Ua . It follows that Pn is an irreducible variety of dimension n.
Definition 2.15. (i) A variety isomorphic to a closed subset of some Pn is called a projective variety.
(ii) A variety isomorphic to a locally closed subset of some Pn is called a quasi=projective variety.
Remark 2.16. (i) Any affine variety is quasi-projective. (ii)
Let S = .A/2(K), e = l' 00 1.
/-class of e, respectively. Then J/-V - RIX
-JJP1.
Let R, J denote the A -class, Also (S/µ)\{0} __ P3.
Definition 2.17. Let X,Y be varieties. A map 0: X -4 Y is a morphism if for any
open set V of Y and f E 0(V), U = 4 1(V) is open in X and foe) E 0(U). dominant if 4)(X) = Y.
tp
is
17
Remark 2.18. (i) The restriction of a morphism to a subvariety is again a morphism.
(ii) For affine varieties X, Y, this definition agrees with that given in Definition 2.1 (iii). (iii) If X, Y are varieties, Ua(a E t) an open covering of X, then a
map 0: X -3 Y is a morphism if and only if 0 1 Ua is a morphism for all a E F. We refer to [34; Theorem 4.4] for the following
Theorem 2.19. Let 4>: X -# Y be a morphism of varieties. Then for any constructible
subset V of X, 4)(V) is constructible in Y. In particular, if X is irreducible, 4)(X) contains a non-empty open subset of 4>(X). See [34; Theorem 6.2] for the following.
Theorem 2.20. Let Y be a projective variety, X any variety.
Then the map
X X Y -, X given by 4>(x,y) = x is a closed morphism.
See [17; Section 4.5, Theorem 2] or [32; Theorems 2.1, 4.3] for the following dimension theorem.
Theorem 2.21. Let
0: X -4 Y be a dominant morphism of irreducible varieties,
dim X = n, dim Y = m. Then
(i) m5n. (ii) For any closed irreducible subset W of Y and any irreducible component V of X with 4>(V) = W, we have dim V >_ dim W + n-m. (iii) There exists a non-empty open subset U of Y contained in 4>(X)
such that for any closed irreducible subset W of Y with W n U * 0 and any
irreducible component V of 01(W) dim W + n-m.
with
4>(V) = W, we have dim V =
18
3 LINEAR ALGEBRAIC SEMIGROUPS
The subject matter of this book is linear algebraic semigroups. We are now in a position to define this concept.
Definition 3.1. A (line az algebraic semigroup S = (S,o) is an affine variety S along with an associative product map, o: S x S -4 S which is also a morphism of varieties. A homomorphism between algebraic semigroups S, S' is a semigroup homomorphism
0: S
S' which is also a morphism of varieties. Isomorphisms and involutions are
similarly defined.
Remark 3.2. If S is an algebraic semigroup, e e E(S), then eS, Se, SeS are closed
subsemigroups of S and hence algebraic semigroups.
Example 3.3. Any finite dimensional algebra over K with respect to multiplication or
the circle operation: aob = a+b - ab is an algebraic semigroup.
Example 3.4.
If S
is any (multiplicative) subsemigroup of '#n (K), then
algebraic semigroup.
Example 3.5. Any finite semigroup is an algebraic semigroup.
5 is an
19
Exam merle 3.6. Let X be a closed subset of Kn. Then S = (A A E ,AGn(K), XA c X) is a closed subsemigroup of An (K).
Example 3.7. Let P e .A,(K)
and let S1 = {AAA E ,AGn(K), AtPA = 01, S2 =
Then S1 is a closed subsemigroup of ,AGn(K), 0 E S
(AAA E ,Aln(K), AtPA = P).
and S2 is a closed submonoid of ,Aln(K).
Example 3.8. Two trivial algebraic semigroups on any affine variety X: (i) xy = y
for all x, y e X, (ii) xy = u for all x,y a X, u a fixed element of X.
Example 3.9. (Rees construction). Let varieties, 4): Y x X -4 S a morphism. Let
S
n= S
be an algebraic semigroup, X,Y affine X X S X Y with
(x,s,Y)(x', s" Y') = (x,s4)(Y,x')s', Y') A
Then S is an algebraic semigroup.
Example 3.10. The map 4): 4n(K) ® 4 (K) - 4np(K) given by 4)(A ® B) = A ® B is a homomorphism.
Example 3.11.
The map
4):
,,* (K) -4 ,Al 2(K) n
given by 4)(A) = A ® A is an
idempotent separating homomorphism.
Example 3.12. Let S = .A6n(K).
Consider the homomorphism 0: S -+ S given by
4)(a) = (det a)a. Then 4)(S) is not closed in S, 4)(S) = S.
Example 3.13. (Semidirect Product). Let S1,S2 be algebraic semigroups. Suppose
for a e S1, b e S21 an element ab a S1 is uniquely determined. Suppose that the
20
map (a,b) -+ ab is a morphism and that for all a,al,a2 E S1, b,bl,b2 E S2 ,
(a1a2)b = abab 1 21
b b a
1
b b 2 = (a 2) 1
Let S = S1 X S2 with multiplication b
(al,bl)(a2,b2) = (a1a2blb2)
Then S is an algebraic semigroup, called the semidirect product of S1,S2.
Problem 3.14.
Generalize the Krohn-Rhodes decomposition theorem for finite
semigroups [39] to linear algebraic semigroups. The more recent approach of Rhodes [103] might also be relevant
The following result is well-known [16].
Theorem 3.15. Let M be a linear algebraic monoid. Then M is isomorphic to a closed submonoid of some ,AGn(K).
Proof. We may assume that M is a closed subset of some Kd.
Since the operation
on M is polynomially defined, there exist morphisms fl-.,fm from M into Kd, gl'"''gm E K[S] such that for all a,b E M, m
ba =
gi(a)fi(b)
i=1 Let V denote the vector space of all maps from S into Kd. If h E V, a,x, E M, let ha(x) = h(xa). If a E M, h E V, let Ta(h) = ha Then Ta E End(V) and Tao Tb =
tab for all a,b e M.
Let I E V denote the identity map. Then for all x E M,
21
M
Ia(x) = xa =
gi(a)f (x).
So
i=1 m
Ia =
gi(a) i
i=1 Thus Ia is in the span of fl,...,fm.
Let Y denote the finite dimensional space
spanned by Ia(a a M). Then each element of Y is a morphism from M into Kd.
If a,b a M, then I'a(Ib) = lab- Hence I'a(Y) S Y. ' If a E M, let I'(a) denote the restriction of ra to Y. Then I': M - End(Y) is a monoid homomorphism. There exist 1 = al,...,an e M such that wi = Ia , i = 1,...,n form a basis of Y. Then
i
mCC
mCC
(1)
I`a( j) = Iaa. = L gi(anj) i = G hij(a) i i=1 i=1
where hij(a) = gi(a)). Clearly each hij is a morphism on M. Extend wl,...,wn to a basis w1,...,wq of < fl,...,fm>. Let q
fi =
,
ct wk
i = 1,...,m
k=1
Then by (1),
q m r (wi ) =
akihij(a)wk , j = 1,...,n
k=1 i=1
m
Let ujk(a) _
qqC
akihij(a).
i=1
So
jk
is a morphism on M and I'a( j) = G jk(a)wk' k=1
22
j = 1,...,n. But Ta(wj) E Y. So n
T (wj) _ k=I
ujk()wk
,
j = 1,...,n
Then 0(a) = (ujk(a))t E An(K) is the matrix of Ta homomorphism of algebraic monoids. Now for all a E M,
So
0: M -+ An(K) is a
n nn
ulk(a)wk(l) = ra(wl)(I) = a
ulk(a)ak =
k=1
(2)
k=1
n
If A = ((3kj) E kn(K), let V(A) _
Rklak E Kd.
Then yr is a morphism and by
k=1
(2), yr(4(a)) = a for all a E M. So (M) = (A E %n(K) ( V(A) E M, O(V(A)) = A) is
closed in 4n(K) and yr =
4_I
on 4(M). This proves the theorem.
Corollary 3.16. Let S be an algebraic semigroup. Then S is isomorphic to a closed subsemigroup of some An(K).
Proof. We may assume that S = (S,o) is a closed subset of some Kd.
Let U E S
and set M = (S x {0}) u {(u,l)} c Kd+1 S' =S x (0) c M. On M define
(a,a)o(b,R) = ((I-(x)(l-{3)(aob) + (a+(3- c43)(a+b-u),a4)
Then M = (M,o) is an algebraic monoid, S =_ S' c M. Theorem 3.15.
We are now done by
23
Remark 3.17. Let M be a closed submonoid of An(K). Suppose M has a zero
If a E M, let 4(a) = a-e. Then : M = O(M) c (1-c)4n(K)(1-e)
e, p(e) = r. __
(K).
Note that e corresponds to the zero of ,Aln__r(K).
The following result was pointed out to the author by W. E. Clark (see [64; Corollary 1.4]).
Theorem 3.18. Let S be a closed subsemigroup of .AGn(K). Then for all a e S, an lies in a subgroup of S.
Proof.
Let
a E S, b = an.
By the Fitting decomposition,
b M e for some
e = e2 a 4n(K). Now S1 = (x a SI ex = xe = x) is a closed subsemigroup of S and b e S 1.
There exists c e An(K) such that ec = ce = c, be = cb = e.
Now for all
ie 71,biS1={xe SlIclxe S1} is closed and
bS1 D b2S1 0 ....
Hence bi S 1 = bi+1 S I for some i e lL+. ebS = bS I. 1
Similarly S 1 = S 1b.
x = ex = cbx = cb = e.
Then S 1 = eS 1 = cibiS I = cibi+l S 1 =
There exists x e S 1 such that b = bx.
Hence e e E(S1).
There exist y,z E SI
So
such that
by = e = zb. It follows that b Re in S.
Corollary 3.19. A closed subsemigroup of 9Jn(K) is a semilattice of groups.
Corollary 3.20. A closed subsemigroup S of ,7n(K) is a semilattice of archimedean semigroups. In particular J2 = J is completely simple for all J E V(S).
Proof. By Corollaries 1.16, 1.17, Theorem 3.18, we may assume that S = 9n(K). a E S, let 4(a) denote the diagonal matrix with the same diagonal as a.
If
Then
24
S -+ .0 (K) is a homomorphism. Let J E Il(S), el, e2 E E(J). Then 4(el) / 4(e2) in
.0 (K).
nilpotent.
Since So
.
is commutative, O(e1) = O(e2).
(K)
u = 1 + e2el- el E
p(el) = p(ele2el) and el M ele2e1
9n(K). in
Clearly
So e2el - el
elu = ele2el.
is
Thus
An(K) and hence in S by Remark 1.3
(iii). Thus ele2 E J and E(J) 2 c J. We are done by Corollary 1.6.
Remark 3.21.
Corollary 3.20 is clearly valid for any snT-subsemigroup of 3n' (K).
Thus any regular subsemigroup of 9n(K) is completely regular.
Definition 3.22. If e is an idempotent in
det(eae + 1-e).
If
IF
kn(K), a E
%(K), then dete(a) =
is a finite set of idempotents in
4n (K),
then
detr(a) = II detf(a).
fEt
Remark 3.23. Let S be a closed subsemigroup of ..f6n(K), e E E(S), a E S. Then by
Theorems 1.4, 3.18, dete(a) * 0 if and only if eae M e in S if and only if eae s e
in S.
Definition 3.24. An algebraic semigroup which is also a group is an al
eg
braic group.
Remark 3.25. Let G be an algebraic group. Then by Theorem 3.15, G is a closed subgroup of some GL(n,K).
So a 1 = (1/det a) adj a for all a r: G, where adj a
denotes the adjoint of a. Hence the map a -+ a 1 is a morphism on G.
Corollary 3.26. Let S be an algebraic semigroup, e E E(S), H the 'V -class of e. Then H is an algebraic group.
Pr f. H = (a E eSe dete(a) * 0) __ G = { (a,a) I a e eSe, a E K, a dete(a) = 1) is an algebraic group.
25
Lemma 3.27.
Let
S
be an algebraic semigroup,
e E E(S).
Then the ideal
I = (a I a E S, a + e) is closed in S.
Proof. Let H denote the S.
class of e.
By Remark 3.23, X = eSe\H is closed in
Let a E I. Then exaye E X for all x,y E X. Conversely let a e S such that
exaye e X for all x,y E S. some x,y E S.
We claim that a E I. Suppose not. Then xay = e for
So exaye = e E H, a contradiction. Thus I = (a E S Iexaye E X for
all x,y E S) is closed. The following result is due to the author [64].
Theorem 3.28. Let S be an algebraic semigroup. Then 2l(S) is a finite partially
ordered set. In particular S has a kernel.
Proof. Suppose the theorem is false. Then there exists an infinite set X c E(S) such that for all e,f a X, e / f implies e = f. For e E X, let I(e) = (a I a E S, a t e) which is closed by Lemma 3.25. We claim that there exists an infinite subset Y of X such that
I(e) n Y I < - for all e E Y. Suppose not. Then there exists f1 E X such
that X1 = I(fl) n X is infinite. There exists f2 E X such that X2 = I(f2) n X1 is infinite.
Continuing, we find a sequence f1,f2,...
in X such that .+l E Xi =
I(f1) n... n I(fi) n X for all i E Z+. Then i+l E I(f1) o...o I(i)' i+l i E 71+.
I(i+1) for all
So we have a strictly descending chain of closed sets,
I(f1)
I(fl) n I(f2)
I(fl) n I(f2) n I(f3) i...
This contradiction shows that there exists an infinite subset Y of X such that I(e) n Y is finite for all e E Y. Choose el a Y.
There exists e2 E Y\I(e1) such
that el * e2. Similarly there exists e3 E Y\(I(e1) u I(e2)) such that e1 * e3, e2 * e3. Thus we find distinct idempotents e1, e2, ... in Y such that ei Iej, ei 1ej for i < j.
26
Since el Ie2, there exist x,y E S such that xely = e2. Let e'2 = elye2xel E E(S). Then
e2 / e2, el >- e2.
el
So
idempotents, e1 > e2 > e3 > ... in
;v-L
Continuing, we find a sequence of
e2.
This is a contradiction since S is a matrix
S.
semigroup.
Remark 3.29. When S is an idempotent semigroup, the above result has also been obtained by Sizer [107]. Theorem 3.28 implies that S has ideals Io c ... C Im = S,
such that Io is the completely simple kernel of S and each Rees factor semigroup I lk
1,
k = 1,...,m is either nil or completely O-simple. See [64]. Kleiman [38] has
shown that the ideals 'k can be chosen to be closed. For generalizations of Theorem 3.28 to snr-matrix semigroups see the author [64], [81] and Okninski [59]. That an sitr- matrix semigroup has a kernel is an early result of Clark [8]. The following result is due to the author [65].
Corollary 3.30. Let S be a closed subsemigroup of lln(K), I an ideal of S. Then n (i) I = (a E S Ian E I) is closed in S.
(ii) r/I is a nil semigroup and an E I for all a E I.
Proof.
We prove only (i), since (ii) follows from it.
(a E S an E SeS). e E E(S).
Let a E I.
Then e r: anS c I.
Then an E I. So
n
I=u
If e E E(S), let X(e) _
By Theorem 3.16, an JVe for some X(e).
By Theorem 3.26, the family
eEE(I)
{ SeS I e E E(S)) and hence the family (X(e) e E E(S)) is finite. Thus we are reduced
to showing that X(e) is closed for all e E E(S).
Fix e E E(S). For f E E(S), let
I(f) = (a E S I a 4 f). Then I(f) is closed by Lemma 3.25. Let F = (f I f E E(S), e 4 f).
Then SeS c I(f) for all f E F. So SeS c I o = n I(f) and I0 is closed. Let I 1 = fEF
(a E San E Io). Then II is closed and X(e) c I. Let a E Il. Then an M h for
some h E E(S) and an E Io Since an E I(h), h e F. So e h an. Thus an E SeS and a E X(e). So X(e) = 1 l is closed.
27
4 LINEAR ALGEBRAIC GROUPS
Let G be a linear algebraic group. We denote the identity element of
G by 1. If X c G, then the normalizer in G of X, NG(X) = (g E G I g 1Xg = X) and the centralizer in G Q X, CG(X) = ( g e G I gx = xg for all x e X). The center
of G, C(G) = CG(G).
Two subsets X, Y of G are conjugate if g1Xg = Y for
some g E G.
Definition 4.1.
Let
G
be a (linear) algebraic group.
component of G containing 1 will be denoted by Gc.
The unique irreducible
Then Gc a G, G/Gc is a
finite group and dim G = dim Gc. G is connected if Gc = G. We refer to [34; Sections 7.4, 7.5] for the following
Proposition 4.2. Let G be an algebraic group. Then (i)
If U is a dense open subset of G, then U2 = G.
(ii) If H is a constructible subgroup of G, then HI = H.
(iii) If H1,...,Hk are closed connected subgroups of G, then the subgroup H of G generated by HI,...,Hk is closed and connected.
Corollary 4.3. Let 0: G -+ G' be a homomorphism of algebraic groups. Then
(i) O(G) is a closed subgroup of G' and the kernel, ker 0 is a closed subgroup of G. (ii) dim G = dim 4(G) + dim ker 4.
28
We refer to [34; Chapter IV], [108; Theorems 4.3.3, 5.2.2] for the following theorem.
Theorem 4.4. Let G be an algebraic group, H a closed subgroup of G.
Then
G/H = (aH I a E G ] can be made into a quasi-projective variety such that the map y:
G -- G/H given by y (a) = all is an open morphism and (i) If Y is any variety, then lxy: Y x G -i Y x G/H is open.
(ii)
If Y is a variety, 4>: G --4 Y a morphism such that 4>(ah) = 4)(a)
for all a e G, h e H, then there exists a unique morphism y. G/H -4 Y such that
$=WoY
(iii) If H a G, then G/H is a linear algebraic group.
Definition 4.5. Let G be a connected group. Then
(i) A maximal closed connected solvable subgroup of G is called a Borel subgroup.
(ii) A closed subgroup P of G containing a Borel subgroup is called a parabolic subgroup. If P * G and if there are no proper closed subgroups between P and G, then P is a maximal parabolic subgrroup.
(iii) A closed connected subgroup T of G is a torus if T =(K) for some n e 7L+..
Remark 4.6. If G = GL(n,K), then
is a maximal torus of G and
(K) is a
Borel subgroup of G. The following result is due to A. Borel. See [34; Theorem 21.3].
Theorem 4.7. Let G be a connected algebraic group, B a Borel subgroup of G.
Then GB is a projective variety. See [34; Corollary 21.3C] or [108; Corollary 7.2.7] for the following.
29
Corollary 4.8. Let 0: G -. G' be a surjective homomorphism of connected groups.
Let T be a maximal torus of G and B a Borel subgroup of G. Then 4(T) is a maximal torus of G' and 4(B) is a Borel subgroup of G'.
Definition 4.9. Let G be an algebraic group, X a variety. Then G acts -n X (on the right) if for each x E X, g e G, there is associated an element x (i) (x
g1)
92 = x
g1g2
g e X such that
for all x e X, g1,g2 a G.
(ii) x- 1 = x for all x e X. (iii) The map, (x,g) -4 x
g is a morphism from X x G into X. If
YcX,Hc0, then Y H= (y hlye Y,he H). The power of Theorem 4.7 is exhibited by the following well-known result [110; p. 68].
Corollary 4.10. Let G be a connected group acting on the right on a variety X. Let
B be a Borel subgroup of G, Y a closed subset of X such that Y B is closed in
X. Then Y G is closed in X. Pr
f. Let y:G
be given by y(a)=aB and let 4=1xy:XxG-4 XxGB.
Then 0 is open by Theorem 4.4 (i). Let F = ((x,g) I x e X, g e G, x
Then F is closed in X x G.
geY
B).
So 4(F) = --$(- F) is closed in X x GB.
Let
p: X x GB - X denote the projection onto X.
By Theorem 4.7, GB is a projective
variety. So by Theorem 2.20, p is a closed morphism. Hence Y G =
is
closed in X.
The next result is due to A. Borel. We refer to [34; Chapter VIII] for proofs.
Theorem 4.11. Let G be a connected group. Then (i) All maximal tori of G are conjugate.
30
(ii) All Borel subgroups of G are conjugate. (iii) If
B
is
a Borel subgroup of
G,
then
NG(B) _
B and G = u x-1Bx. XE G
(iv) If B is a Borel subgroup of G, T a torus in B, then CG(T) is a connected group having CB(T) as a Borel subgroup. If T is a maximal torus, then CG(T) = CB(T) is a nilpotent group.
Theorem 4.11 implies the following result known as the Lie--Kolchin Theorem (see [34; Theorem 17.6]).
Corollary 4.12.
Let G be a closed connected subgroup of GL(n,K).
solvable, then it is conjugate to a subgroup of conjugate to a subgroup of
'9n* (K).
If G is
If G is a torus, then it is
(K).
Definition 4.13. Let G be a closed subgroup of GL(n,K), a E G.
Then a is
unipotent if the only eigenvalue of a is 1. Let Gu = (a E G I a is unipotent). G is unipotent if G = Gu.
Remark 4.14. (i) The above definition is independent of the particular choice of linear
representation of G.
In fact if : G -+ G' is a homomorphism of algebraic groups,
then 4 (Gu) c G. See [34; Theorem 15.3]. (ii) By Theorem 4.11 (iv), any unipotent group is nilpotent.
The next theorem is due to A. Borel. We refer to [34; Theorem 19.3] for a proof.
Theorem 4.15. Let G be a connected solvable group, T a maximal torus of G.
Then U = Gu is a closed normal subgroup of G, G = TU and G/U a T. G is nilpotent if and only if G - T x U.
Moreover
31
Corollary 4.16. Let G be a connected group, H a closed connected normal subgroup
of G. Let T, B be a maximal torus and a Borel subgroup of G, respectively. Then T n H, B n H are a maximal torus and a Borel subgroup of H, respectively.
Proof.
Let To, B 0 be a maximal torus and a Borel subgroup of H, respectively.
Since H a G, we may assume without loss of generality that To c T, B0 c B.
T1 = H n T, B1 = B n T.
Let
Then To c T1 c CH(To) and CH(To) is nilpotent by
Theorem 4.11 (iv). So by Theorem 4.15, To = T. Now Bo c B i and hence B 0 = B 1 a B 1.
So by Theorem 4.11 (iii), Bo = B 11
Definition 4.17. Let G be an algebraic group. A homomorphism x: G - K* is
called a character of G. Let .W(G) denote the group of all characters of G.
Remark 4.18. (i) It is easily seen that %(G) is linearly independent in the vector
space of all K-valued functions on G.
See [34; Lemma 16.1].
(ii) Let T = . (K), xl,...,xn the n projections of T into K*. Then xl,...,xn freely generates .W(T). See [34; Section 16.2]. We refer to [34; Section 16.2], [108; Section 2.5] for the following.
Theorem 4.19. Let T be a torus, dim T = n. Then .%(T) _ (71n,+). Moreover any closed connected subgroup of T is also a torus.
The structure of unipotent groups is more complicated. Even the proof of the following result is not easy (see [34; Theorem 20.5]).
Theorem 4.20. Let G be a unipotent group, dim G = 1. Then G E (K,+).
Definition 4,21. Let G be a connected group, T a maximal torus of G. Then
32
(i) W = W(G) = NG(T)/CG(T) is the Weyl aroun of G.
xCG(T) a W, t e T, then let to = xltx e T.
If a =
Thus W is a subgroup of the
automorphism group of T. (ii)
If x e W M, a E W, then let xa a ,%(T) be given by: Xa(t) _
x(ta) for t e T.
Let £ = .B (T) denote the set of all Borel subgroups of G containing T. If B e , a = xCG(T) a W, then let Ba = Bx, aB = xB, Ba = (iii)
,.
G -1 1 Bo
= x-1Bx. Note that by Theorem 4.11 (iv), CG(T) c B. We refer to [34; Sections 24, 25] for a proof of the following.
Theorem 4.22. Let G be a connected group, T a maximal torus of G. Then (i) W is a finite group. (ii)
If B1, B2 a 2(T), then there exists a unique a E W such that
B 1 = B2.
(iii) I$ (T) I= I W I (iv) I W I = 1 if and only if G is solvable.
Definition 4.23. Let G be a connected group, To a torus in G.
Then To is
regular if $ (To) = (B I B is a Borel subgroup of G containing To) is finite. Otherwise To is sin
.
1
We refer to [34; Proposition 24.2] for the following.
Proposition 4.24. Let G be a connected group, To a torus in G.
Then To is
regular if and only if CG(To) is solvable. In such a case $ (To) = £ (T) and CG(To) c B for all B E
,
(T).
Corollary 4.25. Let : G -+ G' be a surjective homomorphism of connected groups, H = (ker 4))c. Then I W(G) I = I W(H) I
-
I W(G') I
33
Proof. By Corollary 4.8 and Theorem 4.22, we are reduced to the case when G' is
solvable. Let To = T o H.
Then To is a maximal torus of H by Corollary 4.16.
By Theorem 4.22, it suffices to show that
,H(To) = (B o HIB E .G(T0)). B2 n H.
12 (T)
Let B1, B2 a
By Corollary 4.8, (B1) = G'.
connected, we see that G = B
1H.
I ,H(TO) 1. .
By Corollary 4.16,
(To) such that B1 n H =
Hence G = B1(ker 0).
Since G is
By Theorem 4.11 (ii), there exists h e H such that
h-1B 1h = B2. Then h e NH(B I o H) = B 1 n H by Theorem 4.11 (iii). So B 1 = B2
and
I 2G(To) I =
,2I(To) I
is finite.
By Proposition 4.24,
2 (T) = ,lb(To),
completing the proof.
Definition 4.26. Let G be a connected group. Then (i) The maximal closed connected normal solvable subgroup of G is
called the radical of G and is denoted by rad G. The unipotent group, raduG = (rad G) u is called the unipotent radical of G.
(ii) G is reductive if raduG = (1 ). G is semisimple if rad G = 1). (iii) G is simple if G has no closed connected normal subgroups other
than (1 } and G, and is non abelian. (iv)
If T is a maximal torus of G, then the rank of G, rank G =
dim T. The semisimple rank of G, rank ssG = rank (G/rad G).
Remark 4.27. (i) GL(n,K) is a reductive group and SL(n,K) is a simple algebraic group. The direct product of simple algebraic groups is semisimple.
(ii) Let G be a simple algebraic group. Then G need not be simple as an abstract group. However C = C(G) is finite and G/C is simple as an abstract group. See [34; Corollary 29.5]. (iii)
Let 0: G
G' be a surjective homomorphism of connected
groups. Then 4)(rad G) = rad G' and 4)(raduG) = raduG'. a semisimple group and G/raduG is a reductive group.
In particular G/rad G is
34
(iv)
If G is a connected group, then rad G is just the identity
component of the intersection of all Borel subgroups of G.
If H is a closed
connected normal subgroup of G, then rad H a G, raduH a G. Hence rad H c rad G, raduH c raduG.
Definition 4.28. Let G be a group, Hl, H2 subgroups of G.
Then (H1,H2) is the
subgroup of G generated by hIh2h11h21(h1 a Hl, h2 E H2). See [34; Proposition 17.2] for the following.
Proposition 4.29. Let G be an algebraic group, Hl, H2 closed subgroups of G. Then (i)
If H1, H2 are connected, then (H1, H2) is a closed connected
subgroup of G.
(ii)
If H1 or H2 is normal, then
(H1, 112)
is a closed normal
subgroup of G. See [34; Theorem 27.5] for the following.
Theorem 4.30. Let G be a semisimple group. Then G = GI...Gn where G1,...,Gn are the closed normal simple subgroups of G. Moreover, (i) Gi = (Gi,Gi), G = (G,G). (ii) (Gi,G) = ( 1) for
i # j and the product map from G1x...x Gn
onto G has a finite kernel.
(iii) If H is a closed connected normal subgroup of H=Gi
...Gi 1
m
G,
then
for some subset (i1.... ,im] of (1,...,n).
Remark 4.31. Let G be a closed connected subgroup of GL(n,K). Then G/rad G is
a semisimple group. Hence by Theorem 4.30 (i), G = (G,G)rad G. rad G is unipotent, then det a = 1 for all a e G.
In particular if
35
By Theorem 4.30, [108; Proposition 6.15], we have,
Theorem 4.32. Let G be a reductive group. Then C(G) is the intersection of all Borel subgroups of G, rad G = C(G)C is a torus, (G,G) is a semisimple group and G = (G,G)rad G.
Remark 4.33. Let G be a connected group. Then by Remark 4.27 (iii), Theorem
4.32, (G, rad G) c raduG.
Corollary. Let G be a reductive group, C = C(G), G' = (G,G). Let H be a closed normal subgroup of G.
Then H = C'H' .where C' = C n H and H' =
(Hc,Hc) a G'.
Proof. Let G1,...,Gn be the simple components of G'.
Now for each i = 1,...,n,
(H,Gi) c H n Gi a Gi Hence either H n Gi = Gi or else H n Gi c C. Let h E H. Then h = cgl...gn for some c e C, gi a Gi, i = 1,...,n. all g e Gi' giggi
1g-1 =
hgh 1g1
E (H,Gi) c C.
Suppose Gi
H.
Then for
By Remark 4.27 (ii), gi E C.
Hence H c CH' and the result follows.
The next result is known as the Bruhat decomposition.
See [31;
Theorem 28.3].
Theorem 4.35. Let G be a reductive group, T a maximal torus of G.
Then for any
B, B' E ` (T), G is the disjoint union of BaB' (6 E W). See [34; Corollary 28.3] for the following.
Corollary 4.36. Let G be a reductive group, B, B' Borel subgroups of G. Then B o B' contains a maximal torus of G.
36
Definition 4.37. Let G be a reductive group, B, B' Borel subgroups of G.
Then
B, B' are opposite if B n B' is a torus. In such a case, B' is the coosite Borel subgroup of G relative to T = B o B'. See [34; Section 26.2] for the following.
Theorem 4.38. Let G be a reductive group, T a maximal torus of G. Then every
B e 2(T) has a unique opposite B- E £ (T) relative to T. In particular CG(To) is a reductive group for any torus T0 c T and CG(T) = T. Remark 4.39. (i) If B, B
are opposite relative to T, then B, b1BTh are opposite
relative to b1Tb for any b e B. (ii) If G is a connected group, then G/raduG is a reductive group.
Hence for any maximal torus T of G, n 2 (T) = T raduG.
(iii) Let G = GL(n,K), T =
(K), B = On(K), B-= Bt. Then B, B-
are opposite relative to T.
Definition 4.40. Let G be a reductive group, P, P' parabolic subgroups of G. Then
P, P' are Uposite if P n P' is a reductive group. In such a case, if
T c P o P' is a maximal torus, then we say that P' is op
osp
ite to P relative to T.
The following theorem is due to Borel and Tits [5].
Theorem 4.41. Let G be a reductive group, T a maximal torus of G, P a parabolic
subgroup of G containing T. Then P has a unique opposite P relative to T. Moreover P = LU, P = LU
where L = P n P U = radP, U = raduP
In the above theorem, L is called a Levi factor of P.
For the rest of
this chapter, fix a reductive group G, a maximal torus T of G and a Borel subgroup
B containing T.
Let B
denote the opposite of B, relative to T, dim T = m.
Then %(T) __ (7Lm,+) c (IRm,+). We will view £ (T) additively.
37
Definition 4,42. Let X: G -+ GL(V) be a finite dimensional representation. Then
x e £ (T) is a weight of x if Vx = {v e V I2 (t)v = x(t)v for all t e T} * (0).
In
such a case Vx is called the weight space of X. Then since X(T) is diagonalizable,
V=V Q..eV
.
xk
X1
As in the case of Lie groups, the 'tangent space' of G at 1 forms a Lie
algebra Y = .
(G).
Moreover, dim I = dim G.
