Fundamentals of Hopf Algebras

Universitext

Robert G. Underwood

Fundamentals of Hopf Algebras

Universitext

Universitext Series Editors Sheldon Axler San Francisco State University Vincenzo Capasso Università degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus MacIntyre Queen Mary University of London Kenneth Ribet University of California, Berkeley Claude Sabbah CNRS, École polytechnique, Paris Endre Süli University of Oxford Wojbor A. Woyczynski Case Western Reserve University, Cleveland, OH

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Robert G. Underwood

Fundamentals of Hopf Algebras

123

Robert G. Underwood Department of Mathematics and Computer Science Auburn University at Montgomery Montgomery, AL, USA

ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-319-18990-1 ISBN 978-3-319-18991-8 (eBook) DOI 10.1007/978-3-319-18991-8 Library of Congress Control Number: 2015939602 Mathematics Subject Classification (2010): 11Axx, 12E20, 13Axx, 16T05, 16T10, 16T15 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)

to my mother and father

Preface

The purpose of this text is to provide an introduction to the fundamentals of coalgebras, bialgebras, and Hopf algebras and their applications. These topics are at the forefront of modern algebra today. The target audience for this book is graduate students in mathematics who would like to know more about this fascinating subject. Researchers in bialgebras and Hopf algebras may also find this book useful. The prerequisites for the book include the standard material on groups, rings, modules, algebraic extension fields, finite fields, and linearly recursive sequences found in undergraduate courses on these subjects. The book may be used as the main text or as a supplementary text for a graduate algebra course. The reader is referred to [Ch79, La84, Ro02], and [LN97] for a review of this material. That said, it has been the intention to make this book as self-contained as possible. Most of the proofs are given with ample details; the desire was to make them as transparent as possible. A few proofs have been omitted since they are beyond the scope of the book, and for these I have provided references. The book consists of four chapters. In Chap. 1, we introduce algebras and coalgebras over a field K and show that if C is a coalgebra, then its linear dual C is an algebra. On the other hand, if A is an algebra, then A may not be a coalgebra. If we replace the linear dual with the finite dual Aı  A however, then the algebra structure of A yields a coalgebra structure on Aı . In the case A D KŒx, we show that the collection of linearly recursive sequences of all orders over K can be identified with the coalgebra KŒxı . This suggests the novel problem of finding the image of a linearly recursive sequence fsn g under the comultiplication map of KŒxı . In Chap. 2, we treat bialgebras—vector spaces that are both algebras and coalgebras. We show that if B is a bialgebra, then Bı is a bialgebra. For B D KŒx, we show that there are exactly two bialgebra structures on KŒx. It follows that there are exactly two bialgebra structures on KŒxı , and so, we can multiply linearly recursive sequences in KŒxı in two ways: one way is the Hadamard product and the other is the Hurwitz product. We close Chap. 2 with an application of bialgebras to finite automata, formal languages, and the classical Myhill–Nerode theorem of computer science. Specifically, the Myhill–Nerode theorem is generalized to an algebraic setting in which a certain vii

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function plays the role of the language and a “Myhill–Nerode bialgebra” plays the role of the finite automaton that accepts the language. Included are several examples of Myhill–Nerode bialgebras. We introduce regular sequences as generalizations of kth-order linearly recursive sequences over the Galois field GF.pm /. Chapter 3 concerns Hopf algebras, which are bialgebras with an additional map (the coinverse map) satisfying the coinverse property. We give some examples of Hopf algebras and discuss some of their properties. In many ways, the group ring KG, G a finite group, is the prototypical example of a Hopf algebra in that the group ring is cocommutative and its coinverse has order 2. We introduce the space of integrals of a Hopf algebra H and Hopf modules over H, and state the Fundamental Theorem of Hopf Modules which says that a right Hopf module over H is isomorphic to a trivial right Hopf module. We prove a special Rl case of the Fundamental Theorem: if H is a finite dimensional Hopf algebra and H  Rl is the space of left integrals of H  , then H  Š H  ˝H as right Hopf modules, where Rl H  ˝H has a trivial right Hopf module structure. Next we consider Hopf algebras over rings. Many of the properties of Hopf algebras over fields carry over to rings, including under certain conditions, the Fundamental Theorem of Hopf Modules. We define an R-Hopf order H in KG, where R is an integrally closed, integral domain with field of fractions K, G is a finite group, and KG is the group ring K-Hopf algebra. In many ways, R-Hopf orders H in KG play the role of fractional ideals over R in K. We show that an RHopf order in KG is an R-Hopf algebra with structure maps induced from KG, and give a collection of R-Hopf orders in KCp , where Cp denotes the cyclic group of order p. Hopf orders in group rings will have a role to play in the generalization of Galois extensions in Chap. 4 (§4.5). Chapter 4 consists of three applications of Hopf algebras. The first application concerns quasitriangular structures for bialgebras and Hopf algebras. We show that if H is a quasitriangular Hopf algebra, then there is a solution to the Quantum Yang–Baxter Equation (QYBE). We then introduce an infinite group B called the braid group on three strands, whose defining property is the braid relation. A solution to the QYBE translates to quantities that satisfy the braid relation, and consequently, an n-dimensional quasitriangular Hopf algebra H determines an n3 dimensional representation of the braid group  W B ! GLn3 .K/. The second application relates affine varieties and Hopf algebras. We define affine varieties ƒ over a field K and their coordinate rings KŒƒ and give some examples. The Hilbert Basis Theorem is applied to show that an affine variety ƒ can be identified with the collection of K-algebra homomorphisms HomK-alg .KŒƒ; K/, thus we can think of the variety algebraically, through its coordinate ring. When the coordinate ring is a bialgebra we get a monoid structure on the points of the variety; if the coordinate ring is a Hopf algebra, then there is a group structure on the points of the variety. In this way we give an algebraic structure to a geometric object. In the third application, we show how Hopf algebras can be used to generalize the notion of a Galois extension. A Galois extension L=K with group G is equivalent to the notion that L is a Galois KG-extension of K where the KG action on L is induced

Preface

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from the classical Galois action of G on L. It is in this latter form (L is a Galois KG-extension of K) that the concept of Galois extension can be extended to rings of integers—we give necessary and sufficient conditions for the ring of integers S of L to be a Galois RG-extension of R. Significantly, the concept of Galois KG-extension can be generalized to arbitrary K-Hopf algebras (other than the group ring KG) and to other actions (other than the classical action of G as the Galois group). For instance, we show that the splitting field L=Q of the polynomial x3  2 is a Galois QS3 -extension of Q with the classical Galois action of S3 on L, as well as a Galois H-extension of Q in which H is some other Q-Hopf algebra not the group ring QS3 , whose action on L is different from the classical Galois group action of S3 . Moreover, in the cyclic order p case, G D Cp , we find a Hopf order H in KCp and a Galois extensions L=K with group Cp whose ring of integers S is a Galois H-extension of R where the Galois action of H on S is the classical Galois action of Cp on L. Chapters 1–4 begin with a chapter overview which provides a road map for the reader showing what material will be covered, and each section begins with a brief outline of its contents. At the end of each chapter, we collect exercises which review and reinforce the material in the corresponding sections. These exercises range from straightforward applications of the theory to problems designed to challenge the reader. Occasionally, we include a list of “Questions for Further Study” which pose problems suitable for master’s degree research projects. The idea for this book arose as a precursor to the author’s book An Introduction to Hopf Algebras (Springer, 2011), which treats commutative and cocommutative Hopf algebras over commutative rings with unity. An Introduction shows how these Hopf algebras arise as the representing algebras A of representable group functors F D HomR-alg .A; / on the category of commutative algebras over a commutative ring with unity R. The key result is that if A is a commutative R-algebra, then A ˝ A is the coproduct in the category of commutative R-algebras, and so, if A represents F, then A ˝ A represents F  F. Consequently, by Yoneda’s Lemma an algebra map  W A ! A ˝ A (comultiplication) corresponds to a binary operation F  F ! F. If the binary operation admits an identity element and inverses (again given through algebra maps on A satisfying certain conditions), then F is a group and A is an R-Hopf algebra. However, to put a group structure on HomR-alg .A; S/ we only require S to be commutative; the Hopf algebra A can be non-commutative or non-cocommutative (or both: a quantum group). The point is we don’t have to work exclusively in the category of commutative algebras. Thus our approach here is broader—the Hopf algebras in this work are developed directly from the notions of algebras, coalgebras, and bialgebras; they are not necessarily commutative or cocommutative. I owe incalculable thanks and gratitude to the two readers of earlier versions of this book. Their comments, suggestions, and corrections were invaluable to the shaping of the final manuscript. Unquestionably, without the support of my wife, Rebecca, and my son Andre, I would not have completed this book, and to them I express my greatest thanks and appreciation. I would also like to thank some close friends and colleagues who

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have helped me over the years: Professors Lindsay Childs, Nigel Byott, Warren Nichols, Timothy Kohl, Alan Koch, Griff Elder, Paul Truman, James Carter, Enoch Lee, Matthew Ragland, Luis Cueva-Parra, and Yi Wang. Finally, I thank Ann Kostant and Elizabeth Loew at Springer for their support for this book project. From my initial proposal to the final draft, their advice, guidance and encouragement has been critical to the success of this endeavor. Montgomery, AL, USA

Robert G. Underwood

Contents

1

Algebras and Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Multilinear Maps and Tensor Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Algebras and Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 8 21 33

2

Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction to Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Myhill–Nerode Bialgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Regular Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 48 61 64

3

Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1 Introduction to Hopf Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 Integrals and Hopf Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3 Hopf Algebras over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.4 Hopf Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4

Applications of Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Quasitriangular Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Representations of the Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Hopf Algebras and Affine Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Hopf Algebras and Hopf Galois Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 108 121 124 129 135 142

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

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Notation

N Z ZC Q Q R RC R C m AB AB H
Natural numbers D f1; 2; 3; : : : g Ring of integers Positive integers Field of rational numbers Non-zero rationals Field of real numbers Positive real numbers Non-zero real numbers Complex numbers Primitive mth root of unity A is a proper subset of B A is a subset of B H is a proper subgroup of G H is a subgroup of G H is a normal subgroup of G Order of the finite group G Image of a under canonical surjection A ! A=B Residue class group modulo n, residue class ring modulo n 3rd order dihedral group Symmetric group on n letters Klein 4-group Cyclic group of order m Character group of G Degree of L over K Completion of K at the prime ideal P Valuation ring of KP Uniformizing parameter Ring of p-adic integers Field of p-adic rationals Discriminant of M

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MD Fp GF.pn / V V Aı Rl Rr H, H H.i/ B ƒ KŒƒ

Notation

Dual module Finite field with p elements Galois field with pn elements Vector space Dual space Finite dual Space of left, right integrals of H One parameter Hopf order Braid group Affine variety Coordinate ring of ƒ

Chapter 1

Algebras and Coalgebras

In this chapter we introduce algebras and coalgebras. We begin by generalizing the construction of the tensor product to define the tensor product of a finite collection of R-modules, where R is a commutative ring with unity. We specialize to tensor products over a field K and give the diagram-theoretic definition of a K-algebra .A; mA ; A /. We then define coalgebras .C; C ; c / as co-objects to algebras formed by reversing the arrows in the diagrams for algebras. We next consider the linear dual. We show that if .C; C ; C / is a coalgebra, then .C ; mC ; C / is an algebra where the maps mC and C are induced from the transposes of C and C , respectively. The converse of this statement is not true, however: if A is an algebra, then it is not always true that A is a coalgebra with structure maps induced from the transposes of the maps for A. The trick is to replace A with a certain subspace Aı called the finite dual (in fact, Aı is the largest subspace of A for which mA .Aı /  A ˝ A ). Now, if A is an algebra, then Aı is a coalgebra. As an application we show that the finite dual KŒxı can be identified with the collection of linearly recursive sequences of all orders over K.

1.1 Multilinear Maps and Tensor Products In this section we extend the construction of the tensor product M ˝R N of R-modules. We generalize R-bilinear maps to R-n-linear maps to define the tensor product of a set of R-modules M1 ; M2 ; : : : ; Mn as the solution to a universal mapping problem. We show that tensor products can be identified with iterated tensor products in some association. *

*

*

Let n  2 be an integer, let M1 ; M2 ; : : : ; Mn be a collection of R-modules, and let A be an R-module. © Springer International Publishing Switzerland 2015 R.G. Underwood, Fundamentals of Hopf Algebras, Universitext, DOI 10.1007/978-3-319-18991-8_1

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1 Algebras and Coalgebras

Definition 1.1.1. A function f W M1  M2      Mn ! A is R-n-linear if for all i, 1  i  n, and all ai ; a0i 2 Mi , r 2 R, (i) f .a1 ; a2 ; : : : ; ai Ca0i ; : : : ; an /Df .a1 ; a2 ; : : : ; ai ; : : : ; an /Cf .a1 ; a2 ; : : : ; a0i ; : : : ; an /, (ii) f .a1 ; a2 ; : : : ; rai ; : : : ; an / D rf .a1 ; a2 ; : : : ; ai ; : : : ; an /. For instance, an R-bilinear map is an R-2-linear map. Definition 1.1.2. A tensor product of M1 ; M2 ; : : : ; Mn over R is an R-module M1 ˝ M2 ˝    ˝ Mn together with an R-n-linear map f W M1  M2      Mn ! M1 ˝ M2 ˝    ˝ Mn so that for every R-module A and R-n-linear map h W M1  M2      Mn ! A there Q D h, that exists a unique R-module map hQ W M1 ˝ M2 ˝    ˝ Mn ! A for which hf is, the following diagram commutes.

M 1 × M2 × · · · × Mn

h

- A 

@ @ f @

M1

M2

˜ h

...

Mn

We construct a tensor product as follows. Let FhM1  M2      Mn i denote the free R-module on the set M1  M2      Mn . Let J be the submodule of FhM1  M2      Mn i generated by quantities of the form .a1 ; a2 ; : : : ; ai C a0i ; : : : ; an /  .a1 ; a2 ; : : : ; ai ; : : : ; an /  .a1 ; a2 ; : : : ; a0i ; : : : ; an /; .a1 ; a2 ; : : : ; rai ; : : : ; an /  r.a1 ; a2 ; : : : ; ai ; : : : ; an /; for all i, 1  i  n, and all ai ; a0i 2 Mi , r 2 R. Let W M1  M2      Mn ! FhM1  M2      Mn i be the natural inclusion map and let s W FhM1  M2      Mn i ! FhM1  M2      Mn i=J be the canonical surjection. Let f D s . Then the quotient space FhM1  M2      Mn i=J together with the map f (which is clearly R-n-linear) is a tensor product; it solves the universal mapping problem described in Definition 1.1.2. Proposition 1.1.3. FhM1  M2      Mn i=J together with the map f is a tensor product of M1 ; M2 ; : : : ; Mn over R.

1.1 Multilinear Maps and Tensor Products

3

Proof. We show that the conditions of Definition 1.1.2 are satisfied. Let A be an R-module and let h W M1  M2      Mn ! A be an R-n-linear map. There exists an R-module homomorphism

W FhM1  M2      Mn i ! A; defined as .a1 ; a2 ; : : : ; an / D h.a1 ; a2 ; : : : ; an /; 8.a1 ; a2 ; : : : ; an / 2 M1  M2      Mn . Since is R-n-linear, J  ker. /, and so, by the universal mapping property for kernels, there exists an R-module homomorphism hQ W FhM1  M2      Mn i=J ! A; Q 1 ; a2 ; : : : ; an / C J/ D .a1 ; a2 ; : : : ; an / D h.a1 ; a2 ; : : : ; an /. Now for defined as h..a Q all .a1 ; a2 ; : : : ; an / 2 M1  M2      Mn , hs .a 1 ; a2 ; : : : ; an / D h.a1 ; a2 ; : : : ; an /, Q Q and so, hf D h. Moreover, h is unique because f.a1 ; a2 ; : : : ; an / C J W .a1 ; a2 ; : : : ; an / 2 M1  M2      Mn g is a set of generators for FhM1  M2      Mn i=J.



As a consequence of Proposition 1.1.3, we write FhM1  M2      Mn i=J D M1 ˝ M2 ˝    ˝ Mn ; with the coset .a1 ; a2 ; : : : ; an / C J now written as the tensor a1 ˝ a2 ˝    ˝ an . Proposition 1.1.4. Let M1 ; M2 be R-modules and let N1 be an R-submodule of M1 and let N2 be an R-submodule of M2 . Then there is an isomorphism of R-modules M1 =N1 ˝R M2 =N2 Š .M1 ˝R M2 /=.N1 ˝R M2 C M1 ˝R N2 /: Proof. First note that there is an R-bilinear map h W M1  M2 ! M1 =N1 ˝R M2 =N2 ; defined by h.a; b/ D .a C N1 / ˝ .b C N2 /. Since M1 ˝R M2 is a tensor product there exists an R-module map hQ W M1 ˝R M2 ! M1 =N1 ˝R M2 =N2 ; Q Q ˝ b/ D .a C N1 / ˝ .b C N2 /. Now N1 ˝R M2 C M1 ˝R N2  ker.h/, defined as h.a and so by the universal mapping property for kernels, there exists an R-module map ˛ W .M1 ˝R M2 /=.N1 ˝R M2 C M1 ˝R N2 / ! M1 =N1 ˝R M2 =N2 ; with ˛.a ˝ b C .N1 ˝R M2 C M1 ˝R N2 // D .a C N1 / ˝ .b C N2 /.

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Next, let l W M1 =N1  M2 =N2 ! .M1 ˝R M2 /=.N1 ˝R M2 C M1 ˝R N2 / be the relation defined as l.a C N1 ; b C N2 / D a ˝ b C .N1 ˝R M2 C M1 ˝R N2 /; for a 2 M1 , b 2 M2 . We claim that l is actually a function on M1 =N1  M2 =N2 . To this end, let x D a C n, y D b C n0 for some n 2 N1 , n0 2 N2 . Then l.x C N1 ; y C N2 / D x ˝ y C .N1 ˝R M2 C M1 ˝R N2 / D .a C n/ ˝ .b C n0 / C .N1 ˝R M2 C M1 ˝R N2 / D a ˝ bCa ˝ n0 Cn ˝ bCn ˝ n0 C.N1 ˝R M2 C M1 ˝R N2 /; and thus l.a C N1 ; b C N2 /  l.x C N1 ; y C N2 / D N1 ˝R M2 C M1 ˝R N2 . It follows that l is a well-defined function on M1 =N1  M2 =N2 . As one can easily check, l is R-bilinear and since M1 =N1 ˝R M2 =N2 is a tensor product, there exists an R-module map Ql W M1 =N1 ˝R M2 =N2 ! .M1 ˝R M2 /=.N1 ˝R M2 C M1 ˝R N2 /; defined as Ql..a C N1 / ˝ .b C N2 // D a ˝ b C .N1 ˝R M2 C M1 ˝R N2 /. Clearly ˛ 1 D Ql, and thus Ql is an isomorphism.  Let M1 ; M2 ; M3 be R-modules. Then the associative property for tensor products holds. Proposition 1.1.5. There is an R-module isomorphism M1 ˝ .M2 ˝ M3 / Š .M1 ˝ M2 / ˝ M3 . Proof. Let h W M1 M2 M3 ! .M1 ˝M2 /˝M3 be the map defined by .a1 ; a2 ; a3 / 7! .a1 ˝ a2 / ˝ a3 . Then h.a1 C a01 ; a2 ; a3 / D ..a1 C a01 / ˝ a2 / ˝ a3 D .a1 ˝ a2 C a01 ˝ a2 / ˝ a3 D .a1 ˝ a2 / ˝ a3 C .a01 ˝ a2 / ˝ a3 D h.a1 ; a2 ; a3 / C h.a01 ; a2 ; a3 /;

1.1 Multilinear Maps and Tensor Products

5

and so, h is linear in the first component. Likewise h is linear in the other components. Also, h.ra1 ; a2 ; a3 / D .ra1 ˝ a2 / ˝ a3 D r.a1 ˝ a2 / ˝ a3 D r..a1 ˝ a2 / ˝ a3 / D rh.a1 ; a2 ; a3 /; and so h is R-linear in the first component. Likewise, h is an R-linear in the other components. Thus h is an R-3-linear function. Since M1 ˝ M2 ˝ M3 is a tensor product over R, there exists a unique map of R-modules hQ W M1 ˝ M2 ˝ M3 ! .M1 ˝ M2 / ˝ M3 defined by a1 ˝ a2 ˝ a3 7! .a1 ˝ a2 / ˝ a3 . Clearly hQ is an isomorphism. In a similar manner one constructs an isomorphism gQ W M1 ˝ M2 ˝ M3 ! M1 ˝ .M2 ˝ M3 / defined by a1 ˝ a2 ˝ a3 7! a1 ˝ .a2 ˝ a3 /. Let

W M1 ˝ .M2 ˝ M3 / Š .M1 ˝ M2 / ˝ M3 be the composition D hQ ı gQ 1 given by a1 ˝ .a2 ˝ a3 / 7! .a1 ˝ a2 / ˝ a3 . Then

is an isomorphism of R-modules.  By an “iterated tensor product in some association” we mean a tensor product whose factors themselves may be tensor products or tensor products of tensor products, and so on. For example, for the R-modules M1 ; M2 ; M3 M1 ˝ .M2 ˝ M3 / and .M1 ˝ M2 / ˝ M3 are iterated tensor products in some association. For the R-modules M1 ; M2 ; M3 ; M4 ; M5 .M1 ˝ M2 / ˝ .M3 ˝ M4 ˝ M5 / and

M1 ˝ ..M2 ˝ M3 / ˝ M4 / ˝ M5

are iterated tensor products in some association. There is a “natural” isomorphism between tensor products and iterated tensor products in some association. By natural we mean that the map is defined without using any of the properties of the tensor product—we just add parentheses. For example, there is a natural isomorphism M1 ˝ M2 ˝ M3 ! .M1 ˝ M2 / ˝ M3 ;

6

1 Algebras and Coalgebras

defined by a1 ˝ a2 ˝ a3 7! .a1 ˝ a2 / ˝ a3 and a natural isomorphism M1 ˝ M2 ˝ M3 ˝ M4 ˝ M5 ! M1 ˝ ..M2 ˝ M3 / ˝ M4 / ˝ M5 defined by a1 ˝ a2 ˝ a3 ˝ a4 ˝ a5 7! a1 ˝ ..a2 ˝ a3 / ˝ a4 / ˝ a5 : Proposition 1.1.6. Let M1 ; M2 ; : : : ; Mn be R-modules and let S be an iterated tensor product of M1 ; M2 ; : : : ; Mn in some association. Then there is a natural isomorphism M1 ˝ M2 ˝    ˝ Mn Š S. Proof. We proceed by induction on n. The trivial case n D 2 clearly holds. Assume the result holds for any collection of R-modules with M1 ; M2 ; : : : ; Ms with 2  s < n. There exists an integer r, 2  r < n for which S D T ˝ U where T is an iterated tensor product of M1 ; M2 ; : : : ; Mr in some association, and U is an iterated tensor product of MrC1 ; MrC2 ; : : : ; Mn in some association. By the induction hypothesis, S Š .M1 ˝ M2 ˝    ˝ Mr / ˝ .MrC1 ˝ MrC2 ˝    ˝ Mn /; and so, S Š M1 ˝ M2 ˝    ˝ Mn , see §1.4, Exercise 2.



In view of Proposition 1.1.6 we will “ignore the parentheses” and consider tensor products and iterated tensor products with some association as the same objects through the natural isomorphism. We close this section with a few remarks about maps. Proposition 1.1.7. Let M1 ; M2 ; : : : ; Mn ; M10 ; M20 ; : : : ; Mn0 , be R-modules and let fi W Mi ! Mi0 , for 1  i  n, be R-module maps. There exists a unique map of R-modules .f1 ˝ f2 ˝    ˝ fn / W M1 ˝ M2 ˝    ˝ Mn ! M10 ˝ M20 ˝    ˝ Mn0 defined as .f1 ˝ f2 ˝    ˝ fn /.a1 ˝ a2 ˝    ˝ an / D f1 .a1 / ˝ f2 .a2 / ˝    ˝ fn .an / for all ai 2 Mi . Proof. There exists an R-n-linear map .f1  f2      fn / W M1  M2      Mn ! M10 ˝ M20 ˝    ˝ Mn0 defined as .f1  f2      fn /.a1 ; a2 ; : : : ; an / D f1 .a1 / ˝ f2 .a2 / ˝    ˝ fn .an /

1.1 Multilinear Maps and Tensor Products

7

for all ai 2 Mi . Now use the fact that M1 ˝ M2 ˝    ˝ Mn is a tensor product to induce the map hQ W M1 ˝ M2 ˝    ˝ Mn ! M10 ˝ M20 ˝    ˝ Mn0 ; defined as Q 1 ˝ a2 ˝    ˝ an / D f1 .a1 / ˝ f2 .a2 / ˝    ˝ fn .an / h.a Q for all ai 2 Mi . Set f1 ˝ f2 ˝    ˝ fn D h.



As an illustration of Proposition 1.1.7, take n D 2 with M1 D M ˝ M, M2 D M, f1 W M ˝ M ! M10 , f2 W M ! M20 . Then there is an R-linear map f1 ˝ f2 W .M ˝ M/ ˝ M ! M10 ˝ M20 defined as .f1 ˝ f2 /..a ˝ b/ ˝ c/ D f1 .a ˝ b/ ˝ f2 .c/: But in view of our convention to ignore parentheses, this is the map f1 ˝ f2 W M ˝ M ˝ M ! M10 ˝ M20 defined as .f1 ˝ f2 /.a ˝ b ˝ c/ D f1 .a ˝ b/ ˝ f2 .c/: Corollary 1.1.8. Let K be a field and let Vi , 1  i  n, be a finite set of vector spaces over K. Then V1 ˝ V2 ˝    ˝ Vn  .V1 ˝ V2 ˝    ˝ Vn / : Proof. Let fi W Vi ! K, 1  i  n, be a set of K-module maps (that is, fi 2 Vi ; 8i). By Proposition 1.1.7 (with Vi0 D K), there exists a unique K-module map .f1 ˝ f2 ˝    ˝ fn /.a1 ˝ a2 ˝    ˝ an / D f1 .a1 / ˝ f2 .a2 / ˝    ˝ fn .an / Now, f1 .a1 / ˝ f2 .a2 / ˝    ˝ fn .an / D f1 .a1 /f2 .a2 /    fn .an / 2 K;

8

1 Algebras and Coalgebras

since K ˝K K Š K through the map r ˝ s 7! rs. Consequently, V1 ˝ V2 ˝    ˝ Vn  .V1 ˝ V2 ˝    ˝ Vn / :



We remark that we have equality in Corollary 1.1.8 if and only if each Vi is finite dimensional.

1.2 Algebras and Coalgebras In this section we present the diagram-theoretic definition of a K-algebra .A; mA ; A / and compare this definition to the algebras that most readers already know. We discuss quotient algebras and algebra homomorphisms. Next, we define coalgebras .C; C ; c / as co-objects to algebras formed by reversing the arrows in the diagrams for the algebras and give some basic examples. We introduce notation due to M. Sweedler, “Sweedler notation” to write the image of the comultiplication map and we show how Sweedler notation works to simplify computations. We define coideals, quotient coalgebras, and coalgebra homomorphisms. *

*

*

Let K be a field. Definition 1.2.1. A K-algebra is a triple .A; mA ; A / consisting of a vector space A over K and K-linear maps mA W A ˝K A ! A and A W K ! A that satisfy the following conditions. (i) The diagram commutes:

A

A

mA A

A

IA

mA

IA A ?

- A

A

mA mA

A - ?

Here the map IA W A ! A is the identity map and the maps IA ˝mA W A˝A˝A ! A˝A and mA ˝IA W A˝A˝A ! A˝A are defined by a˝b˝c 7! a˝mA .b˝c/ and a ˝ b ˝ c 7! mA .a ˝ b/ ˝ c, for all a; b; c 2 A, respectively. Equivalently, we have for all a; b; c 2 A mA .IA ˝ mA /.a ˝ b ˝ c/ D mA .mA ˝ IA /.a ˝ b ˝ c/

(1.1)

1.2 Algebras and Coalgebras

9

(ii) The diagrams commute:

A

λA

IA

K

- A  6

A



mA   

s2



A ?

 

s1

λA

IA

K

A

Here the maps IA ˝ A , A ˝ IA are defined by a ˝ r 7! a ˝ A .r/, r ˝ a 7! A .r/ ˝ a for all r 2 K, a 2 A, respectively. The map s1 W K ˝ A ! A is defined by r ˝ a 7! ra and the map s2 W A ˝ K ! A is defined as a ˝ r 7! ra. Equivalently, we have for all r 2 K, a 2 A, mA .IA ˝ A /.a ˝ r/ D ra D mA .A ˝ IA /.r ˝ a/:

(1.2)

The map mA is the multiplication map and the map A is the unit map. Property (1.1) is the associative property and property (1.2) is the unit property. Here is the usual definition of K-algebra. Definition 1.2.10 . A K-algebra is a ring A with unity 1A together with a ring homomorphism A W K ! A which satisfies A .r/a D aA .r/ for a 2 A, r 2 R. Then A is a vector space over K with scalar multiplication given as ra D A .r/a D aA .r/

(1.3)

for r 2 K, a 2 A. Our first task is to show that we really don’t have a new definition of K-algebra. Proposition 1.2.2. K-algebras as defined by Definition 1.2.10 coincide with Kalgebras as defined by Definition 1.2.1. Proof. Let A be a K-algebra as in Definition 1.2.10 . Let B W A  A ! A be the map defined as .a; b/ 7! ab, where ab is the multiplication in the ring A. Now B is a K-bilinear (this follows from the associative and distributive properties of multiplication in A). Since A ˝K A is a tensor product, there exists a unique K-linear map mA W A ˝K A ! A defined as mA

X

 X X a˝b D B.a; b/ D ab:

10

1 Algebras and Coalgebras

Let IA W A ! A denote the identity map. The associative property of the multiplication in A implies that mA .IA ˝ mA /.a ˝ b ˝ c/ D mA .mA ˝ IA /.a ˝ b ˝ c/ for all a; b; c 2 A. Thus the map mA satisfies condition (1.1). We next show that the ring homomorphism A W K ! A satisfies condition (1.2). For r; s 2 K, A .rs/ D A .r/A .s/ D rA .s/, so that A is K-linear. From (1.3), mA .IA ˝ A /.a ˝ r/ D aA .r/ D ra; and mA .A ˝ IA /.r ˝ a/ D A .r/a D ra; which shows that (1.2) holds. Thus the triple .A; mA ; A / is a K-algebra. Conversely, suppose that .A; mA ; A / is a K-algebra as in Definition 1.2.1. Define multiplication on A as ab D mA .a ˝ b/, for a; b 2 A. Then A is a ring. From (1.2) for r 2 K, a 2 A, ra D mA .IA ˝ A /.a ˝ r/ D mA .a ˝ A .r// D aA .r/; and ra D mA .A ˝ IA /.r ˝ a/ D mA .A .r/ ˝ a/ D A .r/a; thus A .K/ is in the center of A. Setting r D 1K , 1A D A .1K /, shows that A is a ring with unity, 1A . Setting a D A .s/ for s 2 K shows that A is a ring homomorphism. Clearly the scalar multiplication on A is defined through A , and so, A is a K-algebra in the sense of Definition 1.2.10 .  To simplify notation we will usually write the K-algebra .A; mA ; A / as A. The K-algebra A is commutative if mA  D mA ; where  denotes the twist map defined as  .a ˝ b/ D b ˝ a for a; b 2 A. Example 1.2.3. The field K is an algebra over itself with mK W K ˝K K ! K defined as r ˝ s 7! rs and K W K ! K given as r 7! r for all r; s 2 K. Example 1.2.4. The polynomial ring KŒx is a K-algebra with mKŒx W KŒx ˝K KŒx ! KŒx given by ordinary polynomial multiplication and KŒx W K ! KŒx defined as r 7! r1, for all r 2 K. Example 1.2.5. Let G be any finite group with identity element 1. The group ring KG is a K-algebra with mKG W KG ˝K KG ! KG defined by g ˝ h 7! gh and KG W K ! KG given as r 7! r1, for all g; h 2 G, r 2 K. Since K is a field, the image

1.2 Algebras and Coalgebras

11

KG .K/ is isomorphic to K; KG contains a copy of K through the identification r D r1. The unit map is then given as KG .r/ D r. Example 1.2.6. Let L D K.˛/ be a simple algebraic extension of K. Then L is a K-algebra with mL W L ˝K L ! L given by multiplication in the field L and L W K ! L defined as r 7! r, for all r 2 K. Clearly, the K-algebras of Examples 1.2.3, 1.2.4, and 1.2.6 are commutative, while KG is a commutative K-algebra if and only if G is abelian. Let A; B be K-algebras. The tensor product A ˝ B has the structure of a K-algebra with multiplication mA˝B W .A ˝ B/ ˝ .A ˝ B/ ! A ˝ B defined by mA˝B ..a ˝ b/ ˝ .c ˝ d// D .mA ˝ mB /.IA ˝  ˝ IB /.a ˝ .b ˝ c/ ˝ d/ D .mA ˝ mB /.a ˝ .c ˝ b/ ˝ d/ D .mA ˝ mB /..a ˝ c/ ˝ .b ˝ d// D ac ˝ bd for a; c 2 A, b; d 2 B. The unit map A˝B W K ! A ˝ B is given as A˝B .r/ D A .r/ ˝ 1B for r 2 K. A K-algebra A is a ring with addition given by vector addition and multiplication defined by mA . Proposition 1.2.7. Let A be a K-algebra and let I be an ideal of A. Then the quotient space A=I is a K-algebra. Proof. We need to define a multiplication map mA=I and a unit map A=I . Let s W A ! A=I denote the canonical quotient map. The composition s ı mA W A ˝ A ! A=I is a map of K-vector spaces defined as .sımA /.a˝b/ D abCI. Note that I˝ACA˝I is a subspace of A ˝ A. Let a ˝ b C c ˝ d 2 I ˝ A C A ˝ I for a; d 2 I, b; c 2 A. Since I is an ideal, mA .a ˝ b C c ˝ d/ D ab C cd 2 I, hence I ˝ A C A ˝ I  ker.s ı mA /. Thus by the universal mapping property for kernels, there is a map of vector spaces s ı mA W .A ˝ A/=.I ˝ A C A ˝ I/ ! A=I

12

1 Algebras and Coalgebras

defined as s ı mA .a ˝ b C .I ˝ A C A ˝ I// D ab C I: By Proposition 1.1.4 there is an isomorphism of vector spaces ˇQ W A=I ˝ A=I ! .A ˝ A/=.I ˝ A C A ˝ I/ given as Q ˇ..a C I/ ˝ .b C I// D a ˝ b C .I ˝ A C A ˝ I/: Q The map Let mA=I denote the composition s ı mA ı ˇ. mA=I W A=I ˝ A=I ! A=I; given as .a C I/ ˝ .b C I/ 7! ab C I, now serves as the multiplication map of A=I. As one can check, mA=I satisfies the associative property since mA does. For the unit map of A=I, let A=I be the composition s ı A W K ! A=I. Then it is easily checked that A=I satisfies the unit property. Thus .A=I; mA=I ; A=I / is a K-algebra.  The K-algebra A=I is the quotient algebra of A by I. Let .A; mA ; A /, .B; mB ; B / be K-algebras. A K-algebra homomorphism from A to B is a map of additive groups W A ! B (that is, .a C b/ D .a/ C .b/, for a; b 2 A) for which .1A / D 1B

.mA .a ˝ b// D mB . .a/ ˝ .b//; and .A .r// D B .r/ for a; b 2 A, r 2 K. In particular, for A to be a subalgebra of B (when is an inclusion) we require that 1A D 1B . We now describe objects that are dual (in some sense) to algebras; essentially forming them by reversing the arrows in the structure maps for algebras. These objects are called “coalgebras.” Let C be a K-vector space. The scalar multiplication of C defines two maps s1 W K ˝ C ! C with r ˝ c 7! rc and s2 W C ˝ K ! C with c ˝ r 7! rc, for c 2 C, r 2 K. Definition 1.2.8. A K-coalgebra is a triple .C; C ; C / consisting of a vector space C over K and K-linear maps C W C ! C ˝K C and C W C ! K that satisfy the following conditions.

1.2 Algebras and Coalgebras

13

(i) The diagram commutes:

ΔC

C

C

-

IC

ΔC IC

ΔC

C ?

