Algebras, Rings and Modules Lie Algebras and Hopf Algebras

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Mathematical Surveys and Monographs Volume 168 Algebras, Rings and Modules Lie Algebras and Hopf Algebras

Michiel Hazewinkel Nadiya Gubareni V. V. Kirichenko

American Mathematical Society http://dx.doi.org/10.1090/surv/168

Mathematical Surveys and Monographs Volume 168

Algebras, Rings and Modules Lie Algebras and Hopf Algebras

Michiel Hazewinkel Nadiya Gubareni V. V. Kirichenko

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein

2010 Mathematics Subject Classiﬁcation. Primary 16W30, 05E05, 20C30, 16G20, 17B37, 20G42, 14L05, 81R50, 16W35, 17Bxx; Secondary 05E10, 58B34, 81R60, 81R12, 82C20.

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Library of Congress Cataloging-in-Publication Data Hazewinkel, Michiel. Algebras, rings, and modules : Lie algebras and Hopf algebras / M. Hazewinkel, N. Gubareni, V. V. Kirichenko. p. cm. Includes bibliographical references and index. ISBN 978-0-8218-5262-0 (alk. paper) 1. Hopf algebras. 2. Lie algebras. I. Gubareni, Nadezhda Mikhailovna. II. Kirichenko, Vladimir V. III. Title. QA613.8.H39 2010 512.55—dc22 2010030618

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Preface ix References xi Chapter 1. Lie algebras and Dynkin diagrams 1 1.1. Lie algebras. Definitions and examples 2 1.2. Ideals, homomorphisms and representations 7 1.3. Solvable and nilpotent Lie algebras 11 1.4. Radical of a Lie algebra. Simple and semisimple Lie algebras 15 1.5. Modules for Lie algebras. Weyl’s theorem. Ado’s theorem 22 1.6. Lie’s theorem 29 1.7. The Lie algebra sl(2; k). Representation of sl(2; k)32 1.8. The universal enveloping algebra of A lie algebra 35 1.9. Poincaré-Birkhoff-Witt theorem 38 1.10. Free Lie algebras 43 1.11. Examples of simple Lie algebras 45 1.12. Abstract root systems and the Weyl group 47 1.13. Cartan matrices and Dynkin diagrams 57 1.14. Coxeter groups and Coxeter diagrams 61 1.15. Root systems of semisimple Lie algebras 67 1.16. The Weyl group of a quiver 73 1.17. Reflection functors 78 1.18. Coxeter functors and Coxeter transformations 85 1.19. The Gabriel theorem 87 1.20. Generalized Cartan matrices and Kac-Moody Lie algebras 88 1.21. Historical notes 91 References 94 Chapter 2. Coalgebras: motivation, definitions, and examples 99 2.1. Coalgebras and ’addition formulae’ 100 2.2. Coalgebras and decompositions 102 2.3. Dualizing the idea of an algebra 103 2.4. Some examples of coalgebras 105 2.5. Sub coalgebras and quotient coalgebras 108 2.6. The main theorem of coalgebras 108 2.7. Cofree coalgebras 109 2.8. Algebra - coalgebra duality 112 2.9. Comodules and representations 121 2.10. Graded coalgebras 123 2.11. Reflexive modules 125 2.12. Measuring 126

v vi CONTENTS

2.13. Addition formulae and duality 128 2.14. Coradical and coradical ﬁltration 128 2.15. Coda to chapter 2 129 References 129

Chapter 3. Bialgebras and Hopf algebras. Motivation, deﬁnitions, and examples 131 3.1. Products and representations 131 3.2. Bialgebras 133 3.3. Hopf algebras 138 3.4. Some more examples of Hopf algebras 140 3.5. Primitive elements 146 3.6. Group-like elements 149 3.7. Bialgebra and Hopf algebra duality 152 3.8. Graded bialgebras and Hopf algebras 153 3.9. Crossed products 159 3.10. Integrals for Hopf algebras 162 3.11. Formal groups 167 3.12. Hopf modules 169 3.13. Historical remarks 170 3.14. The Hopf algebra of an algebra 171 References 172