Also, G acts as a group of
automorphisms of -V. This gives rise to the adjoin representation, Ad: G - GL(-V).
The kernel of this representation is just the center of G. We refer to [34; Chapter III],
[108; Chapter 3] for details. The basic example to keep in mind is G = GL(n,K) in
which case 1(G) _ & (K) with [x,y] = xy - yx.
Also Ad(g)(a) = gag 1 for
g e G, a e 'ffn(K).
Definition 4.43.
The non-zero weights (in the additive notation) of Ad: G -,
GL(1(G)) are called the roots of G and denoted by . If a e , let Ta = (ker a)c, Ga = CG(Ta), 1 the weight space of a.
Example 4.44. Let G = GL(3,K).
Then
_ (x1,x2,x3, xl,x2, x3), where xl:
diag(a,b,c) -, a/b, x2: diag(a,b,c) -, b/c, x3:diag(a,b,c) - a/c, x1: diag(a,b,c) -+ b/a, x2: diag(a,b,c) -+ c/b, -x3: diag(a,b,c) -+ c/a.
Note that A = (x1,x2) forms a basis for the
space spanned by 4, and x3 = x1 + x2. See [34; Chapter IX] for the following.
Theorem 4.45. (i) 4) = -4), 4)W = .
(ii) Ta(a E 4)) are exactly the maximal singular tori of G contained in T.
(iii) If a e 4), then W(Ga) = (1, a(,}, $(G(,) = (a,-a), aaa = -a.
(iv) If a e 4), then 1(radu(B n Ga)) is either Xa or .lf.
38
Definition 4.46. (i) If a E 4), then aa: a -+ -a is called a reflection.
TGa
We let
Ta = (t e T I taa = t } c. (ii) If a E (p,
then Ua = radu(B o Ga) if .(radu(B n Ge)) = -Va.
Otherwise Ua = radu(B n Ga). Thus I(Ua) = 'Ya and T C NG(Ua). Uas are called root subgroups. See [34; Theorem 26.3] for the following.
Theorem 4.47. Let a E 4). Then
(i) dim Ua = 1 and there is an isomorphism ea: (K,+) -, U. such that for all t E T, x e K, tea(x)t
1
= ea(a(t)
x).
(ii) For all a e W, alUaa = Uaa' (iii) G is generated by Ua(a E 4)) and T.
Remark 4.48. a e 4) is positive (relative to B) if Ua c B. Let + = (a E 4) a is
Then 4 = - i+ is the set of positive roots relative to B
positive relative to B. ) Moreover
4)
is the disjoint union of 4+ and
4)
Let A = A(B) = -d(B) _
(a E 4+ a is not a non-negative linear combination of
4)+\(a)).
Then A is called
the h= of 4), relative to B. It turns out that A is a basis (over IR) for the span of 4) and every element of
4
is a non-negative integral linear combination of
aa(a a A) are called the simple reflections relative to B.
(B-)= (aa la E A}.
Then
A.
Let 9 = e (B) = e
J = Jeso J = rankssG and W is generated by av.
The map: B -i A(B) is injective and for a E W, A(Ba) = A(B)a. Moreover &(B)W =
. Also Bu is generated by Ua(a E 4+). We refer to [108; Chapter 10] for details. See [34; Proposition 27.2], [110; p. 80] for the following.
Proposition 4.49. Let 4): G -+ GL(V) be a finite dimensional representation such that ker 0 c C(G). Let a E 4), x E £ (T).
in the sum of Vx+ka(k
E
Z+)'
Then for all u E Ua, V E Vx, 4)(u)(v) - v lies
39
The following consequence was pointed out to the author by J. E. Humphreys.
Corollary 4_50. Suppose G c GL(n,K).
Then there exists a e GL(n,K) such that
a 1Ba, a 1B a consist of upper and lower triangular matrices, respectively.
Proof. Let G c GL(V) and let X denote the set of weights of T. If x11 x2 e 9, , define xl <_ x2 if xl -x2 is a non-negative linear combination of
.
Find bases
of Vx(x a ,%) and order them in such a way that if x1 < x21 then a basis vector v1 e Vxl occurs before any basis vector v2 E Vx2
By Proposition 4.49, Ua is
upper triangular with respect to this basis for all a E
and lower triangular for all
ae(
Moreover T is diagonal. Since B is generated by T, Ua(a E
and B
is generated by T, Ua(a e 4), the result follows. The next result is due to Borel, Tits [5].
Theorem 4.51. If (i)
I c e, let WI = < I >, PI = BWIB, PI = B WIB
Then
PI, PI are opposite parabolic subgroups of G relative to T and
W(P1) = W(PI) = WI. (ii)
If P is a parabolic subgroup of G containing B, then P = PI
for some I c eY.
(iii) If I, I' c eY, x e G, x-1PIx a P1, then x e PI, and I c I'.
Definition 4.52. Two parabolic subgroups P, P' of G are of the same tie if they are conjugate. P, P' are of ppposite tune if P' is conjugate to an opposite of P.
Corollary 4.53. Let a e i\(B), U = B. as Y = Yaa and YUa = UaY = U.
Let P = B u BaaB, Y = radu(P).
Then
40
Pr
f.
Since Y a P, aaY = Yaa.
By Theorem 4.51, W(P) =
aaUaa-1 = U
Hence Ua Y.
B.
Now
By Theorem 4.45 (i), dim U/Y = 1.
It
follows that UaY = YUa = U.
Renner [96; Proposition 7.4] derives the following result from the classification of reductive groups [108; Theorem 11.4.3].
Proposition 4.54. G admits an involution
Ua=U
*
such that t* = t for all t E T and
for all aE 4.
Remark 4.55. (i) In the above situation, it is clear that PI = PI.
Thus for any
parabolic subgroup P of G, P, P* are of opposite type. (ii)
If H is a closed normal subgroup of G, then it follows from
Corollary 4.34 that H* = H.
Now assume that (G,G) is simple and let A = A(B), 1 = eY (B). If a,y E A,
then y -yaa turns out to be an integral multiple of a.
This integer is
denoted by < y,a > and called a Cartan integer. The matrix of Cartan integers gives rise to the various possibilities for the root systems: DI(1 Z 4), E6, E7, E8, F4, G2.
AI(1 ? 1), BI(1
2), CI(1 >_ 3),
See [34], [108]. The Weyl group W =
is a
special type of a finite group, called a Coxeter grout. If a, Y E A, let m(ay) denote
the order of aaaY
Then W is completely determined by the relations (aa(,y
)m(a,y) = 1. It turns out that for a *y, m(ay) = 2,3,4 or 6.
as as
Op , if m((x,y) = 4 define as
ay .
If m(a,y) = 3, define
ay , if m((x,y) = 6, define
The possibilities are then given by the following diagrams [12],
[115].
AI: . BI or CI:
.
..
....
_
41
E8:
F4:
Thus the Weyl group does not distinguish between types B1, C1.
The Cartan matrix
can be described completely via the Dynkin diagrams which contain slightly more information than the above (Coxeter) diagrams (see [34; Appendix]). When (G,G) is not simple, its diagram is reducible in that it is the disjoint union of the diagrams of the simple components.
For more details on algebraic groups, we refer to Borel [4], Carter [6], Hochschild [32], Humphreys [34], Springer [108] and Steinberg [110].
42
5 CONNECTED ALGEBRAIC SEMIGROUPS
For algebraic groups, the topological terms 'irreducible' and 'connected'
have the same meaning. For algebraic semigroups, this is not so. example,
S = (diag(a,b) I a2 = b2) c 2(K).
monoids are briefly studied by Renner [101).
Consider, for
Topologically connected algebraic
However, we will use the term
'connected semigroup' to mean that the underlying variety is irreducible.
Definition 5.1.
A connected semigroup
S
is a linear algebraic semigroup whose
underlying variety is irreducible.
Remark 5.2. Let M be a linear algebraic monoid and let M1, M2 be irreducible
components of M containing 1. Then the product map from M1
X
M2 into M
shows that M1M2 is irreducible. Clearly M1, M2 C M1M2 .
Hence M1 = M2.
Thus 1 lies in a unique irreducible component Mc of M.
Clearly Mc is a
connected monoid.
Remark 5.3. Let S be a connected algebraic semigroup, e E E(S).
Then eS, Se,
eSe, SeS are connected semigroups.
Remark 5.4. Let : G0 -i GL(n,K) be a representation of a connected group Go
Then M = M(4) = K4(Go) c kn(K) is a connected monoid with zero.
43
Example 5.5. Let M be a connected algebraic monoid with group of units G.
Let
S' be an irreducible component of S = M\G. Then S' is a connected semigroup which is an ideal of M.
Example 5.6. Let M = K4 with multiplication
(a,b,c,d)(a',b',c',d') = (aa',ab' + bd', dc' + ca', dd').
Then M is a connected monoid with zero. Let e = (1,0,0,0) E E(M).
Then MeM is
not closed. See [65; Example 4.1].
Example 5.7. Let S = K3 with
(a,b,c)(a',b',c') = (aa' + aba'c',b',c).
Then
S
is a connected regular semigroup.
J2 = ((a,b,c) E S a = 0).
Let
J1 = ((a,b,c) E S I a * 01,
Then YC (S) _ (J1,J2), J1 > J.
Let f = (0,1,-I) a E(J2).
Then there is no e E E(J1) with
e
f.
See [65; Example 4.11]. Contrast this
situation with Corollary 6.9. The following result is from the author [67; Theorem 8].
Proposition 5.8. Let S be a linear algebraic semigroup, e e E(S), J,R,L the $-class,
A-class, e-class of e, respectively. Then E(J), E(R), E(L) are closed subsets of S. If S is a connected semigroup, these sets are also irreducible.
Proof. By Corollary 3.16, we may assume that S is a closed subsemigroup of some An(K). Let p(e) = k. If a E S, let S(a) = the sum of products of k eigenvalues of a.
Since S(a) is a co-efficient of the characteristic polynomial of a, S: S -+ K is a
44
morphism. Let X = (fI f E SeS, f2 = f, 6(f) = k).
Then X is closed, E(J) c X. If
f E X, then by Corollary 3.30, f E SeS. So p(f) <_ k. p(f) = k. e >_ f'.
There exist x,y E S such that xey = f.
Since 6(a) = k, we see that
Let f' = eyfxe E E(S).
Then
Since f s f', p(f') = k. So e = f' $ f in S. Hence E(J) = X is closed.
Clearly E(R) = If If E E(S), of = f, fe = e), E(L) = If If E E(S), fe = f, of = e)
are
closed sets.
Now assume that dete(yx) # 0). Then
S
is a connected semigroup, U = ((x,y) I x,y E S,
Then U is a non-empty open (hence irreducible) subset of S x S.
(x,y) -4 (eyxe) -1
is a morphism on U where the inverse is taken in the
a-class of e. Consider the morphism 4>: U -4 E(J) given by 4(x,y) = x(eyxe)-1y. Let f E E(J).
Then xey = f for some x,y r= S. So eyfxe E E(J) and e = eyfxe.
Thus (x,yf) E U, 4>(x,yf) = f. Hence 4(U) = E(J) is irreducible. Let V = (a a E eS,
dete(a) * 0).
Define a morphism yr: V -4 E(R) as yr(a) = (eae) la.
If f E E(R),
then V(f) = f. Hence yr(V) = E(R) is irreducible. Similarly E(L) is irreducible.
The following result of the author [64; Theorem 2.16] turns out to be quite useful.
Theorem 5.9. Let S be a connected semigroup, elf E E(S), e / f. Then there exist e1,e2,f1,f2 E E(S) such that e 52 e1 41 f1 A f and e . e2 . f2 ,I f.
Proof. Let
elf E E(S), e / f.
We claim that there exists el E E(S) such that
e 5E el, elf / f. Suppose not. Let H, H' denote the a-class of e,f, respectively. By Remark 3.23, eSe\H, fSf\H' are closed sets. There exist x,y E S such that xey = f. We have the following closed subsets of eS:
X = (a E eS I fxaf E fSf\H')
Y={aESIaeEeSe\H).
45
Suppose e e X. e e X.
Then fxef e H' and of If.
Hence of $ f, a contradiction. So
Clearly fxeyf = f, whereby ey a X. Also e e Y. We claim that of a Y.
Otherwise efe e H and of I e f, a contradiction. Hence of a Y.
irreducible, we see that eS * X u Y.
Since eS is
So there exists a e eS such that a i X u Y.
Then ea = a, fxaf Af, ae A e. So there exists z e S such that zae = e.
Then
za2 = zaea = ea = a.
Hence a2 s a. By Theorem 1.4 (v), there exists e1 e E(S)
such that a a6 el.
Then e 5E e1.
Also elf of fxaf f.
Hence elf , f,
a
contradiction. Thus there exists el a E(S) such that e 5E e1, elf 0 f. By Theorem 1.4 (vi), there exists fl a E(S) such that
e1 .' fl . f.
Similarly there exists
e2,f2 E E(S) such that e d e2 52 f2 -V f. This proves the theorem. The following result is due to the author [65; Theorem 2.7].
Theorem 5.10. Let S be a connected semigroup. Then Yl (S) is a finite lattice.
Pr f. We can assume by Corollary 3.16 that S is a closed subsemigroup of some An(K).
Let E = E(S). If e e E, let I(e) = (a e S an a SeS J. Then 1(e) is a
closed subset of S by Corollary 3.30. Since
YC (S)
is finite by Theorem 3.28, the
family (SeS Ie a E) and hence the family 4 = (I(e) Ie a E) is finite. By Theorem Since S is a connected semigroup, I(v) = S for some v e E.
3.18, S = u 'I(e).
eeE Then clearly Jv is the maximum element of 2l (S). Since Yl (S) is finite, it suffices
to show that ?l (S) is a A-semilattice. So let e,f e E and let t = (g g e E, e I g,
f g). Let I = u I(g). Let x,y,z a S. Then (xeyfz)n 'g for some g e F.
So
gel'
g e r and xeyfz a I(g).
Define 0: S x S x S -a I as O(x,y,z) = xeyfz.
Since
S x S x S is irreducible and 4 is finite, we see that 4(S x S x S) c 1(h) for some h e l,. Then clearly Je Z Jh, Jf >_ Jh. g e E(J).
Then e 1g, f 1g.
4(x,y,s,t) a I(h). theorem.
Let J e 2e (S) such that Je 2 J, Jf z J. Let
So xey = g = sft for some x,y,s,t a S.
Thus h g and Jh >_ J.
Hence Je A Jf = J.
So g =
This proves the
46
Remark 5.11. The open problem then is to determine all possible 2l (S).
If e is an
idempotent in the maximum $-class of S, then 2l (S) __ 2e (M) where M = eSe. In fact, there exists a connected regular monoid M' with zero such that 2l (M) 2e (M') (see the proof of Theorem 15.1). When 2l (M')\(0) has a minimum element, the possibilities are determined in [89]. The next result is taken from the author [73; Theorem 2.1].
Theorem 5.12. Let S be a connected semigroup with zero 0. Then the following conditions are equivalent.
(i) S is completely regular. (ii) S has no non-zero nilpotent elements.
(iii) S is a monoid and the group of units of S is a torus. (iv) S is isomorphic to a closed submonoid of some
t'In(K)
with 0
being the zero matrix.
Proof. That (iii) => (iv) follows from Theorem 3.15, Corollary 4.12. That (iv) => (i) follows from Corollary 3.19. That (i) => (ii) is obvious. So we are left with showing that (ii) => (iii).
Thus assume that
Corollary 3.16 we can assume that
S
S
has no non-zero nilpotent elements. By
is a closed subsemigroup of some
'11n (K).
Hence an lies in a subgroup of S for all a E S. Let e E E(S). Suppose Se * eSe and consider the morphism 0: Se -4 eSe given by O(x) = ex. dim 41(0) > 0.
By Theorem 2.21,
So there exists x E Se, x * 0 such that ex = 0. Then x2 = xex =
0, a contradiction. Hence Se = eSe.
Similarly eS = eSe. Thus the idempotents of
S lie in the center of S. By Theorem 5.10, 2l (S) has a maximum element J. Let E(J) = {h}.
Then for all a E S, an E ShS = hS.
Suppose S # hS and consider the
morphism yr: S --4 hS given by yr(a) = ha. Then by Theorem 2.21, dim yr 1(0) > 0.
So there exists a E S, a * 0 such that ha = 0. Then an = han = 0, a contradiction.
Hence S = hS and h = 1 is the identity element of S. We may assume that 1 is the
47
identity matrix. We see by Corollary 1.6 that for all a,b E S, a b implies an I bn Now let a E S. Then an A e for some e E E(S). Let S = aS
eS.
Then for all
x E S, a an (ax)n. So un e eS for all u E S 1. Suppose S 1 * eS and consider the surjective morphism
0: S1 -a eS
given by
0(x) = ex.
By Theorem 2.21,
0 1(0) * (0). So there exists b e S 1 such that eb = 0, b * 0. Then bn = ebn = 0, a contradiction. Hence S1 = eS and a E eS. So a 26 e and S is a semilattice of groups by Theorem 1.12.
Let G denote the group of units of dim S
S.
We prove by induction on
that G is a torus. Let e e E(S) such that 1 covers
e.
Consider the
homomorphism y: S -, eS given by y(x) = ex. By the induction hypothesis applied to
eS, eG = y(G) is a torus. Let Sc = yl(e)c. Let V be an irreducible component of y1(e) containing e. By Theorem 2.21, V * (e). So there exists v E V, v * e such
that ev = e. Then v E G. homomorphism det: Se -, K.
So e E Vv 1 c S2. Let Ge = G n Se Consider the Since det 1(0) _ (e), we see by Theorem 2.21 that
dim Se = 1. If Ge is unipotent, then det(Se) = (1,0), a contradiction since Se is a
connected monoid. So by Theorem 4.11 (iv), Ge is a torus. Let T be a maximal
torus of G containing Ge
By Corollary 4.8, y(T) = y(G).
By Corollary 4.3,
G = T, completing the proof.
Most of this book has to do with connected regular monoids M with zero. However, the following is clearly an important open problem.
Problem 5.13. Study connected regular semigroups with zero.
Let S be a connected regular semigroup with zero, e an idempotent in the maximum
$-class. Then M = eSe is a connected regular monoid with zero.
The problem then is to determine the possible S for a given M. point would be to take M = 'kh(K).
A good starting
48
6 CONNECTED ALGEBRAIC MONOIDS
In this chapter we develop the machinery for studying connected monoids. If M is any linear algebraic monoid, then the identity component Me of M is a connected monoid (see Remark 5.2). If G is the group of units of M, then
G is an open subset of M (see Remark 3.23), Me = Gc.
If Go is a connected
group, 0: Go -, GL(n,K) a representation of Go, then M = KO(G0) is a connected monoid with zero. More generally Renner [91] and Waterhouse [116] have shown that
any connected group with a non-trivial character occurs as the group of units of a connected monoid with zero.
Let M be a connected monoid with group of units G. Let X,Y c M.
Then X,Y are conjugate if there exists g e G such that g 1Xg = Y. The centralizes
of X in Y, CY(X) = (y e Y I xy = yx for all x e X) and the normalizer Qf X in Y, N..,(X) = (y e Y IXy = yX). If F c E(M), then the right centralizes Qf IF in X,
CX(I) = (x e X I xe = exe
for all
e e r), the left centralizes Q r
in
X,
CX(I) = (x e X I ex = exe for all e e 17). The center of M, C(M) = (x e M I xy = yx
for all y e M). For e e E(M), let
Me = {ae Mjae=ea=e)c,Ge=GnMe. Let T be a maximal torus of G. Then NG(T) = NG(T) and CG(T) = CG(T).
If
a = xCG(T) e W(G), a e 7, then let a6 = xlax a T. We will also denote W(G) by W(M).
49
The following result is due to the author [67], [83].
Proposition 6.1. Let M be a connected monoid with group of units G and let
a,b E M. Then (i) a s b if and only if MaM = MbM if and only if b E GaG. (ii) a 52 b if and only if aM = bM if and only if b E aG.
(iii) a ' b if and only if Ma = Mb if and only if b E G a.
Proof. Define 0: G x G -, MaM as $(g1,g2) =
glag2.
Then
4)
is a dominant
morphism and MaM is irreducible. So by Theorem 2.19, there exists a non-empty
open subset U of MaM such that U c GaG. Similarly there exists a non-empty
open subset V of MbM such that V c GbG.
So if MaM = MbM,
then
0 # U n V c GaG o GbG. This proves (i). (ii), (iii) are proved similarly.
Proposition 6.2. Let M be a connected monoid with group of units G, dim M = p,
M *G. Let Sl,...,Sk denote the irreducible components of S=W. Then Si is an ideal of M and dim Si = p -1, i = 1,...,k.
Proof. ,,fGn(K).
By Theorem 3.15 we can assume that M is a closed submonoid of some Consider 4): M -4 K given by 4>(a) = det a.
Theorem 2.21, dim Si = p - 1, i = 1,...,k.
Clearly S = 4)-1(0).
By
Now Si c MSiM c S; and MSiM is
irreducible, being the closure of the product map from M x Si x M into S. So Si = MS M and the result follows. r
The following consequence of Corollary 4.10 has been noted in [66], [91]. We follow [91].
50
Proposition 6.3. Let M be a connected monoid with group of units G and let B be
xlffx .
a Borel subgroup of G. Then M = ifG = Gif = u xeG
P r o o f . G acts on M in three ways: a
g = ag, a
g=
gla, a
g = g lag where
gEG,aEM. In each case 11 - B=IT and Gc$ G. By Corollary 4.10, IT
G
is closed in M. It follows that if G = M.
Corollary 6.4. Let M be a connected monoid with zero 0 and group of units G.
Then 0 e T for any maximal torus T of G.
Proof. By Proposition 6.3, we can assume that G is solvable. By Remark 3.17, we
can assume that M is a closed submonoid of some
with 0 being the zero
,Aln(K)
matrix. By Theorem 4.11, we can further assume that G c
(K), T c 2n *(K). Then
M C 9n(K), T c .0 (K). If a e M, then let 4(a) E 91n(K) denote the diagonal matrix with the same diagonal as a.
By Corollary 4.8, 4(G) = 4(T) = T.
So 0 = 4(0) E
(M) = 4(G) c 4(G) = T.
Lemma 6.5. Let M be an algebraic monoid with group of units G. Then Mcg = gMc for all g E G and (ri = McG = GMc.
Proof. Since Gc a G and Mc = Gc, it follows that g1Mcg = Mc for all g E G. Since
G/Gc
is a finite group, there exist
gl,...,gk E G
Gcgl u... u Gcgk. Then
G=Mcglu...uMcgkcMCGcGG=Gi This proves the lemma.
such that
G=
51
The next result is due to the author [73;. Lemma 1.1].
Lemma 6.6. Let M be a connected monoid with group of units G, I a closed
connected right ideal of M, e e E(I).
Let dim I = n, dim eM = m.
Then every
irreducible component of the closed set (a e I I ea = e) has dimension n - m.
Proof. Since I is a right ideal of M, eM c I.
Let Y = (a a eMIa } e in M).
Then Y is a closed set by Lemma 3.27. So V = eM\Y is a non-empty open subset
of eM.
Consider the surjective morphism 0: I -+ eM given by 4)(a) = ea.
By
Theorem 2.21, there exists a non-empty open subset U of eM such that every irreducible component of 01 (u), u e U has dimension n - m. connected semigroup, V o U * 0. Theorem 1.4 (i), e .5 u.
Let u e V n U.
Since eM is a
Then eu = u, e f u.
By Proposition 6.1, eg = u for some g e G. Let
0-1
By
(u) =
Al u ... u At represent the decomposition of 41(u) into irreducible components.
Then dim Ai = n - m, i = 1,...,t. Since
I
If a e I, then ea = e if and only if eag = u.
is a right ideal of M, Ig = I.
It follows that
(e) _ $1(u)g 1 =
Alg1 u ... u Ag1. This proves the lemma. Recall that Me = (a e M ae = ea = e)c, Ge = G o Me The following result and its corollaries are due to the author [66], [67].
Theorem 6.7. Let M be a connected monoid with e e E(M).
Then E(Me) _
(f e E(M) f >_ e) and e is the zero of Me.
Pr f. We may assume that e * 1.
dim M = n.
Let G denote the group of units of M, S = NW,
Let Sl,...,Sk denote the irreducible components of S.
Then by
Proposition 6.2, each Si is an ideal of M and dim Si = n - 1, i = 1,...,k.
Ml = (a e M Iea = e).
Let X be an irreducible component of Ml containing
Let e.
Then by Proposition 6.6, dim X = n -dim e M. Suppose X c S. Then X c Si for
52
some i.
By Proposition 6.6, dim X = n - 1 - dim eM, a contradiction. Thus
X n G * 0.
Xgl
Choose
g e X n G.
C Mi. So e e Mi.
Then eg = e.
So
1,e a Xg 1.
Hence
Let M2 = {a a Ml ae = e}. Then M2 = Me By the
dual of the above argument, e e Me. Now let f e E(M), f >_ e. Then Gf S. Ge So
f e Mf a Me
Corollary 6.8. Let M be a connected monoid with group of units G, e, f e E(M). Then
(i) e , f if and only if
x-1ex
= f for some x e G.
(ii) e 5B f if and only if there exists x e G such that ex = xlex = f. (iii) e -V f if and only if there exists x E G such that xe = xexl = f.
Proof. By Theorem 5.9, (i) will follow from (ii), (iii). By symmetry it suffices to prove (ii). So let e ;R f.
Let Ml = (a a MIae = e). Then GeGf c M. By
Theorem 6.7, e,f a Mi.
Since e 5B f, we see by Proposition 6.1 that there exists
x E Mi n G such that ex = f. Then xe = e and
x-1
ex = f. The result follows.
Corollary 6.9. Let M be a connected monoid, J1,J2 E ?4M).
Then the following
conditions are equivalent.
(i) Jl z J2. (ii) For all e1 a E(Jl) there exists e2 a E(J2) such that e1
e2.
(iii) For all e2 a E(J2) there exists el a E(Jl) such that el >_ e2.
Proof. That (ii) => (i), (iii) => (i) is obvious. Let G denote the group of units of M and suppose J1
J2.
Let e e E(J1), f e E(J2). Then for some x,y a M, xey = f.
Let f' = eyfxe a E(J2). Then e z f'. By Corollary 6.8, there exists x e G such that
f = x-1f'x. Let e' =
x-1
ex a E(J1). Then e' ? f.
53
Corollary 6.10. Let M be a connected monoid with group of units G. Then (i) For any chain r c E(M), there exists a maximal torus T of G
such that F c E(T).
(ii) For any maximal torus T of G, E(M) = u x lE(T)x. xeG
Pr f. Since all maximal tori are conjugate, (ii) follows from (i). So we prove (i) by induction on
I 1' 1.
If I F = 0, there is nothing to prove. So let
the smallest element of F.
I F I >_ 1.
Let e be
By Theorem 6.7, r c Me and e is the zero of Me.
There exists a maximal torus Tl of Ge such that r\(e) c fil. By Corollary 6.4, e e 71. Thus r c fi for any maximal torus T of G containing T1. The following result is due to the author [73; Theorem 1.3].
Theorem 6.11. Let M be a connected monoid with group of units G, e e E(M). Let
Ml = (a e Mlea=e),M2= (ae Mlae=e), M3
=
(ae Mlea=ae=e).
Let
Gi = Min G, i = 1,2,3. Then Mi = Gi, i = 1,2,3.
Proof. First we show that M1 = Gi. Let dim M = n, dim e M = q. Let a e Ml, X
an irreducible component of M containing a. Then by Lemma 6.6, dim X = n - q.
Suppose X c S = M\G.
Then X c S' for some irreducible component S' of S.
By Proposition 6.2, S' is an ideal of M, dim S' = n -1. Now e = ea a S'. So by
Lemma 6.6, dim X = n - 1 - q, a contradiction. So x o G * 0. XnGcG
Thus a E X =
Thus M1 = G1. Similarly, M2 = G2.
Now let a e M3 S .Ml. By Lemma 6.5, there exists g e G1 such that
a e Mig = gMi. Now e e Mi by Theorem 6.7. So e = eg a Mig = gMi. Hence
f = g-lee Mi.
So fe = f, of = e.
f = ue for some u E Gi. a e M3, we see that
Thus f e E(Mi), e .f f. By Proposition 6.1,
So ulgle = e = eulg 1. Since u-lg1 E G3 and
l e M1 n M3 = Min M2. ugla
Considering the monoid Mil
54
we see that Mi n M2 = G1 o G2 S G3. So u 1g-la e G3. Thus a E G3, proving the theorem.
Example 6.12. Let M = ((a,b,c) I a,b,c e K, alb = c2) with pointwise multiplication.
Then M is a connected monoid with zero. Let G denote the group of units of M, e = (0,1,0) E E(M). Then Gl = (x e GIex = e) is not connected. The following result is due to the author [73; Theorem 1.4].
Corollary 6.13. Let M be a connected monoid with group of units G and let
e e E(M). Then GMeG = (aIa E M, ale).
Proof. Let a E M, a (e.
Choose e 1 E E(M), a e 1 I e such that J= Je1 is maximal.
Now xay = e1 for some x,y e M. xe2ay = el.
So el / e2.
Now e2az = e2.
e2a = e2w for some w e G. Let b = for some f e E(M), k E Z+-
a b f e2 jel.
Let z = yelx, e2 = az.
awl.
Thus e2a R e2. By Proposition 6.1,
Then a / b, e2b = e2. Now bk 'V f
Then e2 = e2bk = e2bkf.
By the maximality of J, e2 / f.
f = fe2 = fe2b = if,. So fbk = f.
Then e2 E E(M),
So e2f = e2.
Thus
So e2 .1 f. Since e2b = e2,
Since bk ad f, bk = f.
So fb = bf = f.
f le, there exists by Corollary 6.9, e' e E(M) such that f >_ e', e' , e.
Since
So be' =
e'b = e'. By Corollary 6.8, y4e'y = e for some y E G. So y4bye = ey4by = e.
By Lemma 6.5, Theorem 6.11, ylby e MeG.
Since b = awl, a E GMeG,
proving the result.
Lemma 6,14. Let M be a commutative connected monoid, e E E(M).
Let 0 be a
finite group of automorphisms of M having e as a common fixed point. Then there
exists a closed connected submonoid M1 of M such that e c- M1 and a6 = a for all a E S2, a E M 1.
55
Proof. Let
92 = (al,...,ap).
xa1... xap.
Then V(l) = 1, V(e) = e.
Define a homomorphism yr: M -4M as V(x) =
Let a E Q.
Then 91a = S2, whereby
V(X)CF = Nf(x) for all x E M. Let M1 = yr(M).
Let M = (A ® B I A,B e ,(K) }, e = [0 0] ® [o
Example 6.15.
01
,
f=
Then of = fe = 0 but f e CG(e). Thus CM(e) is not a connected
l00] ®[01 monoid.
The following result is due to the author [67], [69].
Let M be a connected monoid with group of units G and let
Theorem 6.16.
e e E(M). Then (i)
CG (e), CG (e), CG(e)
are closed, connected subgroups of G.
(ii) eM c CG(e), Me c CG (e), eMe c CG(e) . (iii)
If H denotes the A -class of e, then the map r. CG(e) , H
given by
y(a) = ea
I W(Ge) 1.
If T is a maximal torus of G with e E T then Te, eT are maximal tori
is a surjective homomorphism and
I W(CG(e)) I = I W(H)
of Ge' H, respectively.
Pr f. (i) By Corollary 6.10, e E E(T) for some maximal torus T of G.
show that CG(e) is a connected group.
maximal tori of CG(e)c. xy e NG(T) n CG(e).
Let x e CG(e).
First we
Then T, x-1Tx are
So y 1x-1Txy = T for some y E CG(e)c.
So u =
Consider the automorphism a: T -4 T given by as = u1au.