C

C

-

?C

ΔC C

C

Here the map IC W C ! C is the identity map and the maps IC ˝ C W C ˝ C ! C˝C˝C and C ˝IC W C˝C ! C˝C˝C are defined by a˝b 7! a˝C .b/ and a ˝ b 7! C .a/ ˝ b, for all a; b 2 C, respectively. Equivalently, we have for all c 2 C .IC ˝ C /C .c/ D .C ˝ IC /C .c/

(1.4)

(ii) The diagrams commute:

1 −

C

-

Z

− C

Z

K ? IC

C

6

Z Z ΔC Z Z Z Z

1

K C

IC

C

C

C

Here the maps  ˝ 1 and 1 ˝  are defined by c 7! c ˝ 1 and c 7! 1 ˝ c, respectively. Equivalently, .C ˝ IC /C .c/ D 1 ˝ c;

.IC ˝ C /C .c/ D c ˝ 1

(1.5)

for all c 2 C. The maps C and C are the comultiplication and counit maps, respectively, of the coalgebra C. Condition (1.4) is the coassociative property and Condition (1.5) is the counit property. A K-coalgebra C is cocommutative if  .C .c// D C .c/; for all c 2 C.

14

1 Algebras and Coalgebras

We use the notation of Sweedler [Sw69, §1.2] to write C .c/ D

X

c.1/ ˝ c.2/ :

.c/

Since s1 .1 ˝ c/ D c D s2 .c ˝ 1/, condition (1.5) implies that X X C .c.1/ /c.2/ D c D C .c.2/ /c.1/ : .c/

(1.6)

.c/

Sweedler notation needs a bit of explanation. Let C be a coalgebra and let c 2 C. Then  X .IC ˝ C /C .c/ D .IC ˝ C / c.1/ ˝ c.2/ D

X

.c/

c.1/ ˝ C .c.2/ /

.c/

D

X

.c;c.2/ /

c.1/ ˝ c.2/ .1/ ˝ c.2/ .2/ ;

and .C ˝ IC /C .c/ D .C ˝ IC / D

X

X

(1.7)

 c.1/ ˝ c.2/

.c/

C .c.1/ / ˝ c.2/

.c/

D

X

.c;c.1/ /

c.1/ .1/ ˝ c.1/ .2/ ˝ c.2/ :

(1.8)

By (1.4), .IC ˝ C /C D .C ˝ IC /C and so, the expressions in (1.7) and (1.8) are equal. The common value in (1.7) and (1.8) is denoted as X

c.1/ ˝ c.2/ ˝ c.3/ :

.c/

Similarly, the common value of .IC ˝ IC ˝ C /.IC ˝ C /C .c/ D .IC ˝ IC ˝ C /.C ˝ IC /C .c/ D .IC ˝ C ˝ IC /.C ˝ IC /C .c/ D .IC ˝ C ˝ IC /.IC ˝ C /C .c/ D .C ˝ IC ˝ IC /.IC ˝ C /C .c/ D .C ˝ IC ˝ IC /.C ˝ IC /C .c/

1.2 Algebras and Coalgebras

is denoted as

15

X

c.1/ ˝ c.2/ ˝ c.3/ ˝ c.4/ :

.c/

The P coassociative property gives a well-defined meaning to the coproduct .c/ c.1/ ˝ c.2/ ˝ c.3/ , just as the associative property in an algebra gives an unambiguous meaning to the product abc since this product is the common value of a.bc/ and .ab/c. Here are some examples of coalgebras. Example 1.2.9. The field K as a vector space over itself is a K-coalgebra where the comultiplication map K W K ! K ˝ K is defined as K .a/ D a ˝ 1 and the counit map K W K ! K is given as K .a/ D a. The K-coalgebra K is the trivial coalgebra. Example 1.2.10. Let x be an indeterminate and let C D K ˚ Kx be the direct sum of vector spaces. Then C is a K-coalgebra with C W C ! C ˝K C defined on the K-basis f1; xg as C .1/ D 1 ˝ 1;

C .x/ D x ˝ x;

C .1/ D C .x/ D 1: Example 1.2.11. Let x be an indeterminate and let C D K ˚ Kx be the direct sum of vector spaces. Then C is a K-coalgebra with C W C ! C ˝K C defined on the K-basis f1; xg as C .1/ D 1 ˝ 1;

C .x/ D 1 ˝ x C x ˝ 1;

C .1/ D 1;

C .x/ D 0:

Example 1.2.12. Let V denote an n-dimensional K-vector space with basis B D fb1 ; b2 ; : : : ; bn g. Let V W V ! V ˝ V be the K-linear map defined on the basis B as V .bi / D bi ˝ bi ; and let V W V ! K be the K-linear map defined on B as V .bi / D 1: Then as one can easily verify, the triple .V; V ; V / is a K-coalgebra. Example 1.2.13. Let KŒx denote the K-vector space of polynomials in the indeterminate x. Let KŒx W KŒx ! KŒx˝KŒx be the K-linear map defined on the K-basis f1; x; x2 ; : : : g as KŒx .xm / D xm ˝ xm ;

16

1 Algebras and Coalgebras

and let KŒx W KŒx ! K be the K-linear map defined on f1; x; x2 ; : : : g as KŒx .xm / D 1: Then the triple .KŒx; KŒx ; KŒx / is a K-coalgebra. Example 1.2.14. Let KŒx denote the K-vector space of polynomials in the indeterminate x. Let KŒx W KŒx ! KŒx˝KŒx be the K-linear map defined on the K-basis f1; x; x2 ; : : : g as ! m X m i KŒx .x / D x ˝ xmi ; i iD0 m

and let KŒx W KŒx ! K be the K-linear map defined on f1; x; x2 ; : : : g as KŒx .xm / D ı0;m : Then the triple .KŒx; KŒx ; KŒx / is a K-coalgebra. This is the divided power coalgebra. We next discuss the analog of an ideal in a ring for a coalgebra. Let C be a K-coalgebra. A subspace I  C is a coideal of C if C .I/  I ˝ C C C ˝ I and C .I/ D 0. Proposition 1.2.15. Let I  C be a coideal of C. Then the quotient space C=I is a K-coalgebra. Proof. Let s W C ˝ C ! .C ˝ C/=.I ˝ C C C ˝ I/ be the canonical surjection, and consider the composition .s ı C / W C ! .C ˝ C/=.I ˝ C C C ˝ I/: Since C .I/  I ˝ C C C ˝ I, I  ker.s ı C /. Thus by the universal mapping property for kernels, there is a map s ı C W C=I ! .C ˝ C/=.I ˝ C C C ˝ I/ defined as s ı C .c C I/ D C .c/ C .I ˝ C C C ˝ I/ for c 2 C. Let ˛ W .C ˝ C/=.I ˝ C C C ˝ I/ ! C=I ˝ C=I

1.2 Algebras and Coalgebras

17

be the isomorphism in the proof of Proposition 1.1.4, defined as ˛.a ˝ b C .I ˝ C C C ˝ I// D .a C I/ ˝ .b C I/; and let C=I W C=I ! C=I ˝ C=I be the composition C=I D ˛ ı s ı C given as X .c.1/ C I/ ˝ .c.2/ C I/: C=I .c C I/ D .c/

Then C=I is K-linear and satisfies the coassociativity property since C does. Moreover, I  ker.C / and so there is a map C = W C=I ! K, defined as C .c C I/ D C .c/. Put C=I D C . Then C=I is K-linear and satisfies the counit property since C does. Thus .C=I; C=I ; C=I / is a K-coalgebra.  The coalgebra C=I is the quotient coalgebra of C by I. Here is a construction of a quotient coalgebra. Let KŒx be the coalgebra of Example 1.2.13. For m  1, let I be the subspace of KŒx generated by the basis fxm  1; xmC1  1; xmC2  1; : : : g: For i  0, KŒx .xi  1/ D xi ˝ xi  1 ˝ 1 D .xi  1 C 1/ ˝ .xi  1 C 1/  1 ˝ 1 D .xi  1/ ˝ .xi  1/ C .xi  1/ ˝ 1 C 1 ˝ .xi  1/ C 1 ˝ 1  1 ˝ 1 D .xi  1/ ˝ 1 C 1 ˝ .xi  1/ C .xi  1/ ˝ .xi  1/: Hence KŒx .I/  I C KŒx C KŒx ˝ I. Also, C .I/ D 0, and so I is a coideal of KŒx. The quotient coalgebra KŒx=I is a vector space of dimension m on the basis f1; x; x2 ; : : : ; xm1 g. It will become evident that KŒx=I is isomorphic to the group ring Hopf algebra KCm . Let C be a K-coalgebra. A non-zero element c of C for which C .c/ D c ˝ c is a grouplike element of C. Proposition 1.2.16. Let c be a grouplike element of the K-coalgebra C. Then C .c/ D 1. Proof. If c is grouplike, then c D s1 .C ˝ IC /C .c/ D s1 .C ˝ IC /.c ˝ c/ D C .c/c:

by (1.6)

18

1 Algebras and Coalgebras

Thus C .c/c D c D 1c, so that .C .c/  1/c D 0. Now since K is a field and c 6D 0, C .c/  1 D 0, hence C .c/ D 1.  In Example 1.2.12, the grouplike elements of V are precisely the basis B, and in Example 1.2.13, the grouplike elements of KŒx consist of 1; x; x2 ; x3 : : : . Proposition 1.2.17. Let KŒx be the K-coalgebra as in Example 1.2.14. Then 1 is the only grouplike element in KŒx. Proof. Suppose a0 C a1 x C a2 x2 C    C an xn is grouplike. Then a0 D 1 by Proposition 1.2.16. Now .1 C a1 x C    C an xn / D 1 ˝ 1 C a1 .1 ˝ x C x ˝ 1/ Ca2 .1 ˝ x2 C 2x ˝ x C x2 ˝ 1/ ! n X n i C    C an x ˝ xni i iD0 D .1 C a1 x C    C an xn / ˝ .1 C a1 x C    C an xn /; whence, am D 0 for 1  m  n.



Proposition 1.2.18. Let C be a coalgebra and let G.C/ denote the set of grouplike elements of C. Then G.C/ is a linearly independent subset of C. Proof. If G.C/ D ;, then G.C/ is linearly independent. If G.C/ contains exactly one grouplike element, then this element is non-zero, and thus G.C/ is linearly independent. So we assume that G.C/ contains at least two elements. Suppose that G.C/ is linearly dependent. Let m  1 be the largest integer for which S D fg1 ; g2 ; : : : ; gm g is a linearly independent subset of G.C/. Then G.C/nS 6D ;, else G.C/ is linearly independent. Let g 2 G.C/nS. Then there exist scalars ri 2 K with g D r1 g1 C r2 g2 C    C rm gm : Since g 6D 0, ri 6D 0 for at least one ri , 1  i  m. Now, C .g/ D g ˝ g D

m X m X

ri rj .gi ˝ gj /

iD1 jD1

At the same time, C .g/ D

m X iD1

and so,

ri .gi ˝ gi /

1.2 Algebras and Coalgebras

19

m X m X

ri rj .gi ˝ gj / D

iD1 jD1

m X

ri .gi ˝ gi /:

iD1

Since fgi ˝ gj g1i;jm is a linearly independent subset of C ˝ C, ri rj D 0 for i 6D j, 1  i; j  m, and ri2 D ri for 1  i  m. Now, for any ri 6D 0, we have rj D 0 for j 6D i, thus ri 6D 0 for exactly one i, and for this i, ri2 D ri implies ri D 1. Thus g D gi , which contradicts our choice of g. It follows that G.C/ is linearly independent.  The linear independence of grouplike elements will be used in §2.2 when we consider Myhill–Nerode bialgebras. Let C; D be coalgebras. A K-linear map W C ! D is a coalgebra homomorphism if . ˝ /C .c/ D D . .c// and C .c/ D D . .c// for all c 2 C. Proposition 1.2.19. Let C be a K-coalgebra. Then the counit map C W C ! K is a homomorphism of K-coalgebras. Proof. For c 2 C, .C ˝ C /C .c/ D .C ˝ IK /.IC ˝ C /C .c/ X  D .C ˝ IK / c.1/ ˝ C .c.2/ / .c/

D .C ˝ IK /.c ˝ 1/ D C .c/ ˝ 1 D K .C .c//: Moreover, C .c/ D K .C .c//; thus C is a coalgebra homomorphism.



A coalgebra homomorphism W C ! D that is injective and surjective is an isomorphism of coalgebras. Proposition 1.2.20. Let W C ! D be a homomorphism of K-coalgebras. If c is a grouplike element of C, then .c/ is a grouplike element of D. Proof. Since is a coalgebra homomorphism, D . .c// D . ˝ /C .c/;

20

1 Algebras and Coalgebras

and since c is grouplike, . ˝ /C .c/ D . ˝ /.c ˝ c/ D .c/ ˝ .c/: Thus .c/ is grouplike.

 0

Let KŒx be the coalgebra as in Example 1.2.13, and let KŒx be the coalgebra as in Example 1.2.14. In view of Proposition 1.2.20, KŒx and KŒx0 are not isomorphic as coalgebras since the only grouplike element of KŒx0 is 1 (Proposition 1.2.17). The tensor product C ˝ D of two coalgebras is again a coalgebra with comultiplication map C˝D W C ˝ D ! .C ˝ D/ ˝ .C ˝ D/ defined by C˝D .c ˝ d/ D .IC ˝  ˝ ID /.C ˝ D /.c ˝ d/ D .IC ˝  ˝ ID /.C .c/ ˝ D .d//   X D .IC ˝  ˝ ID / c.1/ ˝ c.2/ ˝ d.1/ ˝ d.2/ D

X

.c/;.d/

.c.1/ ˝ d.1/ / ˝ .c.2/ ˝ d.2/ /

.c/;.d/

for c 2 C, d 2 D. The counit map C˝D W C ˝ D ! K is defined as C˝D .c ˝ d/ D C .c/D .d/ for c 2 C, d 2 D. Example 1.2.21. Let C be the K-coalgebra of Example 1.2.10 and let D be the Kcoalgebra of Example 1.2.11. Then C ˝K D is a K-coalgebra on the basis f1˝1; 1˝ x; x ˝ 1; x ˝ xg. Comultiplication C˝D W C ˝ D ! .C ˝ D/ ˝ .C ˝ D/ is defined by C˝K D .1 ˝ 1/ D .1 ˝ 1/ ˝ .1 ˝ 1/; C˝K D .1 ˝ x/ D .IC ˝  ˝ ID /.C .1/ ˝ D .x// D .IC ˝  ˝ ID /..1 ˝ 1/ ˝ .1 ˝ x C x ˝ 1// D .IC ˝  ˝ ID /.1 ˝ 1 ˝ 1 ˝ x C 1 ˝ 1 ˝ x ˝ 1/ D .1 ˝ 1/ ˝ .1 ˝ x/ C .1 ˝ x/ ˝ .1 ˝ 1/;

1.3 Duality

21

C˝K D .x ˝ 1/ D .IC ˝  ˝ ID /.C .x/ ˝ D .1// D .IC ˝  ˝ ID /..x ˝ x/ ˝ .1 ˝ 1/ D .x ˝ 1/ ˝ .x ˝ 1/; C˝K D .x ˝ x/ D .IC ˝  ˝ ID /.C .x/ ˝ D .x// D .IC ˝  ˝ ID /..x ˝ x/ ˝ .1 ˝ x C x ˝ 1// D .IC ˝  ˝ ID /.x ˝ x ˝ 1 ˝ x C x ˝ x ˝ x ˝ 1/ D .x ˝ 1/ ˝ .x ˝ x/ C .x ˝ x/ ˝ .x ˝ 1/: The counit map C˝K D W C ˝K D ! K is given as C˝K D .1 ˝ 1/ D 1;

C˝K D .1 ˝ x/ D 0;

C˝K D .x ˝ 1/ D 1;

C˝K D .x ˝ x/ D 0:

1.3 Duality In this section we consider the linear duals of algebras and coalgebras. We show that if .C; C ; C / is a coalgebra, then .C ; mC ; C / is an algebra, where the maps mC and C are induced from the transpose of C and C , respectively. We ask whether the converse of this statement is true (it isn’t). However, if we replace A with a certain subspace Aı called the finite dual, then Aı is a coalgebra whenever A is an algebra. We show that the finite dual KŒxı can be identified with the collection of linearly recursive sequences of all orders over K. *

*

*

Let C be a K-coalgebra and let C be its linear dual. Proposition 1.3.1. If C is a coalgebra, then C is an algebra. Proof. To show that C is a K-algebra we construct a multiplication map mC and a unit map C and show that they satisfy the associative and unit properties, respectively. Recall that C is a triple .C; C ; C / where C W C ! C ˝ C is K-linear and satisfies the coassociativity property (1.4), and C W C ! K is K-linear and satisfies the counit property (1.5). The transpose of C is a K-linear map C W .C ˝ C/ ! C

22

1 Algebras and Coalgebras

defined as C . /.c/ D

.C .c//;

2 .C ˝ C/ , c 2 C. By Corollary 1.1.8, C ˝ C  .C ˝ C/ . Thus C restricts to a K-linear map mC W C ˝ C ! C defined as for

mC .f ˝ g/.c/ D C .f ˝ g/.c/ D .f ˝ g/.C .c// X D f .c.1/ /g.c.2/ /: .c/

for f ; g 2 C , c 2 C. Let IC W C ! C be the identity map and let IC ˝ mC W C ˝ C ˝ C ! C ˝ C ; be the map defined by f ˝ g ˝ h 7! f ˝ mC .g ˝ h/; and let mC ˝ IC W C ˝ C ˝ C ! C ˝ C ; be the map defined by f ˝ g ˝ h 7! mC .f ˝ g/ ˝ h; for f ; g; h 2 C . We are now ready to show that mC satisfies the associative property. But this follows from the coassociative property of C ! Indeed, for f ; g; h 2 C , c 2 C, mC .IC ˝ mC /.f ˝ g ˝ h/.c/ D C .IC ˝ C /.f ˝ g ˝ h/.c/ D C .f ˝ C .g ˝ h//.c/ D .f ˝ C .g ˝ h//C .c/ X f .c.1/ /C .g ˝ h/.c.2/ / D .c/

D

X

f .c.1/ /.g ˝ h/C .c.2/ /

.c/

D

X .c/

f .c.1/ /

X .c.2/ /

g.c.2/ .1/ /h.c.2/ .2/ /:

1.3 Duality

23

Now, X X X f .c.1/ / g.c.2/ .1/ /h.c.2/ .2/ / D f .c.1/ /g.c.2/ .1/ /h.c.2/ .2/ / .c/

.c.2/ /

.c;c.2/ /

0

D .f ˝ g ˝ h/ @ 0 D .f ˝ g ˝ h/ @ D

.c;c.2/ /

X .c;c.1/ /

X .c;c.1/ /

D

X

1 c.1/ ˝ c.2/ .1/ ˝ c.2/ .2/ A 1 c.1/ .1/ ˝ c.1/ .2/ ˝ c.2/ A

f .c.1/ .1/ /g.c.1/ .2/ /h.c.2/ /

XX .c/ .c.1/ /

f .c.1/ .1/ /g.c.1/ .2/ /h.c.2/ /:

Observe that the coassociativity of C is applied to get us from line 2 to line 3 above. We have established that XX mC .IC ˝ mC /.f ˝ g ˝ h/.c/ D f .c.1/ .1/ /g.c.1/ .2/ /h.c.2/ /: .c/ .c.1/ /

To finish the calculation: mC .IC ˝ mC /.f ˝ g ˝ h/.c/ D

XX .c/ .c.1/ /

D

X

f .c.1/ .1/ /g.c.1/ .2/ /h.c.2/ /

.f ˝ g/C .c.1/ /h.c.2/ /

.c/

D

X

C .f ˝ g/.c.1/ /h.c.2/ /

.c/

D .C .f ˝ g/ ˝ h/C .c/ D C .C .f ˝ g/ ˝ h/.c/ D C .C ˝ IC /.f ˝ g ˝ h/.c/ D mC .mC ˝ IC /.f ˝ g ˝ h/.c/: Thus mC satisfies the associative property. The transpose of the counit map of C is C W K  ! C

24

1 Algebras and Coalgebras

defined as C .f /.c/ D f .C .c// for f 2 K  , c 2 C. Identifying K D K  , we have C W K ! C defined as C .r/.c/ D r.C .c// D rC .c/; for r 2 K, c 2 C. Set C D C and define maps IC ˝ C W C ˝ K ! C ˝ C , f ˝ r 7! f ˝ C .r/, C ˝ IC W K ˝ C ! C ˝ C , r ˝ f 7! C .r/ ˝ f , for f 2 C , r 2 K. We now show that the counit property of C implies the unit property for C . To this end, for f 2 C , r 2 K, c 2 C, mC .IC ˝ C /.f ˝ r/.c/ D C .IC ˝ C /.f ˝ r/.c/ D C .f ˝ C .r//.c/ D .f ˝ C .r//.C .c// X f .c.1/ /C .r/.c.2/ / D .c/

D

X

f .c.1/ /r.C .c.2/ //:

.c/

Thus, X

mC .IC ˝ C /.f ˝ r/.c/ D r

f .c.1/ /C .c.2/ /:

.c/

Proceeding with the calculation: mC .IC ˝ C /.f ˝ r/.c/ D r

X

f .c.1/ /C .c.2/ /

.c/

Dr

X

C .c.2/ /f .c.1/ /

.c/

Dr

X

f .C .c.2/ /c.1/ /

.c/

D rf

X

 C .c.2/ /c.1/

.c/

D rf .c/: Note that the counit property of C is applied to get us from line 4 to line 5 above. In a similar manner, we obtain mC .C ˝ IC /.r ˝ f / D rf :

1.3 Duality

25

Thus C satisfies the unit property and we have established that .C ; mC ; C / is an algebra. We have C .1K /.c/ D C .c/; 8c, and so, C is the unique element of  C for which C f D f D f C for all f 2 C . As we have just seen, if .C; C ; C / is a coalgebra, then .C ; mC ; C / is an algebra. Now suppose .A; mA ; A / is a K-algebra. Then one may wonder if A is a K-coalgebra. The transpose of the multiplication map mA W A ˝K A ! A is mA W A ! .A ˝K A/ . However, if A is infinite dimensional over K, then A ˝K A is a proper subset of .A ˝K A/ . Hence, in the case that A is infinite dimensional, mA may not induce the required comultiplication map A ! A ˝K A . Indeed, here is an example where mA .A / 6 A ˝K A . Example 1.3.2. Let A D KŒx be the K-algebra of polynomials in x. Let mKŒx W KŒx ˝K KŒx ! KŒx be the multiplication map with transpose mKŒx W KŒx ! .KŒx ˝K KŒx/ . Let fsn g be the sequence in K given as 1; 0; 0; 1; 1; 1; 0; 0; 0; 0; : : : (that is, one 1, followed by two 0’s, followed by three 1’s, and so on). Let f be the element in KŒx defined as f D 1e0 C 0e1 C 0e2 C 1e3 C 1e4 C 1e5 C 0e6 C 0e7 C 0e8 C 0e9 C 1e10 C 1e11 C 1e12 C 1e13 C 1e14 C    where ei .xj / D ıi;j , i; j  0. Then mKŒx .f / 62 KŒx ˝K KŒx . In order for the transpose mA to serve as a comultiplication map we must replace A with a certain subspace of A , called the finite dual Aı of A . Let .A; ma ; A / be a K-algebra. Then A is a K-vector space and a ring. An ideal I of A has finite codimension (or is cofinite) if the quotient space A=I is finite dimensional. Let f be an element of A and let S be a subset of A. Then vanishes on S if f .s/ D 0 for all s 2 S. Definition 1.3.3. Let A be a K-algebra. The finite dual Aı of A is the subspace of A defined as Aı D ff 2 A W f vanishes on some ideal I  A of finite codimensiong Example 1.3.4. Let ei 2 KŒx be defined as ei .xj / D ıi;j for i; j  0. Then ei 2 KŒxı since ei vanishes on the ideal .xiC1 / and dim.KŒx=.xiC1 // D i C 1. Proposition 1.3.5. If A is finite dimensional as a K-vector space, then Aı D A . Proof. Exercise.



Our goal is to show that if A is an algebra, then Aı is a coalgebra (Proposition 1.3.9). We need three propositions.

26

1 Algebras and Coalgebras

Proposition 1.3.6. Let I be an ideal of A and let s W A ! A=I be the canonical surjection of vector spaces. Let s W .A=I/ ! A be the transpose defined as s .f /.a/ D f .s.a//, for all f 2 .A=I/ , a 2 A. Then s is an injection. Proof. Let f ; g 2 .A=I/ . Then for a 2 A, s .f /.a/ D s .g/.a/ implies that f .s.a// D g.s.a//. Thus f .a C I/ D g.a C I/, and so f D g.  Proposition 1.3.7. Let f 2 Aı and suppose that f vanishes on the ideal I of A. Then there exists a unique element f 2 .A=I/ for which s .f / D f . Proof. The element f 2 Aı  A is a K-module homomorphism f W A ! K and since f .I/ D 0, I  ker.f /. Let s W A ! A=I denote the canonical surjection. Then by the universal mapping property for kernels there exists a K-module homomorphism f W A=I ! K for which f .s.a// D f .a/ for all a 2 A. Thus s .f /.a/ D f .a/ for all a 2 A.  Let A be a K-algebra with multiplication map mA W A ˝ A ! A. Let mA W A ! .A ˝ A/ be the transpose map defined as mA .f /.a ˝ b/ D f .mA .a ˝ b// D f .ab/: Let Aı  A be the finite dual of A. Proposition 1.3.8. mA .Aı /  Aı ˝ Aı . Proof. Let f 2 Aı . Then f vanishes on some ideal I  A with dim.A=I/ < 1. By Proposition 1.3.7 there exists a unique element f 2 .A=I/ for which s .f / D f . Let mA=I W A=I ˝ A=I ! A=I; .a C I/ ˝ .b C I/ 7! ab C I, be the multiplication map of the K-algebra A=I, cf. Proposition 1.2.7. The transpose of mA=I is mA=I W .A=I/ ! .A=I ˝ A=I/ ; which, since A=I is finite dimensional, becomes mA=I W .A=I/ ! .A=I/ ˝ .A=I/ : Now, mA .f /.a ˝ b/ D mA .s .f //.a ˝ b/ D s .f /.mA .a ˝ b// D s .f /.ab/ D f .s.ab// D f .ab C I/ D f .mA=I ..a C I/ ˝ .b C I///:

1.3 Duality

27

Continuing with this calculation, we obtain mA .f /.a ˝ b/ D f .mA=I ..a C I/ ˝ .b C I/// D mA=I .f /..a C I/ ˝ .b C I// D mA=I .f /.s.a/ ˝ s.b//: Note that mA=I .f / D

Pm

iD1 fi

˝ gi for elements fi ; gi 2 .A=I/ . Thus,

mA .f /.a ˝ b/ D mA=I .f /.s.a/ ˝ s.b//  X m fi ˝ gi .s.a/ ˝ s.b// D iD1

D

m X

fi .s.a// ˝ gi .s.b//

iD1

D

m X

s .fi /.a/ ˝ s .gi /.b/

iD1

D

X m

 s .fi / ˝ s .gi / .a ˝ b/;

iD1

Thus mA .f / D

m X

s .fi / ˝ s .gi /:

iD1

It remains to show that s .fi /; s .gi / 2 Aı . To this end, let q 2 I, then s .fi /.q/ D fi .s.q// D fi .I/ D 0 2 K, and so s .fi / vanishes on I, hence s .fi / 2 Aı . A similar argument shows that s .gi / 2 Aı . Consequently, mA .f /  Aı ˝ Aı .  The transpose of the map IA ˝ mA W A ˝ A ˝ A ! A ˝ A restricted to A ˝ A is the map IA ˝ mA D IA ˝ mA W A ˝ A ! .A ˝ A ˝ A/ : Likewise, the transpose of mA ˝ IA restricted to A ˝ A is mA ˝ IA D mA ˝ IA W A ˝ A ! .A ˝ A ˝ A/ :

28

1 Algebras and Coalgebras

Proposition 1.3.9. If A is an algebra, then Aı is a coalgebra. Proof. Let mA W A ! .A ˝ A/ be the transpose of the multiplication map mA . By Proposition 1.3.8, mA .Aı /  Aı ˝ Aı . Let Aı denote the restriction of mA to Aı . Then Aı W Aı ! Aı ˝ Aı is a K-linear map defined as Aı .f / D mA .f / for f 2 Aı . The first step is to show that the associative property of mA implies the coassociative condition for Aı . For f 2 Aı , a; b; c 2 A, we have .IAı ˝ Aı /Aı .f /.a ˝ b ˝ c/ D .IA ˝ mA /mA .f /.a ˝ b ˝ c/ D mA .f /.IA ˝ mA /.a ˝ b ˝ c/ D mA .f /.a ˝ bc/ D f .mA .a ˝ bc// D f .a.bc// D f ..ab/c/: We have applied the associative property of mA to move from line 5 to line 6 above. Now, .IAı ˝ Aı /Aı .f /.a ˝ b ˝ c/ D f ..ab/c/ D f .mA .ab ˝ c// D mA .f /.ab ˝ c/ D mA .f /.mA ˝ IA /.a ˝ b ˝ c/ D .mA ˝ IA /mA .f /.a ˝ b ˝ c/ D .Aı ˝ IAı /Aı .f /.a ˝ b ˝ c/; and so, the cooassociative property holds for Aı . For the counit map of Aı , we consider the dual map A W A ! K  D K. We let Aı W Aı ! K denote the restriction of A to Aı . For f 2 Aı , r 2 K, Aı .f /.r/ D f .A .r// D f .r1A / D rf .1A / D f .1A /.r/ and so, Aı .f / D f .1A /. We show that the unit property of A implies that Aı satisfies the counit property. For f 2 Aı , r 2 K, a 2 A, .Aı ˝ IAı /Aı .f /.r ˝ a/ D .A ˝ IA /mA .f /.r ˝ a/ D mA .f /.A ˝ IA /.r ˝ a/ D f .mA .A ˝ IA /.r ˝ a// D f .ra/

1.3 Duality

29

D rf .a/ D .1 ˝ f /.r ˝ a/: The reader should note that we have applied the unit property of A to move from line 3 to line 4 above. Thus .Aı ˝ IAı /Aı .f / D 1 ˝ f : In a similar manner, one obtains .IAı ˝ Aı /Aı .f / D f ˝ 1: Thus Aı satisfies the counit condition and it follows that .Aı ; Aı ; A0 / is a coalgebra.  If A is a K-algebra that is finite dimensional as a K vector space, then Aı D A and A is a K-coalgebra. For a; b 2 A, f 2 A , we have the formula f .ab/ D f .mA .a ˝ b// D mA .f /.a ˝ b/  X D f.1/ ˝ f.2/ .a ˝ b/ .f /

D

X

f.1/ .a/f.2/ .b/:

(1.9)

.f /

Here is an application of formula (1.9). Let S be a finite monoid, jSj D m, and let KS be the monoid ring. The K-algebra KS is finite dimensional over K on the basis f1 D g0 ; g1 ; g2 ; : : : ; gm1 g. Now KSı D KS with K-basis fe1 ; eg1 ; : : : ; egm1 g, egi .gj / D ıi;j . Proposition 1.3.10. Let .KS ; KS ; KS / be the coalgebra as in the preceding text. Then X egj ˝ egk ; KS .egi / D gj gk Dgi

for i D 0; 1; : : : ; m  1. Proof. Note that fegj ˝ egk g, 0  j; k  m  1, is a K-basis for KS ˝ KS . Fix i, 0  i  m  1. There exist elements ai;j0 ;k0 2 K for which KS .egi / D

m1 X j0 ;k0 D0

ai;j0 ;k0 .egj0 ˝ egk0 /:

30

1 Algebras and Coalgebras

Let j; k be so that gi D gj gk . Then by (1.9), 1 D egi .gi / D egi .gj gk / m1 X

D

ai;j0 ;k0 egj0 .gj /egk0 .gk /

j0 ;k0 D0

D ai;j;k : If i; j are so that gi 6D gj gk , then 0 D egi .gj gk / D

m1 X

ai;j0 ;k0 egj0 .gj /egk0 .gk /

j0 ;k0 D0

D ai;j;k : It follows that KS .egi / D

X

egj ˝ egk ;

gj gk Dgi

for i D 0; 1; : : : ; m  1.



Also: Proposition 1.3.11. If .A; mA ; A / be a commutative algebra, then .Aı ; Aı ; Aı / is a cocommutative coalgebra. If .C; C ; C / is a cocommutative coalgebra, then .C ; mC ; C / is a commutative algebra. Proof. Let  W A ˝ A ! A ˝ A, a ˝ b 7! b ˝ a, be the twist map. The transpose   restricted to A ˝ A is the twist map on A ˝ A . Now, to prove the first statement, let f 2 Aı , a; b 2 A. Then Aı .f /.a ˝ b/ D f .mA .a ˝ b// D f .mA . .a ˝ b// D   .Aı .f //.a ˝ b/ D  .Aı .f //.a ˝ b/; and so, Aı is cocommutative. The second statement is left as an exercise.



Let KŒx denote the algebra of polynomials over the field K. As in Example 1.3.2, the collection of sequences fsn g1 nD0 over K can be identified with the linear P1 dual corresponds to the infinite sum s D KŒx . Indeed, the sequence fsn g1 nD0 sn en nD0

1.3 Duality

31

where ei .xj / D ıi;j ; 8i; j. The next proposition ties the development of the finite dual to linearly recursive sequences in a rather elegant way. Proposition 1.3.12. Let K be a field. The collection of kth-order linearly recursive sequences over K of all orders k > 0 can be identified with the finite dual KŒxı . Proof. Let fsn g be a kth-order linearly recursive sequence over K of order k > 0 with recurrence relation snCk D ak1 snCk1 C ak2 snCk2 C    C a1 snC1 C a0 sn ;

n  0;

(1.10)

and characteristic polynomial f .x/ D xk  ak1 xk1  ak2 xk2      a1 x  a0 ; for a0 ; a1 ; a2 ; : : : ; ak1 2 K. We identify fsn g with the element s D KŒx . Now, s.f .x// D

X 1

(1.11) P1

nD0 sn en

2

 sn en .xk  ak1 xk1  ak2 xk2      a1 x  a0 /

nD0

D sk  ak1 sk1  ak2 sk2      a1 s1  a0 s0 D 0: It follows that s.g.x/f .x// D 0 for all g.x/ 2 KŒx, and so s vanishes on the principal ideal I D .f .x// of KŒx. One hasP dim.KŒx=I/ D k and so s 2 KŒxı . 1 ı On the other hand, let s D nD0 sn en 2 KŒx . Then s vanishes on an ideal I  KŒx of finite codimension. Since KŒx is a PID, I D .f .x// for some monic polynomial over K of degree k. Then as one can easily check, fsn g is a kth-order linearly recursive sequence over K with characteristic polynomial f .x/.  By Proposition 1.3.9, KŒxı , the linearly recursive sequences over K of all orders, is a coalgebra .KŒxı ; KŒxı ; KŒxı /. We ask: given s D fsn g 2 KŒxı , how does one compute KŒxı .s/ 2 KŒxı ˝ KŒxı ? Suppose s vanishes on the ideal I D .f .x// of KŒx where f .x/ is the characteristic polynomial of s. Let c W KŒx ! KŒx=.f .x// denote the canonical surjection. In view of the method used in the proof of Proposition 1.3.8, we first find s 2 .KŒx=.f .x/// so that c .s/ D s. We next compute mKŒx=.f .x// .s/ D

m X

fi ˝ gi ;

iD1

where fi ; gi 2 .KŒx=.f .x/// . Finally, we have KŒxı .s/ D

m X iD1

c .fi / ˝ c .gi / 2 KŒxı ˝ KŒxı :

32

1 Algebras and Coalgebras

To see how this works in a modest example, take K D GF.2/. Let s D fsn g be the 2nd-order linearly recursive sequence in K with characteristic polynomial f .x/ D x2 C x C 1 and initial state vector s0 D 11 (a Fibonacci sequence). Let ˛ be a zero of f .x/. Then f1; ˛g is a K-basis for KŒx=.f .x// D GF.4/ and f"1 ; "˛ g is a K-basis for .KŒx=.f .x/// D GF.4/ with "˛i .˛ j / D ıi;j , 0  i; j  1. Now, with fei g so that ei .xj / D ıi;j for i; j  0, the sequence s D fsn g can be written as an element of KŒxı : s D 1  e0 C 1  e1 C 0  e2 C 1  e3 C 1  e4 C    Let s D 1  "1 C 1  "˛ . We claim that c .s/ D s. To this end observe that for all j  0, c .s/.xj / D s.c.xj // D s.xj C .f .x/// D s.˛ j / D sj ; and so, c .s/ D s. Our next step is to compute mGF.4/ .s/. But note that mGF.4/ .s/ D mGF.4/ ."1 / C mGF.4/ ."˛ /; and so this depends on the computation of mGF.4/ ."1 / and mGF.4/ ."˛ / To compute mGF.4/ ."1 / and mGF.4/ ."˛ / we use the formula (1.9) and the idea of Proposition 1.3.10. First note that mGF.4/ ."1 / D c0;0;0 ."1 ˝ "1 / C c0;0;1 ."1 ˝ "˛ / C c0;1;0 ."˛ ˝ "1 / C c0;1;1 ."˛ ˝ "˛ /: for some bits c0;i;j , 0  i; j  1. Thus, "1 .ab/ D c0;0;0 "1 .a/"1 .b/ C c0;0;1 "1 .a/"˛ .b/ C c0;1;0 "˛ .a/"1 .b/ C c0;1;1 "˛ .a/"˛ .b/; for all a; b 2 GF.4/. Now since 1 D 1  1 D ˛.1 C ˛/ D .1 C ˛/˛; and ˛ D 1  ˛ D ˛  1 D .1 C ˛/.1 C ˛/; in GF.4/, we conclude that mGF.4/ ."1 / D "1 ˝ "1 C "˛ ˝ "˛ :

1.4 Chapter Exercises

33

By a similar method we also obtain mGF.4/ ."˛ / D "1 ˝ "˛ C "˛ ˝ "1 C "˛ ˝ "˛ : Thus mGF.4/ .s/ D "1 ˝ "1 C "1 ˝ "˛ C "˛ ˝ "1 : Finally, KŒxı .s/ D c ."1 / ˝ c ."1 / C c ."1 / ˝ c ."˛ / C c ."˛ / ˝ c ."1 / D r ˝ r C r ˝ t C t ˝ r; where r D frn g is the Fibonacci sequence in GF.2/ with initial state vector r0 D 10, and t D ftn g is the Fibonacci sequence in GF.2/ with initial state vector t0 D 01.

1.4 Chapter Exercises Exercises for §1.1 1. Let r  1 and let M1 ; M2 ; : : : ; Mr ; MrC1 be R-modules. Show that M1 ˝ M2 ˝    ˝ Mr ˝ MrC1 Š .M1 ˝ M2 ˝    ˝ Mr / ˝ MrC1 2. Let r; s  1 and let M1 ; M2 ; : : : ; MrCs be R-modules. Use induction on s to prove that M1 ˝ M2 ˝    ˝ MrCs Š .M1 ˝ M2 ˝    ˝ Mr / ˝ .MrC1 ˝ MrC2 ˝    ˝ MrCs /: (The trivial case s D 1 is Exercise 1.) 3. Let f1 W V1 ! V10 and f2 W V2 ! V20 be linear transformations of vector spaces and let .f1 ˝ f2 / W V1 ˝ V2 ! V10 ˝ V20 be the map defined as a ˝ b 7! f1 .a/ ˝ f2 .b/. Prove that the transpose .f1 ˝ f2 / W .V10 ˝ V20 / ! .V1 ˝ V2 / restricted to .V10 / ˝ .V20 / is the map f1 ˝ f2 . Exercises for §1.2 4. Let .A; mA ; A / be a K-algebra, let I be an ideal of A, and let .A=I; mA=I ; A=I / be the quotient algebra. Verify that the maps mA=I and A=I satisfy the associative and unit properties, respectively.

34

1 Algebras and Coalgebras

5. Let K be a field, let C be a K-coalgebra, and let c 2 C. Prove that .IC ˝ s1 /.IC ˝ C ˝ IC /

X .c/

D .s1 ˝ IC /.C ˝ IC ˝ IC /

 c.1/ ˝ c.2/ ˝ c.3/

X

 c.1/ ˝ c.2/ ˝ c.3/ :

.c/

6. Let .KŒx; KŒx ; KŒx / be the coalgebra of Example 1.2.14. Verify that the maps KŒx and KŒx satisfy the coassociative and counit properties, respectively. 7. Let .C; C ; C / be a coalgebra with C .C/ D K. Let I D ker.C /. Prove that I is a coideal. 8. Let I and J be coideals of the coalgebra C. Show that I C J D fa C b W a 2 I; b 2 Jg is a coideal of C. 9. Let .C; C ; C / be a K-coalgebra, let I be a coideal of C and let .C=I; C=I ; C=I / be the quotient coalgebra. Verify that the maps C=I and C=I satisfy the coassociative and counit properties, respectively. 10. Let C be a K-coalgebra. Define a coalgebra structure on C ˝ C ˝ C in two different ways. Are the resulting K-coalgebras isomorphic as coalgebras? Exercises for §1.3 11. Prove Proposition 1.3.5. 12. Prove the second statement of Proposition 1.3.11. 13. Let K D GF.2/ and let fsn g be a 3rd-order linearly recursive sequence in GF.2/ with recurrence relation snC3 D snC1 C sn ; characteristic polynomial f .x/ D x3 C x C 1 and initial state vector s0 D 111. Compute KŒxı .s/ 2 KŒxı ˝ KŒxı . 14. Let K D GF.2/ and let fsn g be a 5th-order linearly recursive sequence in GF.2/ with recurrence relation snC5 D sn ; characteristic polynomial f .x/ D x5 C 1 and initial state vector s0 D 10001. Compute KŒxı .s/ 2 KŒxı ˝ KŒxı .

Chapter 2

Bialgebras

In this chapter we consider bialgebras—vector spaces that are both algebras and coalgebras. We give some basic examples and show that if B is a bialgebra, then Bı is a bialgebra. We show that KŒx is a bialgebra in exactly two distinct ways, and so KŒxı is a bialgebra in two distinct ways. Consequently, we can multiply linearly recursive sequences in two different ways, namely, the Hadamard product and the Hurwitz product. Next we give an application of bialgebras to theoretical computer science. We introduce finite automata, and prove the Myhill–Nerode theorem which tells us precisely when a language is accepted by a finite automaton. We then generalize the Myhill–Nerode theorem to an algebraic setting in which a certain finite dimensional bialgebra (a Myhill–Nerode bialgebra) plays the role of the finite automaton that accepts the language. We see that a Myhill–Nerode bialgebra determines a finite automaton and a finite automaton determines a Myhill–Nerode bialgebra. We can think of finite automata and languages in terms of the algebraic properties of their Myhill–Nerode bialgebras. For instance, if two languages determine isomorphic Myhill–Nerode bialgebras, then these languages are related in some manner. In the final section of the chapter, we introduce regular sequences; these are sequences that generalize linearly recursive sequences over a Galois field.

2.1 Introduction to Bialgebras In this section we introduce bialgebras and define biideals, quotient bialgebras, and bialgebra homomorphisms. We show how a bialgebra B can act on an algebra A giving A the structure of a left B-module algebra, and how the bialgebra can act on a coalgebra C so that C is a right B-module coalgebra. We define a certain right action of B on the algebra B , the right translate f ( a of f 2 B by a 2 B. (In fact, the right translate action endows B with the structure of a right B-module © Springer International Publishing Switzerland 2015 R.G. Underwood, Fundamentals of Hopf Algebras, Universitext, DOI 10.1007/978-3-319-18991-8_2

35

36

2 Bialgebras

algebra.) We state an important result which says that dim.f ( B/ < 1 if and only if f 2 Bı . As a consequence we show that if B is a bialgebra, then the finite dual Bı is a bialgebra. We show that KŒx is a bialgebra in exactly two distinct ways, thus KŒxı has two distinct structures as a bialgebra. The resulting multiplications on KŒxı are the Hadamard product and the Hurwitz product of linearly recursive sequences. *

*

*

Definition 2.1.1. A K-bialgebra is a K-vector space B together with maps mB , B , B , B that satisfy the following conditions: (i) .B; mB ; B / is a K-algebra and .B; B ; B / is a K-coalgebra, (ii) B and B are homomorphisms of K-algebras. The requirement that B W B ! B ˝ B be an algebra homomorphism implies that B .ab/ D

X .ab/.1/ ˝ .ab/.2/ .ab/

D B .a/B .b/ 10 1 0 X X a.1/ ˝ a.2/ A @ b.1/ ˝ b.2/ A D@ .a/

D

X

.b/

a.1/ b.1/ ˝ a.2/ b.2/ ;

.a;b/

for a; b 2 B. Let B be a bialgebra. An element b 2 B for which B .b/ D 1 ˝ b C b ˝ 1 is a primitive element of B. Example 2.1.2. Let K be a field, let S be a monoid. Then the monoid ring KS is a K-algebra .KS; mKS ; KS / with multiplication map mKS W KS ˝ KS ! KS defined as mKS .a ˝ b/ D ab and unit map A W K ! KS given as KS .r/ D r for all a; b 2 KS, r 2 K. Let KS W KS ! KS ˝ KS be the map defined as KS

X

 rs s D

X

s2S

rs .s ˝ s/;

s2S

and let KS W KS ! K be the map defined by KS

X s2S

 X rs s D rs : s2S

2.1 Introduction to Bialgebras

37

Then .KS; KS ; KS / is a K-coalgebra. Moreover, as one can easily verify, KS and KS are homomorphisms of K-algebras and so .KS; mKS ; KS ; KS ; KS / is a K-bialgebra called the monoid bialgebra. Example 2.1.3. Let KŒx be the K-algebra of polynomials in the indeterminate x. From Example 1.2.13, KŒx has the structure of a coalgebra, with maps KŒx , KŒx . Since the maps KŒx and KŒx are K-algebra homomorphisms, .KŒx; mKŒx ; KŒx ; KŒx ; KŒx / is a K-bialgebra. Note that KŒx .x/ D x ˝ x, thus this bialgebra is the polynomial bialgebra with x grouplike. Example 2.1.4. Let KŒx be the K-algebra of polynomials in the indeterminate x. From Example 1.2.14, KŒx has the structure of a coalgebra, with maps KŒx , KŒx . Since the maps KŒx and KŒx are K-algebra homomorphisms, .KŒx; mKŒx ; KŒx ; KŒx ; KŒx / is a K-bialgebra. Note that KŒx .x/ D 1 ˝ x C x ˝ 1, thus this bialgebra is the polynomial bialgebra with x primitive. Let B be a K-bialgebra. A biideal I is a K-subspace of B that is both an ideal and a coideal. Proposition 2.1.5. Let I  B be a biideal of B. Then B=I is a K-bialgebra. Proof. From Proposition 1.2.7, we have that B=I is a K-algebra. By Proposition 1.2.15, B=I is a K-coalgebra. One notes that B=I is an algebra map since B is an algebra map. Moreover, B=I is an algebra map since that property holds for B .  Let B; B0 be bialgebras. A K-linear map W B ! B0 is a bialgebra homomorphism if is both an algebra and coalgebra homomorphism. The bialgebra homomorphism W B ! B0 is an isomorphism of bialgebras if is a bijection. Surprisingly, the bialgebra structures on KŒx given in Examples 2.1.3 and 2.1.4 are the only bialgebra structures on KŒx up to algebra isomorphism. Proposition 2.1.6. Suppose the polynomial algebra KŒx is given the structure of a K-bialgebra. Then there is some z 2 KŒx so that KŒz D KŒx and z is either grouplike or z is primitive. Proof. Let KŒx be a bialgebra and suppose that KŒx .x/ D

m X n X iD0 jD0

bi;j xi ˝ xj 2 KŒx ˝ KŒx;

38

2 Bialgebras

for bi;j 2 K. Thus KŒx is a finite sum of tensors bi;j xi ˝ xj in which i is the degree of x in the left factor of the tensor and j is the degree of x in the right factor of the tensor. Let l denote the highest degree of x that occurs in the left factors of the tensors in the sum KŒx . Then bl;j 6D 0 for some j, 0  j  n; let j0 be the maximal j for which bl;j 6D 0. Now, .IKŒx ˝ KŒx /KŒx .x/ 2 KŒx ˝ KŒx ˝ KŒx is a finite sum of tensors of the form cxi ˝ xj ˝ xk , c 2 K; i is the degree of x in the left-most factor in the tensor and k is the degree of x in the right-most factor of the tensor. Note that l is the highest degree of x that occurs in the left-most factors of the tensors in the sum .IKŒx ˝ KŒx /KŒx .x/. Now, .KŒx ˝ IKŒx /KŒx .x/ D .KŒx ˝ IKŒx /

X m X n

 bi;j x ˝ x i

j

iD0 jD0

D

m X n X

bi;j KŒx .xi / ˝ xj

iD0 jD0

D

m X n X

bi;j .KŒx .x//i ˝ xj

iD0 jD0

since KŒx is an algebra homomorphism and so, .KŒx ˝ IKŒx /KŒx .x/ D

m X n X

bi;j

X m X n

i ˝ xj

˛D0 ˇD0

iD0 jD0 2

b˛;ˇ x˛ ˝ xˇ

0

0

l lj j D blC1 l;j0 x ˝ x ˝ x C T;

where T is a sum of tensors in KŒx ˝ KŒx ˝ KŒx of the form cxi ˝ xj ˝ xk with i  l2 . Since bl;j0 6D 0, the highest power of x that occurs in the left-most factors of the tensors in the sum .KŒx ˝ IKŒx /KŒx .x/ is l2 . By the coassociative property of KŒx , one has l2 D l and hence, either l D 0 or l D 1. Now let r denote the highest degree of x that occurs in the right factors of the tensors in the sum KŒx .x/. By the argument above applied to the right factors of the tensors, one concludes that either r D 0 or r D 1. Consequently, KŒx .x/ D b0;0 .1 ˝ 1/ C b0;1 .1 ˝ x/ C b1;0 .x ˝ 1/ C b1;1 .x ˝ x/; for b0;0 ; b0;1 ; b1;0 ; b1;1 2 K.

2.1 Introduction to Bialgebras

39

P Pn i j Put  D KŒx ,  D KŒx . Let y D x  .x/ and let .y/ D m iD0 jD0 ai;j y ˝ y . By comparing the leading coefficients in . ˝ I/.y/ and .I ˝ /.y/ as above, we conclude that ai;j D 0 if i > 1 or j > 1. Since .y/ D 0, we also have a0;0 D 0 and a0;1 D a1;0 D 1. Thus .y/ D 1 ˝ y C y ˝ 1 C ay ˝ y for some a 2 K. If a D 0, then z D y is primitive and KŒz D KŒx. If a 6D 0, put z D 1 C ay. Then z is group-like with KŒz D KŒx.  Let B be a bialgebra, and let A be an algebra and a left B-module with action denoted by “”. Then A is a left B-module algebra if X b  .aa0 / D .b.1/  a/.b.2/  a0 / .b/

and b  1A D B .b/1A for all a; a0 2 A, b 2 B. Let A; A0 be K-algebras. A K-linear map W A ! A0 is a left B-module algebra homomorphism if is both an algebra and a left B-module homomorphism. Let C be a coalgebra and a right B-module with action denoted by “”. Then C is a right B-module coalgebra if X C .c  b/ D c.1/  b.1/ ˝ c.2/  b.2/ .c;b/

and C .c  b/ D C .c/B .b/; for all c 2 C, b 2 B. Let C; C0 be K-coalgebras. A K-linear map W C ! C0 is a right B-module coalgebra homomorphism if is both a coalgebra and a right B-module homomorphism. Let B be a bialgebra. There is a left B-module structure on B defined as .a * f /.b/ D f .ba/; for a; b 2 B, f 2 B . For a 2 B, f 2 B the element a * f is the left translate of f by a. The left translate action endows B with the structure of a left B-module algebra: for f ; g 2 B , a; b 2 B, .a * fg/.b/ D .fg/.ba/ D mB .f ˝ g/.ba/ D .f ˝ g/B .ba/

40

2 Bialgebras

D .f ˝ g/ D

X

 b.1/ a.1/ ˝ b.2/ a.2/

.b;a/

X

f .b.1/ a.1/ /g.b.2/ a.2/ /

.b;a/

D

X

.a.1/ * f /.b.1/ /.a.2/ * g/.b.2/ /

.b;a/

D

X

 .a.1/ * f /.a.2/ * g/ .b/:

.a/

Moreover, .a * 1B /.b/ D 1B .ba/ D B .ba/ D B .b/B .a/ D .B .a/1B /.b/: Likewise, there is a right B-module structure on B defined as .f ( a/.b/ D f .ab/ for all a; b 2 B, f 2 B . For a 2 B, f 2 B , the element f ( a is the right translate of f by a. Note that f ( B D ff ( b W b 2 Bg is a subspace of B . For example, let KŒx be the polynomial bialgebra with x grouplike. There is a right KŒx-module structure on KŒx defined by .f ( xj /.xk / D f .xjCk / for f 2 KŒx , k; j  0. For instance, the right translate ei ( xj (with ei defined by ei .xj / D ıi;j ) is defined as .ei ( xj /.xk / D ei .xjCk / D ıi;jCk : Thus,  ei ( x D j

eij if i  j 0 if i < j:

Note that ei ( KŒx is the subspace of KŒx generated by fe0 ; e1 ; e2 ; : : : ; ei g, hence dim.ei ( KŒx/ D i C 1. Lemma 2.1.7. Let B be a K-bialgebra. Let f 2 B . Then the following are equivalent. (i) dim.f ( B/ < 1. (ii) f 2 Bı . Proof. The proof is beyond the scope of this book. For a proof, the reader is referred to [Sw69], [Ab77, Lemma 2.2.2, Lemma 2.2.5], [Mo93, Lemma 9.1.1]. 

2.1 Introduction to Bialgebras

41

We may not wish to give the proof here, but we can illustrate Lemma 2.1.7 (at least the (i) H) (ii) direction). As above, dim.ei ( KŒx/ D i C 1, and so, ei 2 KŒxı . Indeed, ei corresponds to the .i C 1/st-order linearly recursive sequence 000ƒ‚    …1; ei in K with characteristic polynomial f .x/ D xiC1 and initial state vector „ iC1

vanishes on the ideal .xiC1 / of codimension i C 1. Lemma 2.1.7 is the key to proving the following proposition. Proposition 2.1.8. If B is a bialgebra, then Bı is a bialgebra. Proof. We first show that Bı is an algebra; we need to construct a multiplication map mBı and a unit map Bı that satisfy the associative and unit properties, respectively. By Proposition 1.3.1, B is an algebra with multiplication mB D B and unit map B D B . Let mBı denote the restriction of mB to Bı . We show that mBı W Bı ˝ Bı ! Bı . To this end, let f ; g 2 Bı and let a; b 2 B. Then .fg ( a/.b/ D .mB .f ˝ g/ ( a/.b/ D mB .f ˝ g/.ab/ D .f ˝ g/B .ab/   X D .f ˝ g/ a.1/ b.1/ ˝ a.2/ b.2/ D

.a/;.b/

X

f .a.1/ b.1/ /g.a.2/ b.2/ /

.a;b/

D

X

.f ( a.1/ /.b.1/ /.g ( a.2/ /.b.2/ /:

.a;b/

Moreover, X X .f ( a.1/ /.b.1/ /.g ( a.2/ /.b.2/ / D ..f ( a.1/ / ˝ .g ( a.2/ //.b.1/ ˝ b.2/ / .a;b/

.a;b/

X  X ..f ( a.1/ / ˝ .g ( a.2/ // b.1/ ˝ b.2/ D .a/

.b/

X ..f ( a.1/ / ˝ .g ( a.2/ //B .b/ D .a/

D

X

mB ..f ( a.1/ / ˝ .g ( a.2/ //.b/

.a/

D

X .a/

 .f ( a.1/ /.g ( a.2/ / .b/:

42

2 Bialgebras

Thus fg ( B  span..f ( B/.g ( B//. Observe that dim.span..f ( B/.g ( B/// < 1 since dim.f ( B/ < 1 and dim.g ( B/ < 1 by Lemma 2.1.7. Consequently, dim.fg ( B/ < 1. It follows that fg 2 Bı by Lemma 2.1.7. And so, mBı is a K-linear map mBı W Bı ˝ Bı ! Bı . Also, mBı satisfies the associative property since mB does. Next we need a unit map for Bı . Our choice of course is B W K ! B , but we need to show that B .K/  Bı . Note that B .r/ D rB .1K / D r1B D rB . Now, ker.B / is an ideal of B of finite codimension since the codomain of B is K. Thus B 2 Bı , and so B W K ! Bı . Consequently, we take Bı D B . We conclude that .Bı ; mBı ; Bı / is a K-algebra. By Proposition 1.3.9, Bı is a coalgebra with comultiplication map Bı and counit map Bı . So to prove that Bı is a bialgebra, it remains to show that Bı and Bı are algebra homomorphisms. We consider comultiplication first. Let f ; g 2 Bı , a; b 2 B. Then Bı .fg/.a ˝ b/ D Bı .mBı .f ˝ g//.a ˝ b/ D mBı .f ˝ g/.mB .a ˝ b// D mBı .f ˝ g/.ab/ D .f ˝ g/B .ab/  X D .f ˝ g/ a.1/ b.1/ ˝ a.2/ b.2/ D

X

.a;b/

f .a.1/ b.1/ /g.a.2/ b.2/ /:

(2.1)

.a;b/

Now,

X

f .a.1/ b.1/ /g.a.2/ b.2/ /

.a;b/

D

X

f .mB .a.1/ ˝ b.1/ //g.mB .a.2/ ˝ b.2/ //

.a;b/

D

X

Bı .f /.a.1/ ˝ b.1/ /Bı .g/.a.2/ ˝ b.2/ /

.a/;.b/

X

D .Bı .f / ˝ Bı .g//.I ˝  ˝ I/

 .a.1/ ˝ a.2/ ˝ b.1/ ˝ b.2/ /

.a;b/

D .Bı .f / ˝ Bı .g//.I ˝  ˝ I/.B ˝ B /.a ˝ b/ D .Bı .f / ˝ Bı .g//B˝B .a ˝ b/:

(2.2)

2.1 Introduction to Bialgebras

43

The reader should note that in moving from line 5 to line 6 above, we used the fact that the comultiplication of the coalgebra B ˝ B is defined to be B˝B D .IB ˝  ˝ IB /.B ˝ B /: The transpose of the map B˝B is B˝B W ..B ˝ B/ ˝ .B ˝ B// ! .B ˝ B/ and B˝B restricted to .Bı ˝ Bı / ˝ .Bı ˝ Bı / is the map .B ˝ B /.IB ˝  ˝ IB /; which is the multiplication on Bı ˝ Bı . Hence, .Bı .f / ˝ Bı .g//B˝B .a ˝ b/ D B˝B .Bı .f / ˝ Bı .g//.a ˝ b/ D mBı ˝Bı .Bı .f / ˝ Bı .g//.a ˝ b/ D .Bı .f /Bı .g//.a ˝ b/:

(2.3)

From (2.1)–(2.3) we obtain Bı .fg/ D Bı .f /Bı .g/; and so Bı is an algebra map. Next, we show that Bı is an algebra map. For r 2 K, Bı .fg/.r/ D .fg/B .r/ D .mBı .f ˝ g//.B .r1K / D r.mBı .f ˝ g//.1B / D r.f ˝ g/.B .1B // D r.f ˝ g/.1B ˝ 1B / D rf .1B /g.1B /: From the proof of Proposition 1.3.9, we have the formula Bı .f / D f .1B /; thus rf .1B /g.1B / D rBı .f /Bı .g/ D .Bı .f /Bı .g//.r/:

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2 Bialgebras

Consequently, Bı .fg/ D Bı .f /Bı .g/ and so Bı is an algebra map. We conclude that Bı is a bialgebra.



ı

Proposition 2.1.9. If B is cocommutative, then B is a commutative. If B is a commutative, then Bı is cocommutative. 

Proof. This is just a restatement of Proposition 1.3.11.

Proposition 2.1.10. Suppose that B is a finite dimensional vector space over the field K. Then B is a bialgebra if and only if B is a bialgebra. Proof. Since B is finite dimensional, Bı D B . If B is an bialgebra, then B is a bialgebra by Proposition 2.1.8. Now, if B is a bialgebra, then .B /ı is a bialgebra by Proposition 2.1.8. But since dim.B / D dim.B/ < 1, .B /ı D B (the double dual) which is identified with B. Therefore B is a bialgebra.  As we have seen in §1.3, if KŒx is the polynomial algebra, then the coalgebra KŒxı is the collection of linearly recursive sequences over K of all orders. But there are two coalgebra structures on KŒx as given in Examples 2.1.3 and 2.1.4 making KŒx into a bialgebra. (In fact, by Proposition 2.1.6 there are exactly two bialgebra structures on KŒx up to algebra isomorphism.) And so, by Proposition 2.1.8 there are two bialgebra structures on KŒxı . This means that we can now multiply sequences in KŒxı in two different ways. If KŒx is the polynomial bialgebra with x group-like, then KŒxı is the bialgebra with multiplication defined through thePcomultiplication on KŒx: For fsn g; ftn g 2 KŒxı , g.x/ D liD0 ai xi 2 KŒx, .fsn g  ftn g/.g.x// D .mKŒxı .fsn g ˝ ftn g//.g.x// D .fsn g ˝ ftn g/KŒx .g.x// X  l i D .fsn g ˝ ftn g/KŒx ai x iD0

D .fsn g ˝ ftn g/

l X

ai .xi ˝ xi /

iD0

D

l X

ai fsn g.xi /ftn g.xi /

iD0

D

l X

ai si ti :

iD0

D fsn tn g.g.x//: This is the Hadamard product of the sequences.

2.1 Introduction to Bialgebras

45

The Hadamard product fsn g  ftn g D fsn tn g is a linearly recursive sequence in K. How do we find its characteristic polynomial? We consider the situation in which both fsn g and ftn g are geometric sequences. Let fsn g be the geometric sequence with characteristic polynomial f .x/ D x  ˛, ˛ 2 K, and initial state vector s0 D s0 and let ftn g be the geometric sequence with characteristic polynomial g.x/ D x  ˇ, ˇ 2 K and initial state vector t0 D t0 . Then the Hadamard product is fsn g  ftn g D fsn tn g D fs0 ˛ n t0 ˇ n g D fs0 t0 .˛ˇ/n g; which is a geometric sequence with characteristic polynomial h.x/ D x  ˛ˇ and initial state vector s0 t0 . This is the essential idea behind the following proposition which we give without proof (see, [ZM73, CG93]) Proposition 2.1.11. Let K be a field containing Q. Let fsn g be a kth-order linearly recursive sequence in K with characteristic polynomial f .x/. Let ftn g be an lth-order linearly recursive sequence in K with characteristic polynomial g.x/. Suppose that f .x/; g.x/ have distinct roots in some field extension L=K. Let ˛1 ; ˛2 ; : : : ; ˛k be the distinct roots of f .x/ and let ˇ1 ; ˇ2 : : : ; ˇl be the distinct roots of g.x/. Then the characteristic polynomial of the Hadamard product fsn g  ftn g D fsn tn g is h.x/ D

Y

.x  ˛i ˇj /:

1ik; 1jl; ˛i ˇj distinct

Here is an illustration of Proposition 2.1.11. Example 2.1.12. Let fsn g be the Fibonacci sequence in Q with characteristic polynomial f .x/ D x2  x  1 and initial state vector s0 D .0; 1/. The Hadamard product fsn g  fsn g D fs2n g is 0; 1; 1; 4; 9; 25; 64; 169; : : : : p

The zeros of f .x/ are ˛ D 1C2 5 , 1˛ D of the Hadamard product is

p 1 5 , and so, the characteristic polynomial 2

h.x/ D .x  ˛ 2 /.x  ˛.1  ˛//.x  .1  ˛/2 / D .x  ˛ 2 /.x C 1/.x  .2  ˛// D .x C 1/.x2  3x C 1/ D x3  2x2  2x C 1:

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2 Bialgebras

Indeed, the Hadamard product is a 3rd-order linearly recursive sequence frn g with recurrence relation rnC3 D 2rnC2 C 2rnC1  rn and initial state vector r0 D .0; 1; 1/. If KŒx is the polynomial bialgebra with x primitive, then KŒxı is the bialgebra with multiplication defined through the comultiplication on KŒx. This multiplicaı tion is called Pl the iHurwitz product and is defined as follows. For fsn g; ftn g 2 KŒx , g.x/ D iD0 ai x 2 KŒx, .fsn g ftn g/.g.x// D .mŒKŒxı .fsn g ˝ ftn g//.g.x// D .fsn g ˝ ftn g/KŒx .g.x// X  l i D .fsn g ˝ ftn g/KŒx ai x iD0

! i X i D .fsn g ˝ ftn g/ ai .xj ˝ xij / j iD0 jD0 l X

! i X i D ai fsn g.xj /ftn g.xij / j iD0 jD0 l X

! i X i D ai sj tij : j iD0 jD0 8 9 ! n <X = n D sj tnj .g.x//: : ; j jD0 l X

o nP   n n The Hurwitz product fsn g ftn g D jD0 j sj tnj is a linearly recursive sequence in K. How do we find its characteristic polynomial? Again, we consider the case where fsn g and ftn g are geometric. One has 8 9 ! n <X = n fsn g ftn g D sj tnj : ; j jD0

8 9 ! n <X = n D s0 ˛ j t0 ˇ nj : ; j jD0

2.1 Introduction to Bialgebras

47

8 9 ! n < X n j nj = D s 0 t0 ˛ˇ : ; j jD0 D fs0 t0 .˛ C ˇ/n g; which is a geometric sequence with characteristic polynomial h.x/ D x  .˛ C ˇ/ and initial state vector s0 t0 . Here is the general result, stated without proof, but see [ZM73, CG93]. Proposition 2.1.13. Let K be a field containing Q. Let fsn g be a kth-order linearly recursive sequence in K with characteristic polynomial f .x/. Let ftn g be an lth-order linearly recursive sequence in K with characteristic polynomial g.x/. Suppose that f .x/; g.x/ have distinct roots in some field extension L=K. Let ˛1 ; ˛2 ; : : : ; ˛k be the distinct roots of f .x/ and let ˇ1 ; ˇ2 : : : ; ˇl be the distinct roots of g.x/. Then the characteristic polynomial of the Hurwitz product 8 9 ! n <X = n fsn g ftn g D sj tnj : ; j jD0 is h.x/ D

Y

.x  .˛i C ˇj //:

1ik; 1jl; ˛i Cˇj distinct

Example 2.1.14. Let fsn g be the Fibonacci sequence in Q with characteristic polynomial f .x/ D x2  x  1 and initial state vector s0 D .0; 1/. The Hurwitz product fsn g fsn g is 0; 0; 2; 6; 22; 70; 230; : : : : The zeros of f .x/ are ˛ D of the Hurwitz product is

p

1C 5 , 1˛ 2

D

p 1 5 , and so, the characteristic polynomial 2

h.x/ D .x  2˛/.x  1/.x  .2  2˛// D x3  3x2  2x C 4: Indeed, the Hurwitz product is a 3rd-order linearly recursive sequence frn g with recurrence relation rnC3 D 3rnC2 C 2rnC1  4rn and initial state vector r0 D .0; 0; 2/. For further reading on the Hadamard and Hurwitz products of linearly recursive sequences from a Hopf algebra viewpoint, see [LT90] and [Ta94].

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2.2 Myhill–Nerode Bialgebras In this section we give an application of bialgebras to formal languages and finite state machines (finite automata) of theoretical computer science. After introducing finite automata, and giving some examples, we prove the Myhill–Nerode theorem which tells us precisely when a language is accepted by a finite automaton. We generalize the Myhill–Nerode theorem to an algebraic setting in which a certain finite dimensional bialgebra (a Myhill–Nerode bialgebra) plays the role of the finite automaton that accepts the language. We construct some examples of Myhill– Nerode bialgebras. We see that a finite automaton determines a Myhill–Nerode bialgebra, which in turn determines a (perhaps different!) finite automaton. *

*

*

Let † D fa1 ; a2 ; : : : ; ak g be a finite alphabet and let † denote the collection of words of finite length formed from the letters in †. A word x 2 † is written as x D ai1 ai2 : : : aim ; where i1 ; i2 ; : : : ; im is a finite sequence of integers in f1; 2; 3; : : : ; kg. We assume that † contains the empty word e of length 0. Also, † together with concatenation is a monoid. A language is a subset L  † . Loosely speaking, a finite automaton is a type of computing machine that reads words in † and outputs a state. Here is a formal definition. Definition 2.2.1. A finite automaton is a 5-tuple M D .Q; †; ı; q0 ; F/ consisting of a finite set of states Q, an input alphabet †, a transition function ı W Q  † ! Q, an initial state q0 2 Q, and a set of accepting states F  Q. The finite automaton starts in the initial state q0 . On the input x D ai1 ai2 : : : aim 2 † the machine will move to a new state as follows. The machine reads the left-most letter ai1 and transitions to the state q.1/ D ı.q0 ; ai1 /. The machine then reads the next letter ai2 and moves to the state q.2/ D ı.q.1/ ; ai2 /. Next, the machine reads ai3 and moves to the state q.3/ D ı.q.2/ ; ai3 /, and so on. Eventually, the machine reads the last letter aim and halts at the final state q.m/ D ı.q.m1/ ; aim /. If the finite automaton M is in state q and halts in state q0 upon reading input O x/. In the notation above, x 2 † , then we write q0 D ı.q; O 0 ; ai ai    ai / q.m/ D ı.q 1 2 m O .1/ ; ai ai    ai / D ı.q 2 3 m O .2/ ; ai ai    ai /; D ı.q 3 4 m :: : O .m1/ ; ai /: D ı.q m O a/ D ı.q; a/ for q 2 Q, a 2 †. Observe that ı.q;

2.2 Myhill–Nerode Bialgebras

49

A word x 2 † is accepted by the finite automaton M if on input x, when starting at initial state q0 , the finite automaton halts at a state in F. In other words, the O 0 ; x/ 2 F. A language L is accepted automaton M accepts word x if and only if ı.q by M if the machine halts at an accepting state on all inputs x 2 L. Finite automata can be used to perform many important tasks. For instance, finite automata can be used to decide whether a given word in † has a certain property or not. A “parity-check” machine decides whether or not a given word in f0; 1g has an even number of 1’s. Example 2.2.2 (Parity-Check Automaton). This machine decides whether or not a given word in f0; 1g has an even number of 1’s. We take † D f0; 1g and define the finite automaton M D .Q; †; ı; q0 ; F/ as follows. The set of states of the automaton is Q D fq0 ; q1 g, the initial state is q0 and set of accepting states is F D fq0 g. The transition function ı W Q  † ! Q is given by the table below. q0 q1

0 q0 q1

1 q1 q0

For example, on input 1001, reading from left to right, the Parity-Check Automaton reads 1 and moves from the initial state q0 to the state q1 D ı.q0 ; 1/. The machine then reads 0 and stays in the state q1 D ı.q1 ; 0/, and again reads 0 to remain in the state q1 D ı.q1 ; 0/, and finally reads 1 and halts at the final state q0 D ı.q1 ; 1/. Since q0 2 F, q0 is an accepting state. The machine halted at an accepting state precisely because 1001 has an even number of 1’s. On the other hand, on input 111 starting at state q0 , the machine reads 1 and moves to the state q1 D ı.q0 ; 1/, reads 1 again and move to q0 D ı.q1 ; 1/, and reads 1 a last time and halts at state q1 D ı.q0 ; 1/, a non-accepting state. The machine halted at a non-accepting state because 111 has an odd number of 1’s. In general, on input x 2 † , the Parity-Check Automaton will halt at q0 if and only if x has an even number of 1’s (x is accepted by the Parity-Check automaton). The language accepted by the Parity-Check automaton is precisely the set of words in f0; 1g that have an even number of 1’s. By accepting precisely those words which have an even number of 1’s, the machine “decides” which words have an even number of 1’s. We can represent a finite automaton using a state transition diagram, which is a type of directed graph in which the vertices are the states and the directed edges, labeled with the letters of the alphabet, define the transition function. The transition ı.qi ; a/ D qj for states qi ; qj and letter a 2 † is indicated by labeling with a the directed edge from qi to qj . In Figure 2.1 below, we give the state transition diagram for the Parity-Check automaton. For example, we compute the halting state on input x D 10011 for the finite state diagram in Figure 2.1. Starting at the left arrow going into the initial state q0 , the machine reads digit 1 and moves to state q1 , it then reads the next two 0’s and stays in state q1 , then the machine reads 1 and moves to state q0 , finally the last 1 sends the machine to the halting state q1 .

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2 Bialgebras

Fig. 2.1 Finite state diagram for Parity-Check automaton. Accepting state is q0

Fig. 2.2 Finite state diagram for Check for a String Ending in 11. Accepting state is q2

For our next example, we build a finite automaton that decides whether a given word in f0; 1g ends in the string 11. The language accepted by this automaton will be the set of words in f0; 1g that end in 11. Example 2.2.3 (Check for a String Ending in 11). Let M D .Q; †; ı; q0 ; F/ with states Q D fq0 ; q1 ; q2 g, initial state q0 and set of accepting states F D fq2 g. Define the transition function ı W Q  † ! Q by the table: q0 q1 q2

0 q0 q0 q0

1 q1 q2 q2

The state transition diagram for the automaton is given in Figure 2.2. For instance, x D 01011 ends in 11, and so, on the input 01011 the machine halts in state q2 (an accepting state), as required. On the other hand, on the input 101 the machine halts in state q1 and so, 101 is not a word in the language accepted by M. Here is another example of a finite automaton. Example 2.2.4. We take † D fa; bg. Let M D .Q; †; ı; q0 ; F/ with states Q D fq0 ; q1 g; the initial state is q0 and the set of accepting states is F D fq1 g. The transition function is defined by q0 q1

a q1 q1

b q0 q1

The state transition diagram for the automaton is given in Figure 2.3.