Since W = NG(T)/CG(T) is a finite group, a is of finite order. Clearly ee = e for
all 0 e < a >. So by Lemma 6.14, there exists a closed connected subgroup T1 of T such that e E T1, as = a for all a e T1. So U E CG(Tl) = CG(Tl) c CG(e). But CG(GI) is connected by Theorem 4.11 (iv).
U E CG(e)c. Since y E CG(e)e, x e CG(e)e.
Hence CG(Tl) c CG(e)c. Thus xy =
So CG(e) = CG(e)c is a connected
56
group.
Now let x e CG (e).
Then T, x 1Tx are maximal tori of CG(e)c.
Hence y-1x1Txy = T for some y e CG(e)c.
So u = xy e NG(T) n CG (e).
Clearly NG(T) = NG(T).
So f = ueu1 E T. But ue = eue. So of = f. Since T
is commutative, e = f.
So u e CG(e) C- CG(e)c.
Hence x e CG(e)c.
Thus
CG (e) = CG(e)c is a connected group. Similarly CG(e) is a connected group.
(ii) We may assume that e * 1. Let dim M = n, S = M\G, S1,...,Sk the irreducible components of S.
Then by Proposition 6.2, dim Si = n -1, i = 1,...,k.
Let dim e M = q < n.
0: M -, e M as 4(a) = ea.
Define
Let Oi denote the
restriction of 0 to Si, Vi = (Si), i = 1,...,k. Let i e (1,...,k). If Vi * e M, let Ui = e M\ V. Next suppose Vi = e M. Then Oi is a dominant morphism. So by Theorem 2.21, there exists a non-empty open set Ui of e M contained in 0(Si),
such that for any closed irreducible subset Y of e M with Y n Ui * 0, any irreducible component X of 0 i 1(Y) with O (X) = Y, we have
dimX=dimY+n-1-q
(3)
Let U = U1 n ... n U. Since e S is irreducible, U is a non-empty open subset of e S.
Let x e U. Then x e 4 1(x). Let F be an irreducible component of 01(x)
with x e F. By Theorem 2.21, dim F >_ n -q. Suppose F c S. Then F c Si for some i. Hence x = 4(x) a O(Si) C V. Since x E Ui, we see that Ui n Vi * 0. So Vi = e M.
By (3), dim F = n - 1 - q < n - q, a contradiction. So there exists
ge GoF. Then eg=x. Since ge G,Y=eMx=eMeg is a closed irreducible subset of e M and Mx = M e g is a closed irreducible subset of 41(Y). Let X be an irreducible component of
(Y) containing M x.
Then 4(X) = Y.
So by
Theorem 2.21,
dim X >_ dim Y + n -q
(4)
57
Suppose X c S. Then X c Si for some i. 4 1(Y).
So X is an irreducible component of
Also, X E Y n U c Y o Ul. Thus we have a contradiction by (3), (4). So
X n G * 0. Let X1 = Xg 1. Then Me c Xl, X1 n G * 0, XI is a closed irreducible
So XI = X1 o G.
subset of M.
a E X1 n G.
Let
eag = 4)(ag) E eMeg and ea = eae. Thus a E CG (e).
Then ag a X.
So
So X1 n G c C I (e). Hence
Me c Xl = X1 n G c CG(e) . Similarly e M c CG (e) . Applying this result to the connected monoid
CG (e)
,
eMece
we see that
CG (e)
C
CG(e)
(iii) Clearly y is a homomorphism. By Corollary 4.3 (i), y(CG(e)) is
closed in H. In M,
H c e M e = eCG(e) = y(CG(e)) c y(CG(e))
It follows that H = y(CG(e)). Clearly (ker y)c = Ge So we are done by Corollaries 4.8, 4.16, 4.25.
Corollary 6.17. Let M be a connected monoid with group of units G and let T be a maximal torus of G. Let a E M, F c E(T ). Then
(i) CG (I), CG (I,, CG(r) are closed connected subgroups of G. (ii) If
ea = eae Me for all e e I',
eae X e for all e e r, then a e CG (F) .
then a e CG(F)
.
If ae =
If ea = ae a6 e for all e e t, then
ae CG(I').
Pr f. (i) follows from the repeated application of Theorem 6.16 (i). So we prove (ii). Suppose ae
ea = eae ' e
for all e e F.
We prove by induction on
I IF I
that
CI(I) . If I IF = 0, this is clear. So let e e F, r' = n{e}. Let G' = CG (r'),
58
M' = G' . Then T c G', a E M'. Now ea = eae A e in M and hence in M' by Remark 1.3 (iii). By Theorem 6.16 (iii), there exists u e CG, (e) such that ea = eae = eu.
Then eaul = e. Let M1 = {b a M' deb = e}, GI = M1 o G'. Then by
Theorem 6.11, M1 = G1 . a E CG, (e) = CG (T)
.
Clearly G1 C CG, (e).
Hence au 1 e CG, (e) .
Thus
The other statements are proved similarly.
Corollary 6.18. Let M be a connected monoid with group of units G and let F be a
chain in E(M). Then (i) CG(['), CG(F), CG(F) are connected groups. (ii) If e E IF, f E E(M), then e 5E f implies f e CG (I').
(iii) If e E I', f E E(M), then a.t f implies f E CG(F).
Proof. By Corollary 6.10, r c E(T) for some maximal torus T of G.
Thus (i)
follows from Corollary 6.17. Since (iii) is dual to (ii), it suffices to prove (ii). We
proceed by induction on IF I. If I F I = 0, there is nothing to prove. So let IF I > 0.
If h e IF, let Xh = {f E E(M) If A h}.
Let e be the maximum element of r,
F' = P(e). Let G1 = CG (e), Ml = GI . By Theorem 6.16 (ii), e M c Ml. Xh c e M c M1 for all h E F.
Let G2 = CG(r) = CG (r' ), M2 = G2 .
Thus
By the
induction hypothesis, Xh c M2 for all h E t'. Now let f E X. Then e A f. Let a E Gf .
Then of = f. So for all h E F, ah = afeh = h. Hence a E CG (T).
So
Gf C CG(r). By Theorem 6.7, f E Gf C CG (I, . This completes the proof.
Corollary 6.19. Let M be a connected monoid, e,f a E(M), e / f. Then CG (e) _ CG(f) if and only if e 5E f; CG(e) = CG(f) if and only if e ,91 f.
59
Pr f. First suppose e 5l f. x
By Corollary 6.18 (ii), e,f e CG (e) . By Corollary 6.8,
lex = f for some x e CG (e).
So CG (e) = x 1CG(e)x = CG(xlex) = CG(f).
Next, suppose CG(e) = CG (f). Since e $ f, we see by Theorem 5.9 that there exists
e', f' a E(M) such that e 5B e'
f' A f. By the above, CG(e') = CG(f').
By
Corollary 6.18, e',f' E CG(e') . By Corollary 6.8, there exists y e CG (e') such that
ye'y 1 = f'. But ye' = e'ye'. So f' = e'f' = e'. Hence e A f. The next result is from the author [65], [66].
Theorem 6.20. Let M be a connected monoid with group of units G. Let T be a maximal torus of G.
Then
E(T)
is a finite, relatively complemented lattice.
Moreover, the lengths of the maximal chains in E(T), E(M), 24(M) are all the same.
If M has a zero, then this number is equal to dim T.
Proof. E(T) = V (T) is a finite lattice by Theorem 5.10. Let Jo denote the kernel of M.
Then by Corollary 6.10, J n E(T) = (v) where v is the zero of E(T). 0
Theorem 6.7, E(T) = E(TV).
By
Also, by Corollary 6.9, any maximal chain in ?4M)
gives rise to a maximal chain in E(Mv). Thus we may assume that v = 0 is the zero
of M.
Let t = { 1 > e > ... > 0) be a maximal chain in E(T).
induction on
IFI
We prove by
that dim T = I I' I - 1. Now eT is the group of units of eT = eT
and I" = I\{ 1) is a maximal chain in eT. Thus dim eT = I I" -1 = F I - 2. We have a surjective homomorphism 0: T -, eT given by 0(t) = et.
Clearly
(ker 0)c = Te, Te = Te u {e}. By Proposition 6.2, dim Te = 1. By Corollary 4.3 (ii), dim T = dim eT + 1 = 1171 -1.
By Corollary 6.9, a maximal chain in W(M) gives rise to a maximal
chain in E(M).
By Corollary 6.10, a maximal chain in E(M) is contained in the
closure of some maximal torus and hence by the above, has length equal to dim T.
60
Finally, we show that el,e,f a E(T), eI > e > f. e V h = e 1.
E(T)
is relatively complemented.
Let
We need to find h e E(T) such that eI > h > f, eh = f,
We may assume that e I = 1 (otherwise we work with e 1 I ).
the homomorphism 0: T i eT given by 4>(a) = ea.
By Theorem 2.21, dim 4 (f) > 0.
Consider
Now dim eT < dim T. 4>(f) = f.
So there exists x e T, x * f such that ex = f.
Now x M h for some h e E(T). Then eh = f.
If h = f, then x = fx = efx =
ex = f, a contradiction. Choose h e E(T) maximal with eh = f.
We claim that
e V h = 1.
For suppose ht = e V h * 1. Then by the above, there exists h2 a E(T)
such that
h2 > h, h I h 2 = h.
Then
eh2 = ehlh2 = eh = f, contradicting the
maximality of h. Thus e V h = 1, proving the theorem.
Definition 6.21. Let M be a connected monoid with kernel J0 Theorem 6.20 gives
rise to height function, ht on W(M), E(M) as follows: ht(Jo) = 0, if J, J' E 2C(M)
with J covering J', then let ht(J) = ht(J') + 1. If J e YC(M), e e E(J), let ht(e) _ ht(J).
If ht(1) = p, let ht(M) = ht(E(M)) = p. For i = 0,...,p, let
Vi(M) = 2C(p-i)(M) = (J E YC(M) I ht(J) = i).
Corollary 6.22. Let M be a connected monoid with group of units G and let T be a maximal torus of G. Let el,e2 E E(T), e1 > e2, Ji the /-class of ei, i = 1,2. Then the following conditions are equivalent.
(i) eI covers e2 in E(T). (ii) e1 covers e2 in E(M). (iii) J1 covers J2 in YC(M).
Proof. Clearly (iii) => (ii) => (i). So assume (i). Now el,e2 e F for some maximal
chain t of E(T). Hence JI covers J2.
By Theorem 6.20, Je(e a T) is a maximal chain in V(M).
61
Corollary 6.23. Let S be a connected semigroup. Then the length of any maximal chain in ?1(S) = the length of any maximal chain in E(S).
Proof. By Theorem 5.10, 2l(S) has a maximum element lo. Fix e e E(J0). Let S2
be a maximal chain in
Yt(S).
Then 92' = (J n eSeI J E 0] is a maximal chain in
W(eSe). We can now apply Theorem 6.20. The following result is due to the author [68].
Proposition 6.24. Let M be a connected monoid with group of units G such that a maximal subgroup of the kernel of M is solvable. Then for any maximal chain r of E(M), C6' (r), CG (I) are connected solvable groups.
Proof. By Corollary 6.18, CG (F), CG(I,) are connected. We prove by induction on F
that CG(I) is solvable. If I I' = 1, there is nothing to prove. So let I t I > 1.
Let e be the maximum element of t' = I\(1). Then 1 covers e. Let H denote the A-class of e. Clearly F' is a maximal chain in E(eMe). Hence CH (F') is a solvable group. Define homomorphism,
0: CG (r') - H as 4(a) = ae = eae.
4(CG(I)) c CH(T')
G1 = (a e G ae = e).
is solvable.
Clearly 0 is a
Clearly, the kernel of
4,
Let M1 = Gi and let T be a maximal torus of G1. Then
E(T) = (1,e), e is the zero of T.
So by Theorem 6.20, dim T = 1.
We may
assume that M1 is a closed submonoid of some 'kh(K). Then det a = 1, for all a e G2 = (G?,G1).
So G2 is closed in M1. Since e e T, T G2. Hence G2 is a
unipotent group and therefore solvable. It follows that Gi is solvable. By Theorem
4.22 (iv), Corollary 4.25, CG(I) is a solvable group.
Proposition 6.25. Let M be a connected monoid with group of units G and let T be a maximal torus of G.
Let J,J' a V(M), J ? Y. Let A = J o E(T), A' = J' o E(T).
62
Then
(i) For any e E A, A= tea I a E W) and 1W I= I A i
(ii) For any e e A, there exists e' E A' such that e
I W(CG(e)) I
e'.
(iii) For any e' E A', there exists e E A such that e >_ e'.
Proof. (i) Let e,f E A such that
e / f.
By Corollary 6.8, x-1ex = f for some
x e G.
Since T c CG(f), xTx 1 c CG(e).
CG(e).
So there exists y e CG(e) such that T =
Let a = yxCG(T). Then e6 = x 1y-1eyx =
Thus T, xTx 1 are maximal tori of yxTx-1 y-1
x-1 ex = f.
.
Then yx E NG(T).
Thus W acts transitively
on A. Clearly ea = e if and only if a E W(CG(e)).
(ii) Let e r= A, H the W -class of e. maximal torus of H. f E E(eMe).
By Theorem 6.16, eT is a
By Corollary 6.9, there exists f E E(J') such that e >_ f'. So
By Corollary 6.10, there exists e' a E(eT) such that
f / e'.
Then
e >_ e', e' E A'. (iii) Let e1 >_ e'.
e' E A'.
By Corollary 6.9 there exists e1 E J such that
By Theorem 6.16, Te, is a maximal torus of Ge,. Since e1 E Me,, we
see by Corollary 6.10 that there exists e E E(Te) such that el / e. Then e 'e e', e E A.
Let M be a connected monoid with group of units G and let T be a maximal torus of G.
Definition 6.26. If J E 2l (M), then the wi th of J, w(J) = I J o E(T) 1. If e e E(J), the width of e, w(e) = w(J).
Since all maximal tori in G are conjugate, the above definition is independent of T.
Note that the width of e in CG(e) and CG (e) is
Corollary 6.8. Also note that if e E
then by Proposition 6.25 (i),
1
by
63
w(e)= I(ealae W)I I.
The following result is due to the author [70].
Proposition 6.27. Let M be a connected monoid with group of units G and let
e,f,f' a E(M), f $ f'. Let Jo denote the kernel of M, J the $--class of e. Then
(i) If e >_ f, f', then f,f' a eMe and f $ f' in eMe.
(ii) If e:5 f, f', then f,f' E Me and f $ f' in Me. (iii) W(eMe) = [J0,J] and &(Me) . [J,G].
Proof. (i) is obvious and (iii) follows from (i), (ii). So we proceed to prove (ii). Thus
assume that e <_ f,f'.
f,f' e Me
Then by Theorem 6.7,
x-1f'x = f for some X E G. Then f >_ e, f >_ x1ex.
So e s
CG(f) . Thus there exists y e CG(f) such that y-lx 1exy = e. y-1x-1
f'xy =
y-1
fy = f.
lies in the center of M.
Hence f / f' in CG(e) .
By Corollary 6.8, x-1
ex in fMf c
So xy a CG(e) and
Thus we may assume that e
Let T be a maximal torus of G such that f e T.
Then
e e T and Te is a maximal torus of Ge. By Corollary 6.10, there exists fl E E(Te)
such that f' / f1 in Me- Then f / f1 in M. It suffices to show that f / f1 in Me A
1
Let A = {h e E(fi) I h s f in M}, Al = (h E E(e) I h s f in Me).
c A, f 1 E A.
So it suffices to show that I A I =
I Al 1.
Then
Let G1 = CG (f) _ e
CG(f) o Ge. Then by Theorem 6.16,
I W(G) I = I W(Ge) I- I W(eMe) I
IW(CG(f))I = IW(Gl)I
By Proposition 6.27, I W(G) I = I A I follows that I A I= I A l I.
IW(eMe)I.
I W(CG(f) I
,
I W(Ge) I = I Al I I W(G1) I. It
64
Lemma 6.28. Let M be a connected monoid with group of units G, e e E(M), w(e) = 1.
Then for any Borel subgroups B 1, B2 of G with e e $1 n $2, there
exists y e CG(e) such that y 1B1y = B2:
Proof. There exists x e G such that x1B1x = B2. Now e e T for some maximal
torus T of B2. Also x-1 1 ex a E(B2). By Corollary 6.10, y1x-1exy a T for some y e B2.
w(e) = 1,
Since
y-1 x-1 exy
= e.
Thus
xy a CG(e).
Clearly
y 1x 1B1xy = y 1B2y = B2.
By Lemma 6.14, we have
Lemma 6.29. Let M be a connected monoid with gorup of units G and let T be a maximal torus of G.
Let
T1 = {t e T to = t for all
G E W}c.
Then
E(T1) _ {e a E(T) I w(e) = 1). The following result is due to the author [69], [72].
Theorem 6.30. Let M be a connected monoid with group of units G,J e 2e(M). Then the following conditions are equivalent.
(i) J2 = J (i.e., E(J) 2 C J).
(ii) w(J) = 1.
(iii) E(J) c B for some Borel subgroup B of G.
(iv) E(J) c rad G.
Proof.
(i) => (ii). Let T be a maximal torus of G,
By Theorem 1.4 (i),
of = fe e J2 C J.
e=
f.
e,f e J n E(T). Thus
Then
w(J) =
1.
(ii) => (iii). For e e E(J), let Xe = {f a E(M)je .9l f}, Ye =
[f E E(M) je d f}.
6.7, Xe CM V
Let e e E(J), G1 = {a a GIae = e}c, M1 = G1 .
By Theorem
Clearly Xe is the kernel of M1. Let F be a maximal chain in
E(M1) with e e F. Then G2 = CG(T) is solvable by Proposition 6.24. By Corollary
65
6.18, Xe S_ G2 .
Now G2 S. B for some Borel subgroup B of G. Thus Xe
Similarly there exists a Borel subgroup B1 of G such that Ye S. B1 . e E $n B1, w(e) = 1. B.
Then
By Lemma 6.28, there exists' u e CG(e) such that u 1B1u =
So Ye = u 1Yeu c 11. Let J' denote the $-class of e in B.
h E E(J'), there exists b e B such that h = bleb. Thus Xh, Yh
Then for any C
if.
Now let
e', f' e E(M) such that e Re' .d f' ,R f. Then e' a Xe c $ So f' a Ye' cg. Thus f e Xf, c So Then by Theorem 5.9, there exists
f E E(J).
E(J) c B.
(iii) => (i). By Theorem 5.19, E(J) is contained in a $--class of IT. By Corollaries 3.20, 4.12, E(J)2 C J.
Thus the conditions (i), (ii), (iii) are equivalent. Clearly (iv) => (iii). So
assume that (i), (ii), (iii) hold. Let e e E(J).
We need to show that e e rad G .
Now x1E(J)x = E(J) for all x E G. So E(J) is contained in the closure of every Borel subgroup of G.
Now rad CG(e), being solvable, is contained in some Borel
subgroup B1 of G. If B2 is any other Borel subgroup of G, then by Lemma 6.28, there exists y E CG(e) such that y1Bly = B2.
rad CG(e) c rad G.
e E rad Ge .
So rad CG(e) c B2.
Thus
Since Ge a CG(e), rad Ge c rad G. Thus it suffices to show that
So we may assume that e = 0 is the zero of M.
We proceed by
induction on dim M. Let T be a maximal torus of G, T1 = (t E T I t(7 = t for all 6 E W)C, F = E(T1).
Then 0 E t by Lemma 6.29.
By Theorem 6.20, t is a
relatively complemented lattice. Suppose there exists f e t such that f # 0,1. Then
there exists h e t such that h * 0, 1, fh = 0. Then f E rad Gf c rad G . Similarly
h E rad Gh c rad G . (0,1).
Hence 0 = fh a rad G and we are done. So assume t =
Then w(f) > 1 for all f e E(M) with f * 0,1.
By Remark 4.31, rad G is
not unipotent. So T2 = T n rad G * (1). Let X E NG(T), t e T2. Then x-Itxtl E
T2 o (G, rad G) c T2 o raduG = (1) by Remark 4.33. Thus t6 = t for all t E T2, 6 E W. T h e r e f o r e T2 c T1.
By Theorem 6.20, dim T1 = 1.
Hence T1 = T2 c
66
radG and O E rad G.
Corollary 6.31.
Let M be a connected monoid with group of units
E(rad G) = {e E E(M) I w(e) = 1).
G.
Then
If G is reductive, then E(rad G) is a relatively
complemented sublattice of E(T) for any maximal torus T of G.
Proof. The first assertion follows from Theorem 6.30. Now suppose G is reductive. Then To = rad G is a torus lying in the center of G. So for any maximal torus T of G,
T0 c T and E(T
e E E(T) le6 = e for all a E W)
is a relatively
complemented sublattice of E(T) by Theorem 6.20, Lemma 6.29.
Corollary 6.32. Let M be a connected monooid with zero and group of units G. Then the following conditions are equivalent.
(i) G is solvable. (ii) There exists a maximal chain Y of ?1(M) such that w(J) = 1 for
all JEY (iii)
w(J) = 1 for all J E &(M).
(iv) M is a semilattice of archimedean semigroups.
Proof.
(i) => (iv). This follows from Theorem 3.15, Corollaries 3.20, 4.12.
(iv) => (iii). This follows from Corollary 1.16 and Theorem 6.30.
(iii) => (ii). This is obvious.
(ii) => (i). Let T be a maximal torus of G.
By Proposition 6.25, we can
find a maximal chain F of E(T) such that Y= ( JeIe E F). So w(e) = 1 for all e E F.
By Theorem 6.30, t c rad G . By Corollary 6.10, there exists a maximal torus
TI of rad G such that r c E(T1). By Theorem 6.20, dim T1 = I r -1 = dim T. Thus T1 is a maximal torus of G. Hence G/rad G is a unipotent group. Thus G
67
is solvable.
The following result is due to the author [75].
Theorem 6.33. Let M be a closed connected submonoid of
denote the group of units of M.
.,kn(K)
Let r be a maximal chain in
and let G
E(M).
Then
CG(r)CG(r) = (a E Gidetl.(a) # 01 = (a E Gleae A e for all e E r).
Proof. Let X = (a e G Idetr(a) # 0). Let e E r, H the s-class of e, c1 E CG(r), C2 E CG (r).
Then ecl = ecle E H, c2e = ec2e E H.
c1c2 E X and CG(I')CG(I') c X.
We need to. prove that
Without loss of generality we may assume that 1 E F. I r j.
So eclc2e E H.
If I F I = 1, there is nothing to prove. So let
Thus
X c CG(r)CG(r).
We proceed by induction on IrI
> 1.
Let e denote the
maximum element of r\(1). Let r' =11(e} and let H denote the a-class of e. Let a E X. Then eae E H. By Theorem 6.16 (iii), there exists x e CG(e) such that eae = ex. Let G' = CG(e), M' = G' .
f # 1.
Then e > f.
By Theorem 6.16 (ii), r c M'. Let f E r',
So fxf = fexef = feaef = faf a
f.
By the induction hypothesis
there exists u e CG, (r' ), v E CG, (r') such that x = uv. Then
v E CG(r). Now eae = ex = euv.
Let b =
ulavl.
u e CG(r),
Since u,v E CG(e), we see
that ebe = e. Let el = eb E E(M). Then e A e1. By Theorem 6.16, e1 e CG(r) .
So eby-1 = e.
By Proposition 6.1, there exists y E CG (r) such that eb = e1 = ey. Let
f E r, f # 1.
Then
e >_ f.
So fby 1 = f and by-1
E
CG(I ).
Thus
b e CG(r)CG(r). It follows that a = ubv E CG(r)CG(r), proving the theorem.
Corollary 6.34. Let M be a connected monoid with group of units G, e e E(M).
Then w(e) = 1 if and only if G = CG(e)CG(e) if and only if eGe is the A -class of e.
68
Proof. Let J denote the f-class of e. Suppose w(e) = 1. Then by Theorem 6.30, eGe c J. By Theorem 6.33, G = CG(e)CG(e). Conversely assume G = CG(e)CG(e).
Then eGe c J. So J2 = GeGeG c GJG = J. Also by Theorem 6.16, eGe contains
the A-class of e. This completes the proof. By Theorem 6.33, Corollaries 6.32, 6.34, we have,
Corollary 6.35. Let M be a connected monoid with zero and group of units G. Let IF
be a maximal chain in E(M).
CG(F)CG(T).
Then G is solvable if and only if G =
69
7 REDUCIZVE GROUPS AND REGULAR SEMIGROUPS
In this chapter we wish to begin to consider the situation when the group of units is reductive. The following result is due to the author [72].
Theorem 7.1. Let M be a connected monoid with zero and reductive group of units G.
Let IF be a maximal chain in E(M). Then CG(T) is a maximal torus of G and
CG(fl, C I(I') are opposite Borel subgroups of G relative to CG(I').
Proof.
By Corollary 6.10, t c T for some maximal torus T of G.
Clearly
T c CG(I'). By Proposition 6.24, CG(i) is a connected solvable group. So CG (I) c
B for some Borel subgroup B of G. Let B
denote the opposite Borel subgroup of
G relative to T. Then B o B-= T. So r c B
Now
CB-(r)gB nCG(I-)gBn B=T
(5)
By Corollary 6.35 and (5),
B = CB-( )CB-(I') c CG(f T = CG(F).
Thus B c CG(I'). But CG (I) is a connected solvable group by Proposition 6.24. So B = CG(F). Thus CG(rl) is a Borel subgroup for any maximal chain rl of E(M).
Similarly CG (I'1) is a Borel subgroup for any maximal chain Ti of E(M).
70
In particular B = CG(T). Hence T = B n B = CG(I).
Corollary 7.2. Let M be a connected monoid with zero and a reductive group of units G.
Let e E E(M), H the a-class of e. Then (i) CG(e), Ge, H are reductive groups. (ii)
If B is a Borel subgroup of G with e E E(B), then CB(e), Be,
eBe = eCB(e) are Borel subgroups of CG(e), Ge, H, respectively.
Proof.
H
(i)
Ge a CG(e).
is a homomorphic image of CG(e)
by Theorem 6.16.
Also
So we need only prove that CG(e) is a reductive group. Now e e T
for some maximal chain T of E(M).
By Theorem 7.1, B1 = C6(r) is a Borel
subgroup of G and hence of CG(e). Clearly the width of e in CG (e) is 1. Now rad CG(e) c B2 for some Borel subgroup B2 of CG (e).
Then e E $1 n $2. By
Lemma 6.28, x-1B2x = B1 for some x e CG(e). Hence rad CG(e) c B1 = CG(T). Similarly
rad CG(e) c CG(I).
Thus
rad CG(e) c CG(T),
which is a torus by
Theorem 7.1. (ii)
Let To = rad CG(e) which is a torus by (i). By Corollary 6.31,
e E To Hence CG(e) = CG(TO). Let B be a Borel subgroup of G with e E B.
Let T be a maximal torus of B with e E T. To c T.
Then T c CG(e) and therefore
By Theorem 4.11 (iv), CB(e) = CB(To) is a Borel subgroup of CG(e). So
by Theorem 6.16, Corollaries 4.8, 4.16, Be, eCB(e) are Borel subgroups of Ge, H
respectively. By Theorem 6.16, Corollary 6.34, eCB(e) = eBe.
This completes the
proof.
The following result is due to the author [73] when char K = 0 and Renner [94] for arbitrary characteristic. The proof given here is taken from [77].
Theorem 7.3. Let M be a connected monoid with zero 0 and group of units G. Then the following conditions are equivalent.
71
(i) G is reductive.
(ii) M is regular. (iii) M has no non-zero nilpotent ideals.
(iii) is obvious. Assume (iii). By Corollary 6.31, 0 E rad G.
Proof. That (ii)
rad G is not a torus. Then by Theorem 5.12, rad G has a non-zero
Suppose
Let X E M. Then by Proposition 6.3, x E IT for some Borel
nilpotent element a.
subgroup B of G. Now a E rad G c if. By Remark 3.15 and Corollary 4.12, we may assume that $ c 9n (K) for some n E 71+ and that 0 is the zero matrix. So a is strictly upper triangular. Hence ax is also nilpotent. Thus MaM is a nil ideal of M. It is well-known [35; Chapter VIII, Section 5, Theorem 1] that a nil matrix semigroup
is nilpotent. Hence MaM is a non zero nilpotent ideal of M.
This contradiction
shows that G is reductive.
Now assume that G is reductive. We prove by induction on dim G
that M is regular. Let e E E(M), e * 0,1.
Then by Corollary 7.2 and the induction
hypothesis eMe and Me are regular. Let M be a closed submonoid of dim M = p, S = MSG.
Let T be a maximal torus of G, B, B
subgroups of G relative to T, W = NG(T)/T.
that B, B
'*n (K),
opposite Borel
By Corollary 4.50, we may assume
consist of upper and lower triangular matrices, respectively. Then every
element of B n B
is a diagonal matrix. Let So be an irreducible component of S.
Then by Proposition 6.2, S 0 is an ideal of M and dim So = p -1. We first show
that S 0 is not nil. For suppose otherwise. By Theorem 4.22, B = a1Ba for some
a E W. By Theorem 6.33, B B and hence BaB is an open subset of G. So X =
G\BaB is a closed subset of G, X * G. We claim that So c X. a e X.
Since
0 E if c X, a * 0. a g BOB
So let a E So,
By Theorem 4.35, G is the disjoint union of for any 0 E W, 0 # a.
By Proposition 6.3,
BOB(0 E W).
Thus
M = Gif = iG.
Hence a E $BaB o BaBB c $aB n Bag. Let a = gT, g E NG(T).
72
Then there exist u2b11
bl,b2 E B, ul,u2 E B, such that a = ulgbl = b2gu2.
=g1b21ulg
E
IT n B
Thus
u2b11
is a diagonal matrix.
So
But
u2b11 = g1b21ab11 c Sc (O)- This contradicts the.assumption that S0 is nil. Thus so c X. So S0 c X' for some irreducible component X' of X. Since dim So = p -1 and X' * G, we see that So = X'. This is a contradiction since So c MSG.
Therefore So is not nil. Choose 0 * e E E(So) such that Je is maximal
in 91(S0). Let Y = (a I a E So, ate). Then Y is a closed set by Lemma 3.25. Let
u E So Y.
Then u e. By Corollary 6.13, u E GMeG. Hence u is regular. By the
maximality of Je, u $ e. Thus u e MeM. Hence So = MeM u Y. Since e e Y
and S 0 is irreducible, we see that So = MeM. Now e E I' for some maximal chain F in E(M). Let B1 = CG(I'), B2 = CG (I). Then B1,B2 are Borel subgroups of G by Theorem 7.1. Clearly Ble = eB1e, eB2 = eB2e. Then Bl(eMe)B2 = eMe. Corollary 4.8, GeMeG is closed in M.
By
Since GeG S GeMeG, we see that
so = MeM = GeG c GeMeG. Since eMe is regular, we see that every element of S0 is regular. This proves the theorem.
We now proceed as in the author [73] to treat the case when M does not have a zero.
Theorem 7.4. Let M be a connected monoid with group of units G, e a minimal idempotent of M. Then the following conditions are equivalent
(i) M is regular. (ii) rad G is completely regular. (iii) Ge is a reductive group.
Proof. (i)
(ii).
Let a E rad G. Now axa = a for some x E M. By Propostion
6.3, x E if for some Borel subgroup B of G. Then rad G C B. So a E if. Then
a is a regular element of if. By Corollaries 3.18, 4.12, a A e for some e E E($}.
73
By Remark 1.3 (iii), e e rad G, a ' e in rad G. (ii) z (iii). Since e lies in the kernel of M, we see by Theorem 6.30
that w(e) = 1, e e rad G. Hence (rad G)e is a completely regular monoid with zero e.
By Theorem 5.11, (rad G)e is a torus. Now e e T for some maximal torus T of
G.