2.2 Myhill–Nerode Bialgebras

51

Fig. 2.3 Finite state diagram for automaton of Example 2.2.4. Accepting state is q1

For this automaton, on the input baab the machine halts in state q1 , and so baab is accepted by M. On the other hand, bb is not accepted by this automaton. Can you describe the language accepted by M? As we have seen, a given finite automaton M D .Q; †; ı; q0 ; F/ accepts the language L consisting of words x 2 † for which the machine halts in an accepting state on input x. But given an alphabet and a language L  † , does there exist a finite automaton that accepts L? This question has been settled by the famous Myhill–Nerode Theorem of theoretical computer science [Ei74, Chapter III, §9, Proposition 9.2], [HU79, §3.4, Theorem 3.9]. Let † be a finite alphabet and let L  † be a language. On † we define an equivalence relation L as follows: for x; y 2 † , x L y if and only if xz 2 L precisely when yz 2 L for all z 2 † : The equivalence relation L has finite index if there is a finite number of equivalence classes under L . Proposition 2.2.5 (Myhill–Nerode Theorem). Let † be a finite alphabet and let L  † be a language. Then the following are equivalent. (i) The equivalence relation L has finite index. (ii) There exists a finite automaton that accepts language L. Proof. .i/ H) .ii/. Suppose that L has finite index. We need to construct a finite automaton which accepts L. For x 2 † , let Œx D fy 2 † W y L xg denote the equivalence class of L containing x. Then Q D fŒx W x 2 † g is finite, and will constitute the states of the automaton. Let ı W Q  † ! Q be the relation defined as ı.Œx; a/ D Œxa for a 2 †. Suppose that Œx D Œy. Then x L y and so, x.az/ 2 L exactly when y.az/ 2 L for all z 2 † , whence Œxa D Œya. Now, ı.Œx; a/ D Œxa D Œya D ı.Œy; a/

52

2 Bialgebras

and so ı is well-defined on classes in Q, in other words, ı W Q  † ! Q is a function, which will serve as the transition function for our automaton. Setting q0 D Œe and F D fŒx 2 Q W x 2 Lg yields the finite automaton .Q; †; ı; q0 ; F/ that accepts L. .ii/ H) .i/. Suppose that L is accepted by the finite automaton M D .Q; †; ı; q0 ; F/. Let M be the equivalence relation defined as x M y if and O 0 ; y/. Then M has finite index since there are a finite O 0 ; x/ D ı.q only if ı.q number of states in Q. Note that O 0 ; x/ 2 Fg; L D fx 2 † W ı.q and so L is a union of some of the equivalence classes under M . Now suppose O 0 ; x/ D ı.q O 0 ; y/, and so, ı.q O 0 ; xz/ D ı.q O 0 ; yz/ for all that x M y. Then ı.q z 2 † . Consequently, xz 2 L if and only if yz 2 L for all z 2 † , that is, x L y. Thus the index of L is less than or equal to the index of M .  To determine whether a given language L is accepted by a finite automaton, we compute the number of equivalences classes under L . Example 2.2.6. Let L be the English language, which consists of words built from the alphabet † D fa; b; c; : : : ; zg. We claim that L has finite index. We assume that L is finite, so that there is a word x 2 L of maximal length d. Now, any two words x; y 2 † of length > d are equivalent since xz and yz are not in L for all z 2 † . Thus L admits at most jLj C 1 equivalence classes (the empty word e is not equivalent to any other word in † , and accounts for the additional class.) By the Myhill–Nerode Theorem the English language is accepted by a finite automaton. Example 2.2.7. Let L  fa; bg be defined as L D fxay W x; y 2 fa; bg g: Now, y L e if and only if yz 2 L precisely when z 2 L; 8z 2 fa; bg . Thus Œe D fy W y 62 Lg. Moreover, y L a if and only if yz 2 L precisely when az 2 L; 8z. Since a 2 L and e 62 L, Œe 6D Œa, and so, Œa D fy W y 2 Lg. Thus the set of classes under L is fŒe; Œag, with L D Œa. By the Myhill–Nerode theorem, the language L is accepted by the finite automaton

Example 2.2.8. Let † D fa; bg and let L be the language consisting of all words in fa; bg that contain no consecutive b’s. One has y L e if and only if yz 2 L

2.2 Myhill–Nerode Bialgebras

53

precisely when z 2 L, and so, Œe consists of all y 2 L with the property that either y D e or y ends in a. Now, y L b if and only if yz 2 L precisely when bz 2 L, and so, Œb consists of all y 2 L with the property that y ends in b but not in bb. Finally, y L bb if and only if yz 2 L precisely when bbz 2 L, and so, Œbb consists of all y 62 L. Consequently, the set of classes under L is fŒe; Œb; Œbbg, with L D Œe [ Œb. By the Myhill–Nerode theorem, the language L is accepted by a finite automaton (see §2.4, Exercise 13). Example 2.2.9. In this example, † D fa; bg and L D fan bn W n  0g, that is, L D fe; ab; aabb; aaabbb; aaaabbbb; : : : g: As one can check, the set of classes under L is fŒb; Œe; Œa; Œaa; Œaaa; Œaaaa; : : : g; with L D Œe. Thus L has infinite index, and so there is no finite automaton that accepts the language L. For the moment we let K be a field, let S be any monoid, and let H D KS be the monoid bialgebra with linear dual H  D KS . There is a right H-module structure on H  defined as .p ( x/.y/ D p.xy/ for all x; y 2 H, p 2 H  . For x 2 H, p 2 H  , the element p ( x is the right translate of p by x; p ( H D fp ( x W x 2 Hg is a subspace of H  . Now, let S D † denote the monoid of words in a finite alphabet † and let L  S be a language. Let H D KS denote the monoid bialgebra. Let  W S ! K be the characteristic function of L defined as  1 if x 2 L .x/ D 0 if x 62 L: Then  extends by linearity to an element p 2 H  . Indeed, X  X ax x D ax .x/: p x2S

x2S

Proposition 2.2.10. Let L  S be a language and let p be the element of H  D KS defined as above. Then L has finite index if and only if the set of right translates fp ( x W x 2 Sg is finite. Proof. Let Œx denote the equivalence class of x under L . Define an equivalence relation p on S as follows: x p y if and only if p.xz/ D p.yz/ for all z 2 S. Then

L D p . Let Œxp denote the equivalence class of x under p . Thus fŒx W x 2 Sg D fŒxp W x 2 Sg.

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Define a relation

W fŒxp W x 2 Sg ! fp ( x W x 2 Sg by the rule .Œxp / D p ( x. Suppose that Œxp D Œyp . Then

.Œxp / D p ( x D p ( y D .Œyp /; and so is well-defined on equivalence classes under p . Now assume that

.Œxp / D .Œyp /. Then p ( x D p ( y, hence Œxp D Œyp , and so is an injection. Clearly, is surjective. The result follows.  Proposition 2.2.10 is the point of departure for generalizing the Myhill–Nerode theorem to an arbitrary monoid S: the condition “ L has finite index” gets replaced by the condition fp ( x W x 2 Sg is finite. A certain bialgebra plays the role of the finite automaton that accepts the language, cf. [NU11]. Proposition 2.2.11 (Algebraic Myhill–Nerode Theorem). Let S be a monoid and let H D KS denote the monoid bialgebra. Let p 2 H  . Then the following are equivalent. (i) The set fp ( x W x 2 Sg of right translates is finite. (ii) There exists a finite dimensional bialgebra B, a bialgebra homomorphism ‰ W H ! B, and an element f 2 B so that p.h/ D f .‰.h// for all h 2 H. Proof. .i/ H) .ii/. Let Q D fp ( x W x 2 Sg be the finite set of right translates. For each u 2 S, we define a right operator ru W Q ! Q by the rule .p ( x/ru D .p ( x/ ( u D p ( xu: Observe that the set fru W u 2 Sg is finite with jfru W u 2 Sgj  jQjjQj : We endow the set fru W u 2 Sg with the binary operation “composition of operators” defined as follows: for ru ; rv 2 fru W u 2 Sg, p ( x 2 Q, .p ( x/.ru rv / D .p ( xu/rv D p ( xuv D .p ( x/ruv : Thus ru rv D ruv , for all u; v 2 S. Then fru W u 2 Sg together with composition of operators is a monoid with unity r1 . Let B denote the monoid bialgebra on fru W u 2 Sg over K. Let ‰ W H ! B be the K-linear map defined by ‰.u/ D ru . Then ‰.uv/ D ruv D ru rv D ‰.u/‰.v/:

2.2 Myhill–Nerode Bialgebras

55

Also, B .‰.u// D B .ru / D ru ˝ ru D ‰.u/ ˝ ‰.u/ D .‰ ˝ ‰/.u ˝ u/ D .‰ ˝ ‰/H .u/; and B .‰.u// D B .ru / D H .u/; and so, ‰ is a homomorphism of bialgebras. Let f 2 B be defined by f .ru / D ..p ( 1/ru /.1/ D .p ( u/.1/ D p.u/ Then p.h/ D f .‰.h//, for all h 2 H, as required. .ii/ H) .i/. Suppose there exists a finite dimensional bialgebra B, a bialgebra homomorphism ‰ W H ! B, and an element f 2 B so that p.h/ D f .‰.h// for all h 2 H. Define a right H-module action  on B as b  h D b‰.h/ for all b 2 B, h 2 H. Then for b 2 B, x 2 S, B .b  x/ D B .b‰.x// D B .b/B .‰.x// X  D b.1/ ˝ b.2/ .‰ ˝ ‰/H .x/ .b/

D

X

 b.1/ ˝ b.2/ .‰.x/ ˝ ‰.x//

.b/

D

X

b.1/ ‰.x/ ˝ b.2/ ‰.x/

.b/

D

X

b.1/  x ˝ b.2/  x

.b/

and B .b  x/ D B .b‰.x// D B .b/B .‰.x// D B .b/H .x/: Thus B is a right H-module coalgebra.

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Now, let Q be the collection of grouplike elements of B. By Proposition 1.2.18, Q is a linearly independent subset of B. Since B is finite dimensional, Q is finite. Since B is a right H-module coalgebra with action “”, B .q  x/ D q  x ˝ q  x for q 2 Q, x 2 S. Thus  restricts to give an action (also denoted by “”) of S on Q. Now for x; y 2 S, .p ( x/.y/ D p.xy/ D f .‰.xy// D f .‰.x/‰.y// D f ..1B ‰.x//‰.y// D f ..1B  x/  y/

(2.4)

Let T D fq 2 Q W q D 1B  x for some x 2 Sg In view of (2.4) there exists a function % W T ! fp ( x W x 2 Sg defined as %.1B  x/.y/ D f ..1B  x/  y/ D .p ( x/.y/: Since % is surjective and T is finite, fp ( x W x 2 Sg is finite.



The bialgebras constructed in Proposition 2.2.11(ii) are called Myhill–Nerode bialgebras. The easiest way to construct a Myhill–Nerode bialgebra is to start with a language L  S for which L has finite index. Then by Proposition 2.2.10, the set of right translates fp ( x W x 2 Sg is finite, and so, by Proposition 2.2.11 (i)H) (ii), there exists a Myhill–Nerode bialgebra B, a bialgebra homomorphism ‰ W KS ! B and an element f 2 B so that p.h/ D f .‰.h//, for all h 2 KS. Here are some examples. Example 2.2.12. Let L  S D † D fa; bg be the language given in Example 2.2.7. In this case, the equivalence classes under L are Œe and Œa, with  L D Œa. The characteristic P  Pfunction  W S ! K extends to a function p 2 KS , defined as p x2S ax x D x2S ax .x/. There are exactly two right translates of p: fp ( 1; p ( ag: The set of right operators is fr1 ; ra g. The operators are defined by the table

2.2 Myhill–Nerode Bialgebras

57

p(x p(1 p(a

.p ( x/r1 p(1 p(a

.p ( x/ra p(a p(a

The monoid T D fr1 ; ra g has binary operation given as r1 ra

r1 r1 ra

ra ra ra

The monoid bialgebra KT is the Myhill–Nerode bialgebra. Furthermore, the bialgebra homomorphism ‰ W KS ! KT is given as ‰.x/ D rx .  P Let fer1 ; era g be the basis for KT  dual to the basis fr1 ; ra g for KT. Let h D x2S ax x. Then the element era 2 KT is so that X  X p.h/ D p ax x D ax .x/ D

X

x2S

x2S

ax ıŒa;Œx D

x2S

D era

X

 ax rx

X

ax era .rx /

x2S

  X  D era ‰ ax x

x2S

x2S

D era .‰.h//; as required. Also, KT  is a K-bialgebra with KT  Š K  K, as K-algebras and coalgebra structure defined as KT  .er1 / D er1 ˝ er1 ; KT  .era / D er1 ˝ era C era ˝ er1 C era ˝ era ; KT  .er1 / D er1 .r1 / D 1; KT  .era / D era .r1 / D 0: Example 2.2.13. Let † D fa; bg and let L  fa; bg be the language consisting of all words that contain no consecutive b’s (Example 2.2.8). The equivalences classes under L are Œe, Œb, Œbb with L D Œe [ Œb. The characteristic P  P function  W S ! K extends to a function p 2 KS , defined as p a x D x2S ax .x/. x2S x The set of right translates (states) is Q D fp ( 1; p ( b; p ( bbg. The set of right operators is fr1 ; rb ; rbb ; ra ; rab ; rba g. The operators are defined by the table p(x p(1 p(b p ( bb

.p ( x/r1 p(1 p(b p ( bb

.p ( x/rb p(b p ( bb p ( bb

.p ( x/rbb p ( bb p ( bb p ( bb

.p ( x/ra p(1 p(1 p ( bb

.p ( x/rab p(b p(b p ( bb

.p ( x/rba p(1 p ( bb p ( bb

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2 Bialgebras

The monoid T D fr1 ; rb ; rbb ; ra ; rab ; rba g has binary operation given as r1 rb rbb ra rab rba

r1 r1 rb rbb ra rab rba

rb rb rbb rbb rab rbb rb

rbb rbb rbb rbb rbb rbb rbb

ra ra rba rbb ra ra rba

rab rab rb rbb rab rab rb

rba rba rbb rbb ra rbb rba

The monoid bialgebra KT is the Myhill–Nerode bialgebra and the bialgebra homomorphism ‰ W KS ! KT is given as ‰.x/ D rx . LetPfer1 ; erb ; erbb ; era ; erab ; erba g be the basis for KT  dual to the basis T for KT. Let h D x2S ax x. Then the element 1  erbb 2 KT  is so that X  X p.h/ D p ax x D ax .x/ D

X x2S

D

X

x2S

ax

X

x2S

ıŒy;Œx D

ry 2T; ry 6Drbb

X x2S

ax

X

ery .rx /

ry 2T; ry 6Drbb

ax .1  erbb /.rx / D .1  erbb /

x2S

X

 ax rx

x2S

  X  ax x D .1  erbb /.‰.h//; D .1  erbb / ‰ x2S

as required. Example 2.2.14. Let † D fag, fix an integer i  0 and let Li D fai g  S D † , with characteristic function i W S ! K. Let H D KS and let pi 2 H  D KS be the extension of i . The finite set of right translates of pi 2 H  is Qi D fpi ( 1; pi ( a; pi ( a2 ; : : : ; pi ( ai ; pi ( aiC1 g: The set of right operators on Qi is Ti D fr1 ; ra ; ra2 ; : : : ; rai ; raiC1 g, a monoid. We have, for 0  m; n  i C 1,  ram ran D ramCn if 0  m C n  i C 1 raiC1 if m C n > i C 1 Bi D KTi is the Myhill–Nerode bialgebra, with bialgebra homomorphism ‰ W H ! Bi defined as x 7! rx . The element erai 2 Bi is so that pi .h/ D erai .‰.h//; for all h 2 KS.

2.2 Myhill–Nerode Bialgebras

59

Let B be a Myhill–Nerode bialgebra constructed from a language L with L of finite index (as in Examples 2.2.12, 2.2.13, and 2.2.14). Proposition 2.2.15. The Myhill–Nerode bialgebra B determines a finite automaton .Q; †; ı; q0 ; F/ that accepts the language L. Proof. For the states of the automaton, we let Q be the (finite) set of group-like elements of B; this set is precisely the finite set of right operators Q D frx W x 2 Sg. For the input alphabet, we choose †. As we have seen, the right H-module structure of B restricts to an action “” of S on Q, and so we define the transition function ı W Q  † ! Q by the rule ı.rx ; y/ D rx ‰.y/ D rx ry D rxy ; for rx 2 Q, y 2 S. The initial state is q0 D 1B , and the set of final states F is the subset of Q of the form 1B  x, x 2 S for which p.x/ D f .‰.x// D f .1B ‰.x// D f .1B  x/ D 1 By construction, the finite automaton .Q; †; ı; 1B ; F/ accepts L.



Example 2.2.16. Let B be the Myhill–Nerode bialgebra constructed in Example 2.2.12. The finite automaton is given as .Q; †; ı; 1B ; F/ where Q D fr1 ; ra g, F D fra g, with transition function ı given by the table r1 ra

a ra ra

b r1 ra

The finite automaton determined by the Myhill–Nerode bialgebra B is given in Figure 2.4 below. The finite automaton given in Figure 2.4 accepts the language L of Example 2.2.12. Example 2.2.17. Let B be the Myhill–Nerode bialgebra constructed in Example 2.2.13. The finite automaton is given as .Q; †; ı; 1B ; F/ where Q D fr1 ; rb ; rbb ; ra ; rab ; rba g, F D fr1 ; rb ; ra ; rab ; rba g, with transition function ı given by the table Fig. 2.4 Finite state diagram for Myhill–Nerode bialgebra KT. Accepting state is ra

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2 Bialgebras

Fig. 2.5 Finite state diagram for Myhill–Nerode bialgebra KT. Accepting states are r1 ; rb ; rba ; ra ; rab

r1 rb rbb ra rab rba

a ra rba rbb ra ra rba

b rb rbb rbb rab rbb rb

The finite automaton associated with the Myhill–Nerode bialgebra B is given in Figure 2.5 above. The finite automaton given in Figure 2.5 accepts the language L of Example 2.2.13. Example 2.2.18. For i  0, let Bi be the Myhill–Nerode bialgebra constructed in Example 2.2.14. The finite automaton is given as .Qi ; †; ıi ; 1Bi ; Fi / with Qi D fr1 ; ra ; ra2 ; : : : ; rai ; raiC1 g, F D frai g, with transition function ıi given by the table r1 ra ra2 :: :

a ra ra2 ra3 :: :

rai raiC1

raiC1 raiC1

The state diagram for the automaton is given below. The finite automaton in Figure 2.6 accepts the language L given in Example 2.2.14.

2.3 Regular Sequences

61

Fig. 2.6 Finite state diagram for Bi . Accepting state is rai

2.3 Regular Sequences In the final section of the chapter, we introduce regular sequences; these are sequences that generalize linearly recursive sequences over a Galois field. *

*

*

Let S be a monoid which is countable as a set. Then the elements of S can be listed x0 ; x1 ; x2 ; x3 ; : : : : Let K be a field, let H D KS be the monoid bialgebra, and let p 2 H  . Define a sequence fsn g in K by the rule sn D p.xn / for n  0. Then fsn g is a regular sequence if the set of right translates fp ( x W x 2 Sg is finite. Recall that for x 2 S, the right translate p ( x 2 H  is defined as .p ( x/.y/ D p.xy/; 8y 2 S. Example 2.3.1. Let S D fa; bg and let L be the language consisting of all words that contain no consecutive b’s (see Example 2.2.8). The equivalence classes under

L are Œe, Œb, Œbb, and L D Œe [ Œb. The set S is countable and its elements can be listed as e; a; b; aa; ab; ba; bb; aaa; aab; aba; abb; baa; bab; bba; bbb; aaaa; aaab; : : : Here x0 D e, x1 D a, x2 D b, and so on.PLet  W S ! K be the characteristic  function of L, extending to an element p D 1 nD0 .xn /en 2 H with en .xm / D ın;m . The sequence sn D p.xn / in K is 11111101110110011 : : : The index of L is 3 and so by Proposition 2.2.10, the set of right translates fp ( x W x 2 Sg is finite. Thus fsn g is a regular sequence. Proposition 2.3.2. Assume the notation as above. Then fsn g is a regular sequence if and only if dim.p ( H/ < 1 and the image p.S/ D fp.x/ W x 2 Sg is finite.

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2 Bialgebras

Proof. If fsn g is a regular sequence, then the set of right translates fp ( x W x 2 Sg is finite. Since fp ( x W x 2 Sg generates p ( H, p ( H is finitely generated. Thus dim.p ( H/ < 1. Moreover, for all x 2 S, .p ( x/.1/ D p.x/ and so, fp.x/ W x 2 Sg is finite. For the converse, suppose that dim.p ( H/ D n < 1 and p.S/ D fp.x/ W x 2 Sg is finite. Now, p ( H is an n-dimensional subspace of the K-vector space Map.S; K/, and so, by a standard result there exists elements y1 ; y2 ; : : : ; yn 2 S and a basis ff1 ; f2 ; : : : ; fn g for p ( H for which fi .yj / D ıi;j (see [NU11, Lemma 4.1]). Let x 2 S. Then there exist elements r1 ; r2 ; : : : ; rn 2 K for which p ( x D r1 f1 C r2 f2 C    C rn fn . Evaluation at yj yields rj D p.xyj / 2 p.S/ for 1  j  n. Since p.S/ is finite, there are only a finite number of linear combinations of the fi that represent right translates of p. Hence the number of right translates is finite and fsn g is regular.  As a consequence of Proposition 2.3.2 the notion of regular sequence does not depend on the order in which the elements of S are listed. Proposition 2.3.3. Let K D GF.pm /. Let fsn g be a kth-order linearly recursive sequence in K. Then fsn g is a regular sequence. Proof. Let † be the alphabet consisting of the singleton letter fxg. Then S D † D f1; x; x2 ; : : : g and KŒx with the bialgebra structure of Example 2.1.3 is the monoid bialgebra H DPKS. By Proposition 1.3.12 the sequence fsn g corresponds to an 1 ı m n element p D nD0 sn en in H with en .x / D ın;m . Clearly, sn D p.x / for all n  0. Now by Lemma 2.1.7 dim.p ( H/ < 1. Clearly, p.S/ is finite, and so by Proposition 2.3.2, fsn g is a regular sequence. Alternatively, by the theory of linearly recursive sequences, fsn g is eventually periodic with period r, that is, there exist integers N  0, r > 0 for which snCr D sn for all n  N. Consequently, the finite set p ( 1; p ( x; p ( x2 ; : : : ; p ( xNCr1 ; is a complete list of right translates of p. Thus fsn g is a regular sequence.



Example 2.3.4. Let K D GF.2/. Let fsn g be the 3rd-order linearly recursive sequence over K with characteristic polynomial f .x/ D x3 C x C 1, recurrence relation snC3 D snC1 C sn ; n  0; and initial state vector s0 D 101. Since f .x/ is a primitive polynomial over K (proof?), fsn g is periodic with period r D 23  1 D 7. Indeed, the sequence is 10111001011100 : : : ; which is regular by Proposition 2.3.3. Let sequence in K D GF.2/ of Example 2.3.4 with p D P1fsn g be the regular ı s e 2 KŒx . The finite set of right translates is nD0 n n

2.3 Regular Sequences

63

Fig. 2.7 Finite state diagram accepting language L. Accepting states are r1 , rx2 , rx3 , rx4

fp ( 1; p ( x; p ( x2 ; p ( x3 ; p ( x4 ; p ( x5 ; p ( x6 g: Consequently, Proposition 2.2.11 applies to show that there is a Myhill–Nerode bialgebra B, a bialgebra homomorphism ‰ W KŒx ! B and an element f 2 B for which p.h/ D f .‰.h// for all h 2 KŒx. We construct B as follows. The right operators are T D fr1 ; rx ; rx2 ; rx3 ; rx4 ; rx5 ; rx6 g; with binary operation given as r1 rx rx 2 rx 3 rx 4 rx 5 rx 6

r1 r1 rx rx 2 rx 3 rx 4 rx 5 rx 6

rx rx rx 2 rx 3 rx 4 rx 5 rx 6 r1

rx 2 rx 2 rx 3 rx 4 rx 5 rx 6 r1 rx

rx 3 rx 3 rx 4 rx 5 rx 6 r1 rx rx 2

rx 4 rx 4 rx 5 rx 6 r1 rx rx 2 rx 3

rx 5 rx 5 rx 6 r1 rx rx 2 rx 3 rx 4

rx 6 rx 6 r1 rx rx 2 rx 3 rx 4 rx 5

The Myhill–Nerode bialgebra B is KT Š KC7 . Note that KT  has basis fer1 ; erx ; erx2 ; erx3 ; erx4 ; erx5 ; erx6 g with erm .rxn / D ım;n . There exists a bialgebra homomorphism ‰ W KŒx ! KT defined as ‰.x/ D rx . Let f D er1 C erx2 C erx3 C erx4 : Then p.h/ D f .‰.h//; 8h 2 KŒx, as required. In addition, the sequence fsn g determines the language L  fxg defined as: for n  0, xn 2 L if and only if sn D 1. The language L is accepted by the finite automaton of Figure 2.7.

64

2 Bialgebras

2.4 Chapter Exercises Exercises for §2.1 1. Find an example of a K-bialgebra B and a coideal I of B which is not a biideal of B. 2. Let B be a commutative K-bialgebra. Show that B Š KŒfx˛ g=N, as K-algebras, for some set of indeterminates fx˛ g, and some ideal N of KŒfx˛ g. Under what conditions does this isomorphism extend to an isomorphism of bialgebras? 3. Let B be a bialgebra and let f 2 B . Prove that f ( B is a subspace of B . 4. Let B be a bialgebra and let f 2 Bı . Prove that f ( B is a finite dimensional subspace of B (see §2.4, Exercise 3). 5. Let B be a K-bialgebra and let C be a right B-module coalgebra. Show that C is a left B-module algebra. 6. Let fsn g be the 2nd-order linearly recursive sequence in Q with recurrence relation snC2 D sn and initial state vector s0 D .1; 1/. Let ftn g be the 1storder linearly recursive sequence in Q with recurrence relation tnC1 D 2tn and t0 D .1/. (a) Compute the first 5 terms of the Hadamard product fsn g  ftn g. (b) Compute the characteristic polynomial of the Hadamard product fsn g  ftn g. 7. Let fsn g be the 4th-order linearly recursive sequence in Q with recurrence relation snC4 D snC3  snC2  snC1  sn and initial state vector s0 D .2; 1; 0; 3/. (a) Compute the first 5 terms of the Hadamard product fsn g  fsn g. (b) Compute the characteristic polynomial of the Hadamard product fsn g  fsn g. 8. Let fsn g be the 2nd-order linearly recursive sequence in Q with recurrence relation snC2 D snC1  sn and initial state vector s0 D .1; 2/. (a) Compute the first 5 terms of the Hurwitz product fsn g fsn g. (b) Find the recurrence relation and the initial state vector of the Hurwitz product fsn g fsn g. Exercises for §2.2 9. Let M D .Q; †; ı; q0 ; F/ be the finite automaton with states Q D fq0 ; q1 ; q2 g, input alphabet † D f0; 1g, initial state q0 , set of accepting states F D fq1 ; q2 g and transition function ı W Q  † ! Q given by the table:

2.4 Chapter Exercises

65

q0 q1 q2

0 q1 q0 q2

1 q1 q2 q0

(a) Construct the state transition diagram for M. O 0 ; 01101/; ı.q O 1 ; 1001/. (b) Compute ı.q O 0 ; x/ D q2 . (c) Give a description of all words x 2 f0; 1g for which ı.q 10. Let †0 be a finite alphabet and let L  †0 be a language. Prove that L is an equivalence relation. 11. Let L be the language accepted by the Parity-Check Automaton of Example 2.2.2. Compute the equivalence classes under L . 12. Let L be the language accepted by the Check for a String Ending in 11 automaton of Example 2.2.3. Compute the equivalence classes under L . 13. Construct a finite automaton that accepts the language L of Example 2.2.8. Does your finite automaton coincide with the finite automaton given in Example 2.2.17? 14. Let M D .Q; †; ı; q0 ; F/ be a finite automaton and let M be the relation on O 0 ; x/ D ı.q O 0 ; y/. Prove that M is an † defined as x M y if and only if ı.q equivalence relation. 15. Suppose the language L is accepted by a finite automaton. Can the “only if” part of the Myhill–Nerode theorem be proved using Proposition 2.2.11 (ii) H) (i)? 16. Construct the Myhill–Nerode bialgebra associated with the finite automaton of Example 2.2.2. 17. Construct the Myhill–Nerode bialgebra associated with the finite automaton of Example 2.2.3. 18. Let K be a field, let G be an infinite group, let g 2 G and let eg 2 KG be defined as eg .h/ D ıg;h . Prove that eg has an infinite number of right translates. 19. Let S be a monoid, let H D KS be the monoid bialgebra and let p 2 H  . Suppose there exists a finite dimensional bialgebra B, a bialgebra homomorphism ‰ W H ! B and an element f 2 B for which p.h/ D f .‰.h//; 8h 2 H. Let

W B ! A be an isomorphism of bialgebras. Prove that there exists a bialgebra homomorphism ‰ 0 and an element g 2 A for which p.h/ D g.‰ 0 .h//; 8h 2 H. 20. Let S be a monoid, let H D KS be the monoid bialgebra and let p 2 H  . Suppose there exists a finite dimensional bialgebra B, a bialgebra homomorphism ‰ W H ! B and an element f 2 B for which p.h/ D f .‰.h//; 8h 2 H. Let D ! H be an isomorphism of bialgebras. Show there exists an element q 2 D and a bialgebra homomorphism ‰ 0 W D ! B for which q.d/ D f .‰ 0 .d//, 8d 2 D.

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2 Bialgebras

21. Let K be a field, let S be a monoid, let p 2 KS with p.s/ 2 f0; 1g; 8s 2 S. Prove the following: p 2 KSı if and only if there exists a finite dimensional bialgebra B, a bialgebra homomorphism ‰ W KS ! B and an element f 2 B for which p.h/ D f .‰.h//; 8h 2 KS. In this sense, B “accepts” p; L D fs W p.s/ D 1g is the language accepted by B. Exercises for §2.3 22. Let fsn g be a kth-order linearly recursive sequence over GF.pm / with characteristic polynomial f .x/ 2 GF.pm /Œx. By Proposition 2.3.3, fsn g is a regular sequence. In this sense regular sequences generalize linearly recursive sequences. What is the analog of the characteristic polynomial for regular sequences? Questions for Further Study 1. Let M be the finite automaton given by the state transition diagram below. (Note that the accepting states are q1 ; q3 .)

(a) Compute the Myhill–Nerode bialgebra B associated to the automaton M. (b) Let S D fa; bg , let KS be the monoid bialgebra and let p 2 KS be the functional corresponding to the language L accepted by M. Find f 2 B so that p.h/ D f .‰.h// for all h 2 KS. 2. Let M and M 0 be finite state machines which accept the languages L and L0 , respectively. Suppose M determines Myhill–Nerode bialgebra B and suppose M 0 determines Myhill–Nerode bialgebra B0 . If B is isomorphic to B0 , are the languages L, L0 the same? 3. Can every monoid bialgebra KS, with S finite, be viewed as a Myhill–Nerode bialgebra associated with some finite automaton (or equivalently, some language L with L of finite index)?

Chapter 3

Hopf Algebras

In this chapter we introduce the notion of a Hopf algebra over a field K as a bialgebra with an additional map called the coinverse (or antipode). We discuss some basic features of Hopf algebras and give some initial examples, including the group ring KG. In many respects, KG is the example that is generalized in the concept of Hopf algebra. The coinverse map KG of KG has order 2 under function composition, though this is not the case for every Hopf algebra. If H is a cocommutative Hopf algebra, however, then the coinverse H has order 2, and so, cocommutative Hopf algebras generalize the group ring Hopf algebra KG. We next consider certain subspaces of a Hopf algebra H, called the space of left Rl (or right) integrals. The left integrals H can be described as the set of elements of H for which the action of H by left multiplication is trivial. We then generalize the comultiplication map H to yield the concept of a right comodule over H and show that H  is a right H-comodule. This leads to the concept of a Hopf module over H—a vector space that is both an H-module and an H-comodule. We state the Fundamental Theorem of Hopf Modules, a curious result which says that every right Hopf module over a finite dimensional Hopf algebra H is isomorphic to a trivial right Hopf module. We show that the right H-comodule H  is a right Hopf module and prove a special case of the Fundamental Theorem which Rl Rl says that H  Š H  ˝H, where H  ˝H has the structure of a trivial right H-Hopf Rl module. As a corollary, we conclude that H  is a one-dimensional subspace of H  , and hence H  admits a generating integral. The last two sections of the chapter concern Hopf algebras over rings. Many properties of Hopf algebras over fields carry over to rings, including the notion of integrals and the Fundamental Theorem of Hopf Modules. The resulting isoRl morphism H  Š H  ˝H, valid for Hopf algebras over rings of finite rank, has been used in the classification of Hopf algebra orders in KCp2 , see [Un94], [Un11, Chapter 8]. Next, we introduce Hopf orders in the group ring Hopf algebra KG. Hopf orders are Hopf algebras over rings which in some sense generalize the notion of fractional © Springer International Publishing Switzerland 2015 R.G. Underwood, Fundamentals of Hopf Algebras, Universitext, DOI 10.1007/978-3-319-18991-8_3

67

68

3 Hopf Algebras

ideals over R in K where R is a Dedekind domain. When R is the ring of integers of a finite extension K=Q, we construct a collection of one parameter Hopf orders H.i/ in KCp together with their dual modules, H.i/D . In §4.5, we will show how Hopf orders can be realized as “Galois groups” in a sense that will be made precise.

3.1 Introduction to Hopf Algebras In this section we define a K-Hopf algebra H as a K-bialgebra with an additional map called the coinverse (or antipode) satisfying the coinverse (or antipode) property. We give some examples of Hopf algebras including the group ring KG which in many respects is the prototype Hopf algebra that others are modelled on. We define convolution, a binary operation on linear transformations, and use it to prove that the coinverse is an algebra anti-homomorphism, and consequently, that the coinverse has order 2 under composition whenever H is cocommutative. Using convolution, we also show that the coinverse is a coalgebra anti-homomorphism, a property that can be used to show that the coinverse has order 2 whenever H is commutative. (We will again employ the coalgebra anti-homomorphism property in §3.2.) Next, we consider Hopf ideals, quotient Hopf algebras, and homomorphisms of Hopf algebras. Finally we show that H is a finite dimensional Hopf algebra if and only if H  is a finite dimensional Hopf algebra. *

*

*

Definition 3.1.1. A K-Hopf algebra is a bialgebra over a field K H D .H; mH ; H ; H ; H / together with a K-linear map H W H ! H that satisfies mH .IH ˝ H /H .h/ D H .h/1H D mH . H ˝ IH /H .h/

(3.1)

for all h 2 H. The map H is the coinverse (or antipode) map and property (3.1) is the coinverse (or antipode) property. The field K itself is a K-Hopf algebra (take K D IK ) called the trivial K-Hopf algebra. Here are some other examples of Hopf algebras. Example 3.1.2. Let G be a finite group. Let KG be the monoid (group) bialgebra of Example 2.1.2. Define a coinverse map KG W KG ! KG by KG . / D  1 ; for  2 G. Then KG is a K-Hopf algebra.

3.1 Introduction to Hopf Algebras

69

Example 3.1.3. Let KŒx be the polynomial bialgebra with x primitive (Example 2.1.4.) Define the coinverse map KŒx W KŒx ! KŒx by KŒx .xi / D .x/i ; for i  0. Then KŒx is a K-Hopf algebra. The polynomial bialgebra of Example 2.1.3 with x grouplike, that is, where x 7! x ˝ x under comultiplication, cannot be endowed with the structure of a K-Hopf algebra, see §3.5, Exercise 1. We can modify this bialgebra to yield a Hopf algebra, however. Example 3.1.4. Let KŒx; y=.xy  1/ be the quotient K-algebra. Then KŒx; y=.xy  1/ Š KŒx; x1 : Define comultiplication KŒx;x1  W KŒx; x1  ! KŒx; x1  ˝ KŒx; x1  by making x and x1 grouplike. Define the counit map KŒx;x1  W KŒx; x1  ! K by x 7! 1, x1 7! 1, and define the coinverse map KŒx;x1  W KŒx; x1  ! KŒx; x1  by x 7! x1 , x1 7! x. Then KŒx; x1  is a K-Hopf algebra. The next example is due to M. Sweedler, cf. [Mo93, 1.5.6]. Example 3.1.5. Let K be a field of characteristic 6D 2. Let H be the K-algebra generated by f1; g; x; gxg modulo the relations g2 D 1; x2 D 0; xg D gx: Let comultiplication H W H ! H ˝K H be defined by g 7! g ˝ g; x 7! x ˝ 1 C g ˝ x; let the counit map H W H ! K be defined as g 7! 1, x 7! 0, and let the coinverse map H W H ! H, be given by g 7! g, x 7! gx. Then H is a K-Hopf algebra. A K-Hopf algebra H is commutative if it is a commutative algebra; H is cocommutative if it is a cocommutative coalgebra. The group algebra KG of Example 3.1.2 is cocommutative; it is commutative if and only if G is abelian.