Let B be a Borel subgroup of G with T c B. Now rad Ge c B1 for some
Borel subgroup B1 of G, rad Ge a CG(e). By Lemma 6.28, u-1Blu = B for some u e CG(e).
Hence
rad Ge c T raduG.
rad Ge c B for all B E 2 (T).
By Remark 4.39 (ii),
So (rad Ge)u c (raduG c Ge)c c (rad G)e.
But (rad G)e is a
torus. Hence rad Ge is a torus and Ge is a reductive group.
(iii) * (i). By Theorem 7.3, Me is regular. Let a E M. Since e lies
in the kernel of M, a e. By Corollary 6.13, a e GMeG.
So a is regular. This
proves the theorem. We will need the following result of [78] in Chapter 11.
Proposition 7.5. Let M be a connected regular monoid with zero and group of units G.
Let T be a maximal torus of G, r = {el,...,ek} c E(T), h = el V e2 V ... V ek.
Then CG(r) c CG(h), CG (I') c CG (h) and CG (r-) CC G(h). Proof.
First we show that CG(I') c CG(h).
reductive group CG(I').
sublattice of E(T).
Let To denote the radical of the
Then by Corollary 6.31, r c E(T0) and E(To) is a
Hence h e E(T0).
Since To lies in the center of CG(I'),
CG(F) c CG(h).
We now prove by induction on e = e1V ... V ek 1, f = ek.
show that CG(e,f) c CG (h).
that h A h'.
I II"I
>_ 2,
that CG(T) c CG(h).
Let
Then CG(e1....,ek 1) c CG(e), h = e V f. It suffices to
Let G' = CG(e,f), M' = G'.
We wish to show that h = h'.
Let h' a E(M') such
Let G" = CG,(e,h), M" = G".
Clearly T c G". Since e <_ h, we see by Corollary 6.18, that h' a M".
Let
74
G"' = CrG (f,h)' M"' = G"'. Clearly T c G"'. Since f <_ h, we see that h' e M"'. Now G't 1 9 CG(e,f) n CG(e,f) = CG(e,f) c CG(h). It follows that h = h'. Now let x e G'.
Then by Theorem 5.9, there exists hl,h2 e E(M') such
that h . hl -V h2 A x-1hx. Then h ,5Q h2, h A xh2x-1. By the above, hI = h = xh2x-1.
So h
h2 = x 1hx.
Hence hx = hxh.
Thus G' c CG (h).
Similarly
CG(I') c CG(h). The following result is from the author [77].
Proposition 7.6. Let M be a connected regular monoid with zero and groups of units G.
Let e,f a E(M).
Then there exists maximal tori TI,T2 of G, ei,fi e E(T ),
i = 1,2 such that e 5E e1, e -V e2, f A fI, f .2' f2.
Proof. By Theorem 7.1, CG(e), CG(f), CG (e), CG (f) are all parabolic subgroups of G.
By Corollary 4.36, there exist maximal tori
T19 CG (e) o CG (f), T2 c CG(e) n CG(f).
CG(e), y1 E CG(f), y2 e CG(f)
such that
T1, T2
of
G
such that
By Corollary 6.10, there exists xl,x2 e
el = x11exl, fI = yllfyl E E(TI),
e2 = x21ex2, f2 = y21fy2 E E(T2). The result follows.
Definition 7.7. Let S be a regular semigroup. Then, (i)
S/A is the partially ordered set (eS Ie E E(S)) under inclusion,
S/-V is the partially ordered set (Se le E E(S)) under inclusion. (ii)
If e,f a E(S), then the sandwich g, sand(e,f) _ [h h e fSe, h is
an inverse of of}.
Remark 7.8. (i) The sandwich set sand (e,f) plays a crucial role in the theory of regular semigroups and biordered sets. See [52], Chapter 12. For regular semigroups, sand(e,f) is always non-empty and a rectangular band (i.e., an idempotent semigroup
satisfying the identity xyz = xz).
75
The partially ordered sets
(ii)
S/9, S/ -V are the starting point of
Grillet's theory of regular semigroups [27], [28].
The following result is due to the author [77].
Theorem 7.9. Let M be a connected regular monoid with group of units G. Then (i) M/5¢, M/-V are relatively complemented, complete lattices with all maximal chains having the same finite length ht(M). (ii) If e,f E E(M), then sand(e,f) _ (hI h = e'f'
E(T), T a maximal torus of G, e I e', f 52 f')
for some e',f' E
h h = e' f' = f' e' for some
e',f' E E(M) with e I e', f 5Q f'). (iii) If e E E(M), a E M, then a µ e if and only if a E C(H) where H
is the X -class of e.
Proof.
That all maximal chains in MIA have length ht(M) follows from
(i)
Theorem 6.20. Since M is the maximum element of WA, it suffices to show that M/9B
is a A-semilattice which is relatively complemented. Let eM, fM E M/5Q
By Proposition 7.6, there exists a maximal torus T of G,
where e,f E E(M).
el,f1 E E(T) such that eM = eIM, fM = f1M. Then, clearly eM o fM = eM A fM =
elfIM.
Thus
eM fM
M/5e
is a complete lattice.
Next let
e,f,h E E(M) such that
Without loss of generality, we may assume that e > f > h.
By
Corollary 6.10, there exists a maximal torus T of G such that e,f,h E E(T).
But
E(T)
hM.
is relatively complemented by Theorem 6.20.
e > f' > h such that ff' = h. Then eM f'M
So there exists
f' E E(T),
hM, fM n f'M = fM A f'M = hM.
Hence MIA is relatively complemented. (ii)
e1,f1 E E(T)
By Proposition 7.6, there exists a maximal torus
such that
CG (f) = CG(f1).
e ,I e1, f
.
f1.
T
of G,
By Corollary 6.19, CG (e) = CG(e1),
So by Theorem 6.16, G1 = GG(e) n CG(f) is a connected group and
-
Me = f1Mel 9 G1. Clearly h = e1f1 = f1e1 E sand(e,f) and xlhx E sand(e,f) for
76
all x e G1. Let h' a sand(e,f). Then h' a Me and by Remark 7.8 (i), h , h' in fMe.
{x1 hx
By Corollary 6.8,
h' = x 1hx for some x e G1.
Thus
sand(e,f) _
x e G1 } and the result follows.
(iii)
By Remark 1.21 (ii), we are reduced to the case when
Clearly a µ 1 if and only if a e CG(E(M)).
e = 1.
By Theorem 7.1, CG(E(M)) is the
intersection of all Borel subgroups of G, which is C(G) by Theorem 4.32.
Remark 7.10. (i) Let J/9 = {eMle a E(J)}.
Then M/yE =
u
Let
J/.R.
Je Ye(M)
e e E(J).
Then E(J) = (xexl I x E G). Let x1,x2 e G,
f1M = f2M if and only if x1CG(e) = x2CG(e).
with G/CG(e).
xiexi 1, i = 1,2.
Then
Thus J/.9 is in 1-1 correspondence
But CG (e) is a parabolic subgroup of G by Theorem 7.1. Hence
G/CG(e) is a projective variety by Theorem 4.7. Thus M/.
can be thought of as a
lattice ordered projective variety.
(ii) Theorem 7.9 (iii) shows that M is a central extension of M/µ and hence susceptible to a cohomological approach of [42], [43].
M/µ will be studied
further in Chapters 14, 15.
In the local study of connected regular monoids, Green's relations represent the first step. The ultimate goal is the study of conjugacy classes. For An (K), Green's relations amount to row and column equivalence. Studying conjugacy classes yields the Jordan canonical forms. For algebraic monoids, the problem is much
more difficult. Some advances have been recently made by the author [84]. We will describe the main results of [84] without proofs.
Let M be a connected regular monoid with zero and group of units G.
Let T be a maximal torus of G, W = NG(T)/T. Let e e E(T), a = nT e W. Let
Me,a = eCG(e9 9 e < (Y >)a
Let V=CG(e9I6e ),Y={ge V ge=e). Let
77
H= II Y0,Ge,a=V/H Oe For
x e V, let x* = nxln 1 E V.
Clearly H* = H.
anti-automorphism * on Ge,a . Define 4 = 4e,a' Me,a
So
*
induces an
- Ge,a as follows.
If
a= evn E Me a, v E V, then
4(a) = vH E Ge,a
The main theorem of [84] is the following.
Theorem 7.11. Every element of M is conjugate to an element of some Me,a . a,b E Me a .
Let
Then a is conjugate to b in M if and only if there exists x E Ge,a 4(b).
such that
If F = (l,e0 10 e < a >) , then Me a c FNG(F) and FNG(F) is an inverse semigroup. Thus
Corollary 7.12. M is a union of its inverse submonoids.
Remark 7.13. That An(K) is a union of its inverse submonoids was noted by Schein [1041.
Let e E E(l), a E W.
Then for k e
, Mesa = 0 if and only if
(ea)k = 0. For ,Aln(K), the groups Ge,a are all trivial for nilpotent ea.
Example 7.14. Let Go = (A ® (Al)t I A E SL(3,K)), G = K*Go, M = KGo S = M\G. Then
E(S) = (e®fIe2=e,f2=fE .f1.3(K),eft=fte=0).
Let
78
In particular
010 ®[001 000EE(M) f- [000]
e=[001000000 ]0[000000 01
0
(Me
0
00 001
Also if a =
1 01
a)2 = 0.
®
1
01
1 01
0
001
e W(G), then e6 = f and
(e(Y)2
= 0.
So
The group Ge a can be seen to be the one dimensional torus with
being given by: x -i xl. Thus by Theorem 7.11, the number of conjugacy classes of nilpotent elements of M is infinite. However if C denotes the center of G, then the number of conjugacy classes of nilpotent elements in M/C is finite.
Example 7.15. Suppose char K * 2, n e r x r matrix
1F, n >_ 2.
For r E 71+, let Jr denote the
Let Go consist of all A E SL(2n + 1, K) such that
.
1
lA = Atl 0 02n Thus [34; Section 7.2],
l 1 L 0 02nJ L
Go
is the special orthogonal group of type B.
Let
G = K*Go, M = KG0* Then
000 e =
0 In0 000
0001
f= 0 0 0
E(M).
0 0 In
±1 0
If a =
E W(G), then ea = f and (ea)2 = 0. Thus Me a = 0. It can be
0 J2n seen that Ge,a = PGL(n,K) with the anti-automorphism * on Ge,a given by: A -+ JnAtJn
Thus the conjugacy classes of elements of Me,a in M are in one to one
correspondence with the congruence classes of PGL(n,K).
Two elements A1,A2 E
79
PGL(n,K) are congruent if CtA1C=A2 for some C e PGL(n,K). Contrast the situation for monoids with that for reductive groups, where the number of conjugacy classes of unipotent elements is always finite [44].
80
8 DIAGONAL MONOIDS
By a connected diagonal monoid, we mean a connected monoid T such
that the group of units T is torus. The importance of knowing more about E(T) is clear from the previous chapters. The author [66] was pleasantly surprised to discover that E('I`) is just the face lattice of a rational polytope. However, it should be noted
that the connection between diagonal monoids and rational polyhedral cones is already
clear from the theory of torus embeddings [37], [58]. Our needs are more elementary and we will follow [66].
If X C IRn, then cony X will denote the convex hull of X.
The
convex hull of a finite set in IR' is called a polytope. If P is a polytope, F c P, then
F is a face of P, if for all a,b E P, a E (0,1), as + (1 - a)b E F if and only if a,b E F.
of P.
The set of all faces of P forms a finite lattice ,3(P), called the face lattice The singleton faces are called vertices . Then P is the convex hull of its
vertices. If the vertices are all rational, then P is said to be a rational polytope. We refer to [30] for details.
Let M be a connected monoid with zero 0, group of units G.
By a
character of M we mean a homomorphism x: M -4 (K, ) such that x(1) = 1, x(0) = 0.
Let $ (M) denote the commutative semigroup of all characters on M.
x e , '(M), let x E £ (G) denote the restriction of x tp G.
If
Since M = fs, the
homomorphism x -, z is injective. Thus
$(M) c ,%(G)
(6)
81
By Remark 4.18 (i),
is linearly independent in the vector space of all
,% (G)
K-valued functions on G. By (6), we have,
£ (M) is linearly independent in the vector space of all K-valued
Lemma 8.1.
functions on M.
(K) such that 0 E T in
Let T be a closed connected subgroup of .0 (K).
If
a = diag (al,...,an)
E
.t3n(K),
f
E
K[xl,...,xn],
then let
f(a) = f(al,...,an) E K.
Corollary 8.2. Let a E On(K), a o T. Then there exist monomials f,g E K[x1,...,xn] such that f(b) = g(b) for all b E T, f(a) * g(a).
Proof.
There exists p E K[xl,...,xn] such that p(b) = 0 for all b r= T, p(a) # 0.
Since 0 E T, the constant term of p is zero. So 0 * p = monomials in K[xl,...,xn], m >_ 2.
m
aipi'ai c- K, p1....
i=1 Chose p such that m is minimal. Now each pi,
restricted to T is in £ (T). Hence by Lemma 8.1, pi = pj on T for some i * j. We claim that pi(a) * pj(a).
For suppose pi(a) = pj(a).
obtained by replacing the monomial pi by pj in p.
Let q E K[x1,...,xn] be
Then q(b) = 0 for all b E T
and q(a) = p(a) * 0. This contradicts the minimality of m. Hence pi(a) * pj(a).
Definition 8.3. Let ul,...,un E IRm, u = (ul,...,un), P = conv(ul,...,un). If F E 3(P), then let eF = diag(e1,...,en) where ei = 1 if ui E F, ei = 0 if ui a F.
Let
E(u) _ (eF IF E 3(P)) = 3(P).
If u = al l
... amm E K*.
E 71m, a = diag(al,...,am) E 2n*,1(K), then we let u(a) _
82
Let
Theorem 8.4.
u1.... ,un
a
u
71m,
=
(ul,...,un).
Let
To =
Let T = K*To, T the closure in '0 (K).
{diag(ul(a),...,un(a)) I a a O*1(a)) c . (K). Then E(T) = E(u).
Pr
f.
Let F e
,
Suppose eF 0 E(T).
(P).
Then by Corollary 8.2, there exist
monomials f,g a K[xl,...,xn] such that f(b) = g(b) for all b e T, f(eF) * g(eF). Now f(eF), g(eF) E (0, 1). So we may assume that
f(eF) = 1, g(eF) = 0
i
1
J
(7)
J
Let f = x11 ... xnn, g = x11 ... xnn. Thus il,...,in,
i= i 1 + a i= f(a
...
+ in, j = j 1
1) = g(a
+ ...
+ in
E
71+.
Now for all
ae
1) = ak. Since K is infinite, i = j. Now for all a e .0* (K)
in un(a)
u1(a)11 ... un(a)ln = u1(a) 1 ...
Since K is infinite, we see that n
n
lkuk = k=1
Jkuk
k=1
Let ik = ik'i, jk = jk/j. Since i = j, n
n
n
ikuk k=1
are non-negative integers,
jkuk k=1
n
E ik = E Jk = I. k=1
k=1
K,
83
Now let eF = diag(el,...,en). n E ikuk E F.
Then by (7), ik = 0 for any uk ¢ F.
Hence
n
Also by (7), jP * 0 for at least one uP a F. Hence
k=1
jkuk i F. This k=1
contradiction shows that eF E fi
Now let e = diag(el,...,en) E E(T).
Let
S2 = (k ek = 1 ), F =
conv(uk Ik E Q). We claim that F E Y(P), e = eF. Let 0 r c (1,...,n). Suppose that there exist et E O+(P E I) such that 1 ekuk E F, E eP = 1. kE I'
It suffices to show
PEI'
that then r c Q. N o w there exists 0 * 0' c S2, Sp E IR+ (p E S2') such that
E EP 1 = E Spup, E eP = E 8P pEe' PE PEI' PEI' Q'
(8)
Since ul,...,un E 71m, we see by (8), that eP, Sp(PE F, p e S2') represent a solution to a suitable homogeneous system of linear equations with integer coefficients. But the
solution space has a rational basis. Thus we can find iPip E 7L+ (t E F,p E S2') such that
E ipj =I jPup, I it = I IP pE92' PEI' PEI PEW
Thus
i
II (au,(a)) P = H
for all
aE
.0*n(K),
aE
b = diag(bl,...,bn) E T,
1
H b PP=H
PEI'
(aup(a)) P
pES2'
PEI'
bP.
pES2' P
K.
Since
T = KTo,
we see that for all
84
Since e = diag(el,...,en) E T,
11 eBr 11
fEJT
e =1
PE 92' p
Thus F c 0, proving the theorem. The situation in Theorem 8.4 arises as follows. Let G c GL(n,K) be a 0
reductive group, T0 a maximal torus of Go, To g
(K). Let T = K*To, M = KG
01
G = K*Go Then by Theorem 7.3, M is a connected regular monoid with zero. So
by Proposition 6.1, Corollary 6.10, M = GE(T)G, E(M) = u x1E(T)x.
Thus our
XEG
interest in E(T). Theorem 4.19.
Let dim To = m.
Then To = .Om(K), %(.
So by Remark 4.18 (ii), there exist
ul,...,un E
(K)) __ 71m by 71m
such that
To = {diag(ul(a),...,un(a)) I a E . m(K)}. By Theorem 8.4, E(T) = E(u1,...,un).
Example 8.5. Let Go= (A ®(Al ) t I A E SL(3,K)), M = KGo c * (K). We wish to
compute E(M). Now To = (A ® Al I A E 24(K), det A = 1) is a maximal torus of
Go Let T = K*To, G = K*Go Clearly
To = { diag(a,b,1/ab) ®diag(1/a, l/b,ab) I a,b E K*) = (diag(l,a/b,a2b,b/a,1,ab2,1/a2b,1) a,b E K*).
Correspondingly let ul = (0,0), u2 = (1,-1), u3 = (2,1), u4 = (4, 1), u5 = (0,0),
u6 = (1,2), u7 = (2,-1), u8 = (4,-2), u9 = (0,0). u = (ul,...,u9).
Let
P = conv(ul,...,u9),
Then P is a hexagon with vertices (2,1), (1,2), (-1,1), (-2,4),
(4,-2), (1, 1). Let e1 = diag(1,0,0), e2 = diag(0,1,0), e3 = diag(0,0,1), fl = e2 + e3,
f2 = el + e3, f3 = el + e2.
It is easily verified that E(T) = E(u) _ (0,1) u
{ei ®ej I i,j = 1,2,3, i * j) u (e® ®fIi = 1,2,3) u {fi ® eiI i = 1,2,3).
idempotent in M is of the form A 4 hA 0 Ath'(A4)t where
Now an
h 0 h' E E(T),
85
It follows that Ye(M) = (G,J1,J2,Jo,0) with J1 > Jo, J2 > Jo
A E SL(3,K). E(J1)
Also
{e®fI e2=e,f2=fe J3(K),p(e)= 1,p(f) =2,eft=fte=0),
E(J2) _ (e of l2 = e, f2 = f e
#3(K), p(e) = 2, p(f) = 1, eft = fte = 01,
E(Jo) = (e of le2 = e, f2 = f E #3(K), eft = fte = 0).
Example 8.6. Let Go = (A ® (Al)t I A E SL(3,K) }, M = KGo c *6(K).
Now
(K), det A = 1) is a maximal torus of Go Let T = K*To,
To = (A ® A-1 I A E
G = K*Go Clearly
To= (diag(a,b,l/ab,I/a,l/b,ab) I a,b E K*)
Correspondingly let ul = (1,0), u2 = (0,1), u3 = u6 = (1,1).
u4 = (-1,0), u5 = (0,-1),
Then P is a hexagon with
Let P = conv(u1,.... u6), u = (u1,...,u6).
vertices u1,...,u6.
Let el = diag(1,0,0), e2 = diag(0,1,0), e3 = diag(0,0,1).
Then
E(T) = E(u) = (0,1) u (ei ® ej I i j = 1,2,3, i * j) u {ei ®O I i = 1,2,31 u (0 ® ei Ii 1,2,3). Any idempotent of M is of the form A1hA ®Ath'(A1)t where h ®h' E E(T).
Hence E(M)\(1 ) = {e ® f I e2 = e, f2 = f E 3(K), p(e) <_ 1, p(f) 5 1).
We now wish to prove the converse of Theorem 8.4.
So let T be a
closed connected subgroup of . (K) such that 0 E T in .0 (K). Let dim T = m.
Then by Theorem 4.19, there exists {diag(v1(a),...,vn(a)) I a E an*1(K)}.
0 e conv(vl,...,vn).
(vi/v
E(u).
Zm
such that
T=
Since 0 E 7, we see that 0 e < vl,...,vn >. Thus
vi > 0, i = 1,...,n.
Since vl,...,vn E 71m we can then
We can find v' E Qm such that v
vi) -v'. Then ui
u = (ul,...,un).
E
So it is easy to see [27; Theorem 2.21] that there exists v e IRm
such that the inner product v choose v E 71m.
vl,...,Vn
is a rational point in ( x a IRmI v
v' = 1.
Let ui =
x = 0) = IRm1.
Let
It is routinely verified as in the proof of Theorem 8.4 that E(T) _
Hence we have,
86
such that 0 e T in
Theorem 8.7. Let T be a closed connected subgroup of 'on (K).
Let dim T = m.
Then there exists ul...,un E 71m1 such that with
u = (u1,...,un), P = conv(ul,...,un), E(T) = E(u) = 51(P).
Remark 8.8. If dim T = 2, then P is of course a line and hence I E(T) = 4. We will need the next result in Chapter 10.
Corollary 8.9. Let To c 2(K) be a torus of dimension 1 containing non-scalar matrices. Let T = K*To, E(T)
l,e,f,O).
Let Tl = {t (& t1 It E To) c .d 2(K), n
T2 = K*T1. Then E(T2) = (l,e ®f,f ®e,0).
Pr
f. There exist
nE
71, not all equal such that To = (diag(a...,an) ll t I a E K*).
Let a = max{il,...,iny = min{il,...,in).
Let e = diag(el,...,en) where er = 1 if
it = a, 0 otherwise. Let f = diag(f1,...,fn) where fr = 1 if it = y, 0 otherwise. By Theorem 8.4, E(T) = { l,e,f,0}.
Now T1 = (diag(.... ak....) ( a e K* }, a - y =
max(ik -i1Ik,B = 1,...,n), y -a = min{ik -i,Ik,f = 1,...,n). It follows from Theorem 8.4 that E(T2) = {l,e ®f,f (9 e,0}.
Let the height function ht be as in Definition 6.21.
Corolla. Let T be a connected diagonal monoid with zero 0. Let e,e' E E(T) such that
ht(e) = ht(e') = p > 0.
Then there exists
e = eo, el,...,ek = e',
f ,...,fk E E(T) such that ht(ei) = p, ht(fi) = p -1, ei > i' ei 1 > f . i = 1,...,k. Pr f. Let dim T = m.
Proceeding by induction on m we are easily reduced to the
case when ht(e) = ht(e') = m -1. By Theorem 8.7, E(T) - 5(P) for some polytope P.
Then E(T) is anti-isomorphic to 3(P*) where P* is the dual polytope of P
(see [30; Section 3.4]). Hence the elements of E(T). of height m - 1 correspond to
87
the vertices of P*.
By a result of Balinski [2] (or see [30; Section 11.3]), any two
vertices of a polytope are connected by a sequence of edges. The result follows. By Corollaries 6.10, 6.22, 8.10, we have,
Corollary 8.11. Let M be a connected monoid, J, J' a 2e (M), ht(J) = ht(J') = p > 0. Then there exists J = Jo, Jl,... Jk = J', Jl,...,Jk a ? (M) such that ht(Ji) = p, *
*
ht(Ji)=p-1,Ji>J,Ji
1>Ji,i=1,...,k.
Corollary 8.12. Let T be a connected diagonal monoid with zero. Let 0 * r c E(T) such that (F,5) is a relatively complemented lattice with all maximal chains having
length equal to dim T. Then I' = E(T).
Pr f.
Let
dim T = m.
X = (e a E(I) I ht(e) = m - 1).
We prove the result by induction on
m.
Let
If e e X n IT, then r, = eT o IT satisfies the
hypothesis with respect to eT. Hence E(eT) = I" c IT. Thus it suffices to show that X L; IT.
Suppose not. Then by Corollary 8.10, there exists el e X n I', e2 a X'I' such
that with f = ele2, ht(el) = ht(e2) = m - 1, ht(f) = m - 2. Since
IT
By the above f e IT.
is relatively complemented there exists ei a IT n X such that elei = f.
Hence l,el,ei,e2,f are distinct elements of E(Tf).
By Theorem 6.20, dim Tf = 2.
By Remark 8.8, 1 E(Tf) = 4. This contradiction completes the proof.
Corollary 8.13. Let T be a connected diagonal monoid with zero.
Let a be an
automorphism of E(T) fixing a maximal chain IT in E(T). Then (Y = 1.
Proof. We prove by induction on dim T = m.
Let fI = {e e E(T)Iea = e},
X = (e e E(T) I ht(e) = m -1), IT n X = (f). Then r' = r\( 1) is a maximal chain of E(fT).
Since
fa = f, a restricts to an automorphism of E(fT).
hypothesis, E(fT) c Q.
Suppose S2 * E(T).
By induction
Then by Corollary 8.10, there exist
88
el,e2 e X such that E(e1T) c S2, E(e2T) 4 f2, ht(e3) = m - 2 where e3 = ele2. Then e3 a f . By Remark 8.8, E(Te3) = (1,e1,e2,e3). Since ei = e1 and e3 = e3, we see that e2 = e2.
So e2 a 0. Extend (e2 > e3) to a maximal chain A of
Then A\(e2) S; E(e3T) c E(e1T) c Q.
E(e2T).
Hence A c S2.
By induction
hypothesis, E(eT) c S2, a contradiction. Hence S2 = E(T), completing the proof.
Let Tl c .*1(K), T2 L;
(K) be tori such that 0 E T1 in 9m(K),
0 E T2 in 9ln(K). Let 4): Tl - T2 be a homomorphism such that 4)(1) = 1, 0(0) = 0.
If x e W (T2), let 0: ,%(T)
$(x) a £ (Tl) be given by $(x)(a) = x($(a)).
is a homomorphism.
W(T,)
Next, let
Thus
£ (Tl) be a
ty: £ (T2)
homomorphism. Let Xl,...,Xn e %(T2) denote the n projections of T2 into K. Let jr: Tl - T2 be given by iy(a) = (tVr(X1)(a),ty(X2)(a),...,yv(Xn)(a)). Then V is a n
n
homomorphism, it(1) = 1, yr(0) = 0.
Let T be a connected diagonal monoid with zero, dim T = n.
So by
(6), Lemma 8.1, £(T) is a finitely generated subsemigroup of (71n,+), 0 0 $(T). So ,W(T)
is totally cancellative, i.e. it is cancellative and for all a,b a ,$T), k e 71+,
ka = kb implies a = b.
Conversely let ul,...,um a 71n, 0 E < u1,...,un>.
(diag(ul(a),...,um(a))Ja e 2n (K)) c .9) (K). ,W(T) = < ul,...,un>.
Then
0eT
in
Let T =
'on (K)
and
Grillet [26] has shown that any finitely generated, totally
cancellative, commutative semigroup can be embedded in a free commutative semigroup. Thus we have the following result (see [66] for further details).
Theorem 8.14. There is a contravariant equivalence between the category of connected diagonal monoids with zero and the category of finitely generated, totally cancellative, commutative semigroups without idempotents.
89
9 CROSS-SECTION LATTICES
In this chapter we introduce the central notion of cross-section lattices, due to the author [72], [74], [76].
Definition 9.1.
Let M be a connected monoid with group of units G.
Then
A c E(M) is a weak cross-section lattice if (i)
I A n J I = 1 for all Je 2e (M).
(ii) If e,f a A, then Je z Jf implies e >_ f.
If further A c E(T) for some maximal torus T of G, then A is a cross-section lattice (which is necessarily a sublattice of E(T)).
Example 9.2. In 321(K), A = {[0 1,'
1 0 [0 01' [0 11' [0 01 } is a weak cross-section
lattice which is not a cross-section lattice. The following result is due to the author [72].
Theorem 9.3. Let M be a connected monoid with zero and group of units G.
be a maximal torus of G.
Let T
Then A = (e e E(T) for all f e E(M), e k f implies
f e $} is a cross-section lattice of M for any Borel subgroup B of G containing T.
90
Proof. Let e,f e A such that
e/f
in M.
Then by Theorem 5.9, there exist
e',f ' e E(M) such that e h e' .f f' ,R f. Since e,f e A , e',f ' e if Thus e / f in B. Since B is solvable, we see by Corollary 6.32 that e = f. Hence for all J e
?1(M).
Jl,J2 a U(M), J1 >_ J2. e1 >_ e2.
For
e e E(M), let Xe = (f a E(M)je .`!G f).
IJnA
I
Now let
By Proposition 6.25, there exist ei a Ji o E(T) such that
Extend (el,e2) to a maximal chain t of E(T).
CG(I') is a connected solvable subgroup of G.
B' of G such that CG(F) c B'.
By Proposition 6.24,
Thus there exists a Borel subgroup
By Corollary 6.18, Xe , Xe
Now 2
1
T c B n B'. By Theorem 4.22, there exists u e NG(T) such that u 1B'u = B. Let
f.1 = u 1eiu, i = 1,2.
Then fl,f2 a E(T), f1 Z f2.
Also Xf , Xf L; $ 1
Thus
2
fl,f2 e A. Since i e Ji, i = 1,2, the proof is complete.
Corollary 9.4. Let M be a connected monoid with group of units G and let T be a maximal torus of G.
Then for any chain I' in E(T), there exists a cross-section
lattice A of M such that r c A c E(T).
Proof. Let 11 denote the zero of E(T). Then E(T) = E(T,0). By Theorem 6.16, T1 is a maximal torus of
cross-section lattice of M.
Now F c t'
By Proposition 6.27, any cross-section lattice of M1 is a Hence we may assume that it = 0 is the zero of M.
for some maximal chain
t'
of E(T).
By Proposition 6.24,
CG(F') c B for some Borel subgroup B of G. By Corollary 6.18 and Theorem 9.3,
t' c A for some cross-section lattice A c E(T). The following result is due to the author [70].
Theorem 9.5. Let M be a connected monoid with zero 0 and group of units G.
Then G is solvable if and only if U(M) is relatively complemented.
91
Proof. Let T be a maximal torus of G. Proposition 6.25, Corollary 6.32,
If G is solvable, then by Theorem 6.20,
2l(M) = E(T)
is relatively complemented.
So
assume conversely that 2l(M) is relatively complemented. We prove by induction on
dim M that G is solvable. Let e e E(T), e * 0,1.
By Proposition 6.27, 2l(Me),
24(eMe) are relatively complemented. So by the induction hypothesis, Me, eMe have solvable groups of units. Thus by Proposition 6.27, Corollary 6.32, we have,
if e,fl,f2 E E(T), e:* 0,1, fI "f2 and if (9)
either e >_ i, J= 1,2 or if a <_ i, i=1,2, then f1 = f2
By Corollary 6.32, it suffices to show that w(J) = 1 for all J E 2l (M).
Then choose a maximal J0 E 21 (M) with w(Jo) > 1. e
0
Suppose not.
Let eo,eo E Jo o E(T),
* eo Let J' E 2l(M) such that J' covers Jo Then w(J) = 1. Let J' n E(T) =
(TI).
covers
By Proposition 6.25, 112: eo, n >_ eo So by (9), it = 1. Thus J' = G and G
Jo
So by Corollary 6.22,
1 covers ell e2.
(e E E(T) ht(e) =p-1), Y = (f E E(T) ht(f) =p - 2). Corollary 8.10, there exist
Let
dim T = p, X =
Then eo, eo E X.