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3 Hopf Algebras

The Hopf algebras of Examples 3.1.3 and 3.1.4 are both commutative and cocommutative, while Sweedler’s Hopf algebra of Example 3.1.5 is neither commutative nor cocommutative. A K-Hopf algebra that is neither commutative nor cocommutative is a quantum group. In many ways, the group ring KG of Example 3.1.2 is the canonical example that is generalized to the concept of Hopf algebra. One has KG ı KG D IKG ; so that the coninverse of KG has order 2. We wonder: does every Hopf algebra have coinverse of order 2? The answer is “no.” Consider the Sweedler Hopf algebra H of Example 3.1.5. We have . H ı H /.x/ D H . H .x// D H .gx/ D x; and so, H does not have order 2. (In fact, H has order 4, see §3.5, Exercise 4.) What properties of a Hopf algebra H guarantee that its coinverse map H has order 2, as does the coinverse KG of the prototypical Hopf algebra KG? We shall answer this question in what follows. For the moment we let C be a K-coalgebra and let A be a K-algebra. Let HomK .C; A/ denote the collection of linear transformations

W C ! A. On HomK .C; A/ we can define a multiplication as follows. For f ; g 2 HomK .C; A/, a 2 C, .f g/.a/ D mA .f ˝ g/C .a/ D

X

f .a.1/ /g.a.2/ /:

.a/

This multiplication is called convolution. Proposition 3.1.6. Let C be a K-coalgebra and let A be a K-algebra. Then HomK .C; A/ together with convolution is a monoid. Proof. We show that the axioms for a monoid hold. We first show that is associative. For f ; g; h 2 HomK .C; A/, one has .f .g h//.a/ D mA .f ˝ .g h//C .a/ X D f .a.1/ /.g h/.a.2/ / .a/

D

X .a/

f .a.1/ /

X .a.2/ /

g.a.2/.1/ /h.a.2/.2/ /;

3.1 Introduction to Hopf Algebras

71

which in Sweedler notation equals ciativity of C , X

P .a/

f .a.1/ /g.a.2/ /h.a.3/ / D

.a/

f .a.1/ /g.a.2/ /h.a.3/ /. Now, by the coasso-

XX

f .a.1/.1/ /g.a.1/.2/ /h.a.2/ /

.a/ .a.1/ /

D

X

.f g/.a.1/ /h.a.2/ /

.a/

D mA ..f g/ ˝ h/C .a/ D ..f g/ h/.a/; and so is associative. Next, we show that A C serves as a left and right identity element in HomK .C; A/. For 2 HomK .C; A/, a 2 C, .A C /.a/ D mA .A C ˝ /C .a/ 1 0 X A .C .a.1/ // ˝ .a.2/ /A D mA @ D

X

.a/

A .C .a.1/ // .a.2/ /

.a/

D

X

C .a.1/ /A .1K / .a.2/ /

.a/

D

X

C .a.1/ /1A .a.2/ /

.a/

D

X

.C .a.1/ /a.2/ /

.a/

D .a/ by the counit property. Thus, A C D . In a similar manner, one also obtains A C D , and so, HomK .C; A/ is a monoid under .  Proposition 3.1.7. Let H be a K-Hopf algebra and let HomK .H; H/ be the monoid under convolution . Then H IH D H H D IH H : In other words, H is a left and right inverse of IH under . Proof. Exercise.



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3 Hopf Algebras

Convolution can be used to show that the coinverse map H W H ! H is an algebra “antihomomorphism” (and hence H is an algebra homomorphism if H is commutative). Proposition 3.1.8. Let H be a K-Hopf algebra with coinverse H . Then the following properties hold. (i) H .ab/ D H .b/ H .a/ for all a; b 2 H, (ii) H .1H / D 1H . Proof. For (i), we consider the K-coalgebra H ˝ H and the collection of linear transformations HomK .H ˝ H; H/. We have mH 2 HomK .H ˝ H; H/. We define two additional elements of HomK .H ˝ H; H/ as follows: For a; b 2 H, H mH W H ˝ H ! H;

a ˝ b 7! H .ab/;

D mH . H ˝ H / W H ˝ H ! H;

a ˝ b 7! H .b/ H .a/:

Recall that H ˝ H is a coalgebra with comultiplication H˝H D .IH ˝  ˝ IH /.H ˝ H / P given as a ˝ b 7! .a;b/ a.1/ ˝ b.1/ ˝ a.2/ ˝ b.2/ , and counit H˝H defined by a ˝ b 7! H .a/H .b/. For the proof of (i) we first show that mH D H H˝H D mH ;

(3.2)

mH H mH D H H˝H D H mH mH :

(3.3)

and

We first prove (3.2). For a; b 2 H, .mH /.a ˝ b/ D .mH mH . H ˝ H / //.a ˝ b/ D mH .mH ˝ mH . H ˝ H / //H˝H .a ˝ b// X  D mH mH .a.1/ ˝ b.1/ / ˝ mH . H ˝ H / .a.2/ ˝ b.2/ / D

X

.a;b/

a.1/ b.1/ H .b.2/ / H .a.2/ /

.a;b/

D

X .a/

a.1/ H .b/ H .a.2/ /

by the coinverse property

3.1 Introduction to Hopf Algebras

D

X

73

H .b/a.1/ H .a.2/ /

.a/

D H .b/H .a/1H

by the coinverse property

D H H˝H .a ˝ b/; thus mH D H H˝H . A similar calculation yields mH D H H˝H . We next consider (3.3). For a; b 2 H, .mH H mH /.a ˝ b/ D mH .mH ˝ H mH /H˝H .a ˝ b/ X  D mH mH .a.1/ ˝ b.1/ / ˝ H mH .a.2/ ˝ b.2/ / D

X

.a;b/

a.1/ b.1/ H .a.2/ b.2/ /

.a;b/

D mH .IH ˝ H /H .ab/ D H .ab/1H

by the coinverse property

D H .a/H .b/1H D H H˝H ; thus mH H mH D H H˝H . A similar computation shows that H mH mH D H H˝H . Now, from (3.2) and (3.3) we have mH D mH H mH ; thus . mH / D . mH / H mH H H˝H D H H˝H H mH

D H mH

by Proposition 3.1.6.

Thus (i) is proved. For (ii) observe that 1H D 1K 1H D H .1H /1H D mH .IH ˝ H /H .1H / D mH .IH ˝ H /.1H ˝ 1H / D H .1H /:  We can now show that the cocommutativity of H implies that H has order 2.

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3 Hopf Algebras

Proposition 3.1.9. Let H be a K-Hopf algebra with coinverse H . If H is cocommutative, then H2 D IH . Proof. Let HomK .H; H/ be the collection of linear transformations endowed with convolution . Note that H , H2 , IH , and H H are all elements of HomK .H; H/. For a 2 H, X . H H2 /.a/ D H .a.2/ / H . H .a.1/ //; since H is cocommutative .a/

D

X

H . H .a.1/ /a.2/ /;

by Proposition 3.1.8(i)

.a/

D H .H .a/1H / D H .a/1H

by the coinverse property

by Proposition 3.1.8(ii)

D .H H /.a/: Thus IH . H H2 / D IH .H H / D IH

by Proposition 3.1.6.

On the other hand, IH . H H2 / D .IH H / H2 D H H H2 D H2 and so,

H2

by Proposition 3.1.7

by Proposition 3.1.6,

D IH .



Let H be a K-Hopf algebra with coinverse H . Then H is a coalgebra “antihomomorphism” (in fact, H is a K-coalgebra homomorphism if H is cocommutative). Proposition 3.1.10. Let H be a K-Hopf algebra with coinverse H . Then the following properties hold. (i)  . H ˝ H /H D H H , (ii) H H D H . Proof. We consider the collection of linear transformations HomK .H; H ˝ H/. We note that H 2 HomK .H; H ˝ H/ and define two additional elements of HomK .H; H ˝ H/, they are: X H .a.2/ / ˝ H .a.1/ /; D  . H ˝ H /H W H ! H ˝ H; a 7! .a/

H H W H ! H ˝ H;

a 7!

X . H .a//

H .a/.1/ ˝ H .a/.2/ ;

3.1 Introduction to Hopf Algebras

75

for a 2 H. Recall that H ˝ H is a K-algebra with multiplication mH˝H defined as .a ˝ b/ ˝ .c ˝ d/ 7! ac ˝ bd; and unit H˝H given as r 7! H .r/ ˝ 1H D r1H ˝ 1H , for a; b; c; d 2 H, r 2 K. To prove (i) we first establish that H

D H˝H H D

H

(3.4)

and H H H D H˝H H D H H H :

(3.5)

For (3.4), for a 2 H, .H

/.a/ D mH˝H .H ˝ D mH˝H .H ˝ 0 D mH˝H @ 0 D mH˝H @ D

X

/H .a/ 1 0 X /@ a.1/ ˝ a.2/ A .a/

X

H .a.1/ / ˝

1 .a.2/ /A

.a/

X

1 a.1/ ˝ a.2/ ˝ H .a.4/ / ˝ H .a.3/ /A

.a/

a.1/ H .a.4/ / ˝ a.2/ H .a.3/ /

.a/

D

X

a.1/ H .a.3/ / ˝ H .a.2/ /1H

by the coinverse property

.a/

D

X

H .a.2/ /a.1/ H .a.3/ / ˝ 1H

.a/

D

X

a.1/ H .a.2/ / ˝ 1H

by the counit property

.a/

D H .a/1H ˝ 1H

by the coinverse property

D .H˝H H /.a/: Thus, H D H˝H H . A similar computation shows that H D H˝H H , thus (3.4) holds. A straightforward computation also yields (3.5). It follows that

76

3 Hopf Algebras

.H .

H / H˝H H

/D D.

.H H H / H / H H

D H˝H H H H D H H ;

which yields (i). For (ii): For all a 2 H, H .a/ D H .a/H .1H / D H .H .a/1H / X  D H a.1/ H .a.2/ / by the coinverse property D

X

.a/

H .a.1/ /H . H .a.2/ //

.a/

D

X

H . H .H .a.1/ /a.2/ //

.a/

D H . H .a//

by the counit property. 

The coalgebra anti-homomorphism property of H can be used to show that H2

D IH whenever H is commutative, see §3.5, Exercise 6. We will need the coalgebra anti-homomorphism property of H in §3.2. Let H be a K-Hopf algebra. A Hopf ideal I is a biideal (viewing H as a K-bialgebra) that satisfies H .I/  I. Proposition 3.1.11. Let I  H be a Hopf ideal of H. Then H=I is a K-Hopf algebra. Proof. By Proposition 2.1.5, H=I is a K-bialgebra. Since H .I/  I, H induces a K-linear map H=I W H=I ! H=I defined as a C I 7! H .a/ C I. One has X  mH=I .IH=I ˝ H=I /H=I .a C I/ D mH=I .IH=I ˝ H=I / a.1/ C I ˝ a.2/ C I .a/

X D .a.1/ H .a.2/ / C I/ .a/

D H .a/1H C I D H=I .a C I/1H=I : In a similar manner, one obtains mH=I . H=I ˝ IH=I /H=I .a C I/ D H=I .a C I/1H=I : Thus H=I is a K-Hopf algebra.



3.1 Introduction to Hopf Algebras

77

Let H and H 0 be K-Hopf algebras. A bialgebra homomorphism W H ! H 0 is a homomorphism of Hopf algebras if

. H .a// D H 0 . .a// for all a 2 H. The Hopf homomorphism is an isomorphism of Hopf algebras if

is a bijection. Proposition 3.1.12. Let H be a finite dimensional vector space over the field K. Then H is a K-Hopf algebra if and only if H  is a K-Hopf algebra. Proof. Suppose that H is a K-Hopf algebra. By Proposition 2.1.10, H  is a bialgebra. Let H W H  ! H  be the transpose of the coinverse map H . Then for f 2 H  , a 2 H, .mH  .IH  ˝ H /H  .f //.a/ D mH  ..IH  ˝ H /H  .f //.a/ D ..IH  ˝ H /H  .f //H .a/ D H  .f /.IH ˝ H /H .a/ D f .mH .IH ˝ H /H .a// D f .H .a/1H /

by the counit property.

D H .a/f .1H / D H  .f /H .a/ D H  .f /1H  .a/: In a similar manner one obtains mH  . H ˝ IH  /H  .f / D H  .f /1H  : Consequently, H  D H satisfies the coinverse property and H  is a K-Hopf algebra. Conversely, if H  is a K-Hopf algebra, then H  D H is a K-Hopf algebra.  Example 3.1.13. Let G be a finite group. Then KG is a K-Hopf algebra (Example 3.1.2). Thus KG is a K-Hopf algebra. The subset fp g 2G with p . / D ı ; is a K-basis for KG . The multiplication on KG is defined as .p p /.!/ D p .!/p .!/, hence p p D ı ; p . The unit map of KG , KG W K ! KG is defined by r 7! rKG . The comultiplication of KG is given as KG .p / D

X

p ˝ p! ;

D !

cf. Proposition 1.3.10, and the counit map KG W KG ! K is defined as p 7! p .1/. Finally, the coinverse map KG W KG ! KG is given as p 7! p 1 .

78

3 Hopf Algebras

Let p be a prime number, let n  1 be an integer. Let K be a field containing pn , a primitive pn th root of unity. Let G D Cpn denote the cyclic group of order pn generated by g. Then the linear dual KCpn is a K-Hopf algebra as in Example 3.1.13. Proposition 3.1.14. KCpn Š KCpn , as K-Hopf algebras. pn 1

pn 1

Proof. Let fpi giD0 be the basis for KCpn dual to the basis fgi giD0 for KCpn . Let Ppn 1 ij

W KCpn ! KCpn be the K-linear map defined by pi D p1n jD0 pn gj . Then as the reader can show, is an isomorphism of K-Hopf algebras, see §3.5, Exercise 14. 

3.2 Integrals and Hopf Modules In this section we define the set of left and right integrals of a Hopf algebra over K. Rl Rr We show that the set of left (or right) integrals H (or H ) is a subspace of H and an ideal of H. We compute the ideal of integrals for KG and KG in the case that G is a finite group. Next we specialize to the case where H is finite dimensional and define the concept of a right comodule over H, a vector space that is “opposite” to a right H-module. We show that H  is a right H-comodule. We then consider right Hopf modules over H, which are vector spaces that are both right H-modules and right H-comodules. We state the Fundamental Theorem of Hopf Modules, which says that every right Hopf module over a finite dimensional Hopf algebra H is isomorphic to a trivial right Hopf module of the form W ˝ H for some vector space W. We show that the right H-comodule H  is a right Hopf module and prove a special case of the Rl Rl Fundamental Theorem which says that H  Š W ˝ H, where W D H  and H  ˝H Rl has a trivial right Hopf module structure. We conclude that H  is a one-dimensional subspace of H  , and so H  admits a generating integral. *

*

*

Let H be an K-Hopf algebra. Definition 3.2.1. A left integral of H is an element y 2 H that satisfies xy D .x/y; for all x 2 H. A right integral of H is an element y 2 H that satisfies yx D .x/y; for all x 2 H.

Rl We denote the Rcollection of left integrals of H by H and the collection of right r integrals of H by H . Rl Proposition R r 3.2.2. The set of left integrals H is a subspace of H; the set of right integrals H is a subspace of H.

3.2 Integrals and Hopf Modules

79

Proof. We prove the result for left integrals. We first show that Rl subgroup of H. Let x; y 2 H . Then for z 2 H, we have

Rl H

is an additive

z.x C y/ D zx C zy D .z/x C .z/y D .z/.x C y/; Rl Rl and so H is closed under addition. Moreover, 0 2 H since z0 D 0 D .z/0, and Rl Rl x 2 H since z.x/ D .zx/ D ..z/x/ D .z/.x/. Thus H  H. Now for Rl r 2 K, x 2 H , z.rx/ D .rz/x D H .rz/x D rH .x/x D H .z/.rx/; Rr and so, H is a K-subspace of H. The result for H is proved in the same way and we leave the details to the reader.  Rl Rr Proposition 3.2.3. H is an ideal of H; H is an ideal of H. Rl Rl Proof. We prove the result for H . By Proposition 3.2.2 H is an additive subgroup Rl Rl Rl Rl Rl of H. We show that H H  H and H H  H . Let x 2 H, y 2 H . Then for z 2 H, Rl

z.xy/ D .zx/y D H .zx/y D H .z/H .x/y D H .z/.xy/; Rl and thus, xy 2 H . Moreover, z.yx/ D .zy/x D .H .z/y/x D H .z/.yx/; Rr and thus yx 2 H . The result for H is proved in the same way and we leave the details to the reader.  Rl Rr A K-Hopf algebra H is unimodular if H D H . If H is commutative, then H is unimodular. Commutativity is not a necessary condition for H to be unimodular, however. Rl

Proposition 3.2.4. LetX G be a finite group and let KG be the group ring of G over K. Rl Rr Then KG D KG D K . P

 2G

P Proof. Let L D K  2G  and let a  2G  2 L, aP2 K. Since f g 2G is a K-basis for KG, an element x 2 KG can be written as x D 2G x , for x 2 K. Now ! X  X  X X x a  D  x a

2G

 2G

2G

D

X

 2G

 X  x a 

2G

D KG

X

2G

and so, L 

Rl

KG .

 2G

 X  x a  ;  2G

80

3 Hopf Algebras

Now suppose x D

P

 2G x 

2

Rl

For 2 G, X X x D KG . /x D x D x  D x . / KG .

 2G

 2G

where  W G ! G is a permutation of the elements of G. Thus, x D a for some Rl a 2 K and all  2 G, and so, KG  L. Rr An analogous argument is used to show that KG D L.  Since G is finite, the linear dual KG is a K-Hopf algebra (Proposition 3.1.12), Rl Rr thus KG is unimodular since KG is commutative. We compute KG D KG below. Rl Rr Proposition 3.2.5. KG D KG D Kp1 , where p1 is the element of KG defined by p1 . / D 1 if D 1, p1 . / D 0 if 6D 1. Proof. Recall that P fp W 2 Gg defined by p . / D ı ; is a K-basis for KG . Let ap1 2 Kp1 and let 2G x p 2 KG . Now  X  X  x p ap1 D ax1 p1 D x1 ap1 D KG x p ap1 ;

2G

2G

Rl

and so, Kp1  KG . Rl P Next, assume that  2G x p 2 KG . Then for 2 G, X  X X x p D x p D KG .p / x p D ı ;1 x p ; p  2G

and so x D 0 for allR  6D 1. Hence r argument shows that KG D Kp1 .

Rl

KG

 2G

 Kp1 , and so,

Rl

 2G

KG

D Kp1 . A similar 

For the remainder of this section, all Hopf algebras will be finite dimensional over a field K. Definition 3.2.6. Let H be a K-Hopf algebra and let M be a vector space over K. Then M is a right H-comodule if there exists a K-linear map ‰ W M ! M ˝ H for which (i) .‰ ˝ IH /‰ D .IM ˝ H /‰, (ii) .IM ˝ H /‰.m/ D m ˝ 1K ; 8m 2 M. Example 3.2.7. The K-Hopf algebra H is a right comodule over itself with the comultiplication map H playing the role of ‰. Let M, N be right H-comodules with comodule maps ‰M , ‰N , respectively. A homomorphism of right comodules is a linear transformation W M ! N for which ‰N ı D . ˝ IH / ı ‰M :

3.2 Integrals and Hopf Modules

81

We adapt Sweedler notation for comodules. Let M be a right H-comodule with structure map ‰ W M ! M ˝ H. Extending Sweedler notation for comultiplication, we write X ‰.ˇ/ D ˇ.1/ ˝ b.2/ .ˇ/

for ˇ; ˇ.1/ 2 M, b.2/ 2 H. We shall write ‰.ˇ.1/ / D

X .ˇ.1/ /

ˇ.1/ .1/ ˝ b.1/ .2/ :

Now, .‰ ˝ IH /‰.ˇ/ D .‰ ˝ IH / D

X .ˇ;ˇ.1/ /

X

 ˇ.1/ ˝ b.2/

.ˇ/

ˇ.1/ .1/ ˝ b.1/ .2/ ˝ b.2/ ;

(3.6)

and .IM ˝ H /‰.ˇ/ D .IM ˝ H / D

X .ˇ;b.2/ /

X

 ˇ.1/ ˝ b.2/

.ˇ/

ˇ.1/ ˝ b.2/ .1/ ˝ ˇ.2/ .2/ :

(3.7)

By Definition 3.2.6(i) the expressions in (3.6) and (3.7) are equal. The common value in (3.6) and (3.7) will be denoted as X

ˇ.1/ ˝ b.2/ ˝ b.3/ :

.ˇ/

Similarly, the common value of .IM ˝ IH ˝ /.IM ˝ /‰.ˇ/ D .IM ˝  ˝ IH /.IM ˝ /‰.ˇ/ D .‰ ˝ IH ˝ IH /.IM ˝ /‰.ˇ/ D .IM ˝ IH ˝ /.‰ ˝ IH /‰.ˇ/ D .IM ˝  ˝ IH /.‰ ˝ IH /‰.ˇ/ D .‰ ˝ IH ˝ IH /.‰ ˝ IH /‰.ˇ/

82

is denoted as

3 Hopf Algebras

X

ˇ.1/ ˝ b.2/ ˝ b.3/ ˝ b.4/ :

.ˇ/

We have seen that H is a right comodule over itself. Here is another example of a right comodule. Example 3.2.8. Let G be a finite group and let KG be the K-Hopf algebra of Example 3.1.2, and let KG be the K-Hopf algebra of Example 3.1.13. Then fp g 2G is the basis for KG dual to the basis f g 2G for KG. Let ‰ W KG ! KG ˝ KG be the K-linear map defined by X p p ˝  ‰.p / D  2G

D p ˝ : Then KG is a right KG-comodule with structure map ‰. We can generalize Example 3.2.8 to finite dimensional Hopf algebras. Let fbi gniD1 be a basis for H and let f˛i gniD1 be the basis for H  dualPto the basis fbi gniD1 . We have ˛i .bj / D bj .˛i / D ıi;j (note: bj 2 H  D H) and ˇ D niD1 bi .ˇ/˛i for ˇ 2 H  . Proposition 3.2.9. H  is a right H-comodule with structure map ‰ W H ! H ˝ H defined as ‰.ˇ/ D

n X

˛i ˇ ˝ bi ;

iD1

for ˇ 2 H  . Proof. We show thatPconditions (i) and (ii) of Definition 3.2.6 hold. For (i): first note that an element  ˝ x ˝ y 2 H  ˝ H ˝ H defines a K-linear map H ˝ H ! H by the rule

X

  ˝ x ˝ y . ˝ / D x. /y./;

for ;  2 H  . Let ˇ 2 H  and note that .IH  ˝ H /‰.ˇ/ D

n X

˛i ˇ ˝ H .bi /

iD1

D

n X X iD1 .bi /

˛i ˇ ˝ bi.1/ ˝ bi.2/

3.2 Integrals and Hopf Modules

83

is an element of H  ˝ H ˝ H. Its value at  ˝  is  X n X ˛i ˇ ˝ bi.1/ ˝ bi.2/ . ˝ / ..IH  ˝ H /‰.ˇ//. ˝ / D iD1 .bi /

D

n X X

bi.1/ . /bi.2/ ./˛i ˇ

iD1 .bi /

D

n X

bi ./˛i ˇ

iD1

D ˇ: On the other hand, .‰ ˝ IH /‰.ˇ/ 2 H  ˝ H ˝ H and its value at  ˝  is  X n ‰.˛i ˇ/ ˝ bi . ˝ / ..‰ ˝ IH /‰.ˇ//. ˝ / D iD1

D

X n X n

 ˛j .˛i ˇ/ ˝ bj ˝ bi . ˝ /

iD1 jD1

D

n X n X

bj . /bi ./˛j ˛i ˇ

iD1 jD1

D

X n

bj . /˛j

jD1

 X n

 bi ./˛i ˇ

iD1

D ˇ: which proves condition (i) of Definition 3.2.6. For condition (ii) of Definition 3.2.6: .IH  ˝ H /‰.ˇ/ D

n X

˛i ˇ ˝ H .bi /

iD1

D

n X

H .bi /˛i ˇ ˝ 1K

iD1

D

X n

 H .bi /˛i ˇ ˝ 1K

iD1

D

X n

 bi .H /˛i ˇ ˝ 1K

iD1

D H ˇ ˝ 1K D ˇ ˝ 1K ; since H D 1H  .



84

3 Hopf Algebras

The right H-comodule structure on H  given in Proposition 3.2.9 induces a left H -module structure on H  defined as 

˛  ˇ D s2 .IH  ˝ ˛/‰.ˇ/ D

n X

˛.bi /˛i ˇ;

iD1

for ˇ; ˛ 2 H  . Here, s2 is the map H  ˝ K ! H  given as  ˝ r 7! r , for  2 H  , r 2 K. The induced left H  -module structure is precisely the left multiplication action of H  on itself since ˛ˇ D

n X

X n

˛.bi /˛i ˇ D

iD1

 bi .˛/˛i ˇ D ˛ˇ:

(3.8)

iD1

Let ‰ W H  ! H  ˝ H be the right H-comodule structure map of Proposition 3.2.9. In Sweedler notation, formula (3.8) is now written as ˛ˇ D

X

˛.b.2/ /ˇ.1/ :

(3.9)

.ˇ/

Let M be a right H-comodule with structure map ‰ W M ! M ˝ H. If ‰ respects certain H-module structures on M and M ˝ H, then M is a “right Hopf module” over H. Here is a precise definition. Definition 3.2.10. Let H be an K-Hopf algebra and let M be a vector space over K. Then M is a right Hopf module over H if (i) M is a right H-module with scalar multiplication given as “,” (ii) M is a right H-comodule with structure map ‰ W M ! M ˝ H, (iii) the right H-comodule structure map ‰ W M ! M ˝ H is a homomorphism of right H-modules where M ˝ H is a right H-module with scalar multiplication .m ˝ k/h D

X

m  h.1/ ˝ kh.2/ ;

.h/

for h; k 2 H, m 2 M. Example 3.2.11. Let H be a K-Hopf algebra. Then H is a right module over itself through right multiplication and a right comodule over itself with structure map H . Endow H ˝ H with the structure of a right H-module with scalar multiplication given as .a ˝ b/h D

X

ah.1/ ˝ bh.2/ ;

.h/

for a; b; h 2 H. Then H W H ! H ˝ H is a homomorphism of right H-modules (§3.5, Exercise 17). Thus H is a right Hopf module over itself.

3.2 Integrals and Hopf Modules

85

Example 3.2.12. Let W be a vector space over K and a right H-module with action denoted as “.” Let M D W ˝ H, and on W ˝ H define a right H-module structure by X .w ˝ a/h D w  h.1/ ˝ ah.2/ ; .h/

for w 2 W, a; h 2 H. By §3.5, Exercise 18, W ˝ H is a right H-comodule with structure map IW ˝ H W W ˝ H ! W ˝ H ˝ H: Endow W ˝ H ˝ H with a right H-module structure defined as X .w ˝ a/h.1/ ˝ bh.2/ .w ˝ a ˝ b/h D .h/

D

X

w  h.1/ ˝ ah.2/ ˝ bh.3/ ;

.h;h.1/ /

for w 2 W, a; b; h 2 H. Now IW ˝ H is an H-module homomorphism and so W ˝ H is a right Hopf module over H. Example 3.2.13 (Trivial Right Hopf Module). In Example 3.2.12 we take a vector space W with right H-module action defined as w  h D H .h/w; for w 2 W, h 2 H; W is a trivial right module. Now the right action of H on W ˝ H is X w  h.1/ ˝ ah.2/ .w ˝ a/h D .h/

D

X

H .h.1/ /w ˝ ah.2/

.h/

D

X

w ˝ aH .h.1/ /h.2/

.h/

D w ˝ ah; for w 2 W, a; h 2 H. Likewise, the right H-action on W ˝ H ˝ H is .w ˝ a ˝ b/h D

X

w ˝ ah.1/ ˝ bh.2/ :

.h/

The resulting right H-Hopf module W ˝ H is a trivial right Hopf module.

86

3 Hopf Algebras

Let M, N be right Hopf modules over H. Then the map W M ! N is a homomorphism of Hopf modules if is both a homomorphism of H-modules and a homomorphism of H-comodules. Proposition 3.2.14 (Fundamental Theorem of Hopf Modules). Let H be a K-Hopf algebra, let M be a right Hopf module over H with structure map ‰ W M ! M ˝ H. Let W D fm 2 M W ‰.m/ D m ˝ 1g; and endow W ˝ H with the structure of a trivial right Hopf module over H. Then M Š W ˝ H as right Hopf modules over H. Proof. For the proof, the reader is referred to [Mo93, §1.9].



Our goal is to show that the Fundamental Theorem of Hopf Modules holds in the case M D H  ; we show that there is an isomorphism of Hopf modules H  Š W ˝H where W ˝ H is the trivial Hopf module with W D fˇ 2 H  W ‰.ˇ/ D ˇ ˝ 1g. Of course, we first need to show that H  is a right Hopf module over H. Put D H . We define a right H-module structure on H  by X .ˇ  h/.k/ D ˇ.k .h// D ˇ.1/ .k/ˇ.2/ . .h// (3.10) .ˇ/

and define a right H-module structure on H  ˝ H by X .ˇ ˝ k/h D ˇ  h.1/ ˝ kh.2/

(3.11)

.h/

for h; k 2 H, ˇ 2 H  . The right action of H on H  endows H  with the structure of an “anti” right H-module algebra. Lemma 3.2.15. Let h 2 H, ˛; ˇ 2 H  . Then .˛ˇ/  h D

X

.˛  h.2/ /.ˇ  h.1/ /:

.h/

Proof. Since the comultiplication H  is an R-algebra homomorphism, ..˛ˇ/  h/.k/ D .˛ˇ/.k .h// X D .˛.1/ ˇ.1/ /.k/.˛.2/ ˇ.2/ /. .h//; .˛;ˇ/

for all k 2 H. By Proposition 3.1.10(i), . .h// D

X .h/

.h.2/ / ˝ .h.1/ /;

3.2 Integrals and Hopf Modules

thus X

87

.˛.1/ ˇ.1/ /.k/.˛.2/ ˇ.2/ /. .h// D

.˛;ˇ/

X

.˛.1/ ˇ.1/ /.k/˛.2/ . .h.2/ //ˇ.2/ . .h.1/ //

.˛;ˇ;h/

D

X

˛.1/ .k.1/ /ˇ.1/ .k.2/ /˛.2/ . .h.2/ //ˇ.2/ . .h.1/ //

.˛;ˇ;h;k/

D

X

˛.1/ .k.1/ /˛.2/ . .h.2/ //ˇ.1/ .k.2/ /ˇ.2/ . .h.1/ //

.˛;ˇ;h;k/

D

X

˛.k.1/ .h.2/ //ˇ.k.2/ .h.1/ //

.h;k/

D

X .˛  h.2/ /.k.1/ /.ˇ  h.1/ /.k.2/ / .h;k/

X  D .˛  h.2/ /.ˇ  h.1/ / .k/ .h/



which proves the lemma.

Proposition 3.2.16 (Larson and Sweedler, [LS69]). Let H be a finite dimensional Hopf algebra with linear dual H  . Let fbi gniD1 be a basis for H and let f˛i gniD1 be the basis for H  dual to the basis fbi gniD1 . Let H  be a right H-module through (3.10), and let H  ˝H be a right H-module through (3.11). Then H  is a right Hopf module over H. Proof. By Proposition 3.2.9, H  isPa right H-comodule with structure map ‰ W H  ! H  ˝ H defined by ‰.ˇ/ D niD1 ˛i ˇ ˝ bi . Thus we only need to show that ‰.ˇ  h/ D ‰.ˇ/h, for h 2 H, ˇ 2 H  . For ˛ 2 H  one has   X .h.2/ /h.1/ ; by the counit property ˛.ˇ  h/ D ˛ ˇ  D

X

.h/

.˛  .h.2/ /1H /.ˇ  h.1/ /

.h/

D

X

.˛  .h.2/ /h.3/ /.ˇ  h.1/ / by the coinverse property:

.h/

Now, for k 2 H, P .h/ .˛  .h.2/ /h.3/ /.ˇ  h.1/ /.k/ X D .˛  .h.2/ /h.3/ /.k.1/ /.ˇ  h.1/ /.k.2/ / .h;k/

D

X .h;k/

˛.k.1/ .h.3/ / . .h.2/ ///.ˇ  h.1/ /.k.2/ / by Proposition 3.1.8(i)

88

3 Hopf Algebras

D

X

˛.k.1/ . .h.2/ //1H /.ˇ  h.1/ /.k.2/ /

by the coinverse property

.h;k/

D

X

˛.k.1/ .h.2/ /1H /.ˇ  h.1/ /.k.2/ /

by Proposition 3.1.10(ii):

.h;k/

Continuing with the calculation, we have P ˛.k .1/ .h.2/ /1H /.ˇ  h.1/ /.k.2/ / .h;k/ D

X

˛.k.1/ .h.2/ /h.3/ /.ˇ  h.1/ /.k.2/ / by the coinverse property

.h;k/

D

X

˛.1/ .k.1/ .h.2/ //˛.2/ .h.3/ /.ˇ  h.1/ /.k.2/ /

.h;k;˛/

D

X

˛.2/ .h.3/ /.˛.1/  h.2/ /.k.1/ /.ˇ  h.1/ /.k.2/ /

.h;k;˛/

D

X

˛.2/ .h.3/ /.˛.1/  h.2/ /.ˇ  h.1/ /.k/

.h;˛/

Thus, ˛.ˇ  h/ D

X

˛.2/ .h.3/ /.˛.1/  h.2/ /.ˇ  h.1/ /:

.h;˛/

Now, ˛.ˇ  h/ D

X

˛.2/ .h.3/ /.˛.1/  h.2/ /.ˇ  h.1/ /

.h;˛/

D

X

˛.2/ .h.2/ /.˛.1/ ˇ  h.1/ /

by Lemma 3.2.15

.h;˛/

D

X

˛.2/ .h.2/ /

.h;˛/

D

n XX

X n

 ˛.1/ .bj /˛j ˇ  h.1/

jD1

˛.1/ .bj /˛.2/ .h.2/ /.˛j ˇ  h.1/ /

.h;˛/ jD1

D

n XX .h/ jD1

˛.bj h.2/ /.˛j ˇ  h.1/ /:

by (3.8)

3.2 Integrals and Hopf Modules

89

It follows that ˛i .ˇ  h/ ˝ bi D

n XX

˛i .bj h.2/ /.˛j ˇ  h.1/ / ˝ bi

.h/ jD1

D

n XX

.˛j ˇ  h.1/ / ˝ ˛i .bj h.2/ /bi :

(3.12)

.h/ jD1

Finally, we can show that ‰ respects H-module structure: ‰.ˇ  h/ D

n X

˛i .ˇ  h/ ˝ bi

iD1

D

n XX n X .˛j ˇ  h.1/ / ˝ ˛i .bj h.2/ /bi

by (3.12)

iD1 .h/ jD1

D

n X n XX .˛j ˇ  h.1/ / ˝ ˛i .bj h.2/ /bi .h/ jD1 iD1

D

n XX

˛j ˇ  h.1/ ˝ bj h.2/

.h/ jD1

D

X n

 ˛j ˇ ˝ bj h

jD1

D ‰.ˇ/h: We conclude that H  is a right Hopf module over H.



Example 3.2.17. Let G be a finite group and let KG be the K-Hopf algebra of Example 3.1.2. Then KG is a K-Hopf algebra as in Example 3.1.13. In this example, we illustrate Proposition 3.2.16 by showing that KG is a right Hopf module over KG. Now, KG is a right KG-module through the action .p   /.!/ D p .! KG . // D p .! 1 / D p  .!/: From Example 3.2.8, KG is a right KG-comodule with structure map ‰ W KG ! KG ˝ KG defined as ‰.p / D p ˝ . Now, KG is a right Hopf module over KG, that is, ‰ is a KG-module homomorphism with KG-module structure on KG ˝ KG given as .p! ˝  / D p!  ˝  D p! ˝  :

90

3 Hopf Algebras

We now proceed with the proof of the case M D H  of the Fundamental Theorem of Hopf Modules. Here is our outline for the proof. Step 1.

We show that Rl

H

D W D fˇ 2 H  W ‰.ˇ/ D ˇ ˝ 1g:

Rl Rl Step 2. We give H  the trivial right H-module structure and endow H  ˝H with the trivial right Hopf module structure over H. Rl Step 3. We show that H  Š H  ˝H as right Hopf modules over H. Rl  Proposition 3.2.18. Step 1. H  D W where W D fˇ 2 H W ‰.ˇ/ D ˇ ˝ 1g. Proof. Let fbi gniD1 be a basis for H and let f˛i g be the basis for H  dual to fbi g. Let Rl ˇ 2 H  . Then for all ˛ 2 H  , ˛ˇ D H  .˛/ˇ D ˛.1/ˇ: Thus, ˛i ˇ D ˛i .1/ˇ: Thus, ‰.ˇ/ D

n X

˛i ˇ ˝ bi

iD1

D

n X

˛i .1/ˇ ˝ bi

iD1

D

n X

ˇ ˝ ˛i .1/bi

iD1

D ˇ ˝ 1; and so, ˇ 2 W. Conversely, if ˇ 2 W, then for all ˛ 2 H  , ˛ˇ D ˛.1/ˇ D H  .˛/ˇ Rl by (3.9), and so, ˇ 2 H  . Rl Step 2. We give H  the trivial right H-module structure



ˇ  h D H .h/ˇ; for all ˇ 2

Rl

H ,

h 2 H, give

Rl

H

˝H the structure of a right H-module through

.˛ ˝ k/h D ˛ ˝ kh;

3.2 Integrals and Hopf Modules

91

Rl Rl for ˛ 2 H  , k; h 2 H, and endow H  ˝H with the trivial right H-Hopf module structure Rl Rl .IR l  ˝ H / W H  ˝H ! H  ˝H ˝ H: H

Step 3. Finally, we show that H  Š prove two lemmas. For ˛ 2 H  , let .˛/ D with ‰.˛/ D

X

Rl

H

˝H as right H-Hopf modules. We first

˛.1/  .a.2/ / 2 H  ;

.˛/

P .˛/

˛.1/ ˝ a.2/ .

Lemma 3.2.19. For ˛ 2 H  , h 2 H, .˛  h/ D .h/.˛/. Proof. By Proposition 3.2.16 ‰.˛  h/ D ‰.˛/h D

X

.˛.1/  h.1/ / ˝ a.2/ h.2/ :

.˛;h/

Thus .˛  h/ D

X

.˛.1/  h.1/ /  .a.2/ h.2/ /

.˛;h/

D

X

.˛.1/  h.1/ /  .h.2/ / .a.2/ /

.˛;h/

D

X

˛.1/  ..h.1/ / .h.2/ / .a.2/ //

.˛;h/

D .h/

X

˛.1/  .a.2/ /

by the coinverse property

.˛/

D .h/.˛/:  

Lemma 3.2.20. For ˛ 2 H , ‰..˛// D .˛/ ˝ 1. Proof. We have ‰..˛// D ‰ D

X

X

 ˛.1/  .a.2/ /

.˛/

‰.˛.1/ / .a.2/ /

by Proposition 3.2.16

.˛/

D

X .˛/

.˛.1/ ˝ a.2/ / .a.3/ /

92

3 Hopf Algebras

D

X

.˛.1/  .a.4/ // ˝ a.2/ .a.3/ /

by Proposition 3.1.10(i)

.˛/

D

X

.˛.1/  .a.3/ // ˝ .a.2/ /1 by the coinverse property

.˛/

X .˛.1/  ..a.2/ /a.3/ // ˝ 1 D .˛/

X .˛.1/  .a.2/ // ˝ 1 D

by the counit property

.˛/

D .˛/ ˝ 1:  By Lemma 3.2.20, ‰..H  // D .H  /˝1. Thus by Proposition 3.2.18, .H  /  Rl  H  , and so, there exists a map % W H ! H  ˝H defined as

Rl

%.˛/ D . ˝ IH /‰.˛/ D

X

.˛.1/ / ˝ a.2/ :

.˛/  Proposition 3.2.21 (Larson and Sweedler, R l [LS69]). Let H be a right H-Hopf module as in Proposition 3.2.16 and let H  ˝H be a trivial right H-Hopf module Rl as in Example 3.2.13. Then H  Š H  ˝H as right H-Hopf modules. Rl Proof. Define a map ' W H  ˝H ! H  by

'.˛ ˝ h/ D ˛  h; for ˛ 2

Rl

H ,

h 2 H. The map ' is an H-module homomorphism: '..˛ ˝ h/k/ D '.˛ ˝ hk/ D ˛  .hk/ D .˛  h/  k D '.˛  h/k:

We show that ' is an isomorphism of H-modules by showing that %' D IR l  ˝H H Rl and that '% D IH  . Let h 2 H, ˛ 2 H  . We have %'.˛ ˝ h/ D %.˛  h/ X D .˛  h.1/ / ˝ h.2/ .h/

since ‰.˛  h/ D ˛  h.1/ ˝ h.2/

3.2 Integrals and Hopf Modules

D

X

93

.h.1/ /.˛/ ˝ h.2/

by Lemma 3.2.19

.h/

D

X

.˛/ ˝ .h.1/ /h.2/

.h/

D .˛/ ˝ h

by the counit property.

Now by Proposition 3.2.18, ‰.˛/ D ˛ ˝ 1, and so, .˛/ D ˛. It follows that %'.˛ ˝ h/ D ˛ ˝ h; Rl thus %' is the identity on H  ˝H. Regarding the map '% W H  ! H  , '%.˛/ D '

X

 .˛.1/ / ˝ a.2/

.˛/

 X D' .˛.1/  .a.2/ // ˝ a.3/ D

X

.˛/

.˛.1/  .a.2/ //  a.3/ /

.˛/

D

X

˛.1/  .a.2/ /a.3/

.˛/

D

X

˛.1/  .a.2/ /1

by the coinverse property

.˛/

D

X

.a.2/ /˛.1/

.˛/

D˛ since H  is a right H-comodule. Thus '% is the identity on H  . And so ' is an isomorphism of H-modules. Finally, we show that ' is a homomorphism of right H-comodules, that is, we Rl show that ' preserves right H-comodule structure. For ˛ ˝ h 2 H  ˝H, we have ‰.'.˛ ˝ h// D ‰.˛  h/ D ‰.˛/h D .˛ ˝ 1/h

94

3 Hopf Algebras

D

X

˛  h.1/ ˝ h.2/

.h/

D

X

'.˛ ˝ h.1/ / ˝ h.2/

.h/

D .' ˝ IH /

X

 ˛ ˝ h.1/ ˝ h.2/

.h/

D .' ˝ IH /.IR l  ˝ H /.˛ ˝ h/; H

and so, ' is an isomorphism of right H-Hopf modules.

Rl



Example 3.2.22. Let H D KG and H  D KG . By Proposition 3.2.5, KG D Kp1 . As in Proposition 3.2.21, there is an isomorphism of right KG-Hopf modules ' W Kp1 ˝ KG ! KG defined by '.p1 ˝ / D p1  D p : Its inverse % W KG ! Kp1 ˝ KG is defined by %.p / D . ˝ IKG /‰.p / D . ˝ IKG /.p ˝ / D .p / ˝ D .p  . // ˝ D .p  1 / ˝ D p1 ˝ : CorollaryR 3.2.23. Let H R l be an n-dimensional K-Hopf algebra with space Rofl left l integrals H . Then dim. H / D 1. Consequently, there exists an integral ƒ 2 H for Rl which Kƒ D H . Proof. By Proposition 3.2.21, with H in place of H  , there exists an isomorphism of H  -modules, and thus of K-vector spaces Rl H

˝H  Š H:

Rl Consequently, dim. H / dim.H  / D dim.H/. But n D dim.H  / D dim.H/, and so Rl Rl Rl  dim. H / D 1. Thus there exists an integral ƒ 2 H for which Kƒ D H . Rl Rl Rl An integral ƒ 2 H for which Kƒ D P H is a generating integral for H . As we have seen, for G a finite group, ƒ D  2G  is a generating integral for KG and ƒ D p1 is a generating integral for KG .

3.4 Hopf Orders

95

3.3 Hopf Algebras over Rings In this section we define Hopf algebras over a ring R. Many notions of Hopf algebras over fields generalize to rings, including integrals, comodules, and Hopf modules. If H is a Hopf algebra of finite rank over R, then there is an isomorphism H  Š Rl  H  ˝R H as right H-Hopf modules; if R is also a PID, then one can show that H admits a generating integral. *

*

*

Let R be a commutative ring with unity. Definition 3.3.1. A Hopf algebra H over R is an R-algebra H with structure map H W R ! H, together with additional R-linear maps H W H ! H ˝R H; H W H ! R; H W H ! H;

(comultiplication); (counit); (coinverse):

The maps H , H and H satisfy the identical comultiplication, counit, and coinverse properties, respectively, and the maps H and H are R-algebra homomorphisms. The obvious example of an R-Hopf algebra is the group ring RG where G is any finite group. As we shall see, in the case that R is an integral domain with field of fractions K, an R-Hopf order in KG is also an R-Hopf algebra. Many of the results for Hopf algebras over K carry over to rings. For example, if H is a free R-module of finite rank n, then so is its linear dual H  D HomR .H; R/. Rl The ideal of left integrals H  is defined as before: Rl

H

D fy 2 H  W xy D eH  .x/y; 8x 2 H  g:

Rl In fact, Proposition 3.2.21 holds: H  ˝R H Š H  as right H-Hopf modules. If R is Rl Rl a PID, then the R-submodule H   H  is free of rank l  n. We conclude that H  is a free rank one R-module, that is, there is a generating integral for H  .

3.4 Hopf Orders In this section we introduce Hopf orders in the group ring Hopf algebra KG. We show that Hopf orders are Hopf algebras over rings with structure maps induced from those of the K-Hopf algebra KG. When R is the ring of integers of a finite extension K=Q, we construct a collection of one parameter Hopf orders H.i/ in KCp

96

3 Hopf Algebras

together with their linear duals, H.i/ . In §4.5, we show how Hopf orders can be used to generalize the concept of a Galois group. * * * Let R be an integral domain, let K be its field of fractions, and assume that R is integrally closed in K. Let G be a finite group of order n. The group ring KG is a K-Hopf algebra with comultiplication KG W KG ! KG ˝K KG defined by g 7! g˝g, counit KG W KG ! K, defined by g 7! 1, and coinverse KG W KG ! KG given by g 7! g1 , for g 2 G. Definition 3.4.1. An R-order in KG is an R-submodule A of KG that satisfies the conditions (i) A is finitely generated and projective as an R-module, (ii) A  KG is closed under the multiplication of KG and 1KG 2 A, (iii) A contains a K-basis for KG (equivalently, KA D KG). Proposition 3.4.2. Let A be an R-order in KG. Then every element of A is a zero of a monic polynomial with coefficients in R. Proof. Let ˛ 2 A. Since A is an R-order, ˛A  A. This implies that ˛ satisfies a monic polynomial with coefficients in R (use the “Integrality Theorem” [La84, IX, §1]).  Proposition 3.4.3. Let R be a local integrally closed integral domain with field of fractions K. Let A be an R-order in KG. Then A is free over R of rank jGj. Proof. By definition, A is finitely generated and projective as an R-module, and so A is free over R of rank, say, m. Moreover, m D jGj since KA D KG.  Definition 3.4.4. An R-order H in KG for which KG .H/  H ˝ H is an R-Hopf order in KG. For example, the group ring RG is an R-Hopf order in KG. When G D 1, then the Hopf order R1 D R in K1 D K is the trivial Hopf order. Proposition 3.4.5. Let H be an R-Hopf order in KG where G is a finite group of order n  1. Then the coinverse KG and counit KG satisfy: (i) KG .H/ D H, (ii) KG .H/ D R. Proof. For (i): Let m D mKG , I D IKG ,  D KG . For n D 1 or n D 2, KG is the identity map on KG and so (i) holds. For n  3, let m.n2/ D m.I ˝ m/.I ˝ I ˝ m/    .I ˝ I    ˝ I ˝m/; „ ƒ‚ … n3

.n2/ D .I ˝ I    ˝ I ˝/    .I ˝ I ˝ /.I ˝ /: „ ƒ‚ … n3

3.4 Hopf Orders

97

Then for all g 2 G, m.n2/ .n2/ .g/ D gn1 D g1 ; and so, KG D m.n2/ .n2/ : Now, KG .H/ D m.n2/ ..n2/ .H//  H; 2 D IKG , since m.H ˝ H/  H and .H/  H ˝ H, thus KG .H/  H. Since KG H  KG .H/. This proves (i). For (ii): Recall that the K-algebra structure map KG W K ! KG is defined by KG .r/ D r, for r 2 K. Thus 1K D 1KG in KG. The R-module structure of H is defined through the restriction of KG to R, and so 1R 2 H. Thus R  H, and consequently, R D KG .R/  KG .H/. We next show that KG .H/  R. For h 2 H,

mKG .IKG ˝ KG /KG .h/ D H .h/1KG D H .h/: Consequently, KG .H/  H \ K. Since H is finitely generated as an R-module, KG .H/ is finitely generated as an R-submodule of K. Moreover, for a; b 2 H, KG .a/KG .b/ D KG .ab/ 2 KG .H/; since H is closed under the multiplication of KG. Thus KG .H/ is closed under the multiplication in K. Let s 2 KG .H/. By [La84, IX, §1] s is integral over R, and since R is integrally closed, s 2 R. Thus KG .H/  R, and so KG .H/ D R.  Proposition 3.4.6. Let H be an R-Hopf order in KG, G a finite group. Then H is an R-Hopf algebra. Proof. H is a ring with multiplication mKG since H is closed under the multiplication of KG. Moreover, as an R-submodule of KG, the R-module structure of H is given by the restriction of KG to R. Thus, H is an R-algebra. Since H is closed under the comultiplication of KG, the required comultiplication H can be taken to be KG restricted to H. By Proposition 3.4.5(ii), the counit map H can be taken to be KG restricted to H, and by Proposition 3.4.5(i), the coinverse map H is KG restricted to H. Note that H , H , and H satisfy the comultiplication, counit, and coinverse properties, respectively, since KG is a K-Hopf algebra.  We describe a collection of R-Hopf orders in KG. Let p be a prime number, let G D Cp denote the cyclic group of order p generated by g. For an integer n  1, let

98

3 Hopf Algebras

K D Q.pn / where pn is a primitive pn th root of unity. Then the ring of integers R of K is ZŒpn , and the ideal .p/ factors uniquely as n1 .p1/

.p/ D .1  pn /p

:

Put  D 1  pn . For an integer 0  i  pn1 , let H.i/ denote the free R-module on the basis (      )  g1 2 g  1 p1 g1 ; ;:::; : (3.13) 1; i i i Proposition 3.4.7. For each integer 0  i  pn1 , H.i/ is an R-Hopf order in KCp . Proof. We first show that H.i/ is an R-order in KCp . Since H.i/ is free over R of rank p, it is certainly finitely generated and projective as an R-module. Also, 1 D 1KG 2 H.i/ by construction. Put h D g1 . Then g D i h C 1 so that i 1 D gp D .i h C 1/p

! ! p .p1/i p1 p D h C h C  C  i h C 1: 1 p1 pi p

Hence

! ! p .p1/i p1 p h C  C  h C  i h D 0; 1 p1 pi p

so that

! ! p i p1 p    h D  h .p1/i h: 1 p1 p

    Now  mp mi 2 R for 1  m  p  1 and 0  i  pn1 since p j mp for 1  m  p  1 and mi j p for all 1  m  p  1, 0  i  pn1 . It follows that H.i/ is closed under the multiplication of KCp . Also, KH.i/ D KCp , and so, H.i/ is an R-order in KCp . Now   g1 1 D i .g ˝ g  1 ˝ 1/ KCp i  1 ..g  1 C 1/ ˝ .g  1 C 1/  1 ˝ 1/ i 1 D i ..g  1/ ˝ .g  1/ C .g  1/ ˝ 1  C1 ˝ .g  1/ C 1 ˝ 1  1 ˝ 1/ D

3.4 Hopf Orders

99

1 .1 ˝ .g  1/ C .g  1/ ˝ 1 C .g  1/ ˝ .g  1// i         g1 g1 g1 i g1 C1˝ C ˝ : D 1˝ i i i i D

  Thus KCp g1 2 H.i/ ˝ H.i/. Since KCp is an algebra homomorphism and i H.i/ ˝ H.i/ is closed under the multiplication in KCp ˝ KCp , we conclude that KCp .H.i//  H.i/ ˝ H.i/. Thus H.i/ is an R-Hopf order in KCp .  ˚ g1  Note that H.i/ is generated as an R-algebra by i , thus  H.i/ D R

g1 : i

Observe that H.0/ D RŒg  1 D RŒg D RCp . pn1 Put p D pn . Let CO p denote the character group of Cp generated by  , with  l .gm / D plm , for 0  l; m  p  1. Let tr W KCp ! K denote the trace map defined as tr.x/ D

p1 X

 l .x/:

lD0

 Pp1  m Note that tr D pa0 for am 2 K. As one can check, the map B W KCp  mD0 am g KCp ! K defined as B.x; y/ D tr.xy/ is a symmetric non-degenerate bilinear form on KCp , cf. [CF67, Chapter 1, §3]. With respect to B, we compute the discriminant of RCp , denoted as disc.RCp /. n .p1/

Proposition 3.4.8. With respect to B, disc.RCp / D .p/p D .p

/.

Proof. Using the basis f1; g; g2 ; : : : ; gp1 g for RCp , one computes B.gm ; gn / D tr.gmCn / D pı0;mCn , where m C n is taken modulo p. The result follows.  We next compute disc.H.i// for 0  i  pn1 . n1 i/

Proposition 3.4.9. For 0  i  pn1 , disc.H.i// D .p.p1/.p

/.

Proof. Clearly, H.0/  H.i/ for all i. We first compute the module index ŒH.i/ W H.0/. We find the matrix T in Matp .K/ that multiplies the R-basis (

    )  g1 2 g1 g  1 p1 ; 1; ;:::; ; i i i 

for H.i/ to yield the R-basis n o 1; .g  1/ ; .g  1/2 ; : : : ; .g  1/p1

100

3 Hopf Algebras

for H.0/. One has 0

1 1 0 0  0 B0 i 0    0 C B C 2i B C T D B0 0     0 C B: C : : :: :: A @ :: 0 0 0    .p1/i Thus ŒH.i/ W H.0/ D R det.T/ D Ri 2i    .p1/i D RiC2iCC.p1/i D .p.p1/i=2 /: Now, disc.H.0// D ŒH.i/ W H.0/2 disc.H.i// D .p.p1/i /disc.H.i//; and so, disc.H.i// D .p.p1/i /disc.H.0// D .p.p1/i /.p

n .p1/

p.p1/.pn1 i/

D .

/

by Proposition 3.4.8

/: 

The dual module of H.i/ is defined as H.i/D D fx 2 KCp W B.x; H.i//  Rg: We want to compute disc.H.i/D /, for 0  i  pn1 , and ultimately, show that H.i/D is an R-Hopf order in KCp . We consider the i D 0 case first. Since  B

gm n ;g p





D tr.gmCn =p/ D ım;n ;

g.p1/ 1 g1 g2 ; ; ; ;:::; p p p p

3.4 Hopf Orders

101

is the R-basis for H.0/D dual to the R-basis f1; g; g2 ; : : : ; gp1 g of RCp . We identify H.0/D with the collection of R-linear maps HomR .H.0/; R/: for x 2 H.0/D , a 2 H.0/, x.a/ D B.x; a/. Now, the R-submodule H.0/D of KCp is an R-algebra with multiplication (again, convolution!) defined as .x y/.a/ D .B.x; / ˝ B.y; //KCp .a/ X D B.x; a.1/ /B.y; a.2/ / .a/

D

X

tr.xa.1/ /tr.ya.2/ /;

.a/

for a 2 H.0/, x; y 2 H.0/D . For gl 2 G, 0  m; n  p  1, 

gm gn p p



.gl / D .B.gm =p; / ˝ B.gn =p; //KCp .gl / D B.gm =p; gl /B.gn =p; gl / D tr.gmCl =p/tr.gnCl =p/ D ım;l ın;l D ım;n ın;l   gn .gl /: D ım;n p

n Thus

gm p

o , 0  m  p  1, is the collection of minimal idempotents in the

R-algebra H.0/D . The minimal idempotents in the K-algebra KCp is the set fe0 ; e1 ; : : : ; ep1 g with 1 X nm n  g p nD0 p p1

em D

for 0  m  p  1. Consequently, there is an isomorphism of R-algebras

W H.0/ ! D

p1 M

Rem ;

mD0

 m  defined by g p D em . Note that em 2 H.0/D for 0  m  p  1 and em en D emCn , m C n taken modulo p. Thus .em / D gm .

102

3 Hopf Algebras

Lp1 We identify H.0/D with the R-order mD0 Rem in KCp through , hence, an R-basis for H.0/D is fe0 ; e1 ; : : : ; ep1 g. One has B.em ; en / D tr.em en / D tr.ım;n en / D ım;n ; and so, disc.H.0/D / D R. We can now compute disc.H.i/D / for 0  i  pn1 . Proposition 3.4.10. Let 0  i  pn1 and let H.i/ be an R-Hopf order in KCp . Then disc.H.i/D / D .p.p1/i /. Lp1 Proof. Since H.0/  H.i/, H.i/D  H.0/D D mD0 Rem . Now, disc.H.i/D / D ŒH.0/D W H.i/D 2 disc.H.0/D / D ŒH.0/D W H.i/D 2 D ŒH.i/ W H.0/2 D .p.p1/i /:  Finally, we show that H.i/D is an R-Hopf order in KCp of the form H.j/ for some j, 0  j  pn1 . Proposition 3.4.11. Let 0  i  pn1 , let i0 D pn1  i, and let H.i/ be an R-Hopf order in KCp . Then H.i/D is an R-Hopf order in KCp with H.i/D D H.i0 /, specifically, H.0/D D H.pn1 /. Proof. Since H.0/  H.i/, H.i/D  H.0/D . Endow H.0/D with convolution as multiplication. For a 2 H.0/D , let a a ƒ‚    …a : a D „ l

l

Let e0 ; e1 ; e2 ; : : : ; ep1 be the elements in H.0/D defined as em D 1p Let A be the free R-module on the basis    e  e p1

e1  e0  e1  e0 2 1 0 e0 ; ; : ; : : : ; i0 i0 i0 Now,





g1 B e0 ; i

m  D ı0;m ;

for 0  m  p  1,  B

 e1  e0 l ; 1 D ıl;0 ; i0

Pp1

nm n g. nD0 p

3.4 Hopf Orders

103

for 1  l  p  1, and  B

     p  1 lm g1 m e1  e0 l D ; 2 R; i0 i pn1

for 1  l; m  p  1. Thus, A  H.i/D . Since .el / D gl , we have



e1  e0 l i0



 D

g1 i0

l ;

for 1  l  p  1, thus .A/ D H.i0 /, that is, A Š H.i0 /, as R-algebras. We make the identification A D H.i0 /, thus H.i0 /  H.i/D . Now,  n1 0  disc.H.i0 // D p.p1/.p i / ; by Proposition 3.4.9   n1 n1 D p.p1/.p .p i//   D p.p1/i D disc.H.i/D /

by Proposition 3.4.10:

It follows that H.i0 / D H.i/D . In the case i D 0, H.0/D D H.pn1 /.



We next consider Hopf orders over a local ring. We fix a prime number p and an integer n  1 and assume that K is a finite extension of Q with ring of integers R containing a primitive pn th root of unit pn . Let .p/ D Pe11 Pe22    Pemm be the unique factorization of .p/ into prime ideals of R. Take P D Pi and e D ei for some i and let j jP denote the corresponding absolute value on K. There exists a field extension L of K with the following properties: (i) The absolute value j jP extends uniquely to an absolute value on L, also denoted as j jP , (ii) with respect to j jP , K is dense in L, that is, the closure K D L, (iii) L is complete with respect to j jP , that is, every j jP -Cauchy sequence in L converges to an element in L. The field extension L is the completion of K with respect to j jP , or: the completion of K at the prime ideal P, and is denoted as KP . Note that KP is a finite extension of Qp of local degree ŒKP W Qp . Let  be the uniformizing parameter for KP , and let RP be the valuation ring of KP . Then .p/ D ./e :

104

3 Hopf Algebras

We have p D u e for some unit u 2 RP , and so ord .p/ D e. Since p D v.1p /p1 for some unit v, e D ord .p/ D .p  1/ ord .1  p /: Thus e=.p  1/ is an integer which we denote as e0 ; we have ord .1  p / D e0 . For instance, in the extension Q.pn /=Q of Proposition 3.4.7, e0 D pn1 . Let Cpn denote the cyclic group of order pn generated by g. There is a considerable body of research on the structure of RP -Hopf orders in KP Cpn , see [TO70, Gr92, By93b, Un94, Un96, CU03, CU04, UC05], and [Un08b]. In fact, Hopf orders in KP Cpn have been completely classified in the cases n D 1; 2, see [TO70, Gr92, By93b] and [Un94]. The classification in the n D 1 case is due to Tate and Oort [TO70]. Essentially, an RP -Hopf order in KP Cp looks like the local version of a Hopf order constructed in Proposition 3.4.7. Proposition 3.4.12 (Tate and Oort). Let H be an RP -Hopf order in KP Cp . Then  H D RP

g1 i



for some integer i, 0  i  e0 . Proof. The proof is beyond the scope of this book. The interested reader is referred to [Ch00, Chapter 4]. 

3.5 Chapter Exercises Exercises for §3.1 1. Let KŒx be the polynomial bialgebra with x grouplike. Show that there is no K-linear map KŒx ! KŒx that satisfies the coinverse property. 2. Verify that the maps H , H , and H of Example 3.1.5 satisfy the comultiplication, counit, and coinverse properties, respectively. 3. Let H be a cocommutative K-Hopf algebra and let A be a commutative K-algebra. Prove that f g D g f , for all f ; g 2 HomK .H; A/. 4. Let H be Sweedler’s Hopf algebra of Example 3.1.5. Show that the coinverse map H has order 4. 5. Prove Proposition 3.1.7. 6. Let H be a commutative K-Hopf algebra with coinverse H . Prove that H2 D IH . 7. Give an example of a K-Hopf algebra that does not have a bijective coinverse. 8. Let B be a bialgebra over a field K. Let B W B ! B be the map defined as B .b/ D 0 for all b 2 B. Determine whether B together with B is a K-Hopf algebra.

3.5 Chapter Exercises

105

9. Let S3 D h ; i, 3 D  2 D 1,  D 2  , be the symmetric group on three letters, and let KS3 be the Hopf algebra of Example 3.1.2. Let p and p be dual basis elements in KS3 . Compute KS3 .p / and KS3 .p /. 10. Let H be a commutative, cocommutative K-Hopf algebra. Let mH H W H ! H P be the map defined as h 7! .h/ h.1/ h.2/ . (a) Show that mH H W H ! H is a homomorphism of K-Hopf algebras. (b) In the case that H D KG for G finite, find conditions under which mH H is an isomorphism. 11. Let H be a K-Hopf algebra and view K as the trivial Hopf algebra. (a) Prove that H W H ! K is a homomorphism of K-coalgebras. (b) Is H a homomorphism of bialgebras? (c) Is H a homomorphism of Hopf algebras? 12. Let H be a K-Hopf algebra. The kernel of H is an ideal of H called the augmentation ideal of H. Prove that ker.H / is a Hopf ideal. 13. Let W H ! H 0 be a homomorphism of Hopf algebras. Suppose that h 2 H is grouplike. Show that .h/ is grouplike. Suppose that h 2 H is primitive. Show that .h/ is primitive. 14. Finish the proof of Proposition 3.1.14. Exercises for §3.2 15. Prove that the Hopf algebra H D KŒx; y=.xy  1/ of Example 3.1.4 is unimodular and compute the ideal of integrals of H. 16. Compute the ideal of left integrals for M. Sweedler’s Hopf algebra in Example 3.1.5. Is this Hopf algebra unimodular? 17. Referring to Example 3.2.11, prove that the comultiplication map H W H ! H ˝ H is a homomorphism of right H-modules. 18. Referring to Example 3.2.12, prove that W ˝ H is a right H-comodule with structure map IW ˝ H W W ˝ H ! W ˝ H ˝ H. 19. Let H be Sweedler’s K-Hopf algebra of Example 3.1.5. Rl (a) Compute the ideal of left integrals H  . Rl (b) Give explicit definitions for the isomorphisms % W H  ! H  ˝H and Rl ' W H  ˝H ! H  of Proposition 3.2.21. 20. Let H be a finite dimensional Hopf algebra over a field K. Prove that H  Š Rl H  ˝H as vector spaces over K. Exercises for §3.3 21. Let R be a commutative ring with unity and let H be an R-Hopf algebra. (a) Prove that H .H/ D R. (b) Use (a) to prove that H Š ker.H / ˚ R, as R-modules.

106

3 Hopf Algebras

Exercises for §3.4 22. Let K D Q.27 / with ring of integers R D ZŒ27  and let C3 denote the cyclic group of order 3. Construct the collection of Hopf orders H.i/ in KC3 . 23. Let p be a prime number, let Cp denote the cyclic group of order p, and let K D Q.p /. Prove that up to isomorphism, the only Hopf orders in KCp are ZŒp C3 and ZŒp CpD . 24. Let K D Q.pn /, n  1, with ring of integers R D ZŒpn . Then .p/ D ./e , with e D pn1 .p  1/,  D 1  pn . Let e0 D e=.p  1/ D pn1 , and let i; j be integers 0  i; j  e0 with pj  i. Let Cp2 denote the cyclic group of order p2 generated by g. Prove that  p g 1 g1 H.i; j/ D R ; i j is an R-Hopf order in KCp2 . Questions for Further Study 1. Let K be a finite extension of Q with ring of integers R. Let KT be the Myhill– Nerode bialgebra of Example 2.2.12. (a) Show that RT is an R-order in KT in the sense that the conditions of Definition 3.4.1 are satisfied. Is it true that KT .RT/  RT ˝ RT? (b) Find an example of an R-order A in KT other than RT. Does it hold that KT .A/  A ˝ A? 2. Let K be a finite extension of Q with ring of integers R. Let KT be the Myhill– Nerode bialgebra of Example 2.2.13. (a) Show that RT is an R-order in KT in the sense that the conditions of Definition 3.4.1 are satisfied. Is it true that KT .RT/  RT ˝ RT? (b) Find an example of an R-order A in KT other than RT. Does it hold that KT .A/  A ˝ A?

Chapter 4

Applications of Hopf Algebras

In this chapter we present three diverse applications of Hopf algebras. Our first application involves almost cocommutative bialgebras and quasitriangular bialgebras. We show that a quastitriangular bialgebra determines a solution to the Quantum Yang–Baxter Equation, and we give details on how to compute quastitriangular structures for certain two-dimensional bialgebras and Hopf algebras. We show that almost cocommutative Hopf algebras generalize Hopf algebras in which the coinverse has order 2. We then define the braid group on three strands (or more simply, the braid group) and show that a quasitriangular structure determines a representation of the braid group. For our second application we define affine varieties over a field K and discuss the coordinate ring KŒƒ of an affine variety ƒ. We show that an affine variety ƒ can be identified with the collection of K-algebra maps KŒƒ ! K, and this allows us to think of the geometric object ƒ as an algebraic object through the algebraic structure of its coordinate ring KŒƒ. We show that if KŒƒ is a bialgebra, then ƒ is a monoid, and if KŒƒ is a Hopf algebra, then ƒ is a group. For our third application, we use Hopf algebras to generalize the concept of a Galois extension. We show that the notion of a Galois extension L of K with group G is equivalent to L being a Galois KG-extension of K. In this latter form (L is a Galois KG-extension of K) the notion of a classical Galois extension L=K can be extended to rings of integers. If S is the ring of integers of L and R is the ring of integers of K, we consider when S is a Galois RG-extension of R, and provide an example of when this occurs using the Hilbert Class Field. The notion of a Galois KG-(or RG-) extension can be generalized to Galois H-extensions of rings S=R where H is a Hopf algebra. Moreover, the action of the Hopf algebra H on the ring S need not be induced from the classical Galois action. We give a general result of S. Chase and M. Sweedler which yields a Hopf Galois structure on a ring S in which the action of H on S is not the classical Galois action.

© Springer International Publishing Switzerland 2015 R.G. Underwood, Fundamentals of Hopf Algebras, Universitext, DOI 10.1007/978-3-319-18991-8_4

107

108

4 Applications of Hopf Algebras

Recalling our collection one parameter Hopf orders (§3.4), we show that there exists a ring of integers S in some extension L of K for which S is a Galois H.e0 /-extension (here we do have the classical Galois action), that is, H.e0 / is realizable as a Galois group. Locally, a result due to L. Childs states that every one parameter Hopf order H.i/ is realizable as a Galois group.

4.1 Quasitriangular Structures In this section we introduce almost cocommutative bialgebras and quasitriangular bialgebras. We show that a quastitriangular bialgebra determines a solution to the Quantum Yang–Baxter Equation and show how to compute quastitriangular structures for certain two-dimensional bialgebras and Hopf algebras. We show that almost cocommutative Hopf algebras generalize Hopf algebras in which the coinverse has order 2. *

*

*

Let K be a field. Throughout this section, ˝ D ˝K . Let B be a K-bialgebra and let B ˝ B be the tensor product K-algebra (§1.2). Let U.B ˝ B/ denote the group of units in B ˝ B and let R 2 U.B ˝ B/. Definition 4.1.1. The pair .B; R/ is almost cocommutative if the element R satisfies  .B .b// D RB .b/R1

(4.1)

for all b 2 B. If the bialgebra B is cocommutative, then the pair .B; 1 ˝ 1/ is almost cocommutative. However, if B is commutative and non-cocommutative, then .B; R/ cannot be almost cocommutative for any R 2 U.B ˝ B/ since in this case (4.1) reduces to the condition for cocommutativity. P Write R D niD1 ai ˝ bi 2 U.B ˝ B/. Let R12 D

n X

3

ai ˝ bi ˝ 1 2 B˝ ;

iD1

R13 D

n X

3

ai ˝ 1 ˝ bi 2 B˝ ;

iD1

23

R

D

n X iD1

3

1 ˝ ai ˝ bi 2 B˝ :

4.1 Quasitriangular Structures

109

Definition 4.1.2. The pair .B; R/ is quasitriangular if .B; R/ is almost cocommutative and the following conditions hold: .B ˝ IB /R D R13 R23

(4.2)

.IB ˝ B /R D R13 R12

(4.3)

Clearly, if B is cocommutative, then .B; 1˝1/ is quasitriangular. A quasitriangular structure is an element R 2 U.B ˝ B/ so that .B; R/ is quasitriangular. Let .B; R/ and .B0 ; R0 / be quasitriangular bialgebras. Then .B; R/, .B0 ; R0 / are isomorphic as quasitriangular bialgebras, written .B; R/ Š .B0 ; R0 /, if there exists a bialgebra isomorphism W B ! B0 for which R0 D . ˝ /.R/. Two quasitriangular structures R; R0 on a bialgebra B are equivalent quasitriangular structures if .B; R/ Š .B; R0 / as quasitriangular bialgebras. Example 4.1.3. Suppose that B is a commutative and non-cocommutative bialgebra, for instance, suppose that B D KG for G finite non-abelian. Then .B; R/ cannot be quasitriangular for any R 2 U.B ˝ B/; B has no quasitriangular structures. Example 4.1.4. Let T D f1; ag be the monoid with multiplication table 1 a

1 1 a

a a a

Let KT be the monoid bialgebra. We’ve seen this bialgebra before-it is isomorphic to the Myhill–Nerode bialgebra of Example 2.2.12. As we will show, KT has only the trivial quasitriangular structure R D 1 ˝ 1. Example 4.1.5. Let K be a field of characteristic 6D 2, let C2 be the cyclic group of order 2 generated by g and let KC2 be the group bialgebra. Then there are exactly two non-equivalent quasitriangular structures on KC2 , namely, R0 D 1 ˝ 1 and R1 D

1 .1 ˝ 1 C 1 ˝ g C g ˝ 1  g ˝ g/ : 2

Example 4.1.6. Let H be the Sweedler Hopf algebra of Example 3.1.5, defined over the field K D Q. For a 2 K, let R.a/ D

1 .1 ˝ 1 C 1 ˝ g C g ˝ 1  g ˝ g/ 2 a C .x ˝ x  x ˝ gx C gx ˝ x C gx ˝ gx/ 2

Then R.a/ is a quasitriangular structure for H. Moreover, there are an infinite number of non-equivalent quasitriangular structures of the form R.a/ for H, cf. [Mo93, 10.1.17], [Ra93].