So by
el.... ek+l E X, fl,...,fk E Y such that eo = e1, eo = ek+1'
ei > f , ei+1 > Ii i = 1,...,k. By Corollary 9.4, there exists a cross-section lattice A of M such that eo = el E A c E(T). Suppose p > 2. Then Y * (0). We will obtain a contradiction. There exists fl' E A such that f1 $ fl', el > fl'. fI =f i1 E A.
By (9),
So there exists e2 E A such that e2 .,it e2, e2 > fl. So again by (9),
e2 = e2 E A. Continuing, we find that eo E A.
have a contradiction. Hence p = 2.
Since eo E A , eo $ eo, eo * eo, we
Then by Remark 8.8, E(T) _ (1, eo, eo, 0).
Hence 2l(M) = (G,J,O) is not relatively complemented. This proves the theorem. The following result is from the author [74].
92
Proposition 9.6. Let M be a connected regular monoid and let A1,A2 be two weak cross-section lattices of M.
If A 1 o A2 contains a maximal chain of E(M), then
A 1 = A2.
Proof.
Let G denote the group of units of M and let ht(M) = 0.
(M) -+ Ai be the bijections given by 4i(J) a J, i = 1,2.
dim M that
We prove by induction on
1 = 2. Let t = [I > e > ...) be a maximal chain of E(M) contained
in A 1 n A2. {h a Ai I f >_ h}.
fMf.
Let i: Yl
Let J denote the
$-class of e.
If f e Ai, then let Ai(f) =
Then by Proposition 6.27, Ai(f) is a weak cross-section lattice of
Now eMe is regular and t' = P(1) is a maximal chain in E(eMe) with
t' c Al(e) n A2(e). Hence Al(e) = A2(e). Suppose there exists h E A1\A2. Then there exists e' e E(M), ht(e') = p -1, e' >_ h.
Let J'
denote the
$-class of e'.
Then J * Y. By Corollary 8.11, there exist distinct J = Jo' Jl'"''Jt+l = J' E ?1(M), distinct JO,...,Jt a V(M) such that Jk > Jk' Jk+l > Jk' ht(Jk) = p - 1, ht(Jk) = p -2, k = 0,...,t.
Then
i(Jk) > $i(Jk)' i(Jk+l) > V Y, i = 1,2, k = 0,...,t.
Now
1(Jo) = 2(J0) = e. Let f = 41(J). Then f e Al(e) = A2(e). So f = 42(J*0). Extend (1 > e > f) to a maximal chain Ti of Al(e) =A 2 (e). Let el = Ol(JI)' ei = 42(Jl). Then el > f, ei > f. Since J0 * Jl, we see by Theorem 9.5 that Gf is solvable. By Theorem 7.4, Gf is reductive. Hence Gf is a torus. So by Remark 8.8, IE(Mf)I = 4. I72
Since l,e,el,ei,f E Mf, el = ei.
Let r2 = (I'1\{e)) u {el). Then
is a maximal chain of E(e1Me1) and t2 c A1(el) n A2(el).
hypothesis Al(el) = A2(el).
By the induction
Continuing this process, we see that e' E A2 and
Al(e') = A2(e'). Hence h e A1(e') c A2. This contradiction completes the proof.
Corollary 9.7. Let M be a connected regular monoid. Then every weak cross-section lattice of M is a cross-section lattice and any two cross-section lattices are conjugate.
93
Proof. Let G denote the group of units of M and let A be a weak cross-section lattice of M.
Let r be a maximal chain of A. Then r is a maximal chain of
E(M). By Corollary 6.10, r c T for some maximal torus T of G. So by Corollary 9.4,
IF c A' c T for some cross-section lattice A of M.
By Proposition 9.6,
A = A'. Now let A 1 be a cross-section lattice of M. There exists a maximal chain
rl of A 1
such that for all
e e r,
there exists
f e rl
e / f.
By
x-lrlx = r.
So
such that
Corollary 6.8, Theorem 6.16 (ii), there exists x e G such that
r c A o x 1Alx. Hence A= x-1Alx by Proposition 9.6.
Proposition 9.8. Let M be a connected regular monoid with zero 0 and group of units G.
Let B be a Borel subgroup of G, e e E($). Then the following conditions are
equivalent.
(i) B c CG (e).
(ii) For any f e E(M), e 52 f implies f e E($).
(iii) For any f e E($), e , f in B implies e 5B f.
Proof. For f e E(M), let Xf = (h a E(M) If 51 h). Let T be a maximal torus of B
with e e T. Let IF be a maximal chain in E(T) with e e r. Then B 1 = CG (r) is a Borel subgroup of G by Theorem 7.1. Clearly B 1 c CG (e).
(i) * (ii). B, B1 are Borel subgroups of CG (e) and the width of e in CG (e) is 1.
So by Lemma 6.28, there exists u e CG(e) such that u-1Blu = B.
Then Xe = u-1Xeu c (i)
(iii). Let f e E(B), e / f in B. By Corollary 6.8, xex 1 = f for
some x e B C CG (e). (iii) e 5E xex
So xe = exe and of = f. Thus e R f by Theorem 1.4. (i).
Let x E B.
Then xex' a E($), e s xex' in
IT.
So
Hence exexl = xex' and xe = exe. Thus B C CG (e). (ii) 4 (i). Since T c B o Bl, we see by Theorem 4.22 that x'Blx = B
for some x e NG(T). Let f = x-lex a E(T). By assumption Xe c B. By Corollary
94
6.18,
Xe c $1.
Hence
Xf = u1Xeu c IT.
So e1 ex e c $, fl E Xf L; IT.
Hence
Since B is solvable, we see by Corollary 6.32 that e = f.
Hence
el,f1 E E(M) such that e , R e1
e / f in R.
By Theorem 5.9, there exists
f1 A L
x E CG(e) and B = x-1B Ix c CG (e). This completes the proof. Let M be a connected regular monoid with zero 0 and group of units
Then G is a reductive group. Let T be a maximal torus of G.
G.
As in
Definition 4.21, let £(T) denote the set of all Borel subgroups of G containing T.
If B E 2 (T), then let B E 2 (T) denote the opposite Borel subgroup of G relative
to T, i.e. B n B =T. Let ' (T) = (A I A c E(T) is a cross-section lattice of M).
If
Definition 9.9.
B E .AT),
{e E E(T) I ae = eae for all
4 (B)
then the cross-section lattice of
B,
4(B) =
a e B) and the opposite cross-section lattice of B,
e E E(T) I ea = eae for all a e B). If A E KT), then the Borel subgroup of
A, (3(A) = CG(A) and the opposite Borel sub roup of A, 3 (A) = CG(A).
The fundamental theorem of cross-section lattices due to the author [72], [74], [76] is:
Theorem 9.10. Let M be a connected regular monoid with zero and group of units G.
Let T be a maximal torus of G. Let 2 = 2 (T), 6 = 48(T). Then (i)
If A E le, then R (A) E 2 and (3(A) = R (A).
(ii) If B e (iii)
,
then 4(B), l (B) E Y3 and 4(B1= 4 (B).
(3 = 41 and P-=
1.
(iv) If Al, A2 e 4 then Ai = A2 for some r e W. Proof. Let B E 2 and let A = 4(B). By Theorem 9.3 and Proposition 9.8, A E
s9.
Let I' be a maximal chain in A. Then r is a maximal chain in E(M) and hence
95
by Theorem 7.1, CG (r) E , . B = Q(A) = CG (r).
A' _ (B1.
Clearly B c (3(A) = CG (A) c CG (I') E 2.
Hence B = 13(l;(B)).
Then as above,
Proposition 9.6, A = A'.
So
By Theorem 7.1, B = CG(I).
Let
Clearly F c A'.
By
A' E re, P -(A') = B
Thus 13(A) = 13(A1 and (B-) = 4(B).
A e f f, IF c A a maximal chain. Let B = CG(I) e
,fit
Next let
Clearly r c 4(B).
So by
Proposition 9.6, l(B) = A. By the above, Q(A) = (3(4(B)) = B. Similarly (3(A) E 2
and 4 ((3 (A)) = A. Finally let A1,A2 E W- Then 13(A1), 13(A2) E ,fit By Theorem 4.22, there exists a E W such that 13(A?) = (3(Al)a = 13(A2).
So Aa = A2. This
proves the theorem. The following result of Renner [97] will be needed in Chapter 11.
Corollary 9.11. Let M be a connected regular monoid with zero 0, group of units G.
Let T be a maximal torus of G. Suppose
I W(G)
2, ,$ (T) = (B, B-), U = Bu,
U = Bu Then for any e e E(T) with e g C(M), either e U = U e = (e) or else
Ue=eU-= (e). Proof.
Let A = Z;(B), A = 1;(B1.
Corollary 9.4, E(T)=AvA
By Theorem 9.10, V(T) = (A, A-).
By
Suppose eEA. Then U e = e U e, e U= e Ue.
Since e e C(M), I W(CG(e)) = 1 by Proposition 6.25. Thus CG(e) is a torus. So
by Theorem 6.16, the X -class H of e is a toms. We have a homomorphism 0: U H given by 4(x) = xe = exe.
Since U is unipotent, we see that Ue = (e).
Similarly e U-= (e). In the same way, e e A
implies e U = U -e = (e).
As another application of cross-section lattices, we prove the following result of the author [77].
Corollary 9.12. Let M be a connected regular monoid with zero 0 and group of units G.
Let e,e' E E(M) with ht(e) = ht(e') = p > 0.
Then there exist e = eo,
el,...,ek = e', fl,...,fk E E(M) such that ht(ei) = p, ht(fi) = p - 1, ei > i' ei-1 > f ,
96
i = 1,..,k.
Proof. By Corollary 8.10, the theorem is true when G is a torus. Thus by Corollary
6.10, we are reduced to the case when e $ e'.
By Theorem 5.9, we are then further
reduced to the case when e R e' or e -V e'. By symmetry assume e h e'. Let A be a cross-section lattice of M with e E A. Let B = CG (A), T = CG(A). Then B
is a Borel subgroup of G, T a maximal torus of G, T c B, e e E(T), e' e $ By Theorem 9.10, there exists a cross-section lattice A' c E(T) such that B = CG(A' ).
There exists u E A' such that e $ u in M. Then ht(e) = ht(u). So there exist e = uo, uI ...,uk = U, vl,...,vk E E(T) such that ht(ui) = p, ht(vi) = p - 1, ui > vi,
1 > vi, i = 1,...,k.
By Corollary 6.18, e, e', vl E CB(e,v1). There exists x E CB(e,v1) such that e' = x-1 ex. Let v' = x1v1x e E($). Then vl 9E vl, u1
e' vi.
Now v1, vi, u2 E CB(u1,v1).
There exists y E CB(u1,v1) such that
vi = y 1v1y. Let ul = y luly E E(B). Then u' >_ vi, u1 9E u'. Continuing, we find e' = uo, u...... uk = u', vi,...,vk e E($) such that ht(ui) = p, ht(vi) = p -1,
ui > vi' ui-1 > vi' ui 5 Q ul, vi ,5e vi, i = 1,...,k.,
u' = zuzl for some z E CB (u).
In particular,
u R u'
and
Since B c CG(u), we see that u = u'. This
proves the theorem.
Problem 9.13. Is Corollary 9.12 true without the assumption of regularity?
97
10 9 -STRUCTURE
Let M be a connected regular monoid with zero 0 and group of units G.
Fix a maximal torus T of G. As usual W = NG(T)/T is the Weyl group.
Definition 10.1. (i) 9 = X(M) = (E(T), <-, -) where for e,f a E, e - f if e $ f.
ht 6 = dim T = the length of any maximal chain in
Let
81.
(ii) A bijection a: e -a ea of 6 is an automorphism if e - e6 for all
e e 9, and for all e,f a 9, e
f if and only if ea 2 fa. Let W' = r (i) denote the
group of all automorphisms of (iii) YC' = YC'(M) = 9/-.
If A1,A2 e W', then define A 1 S A2 if
there exist ei E Ai, i = 1,2 such that e1 <- e2. By Proposition 6.25 we have,
Proposition 10.2. YC' is a finite lattice isomorphic to . V.
We refer to 9 as the ,'-structure of M, Yl' __ YC as the V-structure of M. These are finite combinational structures, with 9 determining
2t.
Example 10.3. Let Go = (A ® (A 1)tIA e SL(3,K)), M = KGo c .
(K).
Let
e1 = diag (1,0,0), e2 = diag (0,1,0), e3 = diag (0,0,1), fl = e2 + e3, f2 = el + e3,
f3 = el +e 2* Then by Example 8.5, tt(M), or more precisely the following diagram:
YC',
is represented by
98
el 0 fl, e2 ® f2, e3 0 f3I
®e3
Ifl 0 el,
f2 ®e2, f3
0
e1
e2, e2
0
e 1, e 1
0
e3, e3
0
e 1, e2
0
e3, e3
0
e2
0
Example 10.4. Let Go = (A ® (A1)t1 A E SL(3,K)], M = KGo c ,AG6(K).
Let
el = diag (1,0,0), e2 = diag (0,1,0), e3 = diag (0,0,1). Then by Example 8.6, X(M) is represented by the following diagram:
ffl
ffl
' e2 ®el' el ®e3'e3 ®el' e2 ®e3' e3
2 ®0, e3 ® 01
We refer to [72] for further diagrams of 9--structures.
ED
99
Definition 10.5.
If e e 9, then let ge = (f a c If >_ a), a tf = (f E 9If <_ a).
We
consider 9e, e 9 with 5, -- restricted to them.
If e E 9, H the 2G -class of e, then by Theorem 6.16, e T, Te are maximal tori of H, Ge, respectively. By Proposition 6.27, we have,
Proposition 10.6. If e E 9, then ov(Me) = Ne' '(eMe) = e X. The following result is due to the author [72].
Theorem 10.7. YY- W.
Proof. W acts on 9 faithfully by Theorem 7.1. In this way W c YY. We prove by
induction on dim M that W = YY. X = (f If e
' e - f).
So
e E 6 such that 1 covers
Let
e.
Let
I X = w(e). By Proposition 6.25, W and hence YY acts
transitively on X. By Theorem 6.16, Proposition 6.25,
W = w(e) I W(eMe) I
W(Me)
= w(e) I W(eMe) I
Let YYo= (ae Yea=e}. Then YYI = w(e) I Yo I
If a E YYo, let a E V (e 9) denote the restriction of a to e X.
1, then a fixes a maximal chain in 9 and hence
(Y -+ 6 is a homomorphism. If
by Corollary 8.13, a = 1.
r(e
Thus
l _ I W(eMe) . Hence
Clearly the map
I Yo 1
I YYJ <_
<_
IW.
I YY(e 8) 1.
By the induction hypothesis,
Thus W = YY, proving the theorem.
From now on we identify W with Y.
100
Definition 10.8. (i) If F c 9, V c W, then CV(r) = (a e V I ea = e for all e e F). (ii)
If e e ', a e CW(e), then let ea e W(eMe), ae a W(Me) denote
the restrictions of a to e 9, 9e, respectively.
Proposition 10.9. Let e e 9. Then (i) CW(e) = W(CG(e))
(ii) W(eMe) _ (ea I (;e CW(e) ) (ii') W(Me) = (ae I G e CW(e)) (iv)
If a E CW(e) with ea = ae = 1, then a = 1.
(v) CW(e) - W(eMe) x W(Me).
Proof. Since T Q CG(e), (i) is clear. Define 4): CW(e) -, W(eMe) as 4)(a) = ea,
yr: CW(e) - W(Me) as yr(a) = ae Then a,yf are homomorphisms. If ea = ae = 1, then a fixes a maximal chain in 9 and hence by Theorem 7.1, a = 1. By Theorem 6.16, I CW(e) I = I W(eMe) I
I W(Me) 1.
The result follows.
The following result is due to the author [721.
Theorem 10.10. Let a E W, a * 1. Then a is a reflection if and only if a fixes a
chain I' in 9 of length ht 9-1. In such a case CG(Ta) = CG(F)' Proof. Let m = dim T = ht 9. First suppose that a fixes a chain F in
m -1. Let Ta = (t a TIta = t)c. Then by Lemma 6.14, r c Ta. 6.20, m -1 <_ dim Ta < dim T.
see by Theorem 4.45 that
' of length
So by Theorem
So dim Ta = m -1. Since a E W(CG(T(y)), we
a is a reflection.
Let
To = rad CG(F).
Since
a E W(CG(n), To c Ta. But r c o by Corollary 6.31. So To = Ta by Theorem 6.20. Thus CG(F) = CG(Ta).
Conversely assume that
a
is a reflection.
By Theorem 4.45,
dim Ta = m -1. By Lemma 6.14, 0 e Ta. By Theorem 6.20, there exists a chain r
101
of E(Ta) of length m -1. This proves the theorem.
Let e e 9, a E W, a reflection such that ea = e.
Then by Lemma
6.14, e e Ta. By Theorem 6.20, there exists a chain r c Ta with e E F such that F has length dim T -1. Hence a fixes a maximal chain in either E(Te) or E(eT). Thus by Corollary 8.13, either ae = 1 or ea = 1. Thus we have shown,
Corollary 10.11. Let a E W be a reflection, e e 9 ea = e. Then either ea = 1, ae is a reflection in W(Me) or else ae = 1 and ea is a reflection in W(eMe). The following result is due to Renner [91].
Theorem 10.12.
Let
M, M'
be connected mQnoids with zero, M regular,
M -4 M' a homomorphism such that M' = 4>(M) and X1(0) = (0). Then (i) M' = 4>(M) is regular.
(ii) 0 is idempotent separating. (iii) 0: c(M) __ N(M').
Proof. Let G, G' be the groups of units of M, M', respectively. By Theorem 2.21, dim M = dim M'.
So 4>(G) = G'. By Theorem 7.3, M' is regular. Let T be a
maximal torus of G. 4>(e) = 4>(f), e # f.
First we show that
Then 4>(ef) = 4>(f) * 0.
4>
is
1 1 on E(T).
Suppose e > ef.
So let elf a E(T),
Then since E(T) is
relatively complemented, there exists h E E(T) such that e > h > 0, efh = 0. 0 = 4>(efh) = 4>(eh) = 4>(h), a contradiction. Thus e = ef.
Similarly f = ef.
e = f and 0 is 1-1 on E(T). Now let elf E E(T) such that 4>(ef) = 4>(f) and hence of = f. E(T) = E(4>(T)).
Thus E(T) _ 4>(E(T)) c E(4>(T)).
So M' = G'E(4>(T))G' = 4>(M).
4>(e) >_ 4>(f).
Then
Hence Then
By Corollary 8.12,
Now let elf E E(M), 4>(e) = (f).
By Lemma 7.6, there exists a maximal torus Tl of G, ell f1 E E(T1) such that e 5B ell f 51 f1.
Then 4>(el) 5B 4>(fl) and hence 4>(e1) = 4>(fl).
So el = fl. Thus
102
Similarly e .f f. Hence e = f. Thus 0 is idempotent separating. Finally
e ,9G f.
let e,f a E(T) such that 4)(e) / 4)(f).
Then by Proposition 1.19, e / f. It follows
that X(M) = X(M').
The situation in Theorem 10.12 arises quite often. For example, we have the following from [76; Proposition 2.2].
Corollary 10.13. Let M be a connected regular monoid with zero 0 and group of units
G such that rad G is one dimensional. homomorphism 4): M
Then there exists an idempotent separating
flp(K) such that rad 4)(G) consists of scalar matrices.
Proof. We may assume that M is a closed submonoid of End(V), for some finite
dimensional vector space V over K. We can further assume that M contains the
zero of
End(V).
Let
To = rad G.
Then
dim To = 1, 0 e
o.
Let
xl,...,xm E £(To) denote the weights of To Let Vi = (v e V Itv = xi(t)v for all t e To)
(0), i = 1,...,m. Then V = V1 ®... ® Vm Since To lies in the center of
M, MVi c Vi, i = 1,...,m. Now W(To is a cyclic group. Since 0 e n.
oT ,
there exist
n.
nl,...,nm a Z+ such that x;' = X , i j = 1,...,m. Let Vi = Vi ®... ®Vi denote the ne-fold tensor product of V. If a E M, then a acts on Vi as
a(vl (9 v2 0...) = av 1 0 av2 ®...
Thus a acts on V' = V' ® ... V. Let 4)(a) denote the corresponding element of End(V'). We therefore have a homomorphism 4): M -* End(V').
(0).
Clearly 4 1(0) _
By Theorem 10.12, 0 is idempotent separating. If t e To,
then clearly
n.
4)(t) = xit(t)
1
is a scalar.
Let M be a connected regular monoid with zero 0, group of units G, T
a maximal torus of G. Let s' = (T), 2 = 2 (1), 0, $3 , t,
be as in Definition
9.9. Note that if A c 9, then A E i' if and only if: (i) for all e e 9, there exists a
103
unique f E A such that e - f, (ii) for any e,f E A, a E W, eG ? f implies a >_ f. Thus the family s' is determined by (3 (A) E
cX
Now fix A E 5' and let B = (3(A), B =
,
Definition 10.14. Let e9 = Y (A) = o° (B) = (a I 1 * G E W, B U BaB is a group) denote the set of simple reflections relative to B.
If
Icc
let WI = < I >,
PI = BWIB, Y (B) = (PI I I c es') the set of parabolic subgroups of G containing B. See Theorem 4.51.
We will show in Corollary 10.21 that
QY
is determined within the
system 9.
Lemma 10.15. Let I c e, P = BWIB, t c A.
Then the following conditions are
equivalent.
(i) P = CG (I') (ii)
I = C1(I)
(iii) W(P) = CW(l
Proof. (i) 4 (iii). Let e E F.
Then the width of e - in F is 1.
a E W(P).
So W(P) c CW(I').
CG(>-, = P.
So or W(P) (iii)
Now let a = xT E CW(f').
So e6 = e for all Then X E CG(r) c
(ii). I = eson WI = eson CW(1) = C,(I').
(ii) z (i). Let PI = CG (F) E Y By the above, W(PI) = CW(I),
W(PI)oeso=CW(I'ne3o=C,(I')I. So P1=BWIB=P. Lemma 10.16. Let e E A, eA = (f E Ale >_ f), Ae = (f E A e <_ f } . Then Ae E tRTe)
in Me, eA E V(eT) in eMe.
If a E C ,,(e), then either Ge = 1 and ea E e ° (eA)
or else ea = 1 and ae E eY(Ae).
104
Pr f.
That Ae e Kre ), eA E meT) follows from Proposition 6.27. By Corollary
7.2, Be is a Borel subgroup of Ge and Be = eBe is a Borel subgroup of the
a-class H of e.
Clearly Be c (i(eA), Be c (3(Ae). Thus Be = (i(eA), Be = ROY'
Let a E C,(e), P = B u B a B. Then W(P) = (1,(Y}, Pe = ePe. Since the width of e in P is 1, we see by Theorem 6.16 and Proposition 6.25 that
By Proposition 10.9, W(Pe) _ (l,(;e}, W(Pe)
I W(Pe) I ' I W(Pe) 1.
2=
I W(P) I =
l,ea}.
Since
Be C Pe and Be c Pe, we are done.
Lemma 10.17. Let P e ,P (B) be a maximal parabolic subgroup, To = T o rad P.
Then for any e e E(T0), either p c CG(e) or else P c CG(e).
Pr f. Let A c 4) be the base relative to B. Then -A is the base relative to B See Remark 4.48.
If a E 4), let Ua, aa: a -, -a be as in Definitions 4.43, 4.46.
Then eY = eY(B) = eY(B1 _ (aala e A). By Theorem 4.51, P = BWIB for some
Ice . Then f = I u ((F7) for some YEA. Let e E E(o). Then ea = e for all a r WI. Now P1 = CG(e) is a parabolic subgroup of G by Theorem 7.1. Clearly
T c P1, WI c W(Pl). U--a c P1.
By Remark 4.48, for each a E A, either Ua c PI or
If a * y, then since as E W(Pl) and aaUaaa1 = U., we see that
both Ua, U- c P1. Suppose UY c P1. Let B0 denote the group generated by T, Ua(a E A). Then B is a connected group by Proposition 4.2. Clearly 0 T c B0 c P1 n B.
Now B0 c B 1 for some Borel subgroup B 1 c Pl.
Since
Ua c B 1 for all a e A, we see that each a e A is positive with respect to B
Thus B = B1 c Pl.
Since WI c W(PI), P = BWIB c Pl.
U-Y c Pl. Then Ua c PI for all a E -A.
e E 4(B) = lemma.
(B).
Hence B
CG (e).
I
Next suppose that
So as above, B -c PI = CG(e). Thus
So P = BWIB c CG (e).
This proves the
105
Lemma 10.18. Suppose M c An(K) and that rad G consists of scalar matrices. Let
P E Y(B), P * G. Then there exists e e E(T), e * 0,1 such that p CG (e).
Proof. Let G' = (G,G).
Then G' is semisimple, G = K*G'. Without loss of
generality, we may assume that P is a maximal parabolic subgroup of G, T c(K).
Now T = K*T' for some maximal torus T' of G', B = K*B' for some Borel subgroup B' of G', P = K*P' for some maximal parabolic subgroup P' of G',
T' c B' c P'. Let To = T' o rad P'. Then dim To = 1. Let E(KTo) _ (1,e,f,0). Suppose P bf * fbf.
CG (e), P
Then there exist a,b a P' such that ae * eae,
CG (f).
Consider the representation 4: G' -4 GL(n2,K) given by 4)(g) = g ® (g1)t
Clearly the kernel of 4 is a finite group of scalar matrices. Hence 4(P') is a maximal parabolic subgroup of 4(G') containing 4(To) in its radical. Let M = K4(G').
Then K*4(P') is a maximal parabolic subgroup of K*4(G'), containing
K*4(To) in its radical. By Corollary 8.9, E(K4(To)) = 11, e 0 f, f 0 e, 0).
e1 =e of Now al = a
®(a1)t, b1 = b ®(b 1)t a O(P').
Let
Suppose alel = elatel.
Then
0 * ae 0 (a1)tf = eae 0 f(a1)tf
So
ae = aeae for some a e K*.
Hence ae = eae,
a contradiction. Thus
alel # elatel. Next suppose that elbl l = elbllei. Then 0 # eb1 ®fbt =
eb-1
e ®fbtf
So fbt = afbtf for some a E K*. Hence fbt = Of and bf = fbf, a contradiction.
Thus ate * elatel, ebll
elbllel.
CG (e) or P c CG(f), proving the lemma.
This contradicts Lemma 10.17. Thus P c
106
Corollary 10.19. Let P E
P # G. Then P c CG (e) for some e e E(T), e # 0,1.
Proof. Without loss of generality, we may assume that P is maximal. Suppose the result is false. Then M has no central idempotents other than 0,1. By Corollary 6.31,
rad G is one dimensional. By Corollary 6.13, there exists an idempotent separating homomorphism 4>: M -4 4n(K) such that the radical of G' = 4>(G) consists of scalar
matrices. By Theorem 10.12, M' = 4>(M) _ 4>(M) and 0: X(M) _ X(M').
Since
0-1(0) = (0) we see by Theorem 2.21 that 4 1(1) is a finite group. Hence 4>(P) is
a maximal parabolic subgroup of G. By Lemma 10.18, there exists e E E(T), e # 0,1 such that 4>(P) c CG, (4>(e)) # G. 4>(e) E 4>(A).
Since 4>(P) is maximal, 4>(P) = CG, (4>(e)).
Then
So by Lemma 10.15, P = CG (e). This proves the result. We are now in a position to prove the following result of the author [76].
Theorem 10.20. Let M be a connected regular monoid with zero and group of units G.
Let T be a maximal torus of G, A E d(T), B = Q(A). Let P be a parabolic
subgroup of G containing
B.
Then there exists a chain
r c A such that
P = CG (r).
Proof. We prove by induction on dim M.
We may assume that P # G.
By
Corollary 10.19, there exists e E A such that P c CG(e). There exists I c e such that
P = BWIB.
Hence
ea = e for all a E
I.
By Lemma 10.16, Ae =
(f e A e <_ f) e 5(Te) in Me and eA = (f E Ale? f) E V(eT) in eMe. Let I1 =
(aEII ae=1),12={aEIlea=1},Ii=(ea10EIl),12={aeIGEI2}. Then by Lemma 10.16, 1=11 A21 I' c ey'(eA) = eYl, q c e5'(Ae) = oI2. So by Lemma 10.15
and the induction hypothesis applied to eMe and Me, we see that there exist chains
rl c eA, F2 c Ae such that I' = Ce1(r1), 12 , = C'y2(r2). Let r = IF1 u r2 u (e). Then IF is a chain, I c CQY (r).
Now let a E COY (r). Then ea = e.
10.16, either ae = 1, ea a Ii or else ea = 1 and ae E I.
By Lemma
By Proposition 10.9,
107
Hence I = C
a E I1 U I2 = I.
OYI(I').
By Lemma 10.15, p = CG(F), proving the
theorem.
Corollary 10.21. Let M be a connected regular monoid with zero, A c '(M), a
cross-section lattice. Then cl (A) _ (a E W I a # 1, a fixes a chain of length ht 9 -
1 in A}.
Proof. Let B = (3(A), ht 65 = m.
Let a E W, a * 1. Suppose first that a fixes a
chain t of length m - 1 in A. Then ea = e for all e e t. Let P=C G(F) Then by Lemma 10.15, a e W(P). Since
B.
m -1, we see by Proposition 6.25
I IF
that I W(P) I = 2. Thus P = B u B a B and a E 1(B). Conversely let a e 1(B),
P = B u BaB. such that
IFI
By Theorem 10.20, p = CG(F) for some chain IF c A.
Choose r
is maximal. If there exists e e t, e * 0,1, then proceeding inductively
and using Lemmas 10.15, 10.16, we see that P = G, I W(G) I = 2.
I t = m - 1.
If t = (0,1),
then
By Theorem 10.10, a fixes a chain t' in 6 of length in -1.
Then w(e) = 1 for all
e e t'. Hence I"
consists of central idempotents by
Corollary 6.31. So F' c A.
Let A be a cross-section lattice in a connected regular monoid M with zero.
Definition 10.22. (i) If I c eY (A),
let
AI = (e e Area = e for all a e I),
YCI=(JeleeA,). Let it=(Y(1jIcQY(A)). (ii)
If Ye it, let AY= (e e A1Je e
Remark 10.23. (i) Since all cross-section lattices are conjugate, it is independent of A.
108
(ii) Let I c Y (A), B = 3(A), P = BW1B.
P = CG (I') for some chain I' c A.
Theorem 10.20,
Then P c CG(A1). Then
I' c A1.
By
Thus
P = CG(A1).
If I, I' c c (A), then "I r %,
(iii)
only if
= 11Iul'
Also I c I' if and
111.9 111.
(iv)
is a family of sublattices of
Fl
With respect to inclusion
1'1
11,
closed under intersection.
is a finite Boolean lattice. The importance of 1'1 will be
clear in Chapter 14. (v)
example, if
11
In some cases,
F1
can be determined directly from V.
is a linear chain, then
1'1
is the family of all subchains of
For 11
containing G,(0).
Corollary 10.24.
If
A
is a cross-section lattice,
B = (3(A),
then the map
: Fl - 9(B) given by O (p) = CG (A,) is an inclusion reversing bijection.
109
11 RENNERS DECOMPOSITION AND RELATED FINITE SEMIGROUPS
In this chapter, we wish to generalize the Bruhat decomposition for groups (Theorem 4.35) to monoids. Some initial ideas in this direction were given by
Grigor'ev [25]. However, the correct and complete solution is due to Renner [97]. Throughout this chapter let M be a connected regular monoid with zero 0 and group
of units G. Fix a maximal torus T of G and let
91 = (e a E(T) I ht(e) = 1)
denote the set of minimal non-zero 'diagonal idempotents' of M. We start with:
Proposition 11.1. NG(T) = E(T)NG(T) is a unit regular inverse monoid with group of
units NG(T) and idempotent set E(T). Moreover the fundamental congruence µ on
NG(T) is given by: a µ b if and only if b E Ta.