110

4 Applications of Hopf Algebras

We want to provide some details on how to classify the quasitriangular structures in Examples 4.1.4 and 4.1.5, but first we explain why it is worthwhile to study quasitriangular bialgebras. Pn Let P A be any K-algebra and let R D iD1 ai ˝ bi 2 A ˝ P PA. As before, let R12 D niD1 ai ˝ bi ˝ 1A , R13 D niD1 ai ˝ 1A ˝ bi and R23 D niD1 1A ˝ ai ˝ bi . Proposition 4.1.7 (Drinfeld [Dr90]). Suppose .B; R/ is a quasitriangular bialgebra. Then R12 R13 R23 D R23 R13 R12 :

(4.4)

Proof. One has R12 R13 R23 D R12 .B ˝ IB /.R/ by (4.2)  X n D .R ˝ 1/ B .ai / ˝ bi iD1

D

n X

RB .ai / ˝ bi

iD1

D

n X

B .ai /R ˝ bi

by (4.1)

iD1

D

X n

 B .ai / ˝ bi .R ˝ 1/

iD1

D .B ˝ IB /

X n

 ai ˝ bi .R ˝ 1/:

iD1

Continuing with this calculation, we have  X n .B ˝ IB / ai ˝ bi .R ˝ 1/ D .B ˝ IB /.R/R12 iD1

D . ˝ IB /.B ˝ IB /.R/R12 D . ˝ IB /.R13 R23 /R12

by (4.2)

23 13 12

DR R R :  Equation (4.4) is known as the quantum Yang–Baxter equation (QYBE), see [Mo93, Chapter 10], [Ni12]. Proposition 4.1.7 says that quasitriangular bialgebras determine solutions to the QYBE. In §4.2 and §4.3 we will show that solutions to the QYBE will yield representations of a certain group called the Braid group. For the present, however, we return to the problem of how to construct quasitriangular structures for the monoid bialgebra KT and the group bialgebra KC2 .

4.1 Quasitriangular Structures

111

We need the following proposition which shows that every bialgebra isomorphism W B ! B0 with B quasitriangular extends to an isomorphism of quasitriangular bialgebras. Proposition 4.1.8. Suppose .B; R/ is quasitriangular and suppose that W B ! B0 is an isomorphism of K-bialgebras. Let R0 D . ˝ /.R/. Then .B0 ; R0 / is quasitriangular. Proof. Note that . ˝ /.R1 / D .. ˝ /.R//1 . Let b0 2 B0 . Then there exists b 2 B for which .b/ D b0 . Now B0 .b0 / D B0 . .b// D  . ˝ /B .b/

since is a bialgebra hom.

D . ˝ /B .b/ D . ˝ /.RB .b/R1 /

since .B; R/ is quasitriangular:

Continuing with the calculation, we obtain . ˝ /.RB .b/R1 / D . ˝ /.R/. ˝ /B .b/. ˝ /.R1 / D . ˝ /.R/B0 . .b//.. ˝ /.R//1 D . ˝ /.R/B0 .b0 /.. ˝ /.R//1 D R0 B0 .b0 /.R0 /1 and so, .B; R0 / is almost cocommutative. We next show that condition (4.2) holds. Indeed, .B0 ˝ IB0 /.R0 / D .B0 ˝ IB0 /. ˝ /.R/ X  n

.ai / ˝ .bi / D .B0 ˝ IB0 / iD1

D

n X

B0 . .ai // ˝ .bi /

iD1

D

n X

. ˝ /B .ai / ˝ .bi /

iD1

 X n D . ˝ ˝ / B .ai / ˝ bi iD1

D . ˝ ˝ /.B ˝ IB /.R/:

112

4 Applications of Hopf Algebras

Computing further, one obtains . ˝ ˝ /.B ˝ IB /.R/ D . ˝ ˝ /.R13 R23 / since .B; R/ is quasitriangular  X  X  n n D . ˝ ˝ / a i ˝ 1 ˝ bi 1 ˝ a i ˝ bi iD1

D

X n

iD1

 X  n

.ai / ˝ 1 ˝ .bi / 1 ˝ .ai / ˝ .bi /

iD1

iD1 13

D .. ˝ /.R// .. ˝ /.R//23 D .R0 /13 .R0 /23 ;

thus (4.2) holds for .B0 ; R0 /. In a similar manner we get .IB0 ˝ B0 /.R0 / D .R0 /13 .R0 /12 Thus .B0 ; R0 / is quasitriangular.



To find quasitriangular structures we also employ the following proposition due to Drinfeld [Dr86]. Let s1 W K ˝ B ! B, s2 W B ˝ K ! B be the maps defined as r ˝ b 7! rb, b ˝ r 7! rb, respectively. Proposition 4.1.9 (Drinfeld). Suppose .B; R/ is quasitriangular. Then (i) s1 .B ˝ IB /.R/ D 1, (ii) s2 .IB ˝ B /.R/ D 1. Proof. Put I D IB ,  D B ,  D B . We prove (i). First observe that  X n .s1 ˝ I/. ˝ I ˝ I/. ˝ I/.R/ D .s1 ˝ I/. ˝ I ˝ I/ B .ai / ˝ bi iD1

 X n D .s1 ˝ I/ . ˝ I/.ai / ˝ bi iD1

D

n X

s1 . ˝ I/.ai / ˝ bi

iD1

D

X

ai ˝ bi

iD1

D R: And so, R D .s1 ˝ I/. ˝ I ˝ I/. ˝ I/.R/ D .s1 ˝ I/. ˝ I ˝ I/.R13 R23 / 13

by (4.2)

D .s1 ˝ I/. ˝ I ˝ I/.R /.s1 ˝ I/. ˝ I ˝ I/.R23 /

4.1 Quasitriangular Structures

113

X n

D .s1 ˝ I/. ˝ I ˝ I/

  X n ai ˝ 1 ˝ bi .s1 ˝ I/. ˝ I ˝ I/ 1 ˝ ai ˝ bi

iD1

D

X n

.ai /1 ˝ bi

 X n

iD1

D

X n

iD1

 ai ˝ bi

iD1

 1 ˝ .ai /bi R:

iD1

Thus 1˝

X

.ai /bi D 1 ˝ 1

iD1

and consequently, 1 D s1

X n

 .ai / ˝ bi

D s1 . ˝ I/.R/:

iD1



A similar argument is used to prove (ii).

Now we can show that the monoid bialgebra B D KT of Example 4.1.4 has only the trivial quasitriangular structure. Observe that the linear dual KT  is a K-bialgebra on the basis fe1 ; ea g with ex .y/ D ıx;y , The algebra structure on KT  is given by ex ey D ıx;y ex . By Proposition 1.3.10, comultiplication on KT  is defined by KT  .e1 / D e1 ˝ e1 KT  .ea / D e1 ˝ ea C ea ˝ e1 C ea ˝ ea : The counit map is defined by KT  .e1 / D 1;

KT  .ea / D 0:

There is a bialgebra isomorphism W KT ! KT  defined as .1/ D e1 C ea ,

.a/ D e1 . Proposition 4.1.10. Let KT be the K-bialgebra of Example 4.1.4. Then there is exactly one quasitriangular structure on KT, namely, R D 1 ˝ 1. Proof. Certainly, 1 ˝ 1 is a quasitriangular structure for KT. We claim that 1 ˝ 1 is the only quasitriangular structure. If .KT; R/ is quasitriangular, then .KT  ; R0 /, R0 D . ˝ /.R/, is quasitriangular by Proposition 4.1.8. So, we first compute all of the quasitriangular structures of KT  . To this end suppose that .KT  ; R0 / is quasitriangular for some element R0 2 KT  ˝ KT  . Since

114

4 Applications of Hopf Algebras

KT  ˝ KT  D K.e1 ˝ e1 / ˚ K.e1 ˝ ea / ˚ K.ea ˝ e1 / ˚ K.ea ˝ ea /; R0 D w.e1 ˝ e1 / C x.e1 ˝ ea / C y.ea ˝ e1 / C z.ea ˝ ea / for w; x; y; z 2 K. Put 1 D 1KT  , I D IKT  ,  D KT  ,  D KT  . By Proposition 4.1.9(i), 1 D e1 C ea D s1 . ˝ I/.w.e1 ˝ e1 / C x.e1 ˝ ea / C y.ea ˝ e1 / C z.ea ˝ ea // D we1 C xea and so, w D x D 1. From Proposition 4.1.9(ii), one also has y D 1. Thus R0 D e1 ˝ e1 C e1 ˝ ea C ea ˝ e1 C z.ea ˝ ea / for z 2 K. We want to find all values of z for which .KT  ; R0 / is quasitriangular, necessarily, we require that . ˝ I/.R0 / D .R0 /13 .R0 /23 : Now, . ˝ I/.R0 / D . ˝ I/.e1 ˝ e1 Ce1 ˝ ea Cea ˝ e1 Cz.ea ˝ ea // D .e1 ˝ e1 / ˝ e1 C.e1 ˝ e1 / ˝ ea C.e1 ˝ ea Cea ˝ e1 Cea ˝ ea / ˝ e1 C z..e1 ˝ ea Cea ˝ e1 Cea ˝ ea / ˝ ea / D e1 ˝ e1 ˝ e1 Ce1 ˝ e1 ˝ ea Ce1 ˝ ea ˝ e1 Cea ˝ e1 ˝ e1 C ea ˝ ea ˝ e1 Cz.e1 ˝ ea ˝ ea /Cz.ea ˝ e1 ˝ ea / C z.ea ˝ ea ˝ ea /:

(4.5)

Moreover, .R0 /13 .R0 /23 D .e1 ˝ .e1 Cea / ˝ e1 Ce1 ˝ .e1 Cea / ˝ ea Cea ˝ .e1 Cea / ˝ e1 C z.ea ˝ .e1 Cea / ˝ ea //..e1 Cea / ˝ e1 ˝ e1 C.e1 Cea / ˝ e1 ˝ ea C .e1 Cea / ˝ ea ˝ e1 Cz..e1 Cea / ˝ ea ˝ ea // D .e1 ˝ e1 ˝ e1 Ce1 ˝ ea ˝ e1 Ce1 ˝ e1 ˝ ea Ce1 ˝ ea ˝ ea C ea ˝ e1 ˝ e1 Cea ˝ ea ˝ e1 Cz.ea ˝ e1 ˝ ea / C z.ea ˝ ˝ea ˝ ea //.e1 ˝ e1 ˝ e1 C ea ˝ e1 ˝ e1 C e1 ˝ e1 ˝ ea C ea ˝ ea ˝ e1 C e1 ˝ ea ˝ e1 C ea ˝ ea ˝ e1 C z.e1 ˝ ea ˝ ea / C z.ea ˝ ˝ea ˝ ea //

4.1 Quasitriangular Structures

115

D e1 ˝ e1 ˝ e1 C e1 ˝ ea ˝ e1 C e1 ˝ e1 ˝ ea C z.e1 ˝ ea ˝ ea / C ea ˝ e1 ˝ e1 C ea ˝ ea ˝ e1 C z.ea ˝ e1 ˝ ea / C z2 .ea ˝ ea ˝ ea /:

(4.6)

Thus if .KT  ; R0 / is quasitriangular, then from (4.5) and (4.6) e1 ˝ e1 ˝ e1 C e1 ˝ e1 ˝ ea C e1 ˝ ea ˝ e1 C ea ˝ e1 ˝ e1 C ea ˝ ea ˝ e1 C z.e1 ˝ ea ˝ ea / C z.ea ˝ e1 ˝ ea / C z.ea ˝ ea ˝ ea / D e1 ˝ e1 ˝ e1 C e1 ˝ ea ˝ e1 C e1 ˝ e1 ˝ ea C z.e1 ˝ ea ˝ ea / C ea ˝ e1 ˝ e1 C ea ˝ ea ˝ e1 C z.ea ˝ e1 ˝ ea / C z2 .ea ˝ ea ˝ ea /; and so, z2 D z. Thus either z D 0 or z D 1. If z D 0, then R0 is not a unit in KT  ˝ KT  . Thus R0 D e1 ˝ e1 C e1 ˝ ea C ea ˝ e1 C ea ˝ ea D 1 ˝ 1 is the only quasitriangular structure for KT  . Consequently, if .KT; R/ is quasitriangular, then . ˝ /.R/ D 1 ˝ 1: It follows that R D 1 ˝ 1. Let H be a K-Hopf algebra and let R D

Pn iD1

 ai ˝ bi 2 U.H ˝ H/.

Definition 4.1.11. The pair .H; R/ is a quasitriangular Hopf algebra if .H; R/ is quasitriangular as a K-bialgebra and the coinverse map H is a bijection. Recall this means that the following conditions hold: .H ˝ IH /R D R13 R23 ;

(4.7)

.IH ˝ H /R D R13 R12 :

(4.8)

We note that if H is a finite dimensional Hopf algebra, then its coinverse map H is necessarily bijective (this follows from Proposition 3.2.21, see [Mo93, (2.1.3)]). Thus Hopf algebras that were quasitriangular as bialgebras, yet did not have bijective coinverses, would not include finite dimensional Hopf algebras. A quasitriangular structure is an element R 2 U.H ˝ H/ so that .H; R/ is quasitriangular. Let .H; R/ and .H 0 ; R0 / be quasitriangular Hopf algebras. Then .H; R/,

116

4 Applications of Hopf Algebras

.H 0 ; R0 / are isomorphic as quasitriangular Hopf algebras if there exists a Hopf algebra isomorphism W H ! H 0 for which R0 D . ˝ /.R/. Two quasitriangular structures R; R0 on H are equivalent quasitriangular structures if .H; R/ Š .H; R0 / as quasitriangular Hopf algebras. Our goal is to compute all of the quasitriangular structures on the Hopf algebra KC2 of Example 4.1.5. First, we prove a proposition. Proposition 4.1.12. Suppose .H; R/ is quasitriangular. Then . H ˝ IH /.R/ D R1 . Proof. One has R. H ˝ IH /.R/ D

X n

  X n ai ˝ bi . H ˝ I/ aj ˝ bj

iD1

D

jD1

X n

ai ˝ bi

iD1

D

XX

 X n

 ai H .aj / ˝ bi bj

jD1

ai H .aj / ˝ bi bj

iD1 jD1

D .mH ˝ IH /.IH ˝ H ˝ IH /

X n X n

 ai ˝ aj ˝ bi bj

iD1 jD1

D .mH ˝ IH /.IH ˝ H ˝ IH /.R13 R23 /: Continuing with the calculation, .mH ˝ IH /.IH ˝ H ˝ IH /.R13 R23 / D .mH ˝ IH /.IH ˝ H ˝ IH /.H ˝ IH /.R/ D

n X

H .ai /1H ˝ bi

by (4.7)

by the coinverse property

iD1

D 1H ˝ 1H

by Proposition 4.1.9(i):

Thus . H ˝ IH /.R/ D R1 .



Proposition 4.1.13. Let KC2 be the Hopf algebra of Example 4.1.5. There are exactly two quasitriangular structures on KC2 , namely, R0 D 1 ˝ 1 and R1 D

1Cg 1g ˝1C ˝ g: 2 2

4.1 Quasitriangular Structures

117

Proof. Certainly, R0 D 1 ˝ 1 is a quasitriangular structure for KC2 . Let fp1 ; pg g be the basis for KC2 dual to the basis f1; gg for KC2 . The dual KC2 is a Hopf algebra with comultiplication defined by KC2 .p1 / D p1 ˝ p1 C pg ˝ pg ; KC2 .pg / D p1 ˝ pg C pg ˝ p1 ; counit map given by KC2 .p1 / D 1, KC2 .pg / D 0, and coinverse defined by KC2 .p1 / D p1 , KC2 .pg / D pg . Note that W KC2 ! KC2 defined as .1/ D p1 C pg , .g/ D p1  pg , is an isomorphism of Hopf algebras. If .KC2 ; R/ is quasitriangular, then .KC2 ; R0 /, R0 D . ˝ /.R/, is quasitriangular by Proposition 4.1.8. So, we first compute all of the quasitriangular structures of KC2 . To this end suppose that .KC2 ; R0 / is quasitriangular for some element R0 2 U.KC2 ˝ KC2 /. Then R0 D w.p1 ˝ p1 / C x.p1 ˝ pg / C y.pg ˝ p1 / C z.pg ˝ pg / for w; x; y; z 2 K. By Proposition 4.1.9(i), 1 D wp1 C xpg , and so, w D x D 1. From Proposition 4.1.9(ii), one also has y D 1. Thus R0 D p1 ˝ p1 C p1 ˝ pg C pg ˝ p1 C z.pg ˝ pg /; for z 2 K. Now by Proposition 4.1.12 . KC2 ˝ IKC2 /.R0 / D R0 D .R0 /1 ; and so, z2 D 1. Thus, either z D 1, which yields the quasitriangular structure 1 ˝ 1, or z D 1, which yields the unit R0 D p1 ˝ p1 C p1 ˝ pg C pg ˝ p1  .pg ˝ pg /: As one can verify, .KC2 ; R0 / is quasitriangular. Consequently,

1 .R0 / D R1 D

1g 1Cg ˝1C ˝g 2 2

is a quasitriangular structure for KC2 .



Let H be a K-Hopf algebra. As we have seen, if H is cocommutative, then the pair .H; 1 ˝ 1/ is almost cocommutative. By Proposition 3.1.9, cocommutativity of H implies that H2 .h/ D h for all h 2 H. The generalization of this property for almost cocommutative .H; R/ is due to Drinfeld [Dr90].

118

4 Applications of Hopf Algebras

Proposition 4.1.14 (Drinfeld). Let P .H; R/ be almost cocommutative with R D Pn n a ˝ b 2 U.H ˝ H/. Let u D i i iD1 iD1 H .bi /ai . Then u 2 U.H/ and H2 .h/ D uhu1 ; for all h 2 H. Proof. Let h 2 H. Since .H; R/ is almost cocommutative, RH .h/ D

X n

ai ˝ bi

 X .h/

iD1

D

 h.1/ ˝ h.2/

n XX

ai h.1/ ˝ bi h.2/

.h/ iD1

is equal to H .h/R D 

X

h.1/ ˝ h.2/

 X n

.h/

D

X

ai ˝ bi

iD1

h.2/ ˝ h.1/

 X n

.h/

D



n XX

 ai ˝ bi

iD1

h.2/ ai ˝ h.1/ bi :

.h/ iD1

Consequently, n XX

ai h.1/ ˝ bi h.2/ ˝ h.3/ D

.h/ iD1

n XX

h.2/ ai ˝ h.1/ bi ˝ h.3/ ;

(4.9)

.h/ iD1

where h.3/ is so that H .h/ D .IH ˝ H /H .h/ D

X

h.1/ ˝ h.2/ ˝ h.3/ :

.h/

From (4.9), one obtains n XX

H . H .h.3/ // H .bi h.2/ /ai h.1/

.h/ iD1

D

n XX .h/ iD1

H . H .h.3/ // H .h.1/ bi /h.2/ ai :

(4.10)

4.1 Quasitriangular Structures

119

Now, n XX

H . H .h.3/ // H .bi h.2/ /ai h.1/

.h/ iD1

D

n XX

H . H .h.3/ // H .h.2/ / H .bi /ai h.1/ ;

by Proposition 3.1.8(i)

.h/ iD1

D

n XX

H .h.2/ H .h.3/ // H .bi /ai h.1/ ;

by Proposition 3.1.8(i)

.h/ iD1

D

n XX

H .H .h.2/ /1H / H .bi /ai h.1/

by the coinverse property

.h/ iD1

D

n X

H .bi /ai h by the counit property

iD1

D uh: On the other hand, n XX

H . H .h.3/ // H .h.1/ bi /h.2/ ai

.h/ iD1

D

n XX

H . H .h.3/ // H .bi / H .h.1/ /h.2/ ai ;

by Proposition 3.1.8(i)

.h/ iD1

D

n XX

H . H .h.2/ // H .bi /H .h.1/ /ai

by the coinverse property

.h/ iD1

D

n X

H . H .h// H .bi /ai

by Proposition 3.1.8(ii) and the counit property

iD1

D H2 .h/u: And so, (4.10) yields uh D H2 .h/u;

(4.11)

for all h 2 H. P It remains show that u is a unit in H. To this end, let R1 D m iD1 cj ˝ dj and Pm to1 let v D jD1 H .dj /cj (recall that H is a bijection). Now,

120

4 Applications of Hopf Algebras

uv D u

m X

H1 .dj /cj

jD1

D

m X

.u H1 .dj //cj

jD1

D

m X

. H2 . H1 .dj /u/cj

by (4.11)

jD1

D

m X

H .dj /ucj

jD1

D

m X

H .dj /

jD1

D

n X

H .bi /ai cj

iD1

m X n X

H .bi dj /ai cj

by Proposition 3.1.8(i):

jD1 iD1

Moreover, m X n X

H .bi dj /ai cj D mH . H ˝ IH /

jD1 iD1

X n X m

 bi dj ˝ ai cj

iD1 jD1

D mH . H ˝ IH /.1H ˝ 1H /

since RR1 D 1H ˝ 1H :

D H .1H / D 1H

by Proposition 3.1.8(ii):

Consequently, uv D 1H . Now, 1 D uv D H2 .v/u; by (4.11), and so v D H2 .v/. It follows that v is so that uv D 1H D vu and so u is a unit of H. Consequently, (4.11) is now H2 .h/u D uhu1 ; for all h 2 H.



4.2 The Braid Group

121

4.2 The Braid Group In this section we define a certain infinite group called the braid group on three strands. The braid group is generated by two fundamental braids B1 ; B2 which satisfy the braid relation B1 B2 B1 D B2 B1 B2 . We will use the braid group in the next section. * * * Let R3 denote Euclidean 3-space together with the familiar xyz-coordinate system in the standard orientation. Let ` denote the line containing the point .0; 1; 0/ that is perpendicular to the xy-plane. Let C1 be a smooth curve (a path) in R3 starting at the point .0; 0; 1/ on the z-axis and ending at one of the three points .0; 1; 1/, .0; 1; 2/, .0; 1; 3/ on `; let C2 be a smooth curve in R3 starting at the point .0; 0; 2/ on the z-axis and ending at one of the three points .0; 1; 1/, .0; 1; 2/, .0; 1; 3/ on ` that is not the terminal point of C1 . We assume that C1 and C2 are disjoint. Let C3 be a smooth curve in R3 starting at the point .0; 0; 3/ on the z-axis and ending at one of the three points .0; 1; 1/, .0; 1; 2/, .0; 1; 3/ on ` that is not the terminal point of C1 or C2 . We require that the paths C1 , C2 , and C3 are pairwise disjoint. The resulting collection of paths is a braid on three strands, or more simply, a braid. See Figure 4.1 for an example of a braid. Let B; B0 be braids with paths C1 ; C2 ; C3 and C10 ; C20 ; C30 , respectively. Then B; B0 are equivalent if there exist path homotopies Fi W I  I ! R3 ; I D Œ0; 1, i D 1; 2; 3, for which Fi .x; 0/ D Ci , Fi .x; 1/ D Ci0 and for which i 6D j implies that Fi .x; s/ 6D Fj .y; t/ for all x; y; s; t 2 I. This means that for i 6D j, the path Ci is deformed to the corresponding path Ci0 in such a way that no intermediate path in the deformation of Ci intersects any intermediate path in the deformation of Fig. 4.1 A braid B in R3

z (0, 0, 3) · (0, 0, 2) · (0, 0, 1) ·

·

x

-

C3 C1 C2



· (0, 1, 3)

-

· (0, 1, 2)

-

· (0, 1, 1)

y

·

(0, 1, 0)

122 Fig. 4.2 Equivalent braids in R3

4 Applications of Hopf Algebras

C3

-

C3

-

C1

C2

-

-

C2

C1

-

Cj to Cj0 . In other words two braids B; B0 are equivalent if they are essentially the same in terms of the way the paths are interwoven. In Figure 4.2 below, we given an example of two braids that are equivalent. Braid equivalence is an equivalence relation on the collection of braids in R3 . Consequently, the collection of braids in R3 can be partitioned into a set B of equivalence classes of braids. We let ŒB 2 B denote the equivalence class represented by the braid B. We define a binary operation on B as follows. Let ŒB; ŒB0  2 B. Place B next to B0 so that the right axis (`) of B coincides with the left axis (z-axis) of B0 and then erase the middle axis. The result is a braid which we denote as BB0 (see Figure 4.4 for an illustration); the corresponding equivalence class is ŒBB0 ; the binary operation on B is defined as ŒBŒB0  D ŒBB0 : This is indeed a well-defined binary operation on B: if A 2 ŒB, A0 2 ŒB0 , then ŒAŒA0  D ŒAA0  D ŒBB0 . The braid product on B is easily seen to be associative: for ŒB; ŒB0 ; ŒB00  2 B, one has ŒB.ŒB0 ŒB00 / D .ŒBŒB0 /ŒB00 : In what follows, we simplify matters and identify the braid B with its braid class ŒB in B. The simplest braids are those of the four fundamental braids: B1 ; B2 ; B1 ; B2 , given in the (simplified) braid diagrams in Figure 4.3, below. Amazingly, every braid B in B can be written as the product of finite number of fundamental braids. For example, the braid B of Figure 4.1 is the product B D B1 B22 , as computed in Figure 4.4. Proposition 4.2.1. The collection B of braids in R3 is a group under the braid product.

4.2 The Braid Group

123

Fig. 4.3 The four fundamental braids

3

3

3

3

2

2

2

2

1

1

1

1

B1 3

3

3

3

2

2

2

2

1

1

1

1

B− Fig. 4.4 The braid product B D B1 B22

B2

B−

1

2

3

33

33

3

2

22

22

2

1

11

11

1

3

3

2

2

1

1

Proof. We have already seen that the braid product is associative. Let B0 denote the braid

3

3

2

2

1

1

and let B be any braid. Then B0 B D B D BB0 as one can easily check, and so we choose B0 as the identity element. We need to show that every braid admits an inverse under the braid product. Clearly B20 D B0 . Moreover, B1 B1 D B0 D B1 B1 :

124

4 Applications of Hopf Algebras

For instance, to prove B1 B1 D B0 , one computes

3

33

3

3

3

2

22

2

2

2

1

11

1

1

1

=

3

3

2

2

1

1

By a similar computation, one gets B2 B2 D B0 D B2 B2 : 1 1 Thus B1 0 D B0 , B1 D B1 , B2 D B2 . Now, any braid B can be written as the word

B D Bi1 Bi2 Bi3    Bik ; where i1 ; i2 ; i3 ; : : : ; ik is a finite sequence of integers in f2; 1; 1; 2g. It follows that 1 1 1 B1 D B1 ik Bik1 Bik2    Bi1 ;

and so, B is a group.



The group B of braids given in Proposition 4.2.1 is the braid group on three strands, or more simply, the braid group. Proposition 4.2.2. The braid group B is generated by the fundamental braids B1 ; B2 subject to exactly one relation B1 B2 B1 D B2 B1 B2 :

(4.12)

Proof. As above, every braid B can be written as a product Bni11 Bni22    Bnikk , where i1 ; i2 ; : : : ; ik is a finite sequence of integers in f1; 2g and n1 ; n2 ; : : : ; nk 2 Z. The relation (4.12) follows from the structure of the fundamental braids B1 and B2 . Any relation amongst the braids B1 and B2 reduces to the defining relation (4.12). 

4.3 Representations of the Braid Group In this section we tie together the material in §4.1, §4.2 and present our first application of Hopf algebras. We show that an n-dimensional quasitriangular K-Hopf algebra .H; R/ determines an n3 -dimensional representation of the braid group  W B ! GLn3 .K/. We do this by first showing that the solution of the QYBE (§4.1) yields a braid relation which in turn will determine a group homomorphism  W B ! GLn3 .K/. We give some specific examples of representations determined by the quasitriangular structures of the Hopf algebra KC2 . *

*

*

4.3 Representations of the Braid Group

125

Let .H; R/ be an n-dimensional quasitriangular Hopf algebra over K and let fc1 ; c2 ; : : : ; cn g be a K-basis for H. Then fci ˝ cj ˝ ck g, 1  i; j; k  n, is a 3 K-basis for the n3 -dimensional tensor product algebra H ˝ H ˝ H D H ˝ . The matrices in GLn3 .K/ correspond to the collection of invertible linear transformations 3 3 the matrices in GLn3 .K/P arise from the elements R12 D H ˝ ! H ˝ . Some ofP P 13 23 D i 1 ˝ ai ˝ bi constructed from i ai ˝ bi ˝ 1, RP D i ai ˝ ˝1 ˝ bi , R the element R D i ai ˝ bi 2 U.H ˝ H/. Here is how this happens. For each pair ij D 12; 13; 23, let 3

3

Rij W H ˝ ! H ˝ ; be the map defined by left multiplication by Rij . Also, let ij be the transposition maps: 12 W H ˝ ! H ˝ ;

3

3

x ˝ y ˝ z 7! y ˝ x ˝ z;

13 W H ˝ ! H ˝ ;

3

3

x ˝ y ˝ z 7! z ˝ y ˝ x;

3

3

x ˝ y ˝ z 7! x ˝ z ˝ y:

23 W H ˝ ! H ˝ ; 3

3

Now, define Rij W H ˝ ! H ˝ to be the composition of maps Rij D ij Rij . Note that 3 R12 and R23 are invertible K-linear transformations of H ˝ that (with respect to the K-basis fci ˝ cj ˝ ck g) correspond to matrices in GLn3 .K/. Proposition 4.3.1. Let K be a field and let .H; R/ be a quasitriangular Hopf algebra of dimension n over K. Then the matrices R12 ; R23 in GLn3 .K/ satisfy R12 R23 R12 D R23 R12 R23 : 3

Proof. For x ˝ y ˝ z 2 H ˝ , one has .13 23 R23 13 12 /.x ˝ y ˝ z/ D .13 23 R23 13 /.y ˝ x ˝ z/ D .13 23 R23 /.z ˝ x ˝ y/ X  D .13 23 / z ˝ bi y ˝ ai x D 13 D

X

X

i

 z ˝ ai x ˝ bi y

i

bi y ˝ ai x ˝ z

i

D R12 .x ˝ y ˝ z/;

126

4 Applications of Hopf Algebras

thus 13 23 R23 13 12 D R12 :

(4.13)

Moreover, .12 R13 12 /.x ˝ y ˝ z/ D .12 R13 /.y ˝ x ˝ z/ X  D 12 bi z ˝ x ˝ ai y D

X

i

x ˝ bi z ˝ ai y

i

D R23 .x ˝ y ˝ z/; and so, 12 R13 12 D R23 :

(4.14)

Consequently, R12 R23 R12 D .13 23 R23 13 12 /.12 R13 12 /R12

by (4.13), (4.14)

D 13 .23 R23 /.13 12 12 R13 /12 R12 D 13 .R23 R13 R12 / D 13 .R12 R13 R23 /

by Proposition 4.1.7:

(4.15)

On the other hand, .13 12 R12 13 23 /.x ˝ y ˝ z/ D .13 12 R12 13 /.x ˝ z ˝ y/ D .13 12 R12 /.y ˝ z ˝ x/ X  D .13 12 / bi z ˝ ai y ˝ x D 13 D

X

X

i



ai y ˝ bi z ˝ x

i

x ˝ bi z ˝ ai y

i

D R23 .x ˝ y ˝ z/; and so, 13 12 R12 13 23 D R23 :

(4.16)

4.3 Representations of the Braid Group

127

Moreover, .23 R13 23 /.x ˝ y ˝ z/ D .23 R13 /.x ˝ z ˝ y/ X  D 23 bi y ˝ z ˝ ai x D

X

i

bi y ˝ ai x ˝ z

i

D R12 .x ˝ y ˝ z/; and so, 23 R13 23 D R12 :

(4.17)

Consequently, R23 R12 R23 D .13 12 R12 13 23 /.23 R13 23 /R23

by (4.16), (4.17)

D 13 R12 R13 R23 D R12 R23 R12

by (4.15): 

We now construct our representation of B. Proposition 4.3.2. Let .H; R/ be a quasitriangular Hopf algebra of dimension n over K. Let B denote the braid group. Then there exists an n3 -dimensional linear representation  W B ! GLn3 .K/ defined by: 1 .B1 / D R12 ; .B2 / D R23 ; .B1 / D R1 12 ; .B2 / D R23 ;

and for a braid B D Bi1 Bi2    Bik , where i1 ; i2 ; : : : ; ik is a finite sequence in f2; 1; 1; 2g, .B/ D .Bi1 Bi2    Bik / D .Bi1 /.Bi2 /    .Bik /:

128

4 Applications of Hopf Algebras

Proof. We only need to check that  respects the braid relation (4.12) of Proposition 4.2.2: B1 B2 B1 D B2 B1 B2 , but this is easy since .B1 B2 B1 / D .B1 /.B2 /.B1 / D R12 R23 R12 D R23 R12 R23

by Proposition 4.3.1

D .B2 /.B1 /.B2 / D .B2 B1 B2 /:



Example 4.3.3. Let K be a field with char.K/ 6D 2 and let hgi D C2 denote the cyclic group of order 2. Then .KC2 ; R/ is a quasitriangular two-dimensional K-Hopf algebra with quasitriangular structures R0 D 1 ˝ 1 and R1 D

1 1 1 1 .1 ˝ 1/ C .1 ˝ g/ C .g ˝ 1/  .g ˝ g/; 2 2 2 2 3

cf. Proposition 4.1.13. Now, .KC2 /˝ is a eight-dimensional vector space over K. Choose the basis f1 ˝ 1 ˝ 1; 1 ˝ 1 ˝ g; 1 ˝ g ˝ 1; 1 ˝ g ˝ g; g ˝ 1 ˝ 1; g ˝ 1 ˝ g; g ˝ g ˝ 1; g ˝ g ˝ gg 3

for .KC2 /˝ over K. By Proposition 4.3.2 there are two eight-dimensional linear representations of B. The first is given by the quasitriangular Hopf algebra .KC2 ; R0 / and has the form 0 W B ! GL8 .K/; with

0

0 .B1 / D R12

and

1 B0 B B0 B B B0 DB B0 B B0 B @0 0 0

0 .B2 / D R23

1 B0 B B0 B B B0 DB B0 B B0 B @0 0

0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0

0 0 1 0 0 0 0 0

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 1 0 0

1 0 0C C 0C C C 0C C; 0C C 0C C 0A 1 1 0 0C C 0C C C 0C C: 0C C 0C C 0A 1

4.4 Hopf Algebras and Affine Varieties

129

The second is given by the quasitriangular Hopf algebra .KC2 ; R1 / and has the form 1 W B ! GL8 .K/; with

0

1 .B1 / D R12

and

B 0 B B 1 B 2 B B 0 DB 1 B 2 B B 0 B 1 @ 2 0 0

1 .B2 / D R23

1 2

1 2 1 2 1 2

B B B B B 1 B DB 2 B 0 B B 0 B @ 0 0

1 2

0 1 2

0

0 12 0  12 0 1 0  12 2 0 12 0 1 0 12 2 0 12 0  12 0 12 1 2

 12 1 2 1 2

1 2 1 2

 12 1 2

0 0 0 0

0 0 0 0

1 2

0 1 2

0  12 0 1 2

0

 12 0  12 0 1 0 2  12 0 0 12 0 12 0 12 0  12

0  12 1 0 2 0 12 1 0 2 0 12  12 0 0 12 1 0 2

1 0  12 C C 0 C C 1 C 2 C C; 0 C 1 C C 2 C 0 A 1 2

1 0 0 C C 0 C C C 0 C 1C : 1 1 2C 2 2 C 1 1  2 2  12 C C 1  12 21 A 2 1 1  12 2 2 0 0 0 0

0 0 0 0

As one can check, 0 .B/ Š 1 .B/ Š S3 :

4.4 Hopf Algebras and Affine Varieties In this section we show how K-Hopf algebras are related to affine varieties over K. We show that an affine variety ƒ with coordinate ring KŒƒ can be identified with the collection of K-algebra maps KŒƒ ! K, and this allows us to think of the geometric object ƒ as an algebraic object through the algebraic structure of its coordinate ring KŒƒ. We show that if KŒƒ is a bialgebra, then ƒ is a monoid, and if KŒƒ is a Hopf algebra, then ƒ is a group. *

*

*

Let n  1 be an integer, let K be an infinite field containing Q, and let K n D K     K … denote the cartesian product of n copies of K. Let x1 ; x2 ; : : : ; xn be „ K ƒ‚ n

indeterminates and let KŒx1 ; x2 ; : : : ; xn  denote the ring of polynomials over K.

130

4 Applications of Hopf Algebras

Definition 4.4.1. An affine variety ƒ over K is a subset ƒ of K n consisting of all common zeros a D .a1 ; a2 ; : : : ; an / in K of a finite set F D ff1 ; f2 ; : : : ; fm g of polynomials in KŒx1 ; x2 ; : : : ; xn , that is, ƒ D f.a1 ; a2 ; : : : ; an / 2 K n W f .a/ D 0; 8f 2 Fg: The affine variety ƒ  K n is the set of all simultaneous solutions of the set of equations f1 D 0; f2 D 0; : : : ; fm D 0 if and only if g.ƒ/ D 0 for every g in the ideal N D .f1 ; f2 ; : : : ; fm / generated by f1 ; f2 ; : : : ; fm . Example 4.4.2. ƒ D f0; 1g  K 1 is an affine variety since it is the set of zeros of the polynomial f .x/ D x2  x 2 KŒx. Example 4.4.3. ƒ D f1; 2g  K 1 is an affine variety since it is the set of common zeros of the polynomials fx2  x  2; x3  4x2 C x C 6g  KŒx. Example 4.4.4. ƒ D f.a; a1 / W a 2 K  g  K 2 is an affine variety since it is the set of zeros in K of the polynomial f .x1 ; x2 / D x1 x2  1 2 KŒx1 ; x2 . Example 4.4.5 (Generalization of Example 4.4.4). ƒ D f.a1;1 ; a1;2 ; a2;1 ; a2;2 ; b1 / W ai;j 2 K; b D a1;1 a2;2  a1;2 a2;1 2 K  g  K 5 is an affine variety since it is the set of zeros in K of the polynomial f .x1;1 ; x1;2 ; x2;1 ; x2;2 ; y/ D .x1;1 x2;2  x1;2 x2;1 /y  1 in KŒx1;1 ; x1;2 ; x2;1 ; x2;2 ; y. Example 4.4.6. ƒ D K n is an affine variety since it is the set of zeros of the zero polynomial f .x1 ; x2 ; : : : ; xn / D 0. Let ƒ  K n be an affine variety. The set of polynomials Nƒ D ff 2 KŒx1 ; x2 ; : : : ; xn  W f .a/ D 0; 8a 2 ƒg; is an ideal of KŒx1 ; x2 ; : : : ; xn  called the ideal of ƒ. For example, if ƒ is the affine variety ƒ D f.a; a1 / W a 2 K  g  K 2 , then Nƒ D .x1 x2  1/ and if ƒ D K n , then Nƒ D 0. Let ƒ be the affine variety that consists of the common zeros of the set ff1 ; f2 ; : : : ; fm g of polynomials in KŒx1 ; x2 ; : : : ; xn . Then .f1 ; f2 ; : : : ; fm /  Nƒ . We could, of course, have proper containment. For instance, take K D Q. Then x3  1 defines the affine variety ƒ D f1g with .x3  1/  Nƒ D .x  1/. By the Hilbert Basis Theorem, KŒx1 ; x2 ; : : : ; xn  is Noetherian, see [Wat79, A.5]. Thus Nƒ D .g1 ; g2 ; : : : ; gl / for some polynomials g1 ; g2 ; : : : ; gl 2 KŒx1 ; x2 ; : : : ; xn . Proposition 4.4.7. Let ƒ  K n be an affine variety with ideal Nƒ D .g1 ; g2 ; : : : ; gl /. Let ƒ0 denote the affine variety consisting of the common zeros in K of the polynomials fg1 ; g2 ; : : : ; gl g. Then ƒ D ƒ0 .

4.4 Hopf Algebras and Affine Varieties

131

Proof. By the definition of Nƒ , we have ƒ  ƒ0 . Since ƒ is an affine variety, it consists of points in K n that are common zeros of some set of polynomials ff1 ; f2 ; : : : ; fm g in KŒx1 ; x2 ; : : : ; xn . One has .f1 ; f2 ; : : : ; fm /  Nƒ . Now let b 2 ƒ0 . Then b is a zero of all polynomials in Nƒ , hence b is a common zero of the  polynomials ff1 ; f2 ; : : : ; fm g, and so, b 2 ƒ. Let ƒ  K n be an affine variety. A function W ƒ ! K is regular if there exists a polynomial f 2 KŒx1 ; x2 ; : : : ; xn  for which .a/ D f .a/ for all a 2 ƒ. It is not hard to show that the collection of all regular functions on ƒ is a ring with ring operations defined pointwise, this is the ring of regular functions KŒƒ on ƒ. The ring of regular functions KŒƒ is also called the coordinate ring of ƒ. We want to give a precise description of KŒƒ. Proposition 4.4.8. Let ƒ  K n be an affine variety and let Nƒ denote the ideal of ƒ. Then KŒƒ D KŒx1 ; x2 ; : : : ; xn =Nƒ . Proof. Suppose that g; h 2 KŒx1 ; x2 ; : : : ; xn  with g  h 2 Nƒ . Then for all a D .a1 ; a2 ; : : : ; an / 2 ƒ, 0 D .g  h/.a/ D g.a/  h.a/; and hence g.a/ D h.a/. Thus, g; h define the same regular functions on ƒ. 1





For instance, if ƒ is the affine variety ƒ D f.a; a / W a 2 K g  K 2 , then KŒƒ D KŒx1 ; x2 =.x1 x2  1/ D KŒx; x1  and if ƒ D K n , then KŒƒ D KŒx1 ; x2 ; : : : ; xn . Proposition 4.4.9. Let ƒ  K n be an affine variety with coordinate ring KŒƒ. Then ƒ D HomK-alg .KŒƒ; K/; the identification being a D . 7! .a//, for a 2 ƒ, 2 KŒƒ. Proof. By Proposition 4.4.8, KŒƒ D KŒx1 ; x2 ; : : : ; xn =Nƒ . Consequently, by [Wat79, §1.2, Theorem] HomK-alg .KŒƒ; K/ consists of precisely those points in K n that are common zeros of all polynomials in Nƒ . The result then follows from Proposition 4.4.7.  It is natural to think of the affine variety ƒ  K n in geometric terms as a collection of points in the space K n . We can think about ƒ in algebraic terms and this is done through the algebraic structure of the coordinate ring KŒƒ. We recall our identification ƒ D HomK-alg .KŒƒ; K/ where a 2 ƒ is equated with the K-algebra homomorphism 7! .a/, for all

2 KŒƒ. Now we assume that the coordinate ring B D KŒƒ is not just a K-algebra but is a K-bialgebra. In that case, we obtain a monoid structure on ƒ, as we now show.

132

4 Applications of Hopf Algebras

Proposition 4.4.10. There exists a binary operation ‚ on HomK-alg .B; K/ ‚ W HomK-alg .B; K/  HomK-alg .B; K/ ! HomK-alg .B; K/ defined as ‚.f ; g/.a/ D mK .f ˝ g/B .a/ D for f ; g 2 HomK-alg .B; K/, a 2 B, B .a/ D

X

f .a.1/ /g.a.2/ /;

.a/

P .a/

a.1/ ˝ a.2/ .

Proof. This amounts to showing that the map ‚.f ; g/ W B ! K is a homomorphism of K-algebras. To this end, let a; b 2 B, r 2 K. Then ‚.f ; g/.ra C b/ D mK .f ˝ g/B .ra C b/ D mK .f ˝ g/.rB .a/ C B .b//  X   X D mK .f ˝ g/ ra.1/ ˝ a.2/ C b.1/ ˝ b.2/ D mK D

 X

X

.a/

.b/



f .ra.1/ / ˝ g.a.2/ / C

.a/



rf .a.1/ /g.a.2/ / C



f .b.1/ / ˝ g.b.2/ /

.b/

X

.a/

X



f .b.1/ /g.b.2/ /

.b/

D r‚.f ; g/.a/ C ‚.f ; g/.b/; and so ‚.f ; g/ is K-linear. We next show that ‚.f ; g/ respects multiplication, thus: ‚.f ; g/.ab/ D mK .f ˝ g/B .ab/  X D mK .f ˝ g/ a.1/ b.1/ ˝ a.2/ b.2/ D

X

.a;b/

f .a.1/ b.1/ /g.a.2/ b.2/ /

.a;b/

D

X

f .a.1/ /f .b.1/ /g.a.2/ /g.b.2/ /

.a;b/

D

X

f .a.1/ /g.a.2/ /f .b.1/ /g.b.2/ /

.a;b/

D

X .a/

f .a.1/ /g.a.2/ /

X

f .b.1/ /g.b.2/ /

.b/

D ‚.f ; g/.a/‚.f ; g/.b/: 

4.4 Hopf Algebras and Affine Varieties

133

Of course, as the reader may have noticed, the binary operation of Proposition 4.4.10 is precisely the convolution operation on HomK .B; K/ restricted to HomK-alg .B; K/  HomK .B; K/, see §3.1. Proposition 4.4.11. Let ƒ be an affine variety with coordinate ring KŒƒ. Assume that KŒƒ is a K-bialgebra. Then ƒ D HomK-alg .B; K/ together with convolution is a monoid. Proof. By Proposition 3.1.6, is associative, and so HomK-alg .B; K/ is a semigroup. The composition K B W B ! K serves as a two-sided identity element with respect to , thus: X f .a.1/ /K .B .a.2/ // .f K B /.a/ D .a/

D

X

K .B .a.2/ //f .a.1/ /

.a/

D

X

B .a.2/ /f .a.1/ /

.a/

D

X

f .B .a.2/ /a.1/ /

.a/

Df

X

 B .a.2/ /a.1/

.a/

D f .a/

by the counit property:

In a similar manner one obtains K B f D f . Thus HomK-alg .B; K/ is a monoid under .  We next assume that the coordinate ring KŒƒ is a K-Hopf algebra H. Proposition 4.4.12. HomK-alg .H; K/ together with convolution is a group. Proof. By Proposition 4.4.11 HomK-alg .H; K/ together with is a monoid with twosided identity K H . So we only need to show that each element f 2 HomK-alg .H; K/ has a two-sided inverse under . Note that f H 2 HomK-alg .H; K/. Now, for all a 2 H, X f . H .a.1/ //f .a.2/ / .f H f /.a/ D .a/

Df

X .a/

 H .a.1/ /a.2/ ;

134

4 Applications of Hopf Algebras

since f is an algebra homomorphism. Continuing with the calculation: f

X

 H .a.1/ /a.2/

D f .H .a/1H /

by the coinverse property

.a/

D H .a/1K D K .H .a// D .K H /.a/: In a similar manner one obtains f f H D K H .



The conclusion of Proposition 4.4.12 is the following: If a given K-Hopf algebra H is the coordinate ring of an affine variety ƒ  K n (necessarily, H is a commutative K-algebra), then hƒ; i with ƒ D HomK-alg .H; K/ is a group. In this way we can put a group structure on the geometric object ƒ. Here are some examples. Example 4.4.13. The polynomial ring KŒx is a K-Hopf algebra with x primitive. As one can check, it is the coordinate ring of the affine variety K 1 . Thus hK 1 ; i is a group, where in this case, a b D a C b for all a; b 2 K 1 ; K 1 D HomK-alg .KŒx; K/ is the additive group of K. Example 4.4.14. KŒx; x1  with x group-like is a K-Hopf algebra and it is the coordinate ring of the affine variety ƒ D f.a; a1 / W a 2 K  g. Now, a b D ab, and so, hƒ; i is the multiplicative group of non-zero elements of K. Example 4.4.15. KŒx1;1 ; x1;2 ; x2;1 ; x2;2 ; 1=.x1;1 x2;2  x1;2 x2;1 / is a K-Hopf algebra (§4.6, Exercise 15). It is the coordinate ring of the affine variety ƒ of Example 4.4.5. An element a 2 ƒ can be viewed as the invertible matrix   a a M D 1;1 1;2 : a2;1 a2;2 Convolution is now ordinary matrix multiplication and hƒ; i is the group of matrices GL2 .K/. Example 4.4.16. CŒx=.x3  1/ is an C-Hopf algebra with x group-like; it is the coordinate ring of the affine variety ƒ D f1; 3 ; 32 g  C1 ; hƒ; i Š C3 , in this case. Note: If H is a commutative K-Hopf algebra and K is replaced with any commutative K-algebra A, then HomK-alg .H; A/ is a group, see [Un11, Proposition 3.1.5]. More generally, if H is a K-Hopf algebra (commutative or non-commutative) and A is a commutative K-algebra, then HomK-alg .H; A/ is a group [Ab77, Theorem 2.1.5].

4.5 Hopf Algebras and Hopf Galois Extensions

135

4.5 Hopf Algebras and Hopf Galois Extensions In this section we show that L is a Galois extension of K with group G if and only if L is a Galois KG-extension of K. In this latter form (L is a Galois KG-extension) the concept of a classical Galois extension L=K can be extended to rings of integers. Ultimately, Galois KG-extensions (and hence, classical Galois extensions) can be generalized to Galois H-extensions of rings S=R where H is a Hopf algebra. *

*

*

Let K be a finite extension of Q and let L be a finite extension of K. Let AutK .L/ denote the group of automorphisms of L that fix K and let G be a subgroup of AutK .L/. Let S denote the ring of integers of L. Now, L is a KG-module with scalar multiplication given as X

 X ag g  x D ag g.x/;

g2G

g2G

for ag 2 K, x 2 L. Now, of course, KG is a K-Hopf algebra. Proposition 4.5.1. Let L be a finite extension of K and let G  AutK .L/. Then L is a KG-module algebra. Proof. For each g 2 G one has g  .xy/ D g.xy/ D g.x/g.y/ D .g  x/.g  y/; and g  1L D g.1L / D 1L D KG .g/1L ; P for all x; y 2 L. Thus for h D g2G ag g, X h  .xy/ D ag g.xy/ g2G

D

X

ag .g  x/.g  y/

g2G

D

X .h.1/  x/.h.2/  y/; .h/

and h  1L D

X g2G

ag g.1L / D

X

ag 1L D KG .h/1L :

g2G



136

4 Applications of Hopf Algebras

The field L is a vector space over K and we consider EndK .L/ D HomK .L; L/ the collection of K-linear transformations W L ! L. Over K, EndK .L/ is a vector space with addition given pointwise: . C

/.x/ D .x/ C

.x/;

and scalar multiplication defined as .r /.x/ D r .x/ for r 2 K, x 2 L. As a set of automorphisms of L that fix K, the group G is a subset of EndK .L/. There is a homomorphism of vector spaces | W L ˝K KG ! EndK .L/

(4.18)

defined as | .x ˝ h/.y/ D x.h  y/ for x; y 2 L, h 2 KG. Lemma 4.5.2. The elements of G form a linearly independent set of vectors over L. Proof. Write the elements of G as g0 ; g1 ; g2 ; : : : ; gn1 . If the set fg0 ; g1 ; : : : ; gn1 g is not linearly independent over L, then there exists a smallest positive integer m, 1  m  n, and a set of distinct integers i1 ; i2 ; : : : ; im , 0  i1 ; i2 ; : : : ; im  n  1 for which a1 gi1 C a2 gi2 C    C am1 gim1 C am gim D 0

(4.19)

with a1 ; a2 ; : : : ; am non-zero. Since gim1 6D gim , there exists y 2 L for which gim1 .y/ 6D gim .y/ with gim .y/ 6D 0. For any x 2 L, a1 gi1 .xy/ C    C am1 gim1 .xy/ C am gim .xy/ D 0 thus a1 gi1 .y/gi1 .x/ C    C am1 gim1 .y/gim1 .x/ C am gim .y/gim .x/ D 0; and so, a1 gi1 .y/gi1 C    C am1 gim1 .y/gim1 C am gim .y/gim D 0: Now dividing (4.20) by gim .y/ yields  a1

   gi1 .y/ gim1 .y/ gi1 C    C am1 gim1 C am gim D 0; gim .y/ gim .y/

(4.20)

4.5 Hopf Algebras and Hopf Galois Extensions

137

and subtracting (4.19) gives         gi1 .y/ gim1 .y/ a1  a1 gi1 C    C am1  am1 gim1 D 0: gim .y/ gim .y/  g .y/  am1 6D 0 and so, we have a contradiction of the Note that am1 igm1 im .y/ minimality of m. Thus G is linearly independent.  We give a characterization of Galois extensions. Proposition 4.5.3. Let K be a finite extension of Q, let L be a finite extension of K, and let G be a subgroup of AutK .L/. Then the map | W L ˝K KG ! EndK .L/ is a bijection if and only if L is a Galois extension of K with group G. Proof. Suppose L P is a Galois extension of K with group G. We show that | is a bijection. Let h D n1 iD0 ai gi 2 KG, x 2 L, and suppose that   X  n1 n1 n1 X X | x˝ ai gi .y/ D xai gi  y D xai gi .y/ D 0; iD0

iD0

iD0

for all y 2 L. By Lemma 4.5.2 fg0 ; g1 ; : : : ; gn1 g is linearly independent over L. Thus xai D 0 for all i, and so | is a injection. Since L=K is Galois, jGj D ŒL W K and so dim.L ˝K KG/ D ŒL W K2 D dim.EndK .L//: Thus | is surjective. For the converse, suppose that | is a bijection. Then ŒL W KjGj D ŒL W K2 , and so, jGj D ŒL W K. Put L D K.˛/ with p.x/ D irr.˛; K/ of degree ŒL W K. Since each element of G moves ˛ to some distinct root ˇ of p.x/, L is the splitting field of p.x/ over K, thus L=K is Galois. Let Gal.L=K/ denote the Galois group of L over K. We have G  Gal.L=K/. But since jGal.L=K/j D ŒL W K D jGj, G D Gal.L=K/ and L is Galois over K with group G.  We have shown the following: the notion that L is a Galois extension of K with group G is equivalent to L being a KG-module algebra for which the map | W L ˝K KG ! EndK .L/; defined as | .x ˝ h/.y/ D x.h  y/ for x; y 2 L, h 2 KG is a bijection. We say the L is a “Galois KG-extension of K,” or: KG is “realizable as a Galois group.” Of course, KG is not a group—it is a Hopf algebra—but this terminology makes sense since KG does correspond to the group HomK-alg .KG; K/ as in §4.4.

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4 Applications of Hopf Algebras

In this latter form (involving the bijection | ) the notion of a classical Galois extension L=K can be generalized to rings of integers. Let S be the ring of integers of L, let R be the ring of integers of K; S will play the role of L; RG will play the role of KG. Certainly, S is an RG-module since x 2 S implies that g.x/ 2 S for all g 2 G. In fact, S is an RG-module algebra (proof?) Definition 4.5.4. Let L be a finite Galois extension of K with group G. Let S be the ring of integers of L, let R be the ring of integers of K. Then S is a Galois RG-extension of R if the map | W S ˝R RG ! EndR .S/; defined as | .x ˝ h/.y/ D x.h  y/ for x; y 2 S, h 2 RG is an isomorphism of R-modules. We also say that RG is realizable as a Galois group. The challenge is to construct a Galois RG-extension. Here is a criterion that we can use. Proposition 4.5.5. Let L be a Galois extension of K with group G. Then S is a Galois RG-extension of R if and only if every prime ideal P of R is unramified in S. Proof. The proof is beyond the scope of this book. The interested reader should see [Ch00, §2, p. 18].  The ring of integers R of a finite extension K of Q with ŒK W Q > 1 can never be a Galois ZG-extension since at least one prime p in Z is ramified in R [Ne99, Theorem III.2.17]. So in order to construct Galois RG-extensions with jGj > 1, we must start with a base field K that is larger than Q. Our Galois RG-extension will be the ring of integers of the Hilbert Class Field of K, which is the maximal abelian unramified extension of K. The Hilbert Class Field of K is a Galois extension with Galois group isomorphic to the class group of K, cf. [IR90, Notes, p. 184]. Proposition 4.5.6. Let K be a finite extension of Q for which R is not a PID, that is, for which the class number hR > 1. Then there exists a Galois extension L of K with group G for which the ring of integers S is a Galois RG-extension of R. Proof. Choose L to be the Hilbert Class Field of K. Then each prime P of R is unramified in S, and so by Proposition 4.5.5, S is a Galois RG-extension of R.  p p Example 4.5.7. Let K D Q. 5/ with R D ZŒ 5. The class number of R is 2. The Hilbert Class Field of K is L D K.i/, and the ring of integers of L is p S D RŒ.i C 5/=2. By Proposition 4.5.5, S is a Galois RC2 -extension of R. For more examples of unramified (hence Galois) extensions of number fields, see [He66]. We notice that in the Galois H-extensions given above, with H D KG or H D RG, H is a Hopf algebra and the H-module algebra structure of L is given by the classical Galois action of G on L. We can broaden the notion of Galois extension to include module algebras over Hopf algebras in which the action of the Hopf algebra is not given by the classical Galois action.

4.5 Hopf Algebras and Hopf Galois Extensions

139

Definition 4.5.8. Let R be a commutative ring with unity, let H be a cocommutative R-Hopf algebra which is finitely generated and projective as an R-module, and let S be a commutative R-algebra which is finitely generated and projective as an R-module. Then S is a Galois H-extension of R if S is an H-module algebra with action denoted as h  y for h 2 H, y 2 S, and the map | W S ˝R H ! EndR .S/; defined as | .x˝h/.y/ D x.hy/ for x; y 2 S, h 2 H is an isomorphism of R-modules. We also say that H is realizable as a Galois group and that S is a Hopf Galois extension of R. Here is perhaps the easiest example of a Hopf Galois extension that will yield a non-classical Galois action, cf. [CS69, pp. 35–39]. Example 4.5.9 (S. Chase, M. Sweedler). Let Cn be the cyclic group of order n, generated by g. Let R be a commutative ring with unity and let a be a unit of R. Then the R-algebra RŒz with zn D a, is a Cn -graded R-algebra, that is, RŒz D R ˚ Rz ˚ Rz2 ˚    ˚ Rzn1 ; with Rzi Rzj  RziCj for 0  i; j  n  1. Let RCn be the group ring R-Hopf algebra  j with linear dual RCn and let fpi gn1 iD0 denote the dual basis for RCn , pi .g / D ıi;j . As  j one can check, RŒz is a left RCn -module algebra with action pi .z / D ıi;j zj . The map | W RŒz ˝R RCn ! EndR .RŒz/; defined as | .zi ˝ pj /.zk / D zi pj .zk / D ıi;j ziCk is a bijection, and so, RŒz is a Galois RCn -extension of R. More generally, for G a finite group, a G-graded R-algebra A D ˚ 2G A is a left RG -module algebra, and A is a Galois RG -extension if and only if Ae D R and A is strongly graded, that is, A A D A  for all ;  2 G, see [Ca98, Proposition 8.2.1]. The terminology “strongly graded” is due to Dade [Da80]. We have the following special case of Example 4.5.9. Example 4.5.10. Let n  2, let K be any field containing Q, and let a be an element of K that is not an nth power of an element of K. Then p.x/ D xn  a is irreducible over K. Let ˛ be a zero of p.x/ in C. Then L D K.˛/ is a simple algebraic extension of K of degree n, and L is a Cn -graded K-algebra and a Galois KCn -extension of K with Galois action pi .˛ j / D ıi;j ˛ j . Example 4.5.11. In the case K D Q, n > 2, of Example 4.5.10, we have that Q.˛/ is a QCn -Galois extension of Q; the QCn -module algebra structure of Q.˛/ is not the classical Galois action; Q.˛/ is not a Galois extension of Q. A finite extension of fields K=Q can have more than one Hopf Galois structure.

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4 Applications of Hopf Algebras

Example 4.5.12. Let p.x/ D x3  2 and let K be the splitting field of p.x/ over Q. p 3 Then K D Q.3 ; 2/ and K is a Galois extension of Q with group S3 D h ; i defined by the relations 3 D  2 D 1,  D 2  . Now, by Proposition 4.5.3, K is a Galois QS3 -extension of Q. We want to put a non-classical Hopf Galois structure on K. Consider the group ring KS3 . Let S3 act on KS3 as the Galois group on K and by conjugation on S3 ; denote this action by “.” Thus, for ai 2 K,  .a0 C a1 C a2 2 C a3  C a4  C a5 2 / D .a0 / C .a1 / C .a2 / 2 C .a3 / 2  C .a4 / C .a5 /  and   .a0 C a1 C a2 2 C a3  C a4  C a5 2 / D  .a0 / C  .a1 / 2 C  .a2 / C  .a3 / C .a4 / 2  C .a5 / : Now let KS3S3 D fx 2 KS3 W .x/ D .x/ D xg denote the subset of KS3 fixed by S3 . Then, KS3S3 D fb0 C b1 C  .b1 / 2 C b2  C 2 .b2 /  C .b2 / 2 g; p where b0 2 Q, b1 2 Q.3 / and b2 2 Q. 3 2/. H D KS3S3 is a six-dimensional Q-Hopf algebra which acts on K as a module algebra through , and K is a Galois H-extension of Q. One has H 6Š QS3 and so, the Hopf Galois structure on K is different than the classical Galois structure. For more examples of separable, non-Galois extensions of Q which have (or don’t have) Hopf Galois structures on them, see [CRV14a, CRV14b] and [CRV14c]. In Example 4.5.10 we take n D p, a prime number, K D Q.pm /, for a fixed integer m  1, and a D pm . Then p.x/ D xp  pm is irreducible over K with zero ˛ D pmC1 . L D K.pmC1 / is a KCp -Galois extension of K. By Proposition 3.1.14, KCp Š KCp and so L is a KCp -Galois extension of K with the KCp -module algebra action on L induced from the classical Galois action of Cp on L. (Indeed, L is a Galois extension K with group Cp .) For K D Q.pm / we have R D ZŒpm  and the ideal .p/ has unique factorization m1 .p1/

.p/ D .1  pm /p

:

In this case, e0 D pm1 . Let  D pm 1. We know from §3.4 that there is a collection of Hopf orders in KCp of the form  H.i/ D R

g1 ; hgi D Cp ; i

4.5 Hopf Algebras and Hopf Galois Extensions

141

for 0  i  e0 . We ask: which (if any) of these R-Hopf orders are realizable as Galois groups, that is, for which H.i/ does there exist a degree p extension L=K whose ring of integers S is an H.i/-Galois extension of R? Let i be an integer 1  i  e0 , let i0 D e0  i, and let 0

wi0 D 1 C pi C1 : Then p.x/ D xp  wi0 is irreducible over K. The splitting field of p.x/ over K is p Li0 D K. p wi0 /; Li0 is a Galois extension of K with group Cp , generated by gW

p p p wi0 7! p p wi0 :

Let Si0 be the ring of integers of Li0 . Take i D e0 , so that i0 D 0. Then w0 D 1 C .pm  1/ D pm : One has L0 D K.pmC1 / D Q.mC1 /: with S0 D RŒpmC1  D ZŒpmC1 . Proposition 4.5.13. Assume the notation as above. Then S0 is a Galois H.e0 /extension of R. That is, the R-Hopf order H.e0 / in KCp is realizable as a Galois group. Proof. We apply Example 4.5.9 of Chase and Sweedler. S0 is a Cp -graded R-algebra p with pmC1 D pm , a unit in R. Thus S0 is an RCp -Galois extension of R. But note that RCp Š H.e0 / by Proposition 3.4.11.  What about the remaining R-Hopf orders H.i/ for 0  i < e0 ? It is not so clear which of these (if any) are realizable as Galois groups. They are all realizable as Galois groups if we localize. p Again, suppose that K D Q.pn /, Li0 D K. p wi0 /, with 0

wi0 D 1 C .pn  1/pi C1 ; 0  i  e0 , e0 D pn1 . Let R be the ring of integers of K, let Si0 be the ring of integers of Li0 . Let .p/ D Qe11 Qe22    Qemm be the unique factorization of .p/ in Si0 . Let Q D Q1 and let .Li0 /Q denote the completion of Li0 at Q with valuation ring .Si0 /Q ; let P D ./ and let KP be the

142

4 Applications of Hopf Algebras

completion of K at P with valuation ring RP and uniformizing parameter . There is a collection of RP -Hopf orders in KP Cp of the form  g1 ; HP .i/ D RP i for 0  i  e0 . Now, Childs [Ch87] has shown that each HP .i/ is realizable as a Galois group; Childs also computes the ring of integers that realizes HP .i/. Proposition 4.5.14. Assume the notation as above. Then for each integer i, 0  i  e0 , p 0 (i) .Si0 /Q D RO P Œx with x D . p wi0  1/= i , (ii) .Si0 /Q is a Galois HP .i/-extension of RP , in other words, .Si0 /Q is an HP .i/module algebra and the map | W .Si0 /Q ˝RP HP .i/ ! EndRP ..Si0 /Q /; defined as | .x ˝ h/.y/ D xh.y/ for x; y 2 .Si0 /Q , h 2 HP .i/, is a RP -module isomorphism. Proof. See [Ch87, §14, Proposition 14.3].



4.6 Chapter Exercises Exercises for §4.1

Pn 1. Let .B; R/ be an almost commutative K-bialgebra with Pn R D iD1 ai ˝ bi . Let I be a biideal of B. Prove that .B=I; R/ with R D iD1 .ai C I/ ˝ .bi C I/ is almost commutative. 2. In the proof of Proposition 4.1.10 take z D 0 and compute the image of R0 under the inverse map 1 W KT  ! KT. Does R D 1 .R0 / satisfy the quasitriangular conditions (4.2) or (4.3) with B D KT? Pn  3. Let .B; R/ be a quasitriangular K-bialgebra with R iD1 ai ˝bi . Let f W B ! PD n  B be the map defined as f .˛/ D .˛ ˝ IB /.R/ D iD1 ˛.ai /bi for ˛ 2 B . (a) Prove that f is a homomorphism of K-algebras. (b) Assuming that B is finite dimensional, prove that f is a coalgebra antihomomorphism. P 4. Let .H; R/ be a quasitriangular K-Hopf algebra with R D niD1 ai ˝ bi . Let I be a Hopf ideal of H. Prove that .H=I; R/ with RD

n X .ai C I/ ˝ .bi C I/ iD1

is quasitriangular.

4.6 Chapter Exercises

143

5. Let H denote M. Sweedler’s Hopf algebra of Example 3.1.5. Let RD

1 1 1 1 .1 ˝ 1/ C .1 ˝ g/ C .g ˝ 1/  .g ˝ g/ 2 2 2 2 C x ˝ x  x ˝ gx C gx ˝ x C gx ˝ gx:

Prove that .H; R/ is a quasitriangular Hopf algebra. Exercises for §4.2 6. Draw the braid associated with the braid product B2 B21 B1 2 . 7. Draw the braids that verify the braid relation B1 B2 B1 D B2 B1 B2 . 8. Let S D fB21 ; B22 g be a subset of the braid group B and let hSi denote the subgroup of B generated by S. (a) Prove that hSi G B. (b) Compute B=hSi. 9. Decompose the braid 3

3

2

2

1

1

into a product of fundamental braids. Exercises for §4.3 10. Compute the regular representation  W V ! GL4 .K/ of the Klein 4-group V. 11. Let 1 W B ! GL8 .K/ denote the representation of the braid group given in Example 4.3.3. Prove that 1 .B/ Š S3 . Exercises for §4.4 12. Let B be a K-bialgebra, let A be a K-algebra, and let HomK .B; A/ denote the collection of K-linear maps B ! A. Show that hHomK .B; A/; i is a monoid. 13. Let K be a field and let A D KŒx=.xn /, n  2. Show that A is not the coordinate ring of an affine variety X  K 1 . What if n D 1? 14. Let A D QŒx1 ; x2 =.2  x13 C x22 /. Prove that A is the coordinate ring of an affine variety X  Q2 . 15. Let K be a field and let KŒx1;1 ; x1;2 ; x2;1 ; x2;2 ; y be the K-algebra of polynomials in the variables x1;1 ; x1;2 ; x2;1 ; x2;2 ; y. Let f .x1;1 ; x1;2 ; x2;1 ; x2;2 ; y/ D .x1;1 x2;2  x1;2 x2;1 /y  1: Show that KŒx1;1 ; x1;2 ; x2;1 ; x2;2 ; y=.f .x1;1 ; x1;2 ; x2;1 ; x2;2 ; y//

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4 Applications of Hopf Algebras

is a K-Hopf algebra. Hint: Define comultiplication as .xi;j / D

2 X

xi;k ˝ xk;j

kD1

for 1  i; j  2. Exercises for §4.5 16. Let K D Q.pn /, e0 D pn1 . (a) Find all integers i, 0  i  e0 , for which w D 1 C .pn  1/piC1 is a unit in R D ZŒpn . (b) For anyp i with 0  i  e0 satisfying (a), prove that the ring of integers S of L D K. p w/ is a Galois H.i0 /-extension of R, i D e0  i. Questions for Further Study 1. Find all of the quasitriangular structures for the bialgebra KŒx of Example 2.1.3. 2. Find all of the quasitriangular structures for the bialgebra KŒx of Example 2.1.4. 3. Let K D Q.3 / and let C3 be the cyclic group of order 3 generated by g. Find a non-trivial quasitriangular structure for KC3 . 4. Let K D Z3 and let H denote M. Sweedler’s K-Hopf algebra of Example 3.1.5. Let R D 1 ˝ 1  1 ˝ g  g ˝ 1 C g ˝ g C x ˝ x  x ˝ gx C gx ˝ x C gx ˝ gx: Then .H; R/ is quasitriangular (§4.6, Exercise 5). Compute .B1 / where  W B ! GF64 .Z3 / is the representation given by R. p 5. Referring to Example 4.5.12, let K D Q.3 ; 3 2/ and let H D KS3S3 D fb0 C b1 C  .b1 / 2 C b2  C 2 .b2 /  C .b2 / 2 g; p where b0 2 Q, b1 2 Q.3 / and b2 2 Q. 3 2/. (a) Prove that H is a six-dimensional Q-Hopf algebra. (b) Show that K is a Galois H-extension of Q. (c) Obtain the Wedderburn–Malcev decomposition of both H and QS3 . Conclude that H 6Š QS3 .

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Index

A additive group, 134 affine variety, 130 ideal of, 130 algebra, 8, 9 commutative, 10 homomorphism, 12 Algebraic Myhill–Nerode Theorem, 54 almost cocommutative, 108 antipode, 68 antipode property, 68 associative property, 9 augmentation ideal, 105

coassociative property, 13 cocommutative, 13 cofinite ideal, 25 coideal, 16 coinverse, 68 coinverse (antipode) property, 68 completion, 103 comultiplication map, 13 convolution, 70 coordinate ring, 131 counit map, 13 counit property, 13

B bialgebra, 36 homomorphism, 37 isomorphism, 37 Myhill–Nerode, 56 biideal, 37 braid, 121 equivalence, 121 fundamental, 122 braid group, 124

F finite automaton, 48 accepted language, 49 accepted word, 49 state transition diagram, 49 finite codimension, 25 finite dual, 25 finite index, 51 Fundamental Theorem of Hopf Modules, 86

C characteristic function, 53 Childs, L., 142 coalgebra, 12 cocommutative, 13 divided power, 16 homomorphism, 19 isomorphism, 19 trivial, 15

G Galois H-extension, 139 Galois RG-extension, 138 Galois group realizable as, 138 realizable as a, 139 generating integral, 94 grouplike, 17

© Springer International Publishing Switzerland 2015 R.G. Underwood, Fundamentals of Hopf Algebras, Universitext, DOI 10.1007/978-3-319-18991-8

149

150 H Hadamard product, 44 Hilbert Class Field, 138 homomorphism algebra, 12 bialgebra, 37 coalgebra, 19 Hopf algebra, 77 Hopf algebra, 68 cocommutative, 69 commutative, 69 homomorphism, 77 isomorphism, 77 over a ring, 95 unimodular, 79 Hopf comodule, 80 homomorphism, 80 Hopf Galois extension, 139 Hopf ideal, 76 Hopf module, 84 homomophism, 86 trivial right, 85 Hopf order trivial, 96 Hurwitz product, 46

L language, 48 equivalence relation L , 51 Larson, R., 87, 92 left integral, 78 generating, 94 left translate, 39

M module trivial right, 85 module algebra, 39 homomorphism, 39 module coalgebra, 39 homomorphism, 39 monoid bialgebra, 37 multiplication map, 9 multiplicative group, 134 Myhill–Nerode bialgebra, 56 Myhill–Nerode Theorem, 51 Algebraic, 54

Index N n-linear map, 2

O Oort, F., 104

P primitive element, 36

Q quantum group, 70 quantum Yang–Baxter equation, 110 quasitriangular, 109 Hopf algebra, 115 quasitriangular structure, 109 equivalent, 109 quotient algebra, 12 quotient coalgebra, 17

R R-Hopf order, 96 R-order, 96 regular function, 131 ring of, 131 right integral, 78

S sequence regular, 61 Sweedler, M., 87, 92

T Tate, J., 104 tensor, 3 tensor product, 2 trivial Hopf algebra, 68 twist map, 10

U unit map, 9 unit property, 9