Proof.
NG(T)/T
Since
NG(T)E(7) = E(T)NG(T).
is a finite group,
NG(T) = NG(T)T = NG(T)TE(T) =
Define 0 on NG(T) as: a 0 b if b E Ta. Then 0 is
an idempotent separating congruence on NG(T).
Hence 0 c .t, Let a,b E NG(T),
a µ b. Then a = ex, b = ey for some e E E(T), x,y E NG(T).
e µ exy1. c=
exy-1
Hence e A exy1.
E H.
Let z = xy 1. Then
Let H denote the A -class of
Then for all f E E(eT), a >_ f and so fc µ f µ cf.
e in M,
Hence fc = cf.
110
Thus C E CH(E(eT)) = eT by Theorem 7.1. So exy-1 = et for some t e T and
a=ex=teye Th. Thus .t=d. Definition 11.2.
The Renner monoid of M, Ren(M) is the fundamental inverse
monoid, NG(T)/µ = NG(T)/T.
Remark 11.3. (i) Ren(M) is a finite, unit regular inverse monoid with group of units
W = NG(T)/T and idempotent set E(T). Thus Ren(M) is the subsemigroup of the
Munn semigroup of E(T) generated by
E(T-)
and W.
Thus Ren(M) can be
determined from the 9--structure 81(M) and conversely. (ii)
If M = An(K), then Ren(M) consists of the row and column
monomial matrices in M with 0-1 entries and is thus isomorphic to the symmetric inverse semigroup of degree n. The following result is due to Renner [97].
Theorem 11.4. Let B be a Borel subgroup of G containing T. Then M is the disjoint union of BwB(w a Ren(M)).
Proof. Let M' denote the union of BwB(w E Ren(M)). Then BM'B = M'. Let
B- denote the opposite Borel subgroup of G relative to T, U = Bu, U = Bu. We follow the notation of Definitions 4.43, 4.46, Remark 4.48. Let a E A(B), a = aa.
So W(Ga) = { 1,a). Let w e Ren(M). Then w = eO for some e e E(T), 0 e W.
By Corollary 4.53, we can find a closed connected subgroup Y of U such that U = YUa = UaY, a Y = Ya. So a BwB = a UTwB = a UwTB = a UwB. U = YUa,
aBwB = GYUawB = YGUawB
Since
111
By Corollary 9.11, either Uae = (e) or U
= (e) or e e C(Ga). First suppose
Ue = ( e). Then aBwB = YaUae9B = YaeOB c M'. Next suppose e e C(Ga). By Remark 4.48, either Ua c eBe 1 or else U-a c eBe 1. First let Ua c eBe 1. Then UawB = eUaOBe18 = eeBele = eOB. So
aBwB = YaUawB = YaeOB c M'
Next let U-a c eBe 1.
By Theorem 4.35 applied to Ga, we see that Ga =
UST U UST a U-T. eeBele = eeB. Thus
Now U
TwB = UTeOB = eU ThBe1e =
UTwB = eeB So GawB c wB u U-,TowB = wB u UawB = wB u a UawB.
Since GGa =
Ga, we see that GawB c awB u UawB. Hence
aBwB = YaUawB c YGawB c YawB u YUawB c BawB u BwB c M'
So again oBwB c M'. Finally suppose that Use = (e). Then
U-,TwB = U TeeB = U eTOB = e9B Hence we conclude as above that aBwB c M'.
Thus in all cases aBwB c M'.
Since W is generated by eY(B), we see that WBwB c M'. Hence WM' c M'.
Since G = BWB by Theorem 4.35, GM' c M'. Similarly M'G c M'.
Hence
M = GE(T)G c GM'G c M'. Thus M is the union of BwB(w a Ren(M)).
We now proceed to show that this union
is disjoint.
So let
a,a' E NG(T) such that BaB = Ba'B. There exists e,f E E(T), x,y E NG(T) such
112
that a = ex, a' = fy. Then fy = b1exb2 for some bl,b2 E B. Let J denote the -class of e.
Now a=
By Corollaries 3.20, 4.12, eb1e a J.
xx-1
ex, a' =
yy-1
ey.
Hence of E J.
So the above argument shows that
Thus e = f. x-1
ex = y1ey.
Thus yx 1 E CG(e) n NG(T). By Corollary 7.2, eBe, exBx-1e are Borel subgroups of the
o' -class H of
1 -1 eyx = yx e,
eyx
l
e.
Now eyxl E NH(eT), eyxl = blexb2x 1.
= ebiexb2x-xe
E
Since
(eBe)(exBxle)
By Theorem 4.35 applied to H, eyx-l E eT. Hence ey a exT and a' E aT. This proves the theorem.
Corollary
.
9 is the disjoint union of BwB, w e Ren(M) o IT.
Remark 11.6. The finite monoid Ren(M) n B is only briefly encountered in [97]. It
is a very interesting monoid, worthy of further study.
It is a semilattice of nil
semigroups.
The following is noted in the author [78].
Corollary 11.7. Let J E 2'(M), a E J.
Then there exist e,f E E(T) o J such that
eaf E J.
Proof.
By Theorem 11.4, there exist e e E(T), a E W such that a e BeaB =
Ba((; lea)B. By Corollary 3.20, eBe c J.
Thus ea a J. Similarly of E J, where
f = oleo. By Theorem 1.4, eaf a J. As another application of Theorem 11.4, we obtain some maximal completely simple subsemigroups of M.
This generalizes an informal conjecture to
the author by Francis Pastijn concerning Jn' (K).
113
Let B be a Borel subgroup of G, Jo E
Corollary 11.8.
?l($).
Then
Jo =
(a e M I a Mx for some x e J0) is a maximal completely simple subsemigroup of M.
Pr
f.
Let Eo = E(J0).
By Corollary 3.20, Jo is a completely simple semigroup.
Let a,b E Jo Then a M e, b M e' for some e,e' e Eo Now ee' M f for some
f E E. Let J denote the $-class of e in M. Now ee', b = e'b a J. So by Theorem 1.4, eb = ee' b E J. Now a = ae E J. Thus by Theorem 1.4, ab = aeb E J.
Since ea = a, be' = b, we see by Theorem 1.4 that ab , e A f , ab Y e' . f f.
Hence ab M f and ab E Jo Thus j0 is a subsemigroup of M. Clearly j0 is completely regular and E(J0) = Eo Thus Jo is a completely simple semigroup.
Now let S be a completely simple subsemigroup of M containing Jo
We need to show that E(S) = Eo So let f e E(S). Let T be a maximal torus of B. By Theorem 11.4, there exist bl,b2 E B, e E E(T), U E NG(T) such that f = bleub2.
Now fE0 u Eo c S c J, where J is the $-class of e in M. Let el e Eo o 1'. Then ble1bil a E0, bleIbllf a J. Thus e e a J. Hence el = e. Similarly since I
eu =
uu-1
eu, we see that e1 = u1eu. Hence e e E0, eu = ue. Now f2 = f. So
e = eub2ble and eu-1 = eb2ble a eBe. By Corollary 7.2, eBe is a Borel subgroup of the
' -class H of e and eu 1 e NH(eT).
It follows that eu4 = e.
So
f = beb2 E Jo Hence f e E0, completing the proof.
Problem 11.9. Let J E V(M). Is every maximal closed, irreducible subsemigroup of J of the form described in Proposition 11.8?
Remark 11.10.
Renner [97] obtains several interesting consequences of his
decomposition. We mention a few. (i)
if e E A = 4(B) = l (B-}, then B eB is open and dense in the
,-class J = GeG of e. This generalizes to
J, the big cell (Chevalley's) B-B of
114
G.
(ii) The row echelon form for ,,fln(K) can be generalized to M.
If
x E Ren(M), then it is in 'row-echelon form' if Bx c xB.
For % (K), NG(T) is the semigroup of all row and column monomial matrices. We now present analogues of the row monomial semigroup and the column
monomial semigroup (due to the author [78]) for any connected regular monoid M with zero.
Recall that in this chapter
111
denotes the set of minimal non-zero
idempotents of T.
Definition 11.11. Let M be a connected regular monoid with zero. (i) -Tr = Xr(M) = (a e M I for all e e 91, there exists f e 91 such
that ea = eaf). (ii) %'f = T1(M) = (a e M for all e e 91, there exists f e 91 such
that ae = fae}.
(iii) V = ;=(M)
(a e M IaT c Ta).
(iv) TP =
(a e M I Ta c aT ).
The following result is due to the author [78].
Theorem 11.12. Let M be a connected regular monoid with zero 0. (i)
Then
'r, T_, X't T' are unit regular submonoids of M with group of
units NG(T). (ii) `b'e c --r, -''Q c T,. Xr n -Te = -Ti n Xe = NG(T).
(iii) For all a e
'r,
there exists f e EM such that a .' f in
`,fir
For all a e gl , there exists f e E(T) such that a A f in Fe
(iv) For all
X^ n eMe.
e e E(T), 2r(eMe) = Xr(M) n eMe, gf(eMe) _
115
(v) If e e 91, then for all e' E E(M), e . e' implies e' a Xr', e R
e' implies e' a Xf. Proof. We first prove (iv). Let H denote A -class of e.
torus of H by Theorem 6.16. Let
i31
So eT is a maximal
be the set of minimal elements of E(eT)\(O).
So eg l = Xj u (0). Let a E Xr(eMe), el a 91. If e1 ! g1, then e 1 a = e1ea = 0 =
elael. If e1 E g', then ela = elafl for some f1 a xi c sl. So a e Tr(M) n eMe.
Next let a E Xr(M) n eMe. Let el a 9j. Then ela = elaf for some f e 91.
Let f1 = of e X1 u (0). Then eIa = e1af1 and a e
xr(eMe).
Similarly Tf(eMe) _
9k(M) n eMe.
It is easily checked that
that NG(T) = E(T)NG(T) c .r n T' . Suppose ea * 0.
Tr,
-Tt VQ are submonoids of M and
XI,
Then ea a XT
Let a E Xr, e e 91.
Then ea A e in M. So by Proposition 6.1, ea = ex for some
x e G. Now ex T c Tex. So exTx 1 c Te. Thus exTx1 = exTxle. Hence T, xTx
are maximal tori of CG(e).
So there exists u e CG(e)
such that T =
uxTx-lu-l. Thus ux e NG(T). Hence f = xluleux a 91. Now eaf = exf = exx-1 u-1
eux = eu1eux =
eu-1
Thus a E -Tr
ux = ex = ea.
Hence Ti c `6'r
Similarly W' c Ft We show next that NG(T) = G o Tr.
So let x e G n 'r
Let
6,(x) _ {e a 911 ex = exe), a(x) = 1911 - tfl(x) I. We prove by induction on a(x) that x e NG(T).
Suppose first that a(x) = 0.
7.5, x e CG(E(T)). x e CG(?fl(x)).
Then ,l(x) = 91 and by Proposition
By Theorem 9.10, x e T c Tr
Let e e
61\(T1(x).
So assume a(x) > 0. Then
Then ex = exf for some f e X,. So fle in
M' = CG('1(x)). Then e / f in M'. By Proposition 6.25, there exists u e NG(T) n CG(91(x)) such that
Hence 91(xu 1) x e NG(T).
u-1
eu = f. So ex = exf = exu-1eu.
Thus exu 1 = exu1e.
91(x) u (e) and a(xu1) < a(x). Thus xu1 a NG(T), whereby
Thus NG(T) = G n Xr = G n
r'.
Similarly NG(T) = G n X, =
116
Gn.T'. Next we prove the following claim:
e E E(T), a E Xr, eae c
a in M imply ea = eae
(10)
We may assume that e * 0. Let I' = (f E 91 If S e) = (fl,...,fk).
Since E(eT) is
relatively complemented, we see that e = fl V...V fk. Let a E Tr such that eae X e
in M. Let r(a) = (f a r I fa = faf), y(a) = I I' - Ir'(a)
.
We prove by induction on
Then fa = faf for all f E r. Let
y(a) that ea = eae. First suppose that y(a) = 0.
f E r. Then faf = fa = f(ea) f(eae) fe = f. So faf M f for all f e I'. By Corollary 6.17, a e CG(r ). By Proposition 7.5, CG(I') c CG(e). y(a) > 0. 6.17,
As above, faf 'f for all
a e M'.
f E T(a).
By Remark 1.3, eae N e
Thus ea = eae. So assume
Let M' = CG(I'(a)). By Corollary
in M' and faf M f in M' for all
f E r(a). There exists i E r\r'(a). There exists fi E 9l such that is = iafi. Now
in M', fi Iis = i(ea) fi(eae) i ce = Suppose fi
i.
Thus
fi. / fi in M' and iae = ieae
0.
I. Then f.e = 0 and iae = iafie = 0, a contradiction. So fi E r'.
By Proposition 6.25, 6.27, there exists y E NG(T) o CG(F(a)) n CG(e) such that
yfiy 1 = ff.
So is = iaf i = iayiy
all f E r(a).
Thus flay = iayfi. Clearly fay = fayf for
Also eaye = eaey = (eae)(ey) ad e.
Clearly
r(ay) 2 r(a) u (1i).
Hence y(ay) < y(a) and eay = eaye. Since y E CG(e), ea = eae. This proves claim (10).
Now let a e Tr, J the $-class of a in M. By Corollary 11.7, there exist e,f E J n E(T
such that fae a J.
By Proposition 6.25, f = x-1 ex for some
x E NG(T). Then exae E J. By Theorem 1.4, exae M e. By (10), exa = exae. By Theorem 1.4, xa .. exa = exae.
A-class of e.
So xae = xa and ae = a.
Let H denote the
Then exa = exae E H n Tr(M) = H n $r(eMe) = NH(eT).
By
such that eu = ue = exa.
So
Theorem 6.16, there exists u E CG(e) n NG(T)
117
e = eu-lxa.
Let eo = u 1xa. u-1
e Xeo Since
Since ae = a, we see that eo a E(.Td.
x e NOM, we see that .r is unit regular. Since i c
Clearly and
has NG(T) as the group of units, we see that T' is also unit regular. Similarly T'f,
Xf are the unit regular. Since a .fe in 2r, we see that (iii) is also valid. We have thus proved (i), (iii).
Since NG(T) is the group of units of Xr n Xr, we see that Xr o Xl Let h e E(.Tr n Xe). Then by (iii), there exist e,f a E(1) such
is also unit regular.
that e X h 51
f.
By Theorem 1.4, e / of = fe. It follows that e = f = h. Hence
E(.Tr o T = E(T). Thus Xr n 91= Xr' n 9e = NG(T). This proves (ii). Let e e l. Then by Proposition 6.2, dim eMe = 1. Let V = rad G,
H the a' -class of e in M. Then eV c H, 0 e V. Thus eV = H.
Now let
e' E E(M), e d e'. Let H' denote the '-class of e' in M. By Corollary 6.19, CG (e) =
CG(e').
Thus e' a -'r.
Thus T c CG (e').
So e'T = e'Te' c H' = e'V = Ve' c Te'.
This proves (vi).
Finally we prove (v) by induction on dim M.
Thus we are reduced to
the case when e e j1. Let V = rad G. Then We c T, 0 e V, dim Te = dim T -1.
Thus T = VTe Let a E Tr' (Me). Then aTe c Tea. Since T = VTe, aT c Ta and a E X'r(M). This proves the theorem.
Renner [91; Theorem 4.4.4] has shown that
CM(T) = T.
We now
generalize this result.
Corollary 11.13. Let M be a connected regular monoid with zero. Then CM(E(T)) _
1 = CM(Xl)
Proof. By Theorem 11.12, T c CM(E(T )) c CM(6l) c Xr o T, n CM(6l) = NG(T) n CM(--l) = T.
118
Example 11.14. Let M = .Aln(K).
Then
'fir = Xr' consists of all row monomial
matrices and Xf = -Ti consists of all column monomial matrices.
Example 11.15. Let M = (A e B I A,B E ,A12(Kr)r, deft A = det B), rG the group of r
units of M. Then XrVG = Y X Y where Y = { I y 0J I x,y a K} u { 18 yJ I x,y a K). However it is easy to see that 11 0] ® 11 01 E Xr. Thus Tr * Xr`. Remark 11.16. It follows easily from Theorem 11.12 that 'e(Zr)
;i/.e
&(M),
Tf /5E = E(T).
Remark 11,17. The inverse semigroup NG(T) acts on E(T) on the right as follows:
for e e E(T), a E NG(T), let e a = a- ea. This action extends to -Tr as follows: for e E E(T), a E Tr, let e
a = f where f e E(T) is such that ea 90 f in Xr. The
action of NGCr) on E(T) gives rise to the finite inverse monoid Ren(M). the action of -Tr on E(T) gives rise to a finite fundamental regular monoid
Similarly cr.
We
will obtain this monoid in the next theorem in a slightly different way.
If X is a set then we let .0,7(X) denote the regular semigroup of all partial transformations on X and J(X) the inverse semigroup of all partial one to one transformations on X.
The action is always on the right. See [33; Section 1.4].
The following result is from the author [78].
Theorem 11.18. Let M be a connected regular monoid with zero 0. For a E
-Tr,
define jt(a) E .O 9(91) as follows: if e,f E 91, then e n(a) = f if ea = eaf * 0, en(a) is undefined if ea = 0. Then (i) n: Tr -4 Y 9(X1) is a well defined homomorphism which is one
to one on E. Moreover zt(ta) = x(a) for all t E T.
119
(ii) The kernel it* of it is the largest congruence on -Tr contained
(iii) n* restricted to NG(T) is the fundamental congruence on NG(T) and hence
n(NG(T)) __ Ren(M).
(iv) Pr = tc(Z'I,) is a finite fundamental unit regular monoid having no
non-trivial congruences contained in
..
(v) P ; (r Pr
ff.
Since flf2 = 0 for all fl, f1 E Xl with f1 # f2, it is well defined. Let
(i)
a,b a M, e e 91. First suppose that eab # 0. Then ea # 0, there exists f e 91 such
that ea = eaf.
fb # 0.
So en(a) = f. If fb = 0, then eab = eafb = 0, a contradiction. So
Let ftc(b) = f' E 91.
So
fb = fbf'.
Thus etc(a)rt(b) = f'.
Also,
eab = eafb = eafbf' = eabf' and en(ab) = f'. Next suppose that 7t(a) n(b) is defined
on e and en(a) = f, ftc(b) = f'. So ea = eaf, fb = fbf', ea # 0, fb # 0. So fb A f
in M. Now eab = eafb = eafbf' = eabf'. If eab = 0, then eafb = 0, implying eaf = 0.
Hence ea = eaf = 0, a contradiction. Thus eab # 0 and en(ab) = f'.
It
follows that n(ab) = n(a)n(b). For e E E(T), set 91(e) = {e' E 911 e' <_ a}. Clearly 91(e) is the domain of n(e).
Moreover, since E(T) is relatively complemented, e
is the join of '1(e) in E(T).
Thus the map, e -4 91(e) is injective. Hence it is
1-1 on E(T). That n(ta) = n(a) for a E 9 r, t E T, is obvious. (ii) By (i) and Theorem 11.12, ic* c -V.
Let S be a congruence on
-'r such that S c ... Let a,b E Tr such that a S b. Let e e 91. Then ea ' eb. So ea # 0 if and only if eb # 0. So suppose ea # 0, eb # 0. e n(b) = f'.
Then ea = eaf # 0, eb = ebf' # 0.
Let a n(a) = f,
So in M, f -V ea -V eb
V.
Hence f = f' and n(a) = n(b). Thus 8 c n*. (iii) follows from (i) and Proposition 11.1. (iv) follows from (ii). So we need only prove (v). Now Y and Yr n ,x(91) have the same group of units W. It
is also clear that
r n J(91).
So it suffices to show that E(J n A91)) c
120
E(J). Let h e E(Yr) such that n(h) e
7(tfl). By Theorem 11.12, there exists e E
E(T) such that e ,I h. Then n(e), it(h) e 9(91), it(e) ,I n(h). Since 7(91) is an inverse semigroup, n(h) = n(e) e # This proves the theorem.
Remark 11.19. If M = An (K), then Pr = ,9
Remark 11.20. Let
n(2'_).
Ren(M) n J= 9(91).
Pr' has W as the group of units and
Then
hence is unit regular. Example 11.15 shows that in general Pr * natural congruence which can be defined on 11.18, 0 c it*.
For ,AGn(K), 0 = n*.
There is a
Pr'.
fir: a 0 b if b E Ta.
By Theorem
In general TI O need not even be a finite
semigroup. To see this let M = KG0 c ,M9(K), -where Go = {A 0 (A 1)t A E SL(3,K)1.
Let X
1100
OaO
100
OyO
10 0 0 0 1 0
a, y e K, a + y = -1
.
By Example 8.5,
X c E(M). It is routinely verified that no two elements of X are 0-related.
Problem 11.21. Are the semigroups jr weakly inverse in the sense of Srinivasan [109]?
121
12 BIORDERED SETS
Let S be a regular semigroup. Then what structure does the system E(S)
possess?
The correct answer is provided by the following definition due to
Nambooripad [51], [52].
Definition 12.1. Let E be a set with a partial binary operation. For e, f e E define f
(<1)-1
Suppose that the following conditions and their right-left duals hold for e,f,g a E. (i) <_r, :51 are quasi-orders on E and of is defined if and only if
e<_rf or e<_ff or f
f
(iii) f:5 e, g
r (iv) g
For e,f E E, let fE _ (g E El g
if eg
(h a fEe I g { h for all
g E fEe).
Assume that the following condition and its
right-left dual holds for all e,f,g E E: (v) f
Then E is called a i r r
§e,t.
E is a regular biordered = if further,
(vi) sand(e,f) * 0 for all e,f E E.
122
Remark 12.2 (i) If e R f then e e sand(e,f). If e . f, then f e sand(e,f).
(ii) Let S be a semigroup, e,f a E(S). If of = f, define e o f = f, f o e = fe a E(S).
If fe = f, then define f o e = f, e o f = of a E(S). Then it is
routinely verified that E(S) = (E(S),o) is a biordered set. If S is regular, then E(S) is a regular biordered set.
(iii) The basic theorem of Nambooripad [51], [52] is that every regular biordered set is the biordered set of idempotents of some regular semigroup. Recently David Easdown has shown that any biordered set is the biordered set of idempotents of some semigroup [20]. (iv)
The axioms for a biordered set are quite complicated. However,
considering the general nature of semigroups, it is rather surprising that such a finite
axiomatization is even possible.
For connected regular monoids with zero, the
quasi-orders <_r, <_f determine the biordered set E (Corollary 12.5). This is not so in general [52; Example 1.1].
(v) In the synthetic construction of biordered sets, one often starts with two quasi-orders <_r, <_$
Then, whenever f
f <_e e, one must define ef.
This partial binary operation would then first have to be
shown to be well defined.
(vi) A regular biordered set with
Definition 12.3. Let E be a biordered set.
If e,f a E, then e 9' f if for all el EE with a Ael, there f1 e E with e1 90 f1 A f, and for all fl E E with fl A f, there exists (i)
exists
elEE
e
el
exists f1 e
e .2 e1 ,R f1.
. c A' , I c
el
e
e E with
f1
Clearly
A', .I'
are equivalence relations on E
21.
(ii) E is reduced if A= A', 1= -'.
and
123
(iii) E is locally reduced if eEe is reduced for all e E E.
(iv) A semigroup S is reduced if E(S) is reduced.
S
is locally
reduced if E(S) is locally reduced, i.e. eSe is reduced for all e e E(S). The following result is due to the author [86].
Theorem 12.4. Let E be a locally reduced biordered set, e,f E E, f
Then
f1 = fe E E is characterized within the system (E, Sr, S f) by the following threee properties. (a) f ,%l f1 <_ e.
(b) For all h E E with fl I h 5 e, there exists h' E E such that f -V h' A h.
(c) For all h' E E with f ,I h'
h' Ah If1. Similarly if (E, <_r, <_2)
f <_f e,
then
of
is characterized within the system
by the dual conditions. In particular, the system (E,
uniquely
determines the biordered set E.
Proof. Let e,f E E, f
fe Y h < e. Then by 12.1 (v) and Remark 12.2 (i), h E sand (fe,h) = sand (f,h) e. So
there exists h' E sand (f,h) such that h = h'e. By 12.1 (ii), h A h'. Now h' <_2 f,
h'
<_r
e, f <_r e.
So by 12.1 (iii), (fh')e = (fe)(h'e) = (fe)h = fe.
Hence
Thus fh' = f and f I h'. So fe satisfies (a), (b). Now let V E E with f I h'
(iii), fe 2 h'e = h. Hence fe satisfies (a), (b), (c). Finally let fl, f2 E E, satisfying (a), (b), (c). Then by (a), fl,f2 E eEe,
fl 9t f2. Let h1 E eEe such that fl 2 hl. Then by (b), there exists h' E E such
that f I h' 9 h. So h'
It follows that fl .,I' f2 in eEe. Since E is
124
locally reduced, f 1
f f2. Thus f I= f2.
Corollary 12.5. Let M be a connected regular monoid with zero. Then M is locally
reduced and hence the biordered set E(M) is completely determined by the system (E(M),
Proof. For e e E(M), eMe is also a connected regular monoid with zero.
So it
suffices to show that M is reduced. Let G denote the group of units of M. Let e,f a E(M) such that e 5E' f. By Proposition 7.6, there exists a maximal torus T of G,
There exists h e E(M) such that
el,f1 E E(T) such that e 51 el, f 52 f1.
el . h 52 f.
Since f 52 fl, we see by Theorem 1.4 (vi) that e1 51 fIe1.
fle1 = e1f1.
So by Theorem 1.4 (i), e1 = f1.
Similarly
=
Hence
But
e A f and 51 = A'.
f'.
Definition 12.6. Let E be a regular biordered set, eEe -4 fEf is an isomorphism).
Define
f'yr(x) = yr' (xe') for all x e eEe. For (e,yr,f) e N' (E), let
[e,yr,f]
N' (E) = ((e,N,f) I e,f a E, yr:
(e,yr,f) - (e',yr',f') if e 51 e', f 91 f',
Then - is an equivalence relation on ,N' (E).
denote its - -class. Let .N(E) = .iY' (E)/
If
a = [e,yr,f], b = [e',yr',f'] e .N(E), h e sand (f,e'), then let
ab = [W 1(fh), X, yr'(he')]
where A.(x) = yr'(hyr(x))e'). .N(E).
This is a well-defined associative binary operation on
,N(E) is called the Nambooripad semi
up of E.
Nambooripad's analogue of the fundamental representation (see Remark 1.25) is [51], [52]:
125
Theorem 12.7. (i) If E is a regular biordered set, then
.N(E) is a fundamental
regular semigroup with biordered set of idempotents E. (ii) Let S be a regular semigroup with biordered set of idempotents
E = E(S). For a E S with inverse a-, let 9S(a) = [aa,y a a] where y(x) = a xa.
Then 9S: S -> X (E) is a well-defined, idempotent separating homomorphism with kernel µ.
Definition 12.8. Let E be a biordered set. Then a map *: E -+ E is an involution if
(i) (e*)* = e for all e E E. (ii)
If
e,f E E, with
of
defined, then
f*e*
is defined and
(ef)* = f*e*.
Corollary 12.9. Let M be a fundamental unit regular monoid with group of units G,E = E(M).
Let *: E -+ E be an involution. Then * extends to an involution of M
if and only if for all x E G, there exists y E G such that (x-lex)* = y e* y1
Proof. The necessity of the condition being obvious, we prove sufficiency. By general
considerations, * extends to an involution of .N(E).
Since M is fundamental, we
may assume by Theorem 12.7 that M is a submonoid of .N(E). Let X E G. Then
there exists y E G such that
ye*y_1
= (x-lex)* =
x*e*(x*)_1
for all e E E.
By
Remark 1.21 (ii), x* = y E G. Since M = EG, M* = M.
Definition 12.10. A semigroup S is locally inverse if S is regular and eSe is an inverse semigroup for all e E E(S). The next result is from Nambooripad [52; Theorem 7.6].
Corollary 12.11. equivalent.
Let E be a biordered set.
Then the following conditions are
126
(i) eEe is a semilattice for all e E E. (ii) I sand (e,f) ( = 1 for all e,f E E.
(iii) E = E(S) for some locally inverse semigroup S.
Definition 12.12. A local semilattice is a biordered set E such that I sand (e,f) = 1 for all e,f E E.
Remark 12.13. (i) A related, more general system (which does not always yield a
semigroup) was considered by Schein [105]. Local semilattices were first considered by Nambooripad [52]-{55]. See also [46], [47].
Locally inverse semigroups have been much studied.
(ii)
See for
example [45], [60], [119]. The following definition is due to the author [79].
Definition 12.14. Let SZ = (04) be a A-semilattice with a minimum element 0. Let
1 be a symmetric relation defined on S2 such that 0 1 0.
Then 52 = (52,1) is a
parabolic semilattice if the following conditions hold. (i) a52 = {(3 E 52 I R <_ a} is finite (and hence a lattice) for all a E 52.
(ii)
If 7,a1,a2,01,02E52,a110'2,0 1102,a1?01,Y?a2,
Q2, then a2 >_ R2. (iii)
If al, a2, PI E 52, al 1 a2, al ? (3l, then there exists
(unique by (ii)) such that a2 >_ R2' R1 1 R2' (iv) If a, al, a2, (3. (3l, R2 E 52, a >_ ai, Q
(32 E 52
Pi, ai L. Pi, i = 1,2, then
(alVa2)1(Q1V(32). Remark 12,15.
subgroups of G.
Let G be a reductive group, 52 = 52G the set of all parabolic If P11 P2 E 52, define P1 <_ P2 if P2 c Pl. Define Pl 1 P2 if
P12 P2 are opposite (see Definition 4.40). Then (52,.x) is a parabolic semilattice.
127
Definition 12.16.
Let
S2 = (52,1)
be a parabolic semilattice,
set EQ =
((a,(X') I a,a' a 0, a 1 a'). If e = (a,a'), f = (13,13') E E., then define f 5r e if f3 <_ a, f 5f e if (3' S a'. If f <_r e then let of = f, fe =
where R E 52 is
such that R 1 (3 , 0-:5 a'. If f <_f e, then let fe = f, of =((31,(3') where R1 E 52 is such that R1 1 R' amd R1 <_ a. The following result is due to the author [79].
Theorem 12.17. Let SZ be a parabolic semilattice. Then EQ is a local semilattice with an involution.
Proof. The involution on E = EQ is given by the map:
e = (a,(x'), f = ((3,(3') E S2 such that e
Definition 12.14 (iii), there exists 0' <_ R
Q
E SZ
(a,a') - (a',a).
Then a
Let
:5 cc'.
such that a' <_ Q
R13
By
Then
By 12.14 (ii), R = R'. Hence a' = (3'. So again by 12.14 (ii), a = P.
So e = f. It now follows easily that the partial binary operation of Definition 12.16 is well-defined. Let < =
(ii) hold. Let e = (a,a), f g Sf f.
ge <_ fe.
Clearly 12.1 (i),
g = (yy-) E E. Suppose first that f
Then 0 <_ a, y5 a, y-:5 R
Now if ge = (y,y' ), fe
1.
We see by 12.14 (ii) that 'Y:5 P.
then
<_ a
So g <_ f.
By 12.14 (ii), y'
So
Hence
f Sr e, g <_r e, g <_ f f imply g:5 f, ge <_ fe
In particular 12.1 (iii) holds. Next assume that g
Let (ge)f = (y,y').
Then 7:5 (3 <_ a.
Then y" <_ (3
If
So by definition,
gf = (y,y") = (ge)f. Hence 12.1 (iv) also holds.
Now let e = (a,a-), f = ((3,(31 e E. Then fEe c (352 x a-0 is finite. Let f E e = (hl,...,hk), hi = (yi,y?, i = 1,...,k. Then y i 5 (3, y l <_ a , i = 1,...,k. So
128
Y = Y1 V ... V yk 1 Y = Y1 V ... V Yk by 12.14 (iv). Clearly h = (Y,Y-) E fEe, hi <_ h,
i = 1,...,k.
It follows that sand (e,f) = (h). Now let g e E and suppose that e
f
So by (11), h < e, hg 5 eg.
Also, hg 9 h
Hence hg a fg E eg. Let sand (eg,fg) = (h'). We claim that
whereby hg
h' = hg. Now h Ahg<- h' < rfg ,Rf
So h'
So by (11), h' <- eg .9 e. Thus h'
Now We A h'
h' ,R h'e <- h and h'
Definition 12.18. Let G be a reductive group,
f1
the parabolic semilattice of all
parabolic subgroups of G. Set EG = EQ. We call EG the local semilattice of G.
Definition 12.19. Let E be a biordered set. (i) Let < E >- denote the subsemigroup of .N(E) generated by E. (ii)
Let - denote the transitive closure of .% o .. on E and let
Y1(E) = 9,l(< E >-) = E/-.
Remark 12.20.
(i)
For e e E, let [e] denote the - -class of e.
By Nambooripad [52],
< E >-
is the unique fundamental,
idempotent generated regular semigroup having E as the biordered set of idempotents.
By [21; Lemma 2.2], - is just the Green's relation
.
on -< E } restricted to E.
(ii) Let E be a local semilattice. Suppose that - restricted to eEe is equality for all e e E.
for all e e E.
See [54].
Then it follows from (i) that e -< E >- e is a semilattice
129
13 TITS BUILDINGS
The collection of parabolic subgroups of a reductive group has a rich geometric structure. This structure is rather amazingly preserved, via Tits buildings, in
a much larger class of groups, including all finite simple groups of Lie type. In this chapter we will briefly describe the theory of Tits buildings [115] and explain how a local semilattice can be naturally associated with such a system.
By a complex is meant a A-semilattice f2 = (f2,<_) with a minimum element 0 such that for all a E f2, af2 = (P e 01 Q <_ (X) is a finite Boolean lattice.
The minimal elements of S2\(0) are called vertices. If a E S2, then the rank of a is defined to be the number of vertices in aft. The maximal elements of 0 are called
chambers. We will assume that all chambers are of the same rank d and that every
element of fI is
<_
a chamber. We define the rank of f2 to be d.
chambers. We will assume that
f2
Let a, a' be
is connect s i.e. there exist chambers a =
ao,al, ...,am = a' such that ai A ai+l has rank d -1 for i = 0,...,m - 1. If m is minimal, then we set dist(a,a') = m. A non-empty subset 0' with the property that
aft' c 0' for all a e f2'
is said to be a subcomplex.
fI is said to be thick if every
element of rank d -1 is less than at least three chambers.
Q is said to be thin if
every element of rank d -1 is less than exactly two chambers.
Definition 13.1. A
its building is a pair f2 = (0,A where f2 is a complex, ,A a
family of finite subcomplexes called apartments such that
(i) f2 is thick.
130
(ii) Each apartment E is thin. (iii) Any two elements of 92 belong to an apartment. (iv)
If
E, E' e , e and if a,(3 E E n E', then there exists an
isomorphism 0: E -4 E' such that 4)(y) = y for all y E aQ u 0S2.
Example 13.2. Let G be a reductive group, 0 the set of all parabolic subgroups of
G ordered by reverse inclusion. {P E Q I T c P}.
If T is a maximal torus of G, let E(T) =
Let ,9 = (E(T) I T is a maximal torus of G). Then (S2, 4) is a
building.
We briefly list some basic facts concerning buildings. We refer to [111],
[115] for details. Let E E
,e, a a chamber in
E.
Then there exists a unique
retraction na: E -+ aE, i.e. (i) 7ta(0) = 0 for all (3 a aE, and (ii) 'ta restricted to a'E is an isomorphism for any chamber a' E E. If (3,(3' E E, then (3,(3' are said to
be of the same Me, type (0) = type ((3'), if
7ta((3) = 7ta(G3').
This concept is
independent of the choice of the chamber a. If a E E is a chamber, then there exist
a unique a' E E called the gpposite of a in E such that dist((x,a') is maximum. There is a unique automorphism 8: E - E such that for any chamber a of E, a and 8(a) are opposite. We then define (3 and 8((3) to be opposite for any (3 E E. Now let (3,(3' e Q. Then we define (3,(3' to be of the same fig, type ((3) = type ((3') if they
are of the same type in some (hence every) apartment containing them.
(3,(3'
are
defined to be opposite ({3 1 (3') if they are opposite in some (hence every) apartment containing them. (3'.
(3,(3'
are of opposite We if
(3
is of the same type as an opposite of
In the situation of Example 13.2, these definitions agree with those given in
Definitions 4.40, 4.52. If a,a', (3,(3' E S2 and if a 1 a', (3 type ((3) if and only if type (a') = type (0').
.
13',
then type (a) =
If a,(3 E 0, then by [115; Proposition
3.30], type ((x) = type ((3) if and only if there exists y E S2 with a
y, 0 1 y.
If sZ
is of rank 1, then any two non-zero elements have the same type and any two non-zero unequal elements are opposite.
131
It is routinely verified that
(52,1)
is a parabolic semilattice (see
Definition 12.14). Hence by Definition 12.16, Theorem 12.17, we can construct a local semilattice E = EK2.
We follow the notation of Definition 12.19. By the preceding
discussion on buildings, we see that - = A o d o .5 = d o R o d and that for any e = (a,(x'), f = ((3,(3') E E, e - f if and only if type (a) = type (1i) if and only if type (a') = type ((3').
In particular if e,fl,f2 E E, e 2 f., i = 1,2 and if f1 - f2, then
fI = f2. Thus by Remark 12.20 (ii), e j E } e is a semilattice for all e E E. Also note that
?t(-< E Y) = E/- is a finite Boolean lattice. Thus we have shown the
following result of the author [79].
Theorem 13.3. (i) E = E0 is a local semilattice.
(ii) e -< E >- e is a semilattice for all e E E. In particular x2 = x3 for
all xE -<E>.
(iii) -= Ao do 5 = do Ao donEand ?1({E })-E/-
is a finite
Boolean lattice.
Let a E 52 be a chamber, 0 E 52. Then by [115; Section 3.19], there exists a unique chamber a' E S2 such that a' >- 0, dist(a,a') is minimum.
denoted by proj1(a).
Let a,a' E 52 be chambers,
(3,13' E S2
a' is
be of rank d - 1.
Suppose a > 13, a' > 13', 0 1 1i'. Then by [115; Proposition 3.29], a 1 a' if and only
if proj1((x') * a. Let E = Ej2, Emax the set of maximal elements of (E4).
Lemma 13.4. Let e = (a,al E Emax, h = (13,13) E E, e covers
h.
Then there
exists a unique f* = f*(e,h) E Emax such that ef* = f*e = h in < E >. Moreover f* = (proj1(a-), proj1-(a)).
Let f E Emax' f >- h.
f 9G f*, and fe = h if and only if f d f*.
Then of = h if and only if
132
Since a 1 a
Proof.
we see that projR-((x) * (x-.
a-= projR-proj1(aj.
Hence
projR(a-) 1 projR-((x).
f* = (proj1(a-),
So
Let f = ( y , y l E Emax, f > h. Then y > (3, y > R
projR-(a)) E Emax.
f A f*. Then y # proj1(aj. So y 1 a
Hence of $ e
But by [115; Theorem 3.28],
Hence e = (a,al .v(y,a-) A (y,y-) = f.
Thus of # h.
in < E >-.
Suppose
Next suppose that
f R P.
Then
y = proj1(a-). So y is not opposite to a-. It follows that there is no el E E with e
e
l 5B f.
So of
e.
Now efh = hef = h.
Since Je covers Jf , of s h. It
follows that of = h. This proves the lemma.
If e,f E Emax, then define e S f if in
E }, of = fe is covered by e.
Let 6* denote the transitive closure of S. Let p(e) _ (h E E I h <_ f, f S*e for some fE
Emax}.
Now let e = (a,al E Emax*
containing a,a
If R E S2, then let
Let E
E E).
R
Let E be the unique apartment of 0 denote the unique opposite of R in E.
By Lemma 13.4, p(e) =
E.
So (p(e,)<_) = E.
E - E/ R denote the natural map. Let A' = {2,(p(e)) I e E
Emax}.
Let
?,:
We have shown:
Theorem 13.5. (E/,9, A) is a building isomorphic to (91,.x.
Definition 13.6. Let E = EQ. (i)
Aut* E is the group of all automorphisms a of E such that
e - e(Y for all e E E.
(ii) Aut* < E )- is the group of all automorphisms a of the semigroup
< E r such that a $ as for all a E < E }. (iii) Aut* 12 is the group of all automorphisms a of fl such that type
(a) = type (a(y) for all a E Q. The following result is due to the author [79].
Theorem 13.7. Ant* EQ = Aut* < ES2} __ Aut* Q.
133
Proof.
That Aut*EQ = Aut* { EQ} follows from Theorem 12.7. So we need to
Then a E Aut*E where
First let a E Aut*S2.
show that Aut*E __ Aut*S2.
(a,a) 6 = (a(y,a'a). Conversely let 0 E Aut*E. Then for all e,e' E E, e A e' if and only if eO 51 e'O, and e G1,G2 E Aut* S2
GI = a2.
e' if and only if e9 X e'O.
It follows that there exist
such that (a,0)0 = (aa1,3a2) for all (a,(3) E E.
For suppose aal
aa2 for some a E Q. Then aala2l # a. Now a,
aa1a21 E E for some apartment aa1a21 is not opposite to
We claim that
0.
E.
So a 1 R for a unique
So aa1 is not opposite to
13a2.
Q E E.
Then
Thus (a,(3) E E,
(aa1,(3a2) a E. But (aa1,(3a2) = (a,(3)9 E E, a contradiction. Hence al = a2.
It
follows that Aut* E = Aut* S2, proving the theorem.
Definition 13.8. A Tits system is a quadruple (G,B,N,S) where G is a group, B,N
are subgroups of G generating G, T = B n N a N, S is a generating set of order 2 elements of W = N/T such that
(i) 9B9 * B for any 0 E S. (ii) aB9 c BaB u Bo-9B for all a E W, 0 E S.
Remark 13.9. Let G be a reductive group, T a maximal torus of G, B a Borel
subgroup of G, N = NG(T), ol the set of simple reflections relative to B. (G,B,N,QY)
Then
is a Tits system. Moreover any finite simple group of Lie type admits a
Tits system. See [111], [115].
Let (G,B,N,S) be a Tits system and assume that the Wvl groan W =
N/T is finite. If I c S, let Wl = , Pl = BWJB.
subgroups of G containing B.
Let E _ (a4Plala E W, I c S).
(x-1YxJx E G), S a = (x1PlxJx E G, I c S). If P
P1 c P2.
Then the Pi's are exactly the Let
4=
then define P1 > P2 if
Then by [115; Theorem 3.2.6], S2 = S2G = (52,4 is a building. The
elements of Sa are called parabolic subgroups of G. If P E 52, then P is conjugate
to a unique P1, I c S. Define type (P) = I. Then two parabolic subgroups are of the
134
same type if and only if they are conjugate. The conjugates of B are called Borel subgrouis. We call EG = EQ the local semilattice of G.
Problem 13.10. Let G be a finite simple group of Lie type. Then G acts on EG by
conjugation. So G c ,r(EG). Let M be the submonoid of .N(E 1) generated by G and EG. Then G is the group of units of the fundamental finite regular monoid M and E(M) = EG.
Study this monoid M. For example, it is always unit regular [85].
135
14 SYSTEM OF IDEMPOTENTS
Many seemingly unrelated ideas, from the previous chapters, come together in this chapter. Let M be a connected regular monoid with zero 0 and group
Let E = E(M) denote the biordered set of idempotents of M.
of units G.
By
Corollary 12.5, E is completely determined by the system (E, 5r
the structure of M is contained in E ? e E E, let [e]
denote the - -class of
Let -, 2t(E) be as in Definition 12.19. If e.
If el, e2 E E, define [el] >- [e2] if
e1 >_ e2 for some e2' e [e2]. By Theorem 5.9, 2t = Ye(E) is a lattice, isomorphic to 21(M).
To continue our discussion, we will need the following result from the author [79]. The proof is new.
Lemma 14.1. Let e,h,f E E such that e covers h covers
f.
Then there exists a
unique h* = h*(e,f) E E such that e > h* > f and hh* = h*h = f. e > hl > f.
Let h1 E E,
Then h1 * h* if and only if there exists h2 E E, e > h2 > f, such that
either h ,R h2 d h1 or h
h2 ,5E h1.
In particular, h* is determined within E.
Proof. By Theorem 6.7, we are reduced to the case when e = 1, f = 0. By Remark
8.8, we may assume that G is not a torus. Let T = CG(h),E(T) = (l,h,h*,0), W = (1,,u), B = CG (h) = CG(h*), B = CG (h) = CG(h*). such that h1h = hh1 = 0. By Theorem 11.4,
Let hl E E(M), hl # 0,1
136
h1 e BhB u Bh aB u BahB u Bh*B
Since Bh = hBh, 0 e h(BhB u BhaB), 0 e (B(YhB)h = BaBh.
Since hhl = h1h = 0,
we see that hl E Bh*B = Bh*Bh*.
hl ..
So by Theorem 1.1 (i),
Similarly,
h*.
since
hl E B-hB-v B-haB u B ahB v B-h*B , we see that h1 R h*. Hence hl = h*. So if h1 E E(M), h1 o 0,1,h*, then either
hhl / h or hlh / h. We are done by Theorem 1.1 (vi). Back to our discussion. Recall that a subset A of E is a cross-section lattice if: (i) for all e E E, there exists a unique e' E A such that e - e', (ii) for all
e,f E A, [e] E.
[f] implies e >_ f.
See Corollary 9.7. Fix a cross-section lattice A of
Let B = CG(A), B-= CG(A), T = CG(A). Let v(A) denote the smallest subset
of E containing A and having the property that if e,h,f E v(A) with e covering h covering f in E, then h* = h*(e,f) (of Lemma 14.1) is in v(A). Then by Corollary 8.10, v(A) = E(T). Then 9 = SA = (v(A), <_,-) is just the 61-structure of Chapter 10.
The point is that we have determined 9 completely from E. By Theorem 10.7, the
Weyl group W is recovered from 6 as its group of automorphisms. By Corollary 10.21, the set of simple reflections relative to B is recovered as
eY = * A) =
(a E W I a # 1, a fixes a chain of length ht X-1 in A). Let Tl be as in Definition 10.22.
Definition 14.2. (i) If YE Tl, A a cross-section lattice of E, then yp of AY= Y.
Recall that AY= (e E AI[e] E
(ii) E = (AAA is a cross-section lattice of E, YE TL}. If A,A' E
E,
then A :5 rA' if for all e E A, there exists e' E A' such that e A e'.
Similarly
define A <_B A' if for all e e A, there exists e' E A'
e'.
such that e .
Let
137
5=<_rn:1. (iii) If e e E, let A(e) E Tl denote the intersection of all YE Tl with
[e] E Y The following result is due to the author [79].
Theorem 14.3. (i) E is a local semilattice and 0: E _ EG where 0(A) = (CG (A),
CG(A)) (ii)
A
If A,A' E E, then A - A' if and only if type(A) = type(A'). In
A
particular Yl (E) - Tl. (iii)
If e e E, A a cross-section lattice of E containing e, then
CG (e) = CG(AX (e)), CG (e) = CG(A),(e)).
If A' is any other cross-section lattice of
E containing e, then A'X(e) = AX(e).
Proof. (i) If A E E, then CG (A) n CG (A) = CG(A) is a reductive group, whereby
0(A) E EG.
Now let P, P be opposite parabolic subgroups of G relative to a
maximal torus T of G. Let B c P be a Borel subgroup of G containing T, B its opposite, relative to T.
By Theorem 4.51, there exist I c eY = &(B) such that
P = BWIB, P = B WIB
There exists, by Theorem 9.10, a cross-section lattice
A c E(T) such that B = CG(A), B-= CG (A). Let Y= (J E Vie 0 = e for all 6 E I, e E J o A). Then by Remark 10.23 (ii), p = CG(AO, P =CL(AY). It follows that
0 is surjective. A
Next let A,A' E E, Y = type(A), Y ' = type(A').
Now A = A Y
Y' for some cross-section lattices A,A' of A. By Corollary 9.7, xAx1 = A' for some x e G. Suppose that A Sr A'. Then Yc Y' and xAx4 c A'. So A' = A'
xex4
,
e for all e e A.
Hence xe = exe for all e E A.
CGr (A') c C G r (xAx1) = xC rG(A)x4 = C rG(A). conversely that 0(A) <_r 0(A').
So X E CG (A) and
Thus 0(A) <_ r 0(A').
Assume
So x CG(AY,)x 1 = CG(A' Y,) = CG (A') C CG(A)
138
CG(A Y).
e A xex
By Theorem 4.51 (iii), x E CG(A ,) and Y c V.
1 E A' Y c NY
Hence A= A Y
So for all e E AY,
Similarly, A
only if 0(A) <e 0(A'). In particular 0(A) = 0(A') implies A = A'. Hence 0 is an isomorphism. A
(ii) Let A,A' E E, type (A) = type(A') = Y Then A = A Y A' = A' Y for some cross-section lattices A, A' of E. By Corollary 9.7, x1Ax = A' for some x r: G.
x4Ax = A'
So
and
x4CG(A)x = CG (A').
By Theorem 13.3,
0(A) - 0(A'). So A - A' by (i). (iii) Let e E A, A a cross-section lattice of E. Then clearly X (e) _
(f E Alfa = f for all GE W with e 6 = e} and O(AX(e)) = (CG (e), CG (e)) by Corollary 10.24. The second statement now follows from (i). A
Definition 14.4. (i) Let 21 = (21,X,EG) where ?: 2e -4 2l (EG) is given by A,(J) _ [CG (e), CG(e)] E 2l(EG). (ii) Let EAR = {(u,h)Ju E 2l, h E E^., X(u) = [h]). a2 = (u2,h2) E EAA
Let a1 = (ul,hl),
a2 if
Define al Aa2 if uI = u2 and hl ,R h2. Define a1
u1 = u2 and h1 i h2. Define aI < a2 if u1 <_ u2 and there exists h EEG such that h 1
<_ h, h2 <_ h.
Define
Remark 14.5. By Theorem 14.3, the system 2e is determined for E. The following result is due to L. Renner and the author [89].
Theorem 14.6.
E AA
uniquely determines a biordered set and E
isomorphism is given by: e
EA
.
The
(Je, CG (e), CG(e)).
Proof. Let yr: E i E A be given by yr(e) = (Je, CG(e),GG(e)). Let (J,P,P-) E E ^u . Then there exists e E E(J) such that p = CG (e). Then CG(e) is opposite to P. So
there exists x E P such that xCG(e)x 1 = P . So (J,P,P-) = yr(x-lex) and yr is
139
surjective. Let e,f E E. By Corollary 6.19, e A f if and only if yr(e) A W(f), and e ,V f, if and only if V(e) ,V yr(f).
In particular, yr is bijection. Next assume that
e ? f. Then Je = Jf. Let P = CG(e,f), P = CG(e,f). (CG(e),CG(e)), (P,P1 >- (CG(f),CG(f)).
Then (P,P1 e EG, (P,P1 >_
Hence V(e) ? yr(f).
Finally assume that
Then Je >- if and there exist opposite Borel subgroups B,B
V(e) >_ yr(f).
of G
such that B C CG (e) o CG (f), B-c CG(e) n CG (f). Now B = CG (A), B-= CG (A)
for some cross-section lattice A of E.
Let T = CG(A) = B n B
x E CG(e), y E CG(f) such that e1 = xex-1, f1 =
yfy-1
E E(T).
f A fl. So by Corollary 6.19, CG(e) = CG(el), CG(f) = CG (fl).
e1,f1eA.
There exist
Then e ,R el, By Theorem 9.10,
So {el}=JenA,(fl)=JfoA. Since Je _ Jf,el?fl.
So f<_re.
Similarly, f:5 f e and f<_ e. This completes the proof.
Definition 14.7. Let Aut*E denote the group of all automorphisms 6 of E such
that e - ea for all e E E. The next result is due to the author [79]. Its proof is simplified due to Theorem 14.6.
Corollary 14.8. Aut*E = Aut*EG = Aut*f G.
Proof. If 6 E Aut*S2G, then clearly (J,P,P-) -4 (J,P6,P 6) is an automorphism of E AA __ E.
n
E=EG*
If 6 E Aut*E, then the map: A -4 A6 = (e6 e e A) is an automorphism of The result now follows from Theorem 13.7.
Example 14.9. Let M = ,,f6n(K).
A
Then E is the local semilattice of all chains of
idempotents in M containing 0,1 and ?,: 24(M) - 24(EG) linearly orders the maximal parabolic subgroups of G.
Renner [96] has shown that any normal connected regular monoid with zero admits an involution. See Corollary 16.14. Without the assumption of normality,
140
we now prove a weaker result.
Corollary 14.10. Let M be a connected regular monoid with zero and group of units G.
Let T be a maximal torus of G.
Then M admits an abstract semigroup
involution *, such that t* = t for all t E T.
Proof. By Proposition 4.54, G admits an involution * such that t* = t for all t E T
and so that P* is opposite to P for any parabolic subgroup P of G containing T. Hence P, P* are parabolic subgroups of opposite type for any parabolic subgroup P
of G. We therefore have an involution * on EA given by (J,P,P-)* = (J,(P1*,P*). By Theorem 14.6, we have an involution * on E. If e E E, then CG(e*) = CG(e)*, CG(e*) = CG (e)*, e - e*.
(x-1
It is also clear that e* = e for all e E E(T) and that
ex)* = x*e*(x*) 1 for all x E G, e E E
(12)
By Corollary 12.9, * extends to an involution of the monoid M/µ. The problem now is to lift the involution to M. First let e,f E E such that e R f.
By Theorem 7.1, e,f E B for some
Borel subgroup B of G. Let B 1 = (x E B I xe = e) c. Then e,f E B 1 by Theorem 6.7.
Since BI is solvable, we see by Corollary 6.8, that there exists a unipotent
element u E B1 such that eu = f. Now u* E CG (e)* = CG (e*) is unipotent. So u*e* is a unipotent element of the M-class of e* and f* g u*e*. By Theorem 7.9 (iii), f* = u*e*. Hence for e,f E E, e 5C f, implies that there exists u E G such that
eu = f, ue = e, f* = u*e*.
(13)
Now let e E E(T). Then CG(e)* = CG(e), X = {y E G I ye = ey = e} is
a closed normal subgroup of CG(e).
Hence X* = X by Remark 4.55 (ii). Now let
141
x E G such that xe = e.
Then f = ex E E, e R f.
u e G with ue = e, u*e = f*. e = (x*)
l
u*e = (x*) -l f*.
So by (13), f = eu for some
Then ux 1 e X and hence (ux 1)* a X. Thus
l
1
But f = xex.
So by (12), f* = x*e(x*).
Hence
e = ex*. So using (12) we have,
xe = e, e e E, x e G imply a*x* = e*
(14)
Now let e,f a E, x,y e G such that ex = fy.
So e 52 f. By (13),
there exists u e G such that f = eu, u*e* = f*. So euyx1 = e, and by the right-left
dual of (14), x*e* = y*f*.
Let a E M.
Then there exists e e E, x e G such that
a = ex. Define a* = x*e*. By the preceding argument, this is well-defined. By (12), (xay)* = y*a*x* for all x,y a G, a E M. Let e e E(M), x E G such that exe Me. Then
exe = ey for some y e CG(e).
e 52 exy1.
So
Then exy1e = e, CG(e*) = CG(e)*,
e* .2' (exy1)* = (y4)*x*e*.
(ey)* = y*e* = e*x*e*.
Thus
e* = e*(y4)*x*e* and
So (exe)* = e*x*e* if exe Me, x E G.
(ab)* = b*a* for all a,b e M with a
b
ab.
It follows that
Now it suffices to show that
(ef)* = f*e* for all e,f a E. There exist hl,h2 e E such that hI A of ,2 h2. Let J
denote the /-class of ef. Now of = (h1e)(fh2), hle, fh2 e J. So by the above
(ef)* = (fh2)*(hle)*
Now hI
So hI <_Q e* and (hle)* = e*hl (since
*
is an involution of E).
Similarly (fh2)* = h2f*. So
(ef)* = h2f*e*hI
Since * is an involution of M/µ, hI
completing the proof.
f*e* 52 h2.
It follows that (ef)* = f*e*,
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Conjecture 14.11. Let M be a connected regular monoid with zero. Then there exists
a connected regular monoid M' with zero such that (E(M), (Sr)
(E(M'),
Examples 8.5, 8.6 form such a pair.
<_r, <_f).
The next result is due to the author [79].
Theorem 14.12. Let M be a connected regular monoid with zero and let S = M\G. Then 21(S) is a Boolean lattice if and only if E(S) = EG.
In such a case, S is a
locally inverse semigroup.
Proof. Suppose first that 21(S) is a Boolean lattice. Then by Theorem 9.5, eMe has
a solvable group of units for all e e E(S).
Being regular, eMe is commutative.
Hence S is a locally inverse semigroup. Moreover, for any e, fl,f2 E E(S), e >_ fl,
e >_ f2, f1 "' f2 imply fl = f2. Define 0: E(S) e,f r: E(S).
Suppose f <_r e.
EG as 4)(e) = (CG (e), CGf (e)).
Then by Theorem 6.16, e,f e CG (e).
Borel subgroup of CG (e) and let T be a maximal torus of B. B = CG(A) for some cross-section lattice A c E(T).
Let
Let B be any
By Theorem 9.10,
There exists a E CG (e) such
that e' = aeal E E(T). Then e 5C e'.
So by Corollary 6.19, CG(e) = CG(e'). So
B c CG(e').
Let f ' E Jf n A. Then since Je >_ Jfl we
have e'
By Theorem 9.10, e' E A.
Since f _ f ', f ' s fe' in M.
f '.
Thus f ' = fe. Hence f R f '. By Corollary 6.19, B c CG(f ') = CG (f).
Since B is
an arbitrary Borel subgroup of CG(e), we see by Theorem 4.11, that CG(e) c CG (f). So 0(f)
Assume now that e,f a E(S), 4(f)
of
Let J denote the maximum element
9,((S), A = (h e J n E(T)jh >_ e) = {h1,...,hk}.
complemented, e = hl...hk. Then
f A f '.
Since
E(T)
is relatively
There exists a E CG (f) such that f ' = afa 1 E E(T).
Let h E A.
cross-section lattice A c E(TT)'
Let
Then
h >_ e.
By Corollary 9.4, there exists a
such that e,h e A. So B = CG (A) c CG (e) c CG(f) _
143
CG (f ').
Since J z If, we see that h z f '.
By Theorem 9.10, f ' e A.
hl...hk >_ f '. particular,
4)
Hence f
So e =
Similarly, 4)(f)
is injective. Now let (P,P7) E EG.
In
Then by Theorem 10.20, there
exists a chain r in E(S) such that p = CG(I ), P = CG(I').
Let e denote the
maximum element of F. Then by the above, (P,P-) = (CC (e), CG (e)) = 4)(e). Hence E(S) = EG.
Conversely, suppose E(S) _ EG.
Then by Theorem 5.9,
W(S) __ YG(EG).
But V(EG) is a Boolean lattice by Theorem 13.3. This proves the theorem.
Remark 14.13. (i) The monoid in Example 8.6 satisfies the hypothesis of Theorem 14.12.
(ii) Let Go be a reductive group, 4): Go -, GL(n,K) a representation. Let M(4)) = K4)(Go) c An(K). Then M($) is a connected regular monoid with zero. Let E(4)) = E(M(Q)). Then E(4)) is a geometrical object, which in light of Theorems
14.3, 14.6, may be viewed as a generalized building. One would conjecture that there
exists a representation 0 of G0 such that M(4)) satisfies the hypothesis of Theorem 14.12. In such a case E(4)) is (essentially) the building of G. Theorems 13.3, 14.6 suggest the following problem: Find all biordered sets naturally constructible from a Tits building. The answer is given by the theory of monoids on groups, developed by the author [85]. We briefly describe some aspects of the theory (without proofs).
Let (G,B,N,S) be a Tits system. A monoid M is a monoid on G if M has G as the group of units and the following three axioms on E = E(M) are satisfied: (1)
If e,f a E, e
f,
then
CG(e,f) and
CG(e,f)
are opposite
parabolic subgroups of G.
(2) If e,f e E with e I l f or e
f,
then x 1ex = f for some x e G.
(3) M = < E,G >
For e,f a E, define e = f if x -1 1 ex = f for some x e G. Let
144
2/ = 2/(M) = Eh
For e,f a E, define [e] 2 [f] if e >_ fI for some f1 E [f]. The axioms imply that (24<_)
is a partially ordered set. Consider the type map ? = A.(M): 2/ -4 2S given by
),([el) = type(CG(e)) Conversely let 2/ be any partially ordered set with a maximum element 1.
Let X: 2/ -1 2S be any map such that X(1) = S.
Call X transitive if for all
Jl,J2'J3 E 2/ with J1 <_ J2 <_ J3, we have that any connected component of X(J2) is
either contained in X(Jl) or in )L(J3). Let
E(k) _ ((J,P,P1 I J E 2/, P,P are opposite parabolic subgroups of G, X(J) = type(P) )
Let f = (J,P,P1, e = (J',Q,Q-) a E(A,). Define f
Q (i.e. P o Q is parabolic). Define f
0
be a Borel subgroup of G such that Bo C P n Q.
Let Bo be an opposite of B0 contained in P. It turns out that there is a unique opposite P' of P (independent of B0,B_ such that
B' = proj -proJQBo c P' Q
Then define
of = f, fe = (J,P,P')
If f <_B e, then fe = f, of are defined dually. The following result is due to the author [85].
145
Theorem 14.14. Let Yl -+
Yl
be a partially ordered set with a maximum element 1, ?:
2S such that A,(1) = S and X is transitive. Then E(X) - E(M) for some monoid
M on G. Conversely every E(M) is obtained in this manner. Actually, many of the results from the theory of linear algebraic monoids
(for instance, Renner's decomposition) can be generalized to monoids on groups. We refer to [85] for details.
146
15 f--RREDUCIBLE AND 0-CO-IRREDUCIBLE MONOIDS
In this chapter, we wish to restrict the lattice of
$-classes
W.
Fix a
connected regular monoid M with zero 0 and group of units G. The following result is due to Renner [94].
Theorem 15.1. representation
M 4)
has a completely reducible, idempotent separating linear
such that 0(0) = 0 and for any maximal torus T of G, 0 17-
T -- 4)(T) is an isomorphism.
Proof. By Remark 3.17, we may assume that M is a closed submonoid of some
%n(K) containing the zero matrix. Let A denote the linear span of M in
k (K).
Then A is a finite dimensional algebra over K. Let N denote the (nil) radical of A.
Then N n M = (0). Let yr: A -4 A/N denote the natural algebra homomorphism. Let
4) denote the restriction of W to M. Then Q-1(0) = (0). By Theorem 10.12, 4) is
idempotent separating. Let T be a maximal torus of G, T1 the linear span of T. Then
TI
Tl - yr(TI).
is a diagonalizable algebra and hence T1 o N = (0).
Since T c T1,
IT. T =_ 4)(T) = O(T).
So
WIT1:
Finally, AN is a finite
dimensional semisimple K-algebra and 4)(M) spans AN. Hence 0 is completely reducible.
Definition 15.2. M is
-irreducible if I Y&I(M) I = 1. M is %-co-irreducible if
te(l)(M) I = 1 (see Definition 6.21). M is $-linear if te(M) is a chain.
147
The following result is due to Renner [96; Corollary 8.3.3].
Corollary 15.3. M is $-irreducible if and only if M has an irreducible, idempotent separating linear representation. In such a case dim C(G) = 1.
Proof.
Suppose first that M is $-irreducible. Let V1(M) = (JO), To = rad G.
Then by Corollary 6.31, 0 e To Suppose dim To > 1. Then by Theorem 6.20, there
of = 0, e # 0, f * 0.
exist e,f E E(To) such that
Proposition 6.25,
e >_ e0, f >_ eo
So
If eo a Jo o E(T), then by
0 = of >_ eo,
a contradiction.
Hence
Now by Theorem 15.1, M has an idempotent separating, completely
dim To = 1.
reducible linear representation 0. irreducible. Let eo E E(J0).
Now
= Ol ®...
Then 4)(eo) # 0.
where each
So 4i(eo) # 0 for some
Oi
i.
is So
Oi(e) # 0 for all e e E(J0). Let f e E(M), f # 0. Then Jf > Jo So f >_ e for some e E E(J0).
Hence 4i(f) * 0. It follows that 4i 1(0) = (0). By Theorem 10.12, 4i is
idempotent separating.
Conversely let 0: M - End(V) be an irreducible, idempotent separating
representation of M, where V is a finite dimensional vector space over K Then 4)(M) (4)(0)V) = 4)(0)V
ei E E(Ji), i = 1,2. O(M)VI C V1.
and so 0(0) = (0).
Let VI
Let J1,J2 E ?1I(M), Jl * J2.
denote the span of 4)(Me1)V.
Let
Then VI * (0),
So VI = V. But e2Me1 = (0). So 4)(e2)V = (0), a contradiction.
Remark 15.4. (i) M is s-co-irreducible if and only if S = MSG is a connected
semigroup. In such a case, we see as in the proof of Corollary 15.3 that dim C(G) = 1.
(ii) Let e e E(M), e * 0,1. If M is -irreducible, then so is eMe. If M is s-co-4rreducible, then by Proposition 6.27, so is Me
Definition 15.5. G is nearly semisimple if dim C(G) = 1. dim C(G) = 1
and (G,G)
G is nearly simple if
is a simple algebraic group (i.e., G/C(G) is a simple
148
group).
Remark 15.6. G is nearly semisimple if and only if ranks G = ht(M) -1.
Lemma 15.7. Suppose G is nearly semisimple, e E E(M), e # 0,1, H the A-class
of e. Then CG (e) is a maximal parabolic subgroup of G if and only if Ge, H are both nearly semisimple.
Proo
By Therorem 6.16, ranks CG(e) = rank ssCG(e) = ranks H + rankssGe <_
ht(e M e) + ht(Me) -2 <_ ht(M) -2. The result follows from Theorem 4.51.
Corollary 15.8.
Let
e E E(M), e # 0,1, H the Ai -class of
e.
If M is
/-irreducible, then CG (e) is a maximal parabolic subgroup of G if and only if Ge is nearly semisimple. If M is J- co-Irreducible, then CG (e) is a maximal parabolic subgroup of G if and only if H is nearly semisimple. The following result is due to L. Renner and the author [89].
Proposition 15.9. Suppose that M is either O-irreducible or /-co-irreducible. Let e,f E E(M)\{0,1).
Then CG(e) = CG(f) if and only if e 5E f; CG(e) = CG (f) if and
only if e -V f.
Proof. We assume that M is J-irreducible, the other case being similar.
If e A f,
then CG(e) = CG (f) by Corollary 6.19. So assume conversely that CG (e) = CG(f).
Let T be a maximal torus of CG(e) such that e E E(T). There exists x E CG(f) such that
f'=
xfx-1
E E(T).
Then
f A f ' and hence CG(e) = CG (f).
2(l(M) = {J0), X = {huh E JO n E(T), h 5 e) = (h1,...,hk). complemented, e = hI V ... V hk. AE
' (T) with e,h E A.
Let h E X.
Let
Since E(T) is relatively
By Corollary 9.4, there exists
Then B = CG(A) c CG (e) = CG (f ').
So f ' E (B) = A.
149
Since
J
It follows that f ' >_ e. Similarly, a >_ f '. So e = f ' SQ f.
Corollary 15.10. If M is ,4rrreducible and /-co-irreducible, then M is s-linear.
Proof. Let A be a cross-section lattice of M, T = CG(A), B = CG (A). By Corollary 15.8, Proposition 15.9, CG (e) (e a A\(0,1)) are distinct maximal parabolic subgroups
of G containing
B.
But by Theorem 4.51, G has exactly rankssG maximal
parabolic subgroups containing B.
But by Remark 15.6, rankssG = ht(M) -1.
Hence I A I - 2 = ht(M) -1 and A is linear.
Remark 15.11. Let n = ht(M). The author [77] has shown that M is /-linear if and
only if there exists a e M such that an = 0, a° 1 # 0.
It is also shown there that in
such a case, G is nearly simple of type Af , BI , C1, F4 or G2.
Lemma 15.12. Suppose G1, G2 are closed, connected normal subgroups of G such that G = G IG2, (G 1,G2)
1) and rad G c G 1. Let M1 = Gl. Then
(i) E(Ml) _ {e a E(M) IG2 c CG(e)). (ii) If T1 is a maximal torus of Gl, e,f a E(T1), then the join of elf
in E(T1) is the join of e,f in E(M).
Then x = yz for
Proof. (i) Let e e E(M) such that G2 C CG(e). Let x e CG(e).
some y e G1, z e G2.
Then y e CG(e).
So
CG(e) = CG (e)G2, rad G2 c 1
rad G c GI.
So
rad CG(e) = rad CG (e) rad G2 c Gl.
By Corollary 6.31,
1
e e G1= M1. (ii)
Let h denote the join of e,f in CG(e,f).
By Corollary 6.31, it
suffices to show that h e M1. By Proposition 7.5, G2 c CG(e) n CG(f) c CG(h). By
(i), h e M1.
150
Definition 15.13. (i) Let S1,S2 be semigroups with zero. Then we denote by S =
S 1 A S2 the Rees factor semigroup, S 1x S2/I where I = S 1 x (0) u (0) x S2. (ii)
Let EI,E2 be biordered sets with identity elements. Then E =
E1 v E2 is the sub-biordered set (EI\(1)) x (E2\(1)) u ((1,1)) of El x E2.
Theorem 15.14.
Suppose
G1,G2
are closed connected normal subgroups of G
containing rad G such that G = G1G2, (G 1,G2) _ 1). Let Mi = Gi, i = 1,2. Then (i)
If M is /-co-irreducible, then MI,M2 are /-co-irreducible,
E(M) = E(MI) V E(M2) and W(M) = V(MI) V 11(M2)' (ii)
If M
is
$ -irreducible, then
MI,M2
are / -irreducible,
M/µ =_ MI/p. A M2/µ, 11(M) = 14M1) A 11(M2), and M = MIM2.
Proof.
(i)
Let Jo
1I1)(M), E = E(M), El = E(MI), E2 = E(M2), E' = E\(1 },
Ei = EI\(1 ), E2 = E2\{1}. Let ei a Ei, i = 1,2. Let Ti be a maximal torus of Gi such that ei E E(T), i = 1,2.
Since TIT2 is a maximal torus of G, e1 V e2 a E.
There exists h E E(Jo) such that e1 <_ h. There exists e2 a E, e2 $ e2 such that e2 <_ h.
There exists y E G such that y le2y = e2.
yi e Gi, i = 1,2.
So e2 = Y21e2Y2, eI = y21e1y2.
Now y = Y1Y2 for some
So y2hy21
e2, Y2hy21 ? eI.
Hence e1 V e2 * 1 and eI V e2 a F. Now let e e E, ht(e) = 1. Then dim eMe = 1 by Proposition 6.2. By Corollary 15.8, p = CG (e) is a maximal parabolic subgroup of G.
By Theorems 4.30, 4.51, either there exists a maximal parabolic subgroup P1 of
G such that G = P IG2 or else there exists a maximal parabolic subgroup P2 of G2
such that G = GIP2.
By Theorem 10.20, either there exists e1 E Ei such that
CG (e) = CG (e1) or else there exists e2 E E2 such that CG (e) = CG(e2).
Suppose
that there exists e1 a Ei such that CG(e) = CG(e1). By Proposition 15.9, e 5l el. Since G = GIG2, (GI,G2) = 1, we see by Corollary 6.8 that e e Ei. Thus
(e E E ht(e) = 1) c Ei u E2.
151
Since E(T) is relatively complemented for any maximal torus T of G, we see by Lemma 15.12, that
E'= {el Ve2jeie Ei,i= 1,2) Now let ei, i E El, i = 1,2, e = e1 V e2, f = f1 V f2. By Proposition 7.6,
there exist maximal tori Ti of Gi, ei,f.' E E(Ti) such that ei 52 ei, f. 52 Pi, i = 1,2.
There exists x e CG (el) such that x-1elx = ej. Since T1T2 is a maximal torus of G, we see by Proposition 7.5 that x e CG(e1) n CG(e2) S CG(el V e2). ej V e2 = x 1(el V e2)x . el V e2. Similarly ei V e2 A ei V e2. e1 V e2 5£ ei V e2, fl V f2 51 fi V ff.
and e 51 f.
So So
So if ei 51 i, J= 1,2, then ej = fi, e 2 = f2
Conversely suppose e 5Z f.
Then ei V e2 = fi V f2 E E(T) where
Suppose ei ? fi. Then since E(Tl) is relatively complemented, we can
T = T1T2.
find h e E(T1) such that 1 > h >_ ei, h V fi = 1.
So by Lemma 15.12, h V e2 =
h V fi V ff = 1, a contradiction. So ei ? fi. Similarly fi >_ ei and ej = fi. For the same reason e2 = f2. Hence e1 5B fl, e2 51 f2.
el -V fl, e2
f2.
Similarly e 91 f if and only if
In particular, e = f implies el = fl, e2 = f2.
Next suppose e,f E E' such that e >_ f.
Then e,f E T for some
maximal torus T of G. It follows from the above that there exist ei,fi E E(T) n Mi, i = 1,2 such that e = el V e2, f = f1 V f2.
Suppose e1 >_ f1.
h E E(Tl) such that
Then 1=hVf5hVe=hVe2, a
I
contradiction. Hence e1 >_ f1.
Similarly e2 >_ f2.
ei >_ fi, i = 1,2, then clearly e1 V e2 >_ fl V f2. that
E
E1 V E2.
In particular,
M1,M2
Then there exists
If conversely, ei,fi E Ei with
It now follows from Corollary 12.5
are / -co-irreducible and
&(M)
2Z(Ml) V Yl(M2).
(ii) Let E = E(M), Ei = Ei(M), E* = E\(O), Ei = El (O), M* = M\{0),
Mi = Mi (O), i = 1,2. By the proof of (i), E* = ElE2 and for any ei,fi E Ei, i = 1,2,
e1e2 A fi'2 implies ei 5E f, i = 1,2; e I e2
flf2 implies ei d fi, i = 1,2.
In
152
particular M* = M1M2. Now let ai,bi E Mil i = 1,2 such that a1a2 µ blb2. Now ai
,
ei, bi
b1b2 5E f1f2.
Similarly
.
fi
in
Mi
*
for some ei,fi a Ei, i = 1,2.
Then a1a2 52 e1e2,
So by the above, el A f1, e2 A f2. Hence ai A bi in Mi, i = 1,2.
ai . bi
in Mi, i = 1,2.
So ai 0 bi
Then xalya2 = xala2y .t xblb2y = xblyb2.
in Mi., i = 1,2.
Let x,y a M1.
So xaly = 0 if and only if xbly = 0.
If xaly * 0, then by the above, xaly V xb1y. Hence al µ b1 in M1. Similarly
a2.t b2 in M2. It follows that M/µ = Ml/µ A M2/µ. In particular M1,M2 are f-irreducible and ?4M) a ?1(M1) A 9.l(M2).
Example 15.15. Let M = (A ® B I A,B e A 2(K), det A = det B) , G the group of
units of M.
Let Gl = {A ® B E GSA is diagonal), G2 = (A ® B I B is diagonal),
Mi = Cii, i = 1,2. Then M is $- co-irreducible but M * M1M2.
Remark 15.16. Suppose M is $ -irreducible,
e e E(M), H the A -class of e.
L. Renner has recently shown that C(H) = e C(G). It follows from Theorem 7.9 that
for a,b a M, a µ b if and only if b = eta for some a e C(G).
Definition 15.17.
Let k e
II+, k 5 ht(M).
Then M is k-fold s-irreducible if
Vi(M) = 1 for i = 1,...,k. M is k-fold f-co-irreducible if
12l(1)(M)
= 1 for
By Theorems 4.30, 15.14, we have,
Corollary 15.18.
Suppose that
M
is either 2-fold
$ -irreducible or 2-fold
f-co-irreducible. Then G is nearly simple.
Corollary 15.19. Suppose that M is 3-fold $-co4rreducible. Then I Y11(M) 5 2. Moreover,
153
(i) M is 94inear if and only if G is of type AB , B I , C1, F4 or G2.
(ii)
If M is 4-fold j-co-irreducible, then G is of type D1 if and
only if J Yll j =2, I Yei =1 for i>1.
Proof. Let te")(M) = (Jo) , A a cross-section lattice of M, T = CG(A), B = CG (A), (Y= &'(B), F = {e e AjCG(e) is a maximal parabolic subgroup of G). If e e IF, let
W(CG(e)) = (1,(x(e)).
Then
a: F - c is a bijection by Theorem 10.20 and
Proposition 15.9. If e e I', then by Lemma 10.16,
CG(e). Let simple).
QYo = ((Y e cYl QY\{a}
Then
a(F0) = e o
eY\(a(e)) is
(CB(e)) in
is irreducible), F0 = (e a I I(CG(e),CG(e)) is
Let A n JO = (eo).
Then eoMeo is 2-fold
/-co-irreducible and hence by Corollary 15.18, has a nearly simple group of units.
Let I'o = ro (eo). Then F = {e e A ht(e) = 1). Now by the Dynkin diagrams, I QYo I = 2 or 3 with I ego I being 2 exactly when G is of type A, , Be , C1, F4 or G2.
Thus
I' I = 1 or 2. If ro = 1, then Yl is linear by Corollary 15.10. This
proves (i). Now suppose
I aYo
3 and that M is 4-fold $-co-irreducible. Then
ht(M) > 5 and G is of type De if and only if o\(a) is of type Al 1 for exactly two elements a of QYo
/-linear by (i).
Now eoMeo is 3-fold s4rreducible and hence is not
So if G is of type D1 , then Ge is of type Ae 1 for e e To
Hence by (i), Me is s-linear for e e I'o Since r, = (e e A ht(e) = 1)
and
I ro I = 2, we are done by Corollary 8.11.
Remark 15.20. If M is $-irreducible, the possible W(M) are determined in [89].
The following result is due to the author [77; Theorem 2.7].
Theorem 15.21. Suppose that M is either ,-co-irreducible or 2-fold $-irreducible. Let S = M\G, dim M = n.
Then every element in S is a product of 2n + 6
154
idempotents in S.
Proof. Let a E S. Then a = e'x for some x e G, e' E E(S). Now e 2 e' for some e e E(S) with
1
covering e.
Then a = e'ex.
Let H denote the M -class of e.
By Theorem 5.9, there exist el,f1 E E(S) such that e e1x-1
Theorem 1.4, aE
ex M ex.
,
eI I-° fl yQ x-1 ex.
So by Remark 1.3 (viii), ex E H
e1x-1
By
ex
and
e'He1x1ex. Thus it suffices to show that every element of H is a product of
2n + 3 idempotents in S.
So let e e E(S), 1 covering e, H the 2B-class of e. Let Jo denote the
,-class of e, E0 = E(J0). By Proposition 5.8, E0 is a closed irreducible subset
of M. We have the product maps: E0 x ... x E0 -1 E. Thus Eo is irreducible for all k E Z+. Clearly
EocE2cE3c o- 0- ... So there exists m e II+ , in <- n such that Eo = Eo for all f E 1L, I ? m. S 1= E o.
Let
Then S 1 is a connected semigroup, e S l e is a connected monoid. Let
H 1= H n e S 1
e denote the group of units of
e S 1 e.
If X E CG(e), then
x-1e S1 ex c e S1 e. So x1 HIx c H1. Since H = eCG(e) by Theorem 6.16, we
see that H1 is a closed connected normal subgroup of H. Let X = E 0 x...x Eo, the m-fold direct product. Define 0: X -i eSIe as v}(el,...,em) = eel ... eme.
4(X) = eEoe.
So 4(X) = eS1e.
Thus
By Theorem 2.19, 4(X) contains a non-empty
open set U of eSle. Now H1 is an open subset of eSle by Remark 3.21.
So
By Proposition 4.2, U1 =U o H1 is a non-empty open subset of Hl. Thus it suffices to show that Hl = H. H1 =U 21 CU 2 C E2m+3 o
155
First assume that M is $ -co-irreducible. Then by Theorem 6.20, E(S) = E(S1).
Hence E(eSe) = E(eSle).
So again by Theorem 6.20, the maximal
tori of H, H1 have the same dimension. So by Corollary 4.34, H = H1.
Finally assume that M is not f -co-irreducible. Then ht(M) > 2 and
M is 2-fold $-irreducible. Let VI(M) = {J1). e > h.
There exists h e E(J1) such that
By Proposition 6.27, Mh is f-irreducible. So by Corollary 15.3, Gh is
nearly semisimple. By Corollary 6.31, the width of e in Mh is greater than 1. So
there exists f e E(J0) such that of = fe, e * f, f > h.
Hence of E J0, of * 0. Now
of a eS1e = ffl. So dim Hl > 1,H l * (H1,H1). Now eMe is 2-fold f-i reducible
and hence by Corollary 15.18, H is nearly simple. So by Corollary 4.34, H = Hl, proving the theorem.
Remark 15.22. That non-invertible n x n matrices over a field can be expressed as products of idempotent matrices was first noted by Erdos [22]. See also [13].
Example 15.23. Let M = ( A ®B I A,B E ,A2(K)).
Then M is f-irreducible but
[ 0 01 ®10 01 is not expressible as a product of idempotents of M.
Remark 15.24. Recently the author [87] has shown that if M is f-irreducible, then the non-units of M are products of idempotents if and only if the group of units G is nearly simple.
156
16 RENNER'S EXTENSION PRINCIPLE AND CLASSIFICATION
In this chapter, we will need some additional algebraic geometry.
Definition 16.1. Let X,Y be irreducible affine varieties, 0: X -+ Y a dominant
morphism. Then (i) 0 is finite if K[X] is an integral extension of 4)*(K[Y]) e K[Y]. (ii) 0 is birational if there exists a non-empty open subset U of Y
such that 0
10-1
(U) : 4 1(U) - U is an isomorphism.
Remark 16.2. (i) If 4) is finite, then it is closed (and hence surjective) and the inverse
image of any point in Y is a finite set. See [34; Proposition 4.2].
(ii) Let M,M' be connected regular monoids with zero, 4): M -a M' a dominant homomorphism. If 4) is finite, the clearly 4)
(0) = (0). The converse has
been shown by Renner [91; Proposition 3.4.13]. Equivalently
4)
is idempotent
separating (Theorem 10.12). The homomorphism of Example 3.12 is not finite.
Definition 16.3. Let X be an irreducible affine variety. Then
(i) X is normal if K[X] is integrally closed, i.e., the integral closure
of K[X] in K(X) is K[X]. (ii) The normalization of X is X =
where k is an irreducible
normal affine variety, 4): X -+ X is a finite birational morphism.
157
Remark 16.4. Normality is a local property: X is normal if and only if for all x E X, the ring 0x = {f/gI f,g a K[X], g(x) * 0) is integrally closed. In particular if X is
normal, then any affine open subset of X is also normal. When X is not normal, the
integral closure of K[X] in K(X) is a finitely generated algebra, giving rise to the
normalization X of X. See [106; Chapter II, Section 5.2] for details.
Proposition 16.5. Let X be an irreducible affine variety. Then X admits a unique normalization k = (X,O).
Moreover, any dominant morphism from a normal variety
into X factors through 0 uniquely. The following result is due to Renner [91].
Corollary 16.6.
Let M be a connected regular monoid with zero. Then the
normalization M = (M,4) is also a connected regular monoid with zero and 0 is a finite birational homomorphism.
Pref. We have the product map p: M x M -4 M. Hence p o (0 x 0) : M x M - M. By Proposition 16.5, we have the product map p: M x M , M.
Using Proposition
16.5, again, we see that p is associative and that- 0 is a homomorphism. Now
1(1) is a finite group and hence
1(1)c = ( 1).
Thus the group of units of M is
reductive. By Theorem 7.4, 11 is regular. Now 1(0) is a finite ideal of i I and hence the zero of M.
Remark 16.7. (i) Let 7 be a connected diagonal monoid with zero.
Then T is
normal if and only if, for all x E £ (T), n E Z+, n x E $(T) implies x E ,S(T). See [37]. In the multiplicative notation this means that the commutative semigroup 'W(T)
has the following property: if a,b E £ (T), i E 1[+, then al b' implies a b. (ii)
T = I diag (a,b,c) I a,b,c E K, a2b = c2)
is not normal.
Its
normalization is T1 = {diag (a,b,c) I a,b,c e K, ab = c) with 0: Ti - TI given by
158
4)(a,b,c) = (a,b2,c).
We will need the following consequence of Zariski's Main Theorem (see [108; Exercise 4.28(3)]).
Proposition 16.8. Let 0: X -# Y be a bijective, birational morphism of irreducible
affine varieties such that Y is normal. Then 0 is an isomorphism.
Definition 16.9. Let M be a connected regular monoid with zero, A a cross-section
lattice of M. Then (i) Amax denotes the set of maximal elements of A\( 1). (ii)
If e E Ama, then M(e) = M(e,A) = (a E M I faf M f for all
f E eA } _ (a c- M deteA(a) # 0) where deteA is as in Definition 3.22.
Remark 16.10. Let G be a reductive group, B, B
opposite Borel subgroups of G
relative to a maximal torus T. Let U = Bu, U =B u. Then BB is an affine open
subset of G and is called the hiLgcell of G.
Moreover, the product map from
U x T x U onto B B is an isomorphism of varieties. See [34] for more details on Chevalley's big cell. The open sets M(e) are due to Renner [96]. They will be used in Theorem 16.13 below. The following result is due to Renner [96; Theorem 4.4].
Theorem 16.11. Let M be a connected normal regular monoid with zero and group of
units G. Let A be a cross-section lattice of M and set B = CG(A), B-= CG (A), T = CG(A), U = Bu, U-= BU. Then for any e E Amax' we have,
(i) M(e) is an affine open subset of M and M(e) c G u Je (ii) T(e) = T u eT is an affine open submonoid of T.
(iii) The product map, p: U x T(e) x U -4 M(e) given by p(x,t,y) _ xty is an isomorphism of varieties.
159
Proof. (i), (ii) being obvious, we prove (iii). Let H denote the M -class of e.
Theorem 6.16, Lemma 10.16, Be = eCB(e) = CH(eA), eB
By
= eCB-{e) = CH(eA),
Ue = (Be)u, eU = (eB)u Let f e eA, x e U-1 t e T(e), y e U. Then fx = fxf, yf = fyf. So fxtyf = (fxf)(ft)(fyf) X f. Hence xty a M(e). then by Theorem 6.33, a e BB = p(U x T x U).
By Theorem 6.33, eae e eBBe.
eae a M(e) n H.
CB-(e) such that eae = eyxe. Let b = y
ax
Let a E M(e).
If a E G,
Next suppose a e 'e
Then
So there exist x e CB(e), y e
Then ebe = e. So f = eb a E(M),
e 5P f. By Proposition 9.8, f e if. By Proposition 6.1, there exists x1 E B such that exI = f = eb. Let c = bx11. Then ec = e. So ce e E(M), e .. ce. By Proposition
9.8, ce e B
By Corollary 6.8, there exists yI e B
Let d= y l 1c.
Then de = e, ed = ey 1 Ic = ec = e.
Theorem 1.4 (i) that d = e.
such that yle = ce, eyI = e.
Since d $ e, we see by
Hence a = yylexlx E B eB c p(U x T(e) x U). It
follows that p is surjective.
By Remark 16.10, p restricted to U x T x U is an isomorphism onto
the open set B B of M(e). Hence p is birational. We show next that
p
is
injective. So let ul,ui e U , u2,u2 e U, t,t' e T such that uIetu2 = uiet'u'2. Then
uiluie
=
etu2(u2)-1(t')l.
u11uiee Be.
So u1luie = eu11uie = eu1lul a eU
Similarly
Thus uj uiee eU nBe=eU oeT= (e). Now Ge=TecT.
So { z E CG(e) I ez = ze = e), being a closed normal subgroup of CG(e) is contained
in T by Corollary 4.34. In particular, u11ui e To U = (1). Similarly u2 = u2. M(e) is normal.
Thus uI = ui.
So et = et'. It follows that p is a bijection. By Remark 16.4, So by Proposition 16.8, p is an isomorphism. This proves the
theorem.
See [29; Section 5, Lemma 1] for the following.
Proposition 16.12. Let X,Y be irreducible affine varieties, X normal. Let U be a non-empty open subset of X such that dim(X\U) <_ dim X -2. Then any morphism
0: U Y extends to a morphism ij X Y.
160
The extension principle below is due to Renner [96; Corollary 4.5].
Theorem 16.13. Let M be a connected normal regular monoid with zero and group of
units G. Let T be a maximal torus of G. Let S be an algebraic semigroup, 01: G -4 S, 02: T -# S homomorphisms such that 01 T = 2 IT. Then there exists a unique
homomorphism 0: M -+ S extending 01'02'
Proof. Since G = M, it suffices to extend 11 02 to a morphism 0: M -4 S.
Let
A c E(T) be a cross-section lattice, B = CG (A), B = CG (A), U = Bu, U-= Bu. By Corollary 3.28, every ideal of M is closed. Hence by Proposition 6.2, the irreducible
components of MSG are exactly MeM(e E Amax).
M(e) u G (e E
Amax).
Let M' denote the union of
Let X be an irreducible component of "'.
Then
X c MeM\M(e) for some e E Amax' Then dim X:5 dim(MeM\M(e) < dim MeM < dim M. Hence dim(M\M') <_ dim M - 2.
By Proposition 16.12, it suffices to extend
01, 02 to M'. Now for e,f E Amax' e * f, we have M(e) n M(f) c G. Hence by Remark 2.17 (iii), it suffices to find, for e E Amax, a morphism 9: M(e) -4 S such that 0 1 T ( e ) = 02 1 T(e) and 0 BB _ 11 BB. D e f i n e y . U x T(e) x U y(ul,t,u2) = 4)1(ul)4)2(t)4)1(u2).
S as
Let 0 = y o p 1, where p: U -x T(e) x U = M(e) is
as in Theorem 16.11.
The next result due to Renner [96; Theorem 8.2] follows from Proposition 4.54 and Theorem 16.13.
Corollary 16.14. Let M be a connected normal regular monoid with zero, group of
units G.
Let T be a maximal torus of G. Then M admits an involution * such
that t* = t for all
t
T.
Renner [96] uses his extension principle to establish the following classification theorem.
161
Theorem 16.15. Let G be a reductive group with a maximal torus T and Weyl
group W.
Let T be a normal connected diagonal monoid with zero, having T as
the group of units. Suppose the action of W on T extends to T. Then there is a unique connected normal regular monoid M with zero having G as the group of units such that T is the closure of T in M. This provides a discrete geometrical classification since normal torus embeddings T --+ fi can be viewed geometrically [37], [58]. Let dim T = n, so that ,W (T) = V. p C IRn.
Let C = i (T) v (0). Then C = P n 71n for some polyhedral cone
To obtain a monoid, P must be W-invariant. We refer to [95], [96] for
details, where the classifying invariants are axiomatized into the concept of a polyhedral root system.
Example 16.16. Consider G = GL(2,K).
So W (T) = 712.
induces the automorphism a: (i,j) -+ (j,i) on
712.
If W = { 1,(Y},
then a
Also G has an automorphism:
A -4 (A 1)t which induces the automorphism: x -4 -x of 7[2. Thus Renner's theorem
says that there is a one to one correspondence between connected normal regular monoids with zero having G as the group of units and symmetric rational cones (i.e. invariant under (Y) in the plane o 2, and with a cone and its negative identified. See [95] for details.
How about connected regular monoids without zero? By Theorem 7.4, the closure of the radical of the group of units is a completely regular monoid. So the first step is to classify connected regular monoids with a solvable group of units. So let
M be a connected completely regular monoid such that the group of units G is
solvable. Let G = UT where U is the unipotent radical of G and T a maximal Let G1 = UTe, M1 = G1. and E(M) = E(Ml). Then by Theorem 6.7 and Corollary 6.8, e is the zero of torus. Let e denote the minimum idempotent of E(T).
e
The following classification theorem is due to Renner [102].
162
Theorem 16.17. Let G = UT be a solvable group. Let yr denote the set of non-zero weights with respect to the action of T on the Lie algebra of U.
connected diagonal monoid with zero having T
Let T be a normal
as the group of units, such that
yr c X(T) u -%(T). Then there exist a normal connected completely regular monoid
M such that T is the closure of T in
Conversely any normal connected
M.
completely regular monoid with G as the group of units and T having a zero, is obtained in this manner.
Remark 16.18. If G is not nilpotent and dim T r=r 1, then there are exactly two such monoids.
For example, the monoids
M1 = {
I
bl a E K*, b E K}, M2 =
Oa E K*, b E K} have isomorphic groups of `units.
Ilb a 111
Problem 16.19. monoids.
1
J
Study the system of idempotents of connected completely regular
163
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INDEX
adjoint representation 37 affine variety 14 algebraic group 24 algebraic semigroup 18 archimedean semigroup 7
big cell 113, 158 biordered set 121 Borel subgroup 28 Bruhat decomposition 35 center 27, 48 centralizer 27, 48 character 31, 80 closed set 12 completely regular semigroup 6 completely simple semigroup 4 completely 0-simple semigroup 4 congruence 1 conjugate 27, 48 connected semigroup 42 connected group 27 constructible set 13 Coxeter group 40 cross-section lattice 89 diagonal monoid 80 dimension 15 dominant morphism 12 Dynkin diagram 41
'-structure 109
face lattice 80 finite morphism 156 fundamental congruence 10 fundamental regular semigroup 10 Green's relations 2
height 60, 97 homomorphism 1, 18 ideal 4 idempotent separating 9 identity component 27, 42 inverse of an element 1 inverse semigroup 10 involution 1 irreducible component 13
/-co-irreducible 146 ,$-irreducible 146
/-linear 146 kernel 14
left centralizer 48 Levi factor 36 local semilattice 126, 128 monoid 1 morphism 12 Munn semigroup 10
Nambooripad semigroup 124 nearly semisimple group 147 nearly simple group 147 normal variety 156 normalizer 27, 48 opposite Borel subgroups 36 opposite cross-section lattice 94 opposite parabolic subgroups 36 opposite type 39, 130 parabolic semilattice 126 parabolic subgroup 28 polytope 80 projective variety 16
171
quasi-projective variety 16 radical 33 rank 33 reductive group 33 Rees factor semigroup 7 reflection 38
regular f-class 4
regular semigroup 1 Renner monoid 109 right centralizer 48 root subgroups 38 root system 37, 40 sandwich set 74, 121 semilattice union 6 semisimple rank 33 semisimple group 33
simple algebraic group 33 simple reflections 38 strongly n-regular semigroup 1 symmetric inverse semigroup 10 Tits building 135 Tits system 133 torus 28 type 39, 130 unipotent group 30 unipotent radical 33 unit regular 1
weak cross-section lattice 89 Weyl group 32, 133 weight, weight space 37 width 62