Chapter 4. The Hopf algebra of symmetric functions 175 4.1. The algebra of symmetric functions 175 4.2. The Hopf algebra structure 184 4.3. PSH algebras 185 4.4. Automorphisms of Symm 193 4.5. The functor of the Witt vectors 194 4.6. Ghost components 197 4.7. Frobenius and Verschiebung endomorphisms 199 4.8. The second multiplication of Symm 202 4.9. Lambda algebras 203 4.10. Exp algebras 210 4.11. Plethysm 213 4.12. The many incarnations of Symm 214 References 215

Chapter 5. The representations of the symmetric groups from the Hopf algebra point of view 217 5.1. A little bit of ﬁnite group representation theory 217 5.2. Double cosets of Young subgroups 221 5.3. The Hopf algebra 223 5.4. Symm as a PSH algebra 226 5.5. The second multiplication on RS 227 5.6. Remarks and acknowledgements 229 References 229

Chapter 6. The Hopf algebra of noncommutative symmetric functions and the Hopf algebra of quasisymmetric functions 231 CONTENTS vii

6.1. The Hopf algebra NSymm 232 6.2. NSymm over the rationals 234 6.3. The Hopf algebra QSymm 235 6.4. Symm as a quotient of NSymm 238 6.5. More on Shuﬄe and LieHopf 241 6.6. The autoduality of Symm 248 6.7. Polynomial freeness of QSymm over the integers 250 6.8. Hopf endomorphisms of QSymm and NSymm 254 6.9. Verschiebung and Frobenius on NSymm and QSymm 255 References 260

Chapter 7. The Hopf algebra of permutations 263 7.1. The Hopf algebra of permutations of Malvenuto, Poirier and Reutenauer 263 7.2. The imbedding of NSymm into MPR 267 7.3. LSD permutations 271 7.4. Second multiplication and second comultiplication 274 7.5. Rigidity and uniqueness of MPR 275 References 276 Chapter 8. Hopf algebras: Applications in and interrelations with other parts of mathematics and physics 277 8.1. Actions and coactions of bialgebras and Hopf algebras 277 8.2. The quantum groups GLq(n, C) and multiparameter generalizations 285 8.3. The quantum groups Uq(sl(n; k)) 294 8.4. R-matrices and QIST: Bethe Ansatz, FCR construction and the FRT theorem 296 8.5. Knot invariants from quantum groups. Yang-Baxter operators 311 8.6. Quiver Hopf algebras 324 8.7. Ringel-Hall Hopf algebras 333 8.8. More interactions of Hopf algebras with other parts of mathematics and theoretical physics 347 8.8.1. Capelli identities and other formulas for matrices and determinants with mildly noncommuting entries 347 8.8.2. Quantum symmetry 355 8.8.3. Hopf algebra symmetries in noncommutative geometry 369 8.8.4. Hopf algebras in Galois theory 371 8.8.5. Hopf algebras and renormalization 372 8.8.6. Quantum calculi 372 8.8.7. Umbral calculus and Baxter algebras 376 8.8.8. q-special functions 380 References 381 Index 407

Preface

This volume, the third in our series, is about Hopf algebras (mostly) and Lie algebras. It is independent of the ﬁrst two volumes; though, to be sure, there are some references to them, just as there are references to other books and journals.

The ﬁrst chapter is devoted to Lie algebras. It is a fairly standard concise treatment of the more established parts of the theory with the exceptions that there is a bit more emphasis on Dynkin diagrams (also pictorially) and that the chapter includes a complete treatment of the correspondence, initiated by Peter Gabriel, between representations of quivers whose underlying graph is a Dynkin diagram and representations of the Lie algebra with that Dynkin diagram, Gabriel, [4], [3]; Bernstein-Gel’fand-Ponomarev, [1]; Dlab-Ringel, [2]. The treatment is via the very elegant approach by Coxeter reﬂection functors of [1].

The remaining seven chapters are on Hopf algebras. The first two of these seven are devoted to the basic theory of coalgebras and Hopf algebras paying special attention to motivation, history, intuition, and prove- nance. In a way these two chapters are primers1 on their respective subjects. The remaining five chapters are quite different.

Chapter number four is on the symmetric functions from the Hopf algebra point of view. This Hopf algebra is possibly the richest structured object in mathematics and a most beautiful one. One aspect that receives special attention is the Zelevin- sky theorem on PSH algebras. The acronym PSH stands for ’positive selfadjoint Hopf’. What one is really dealing with here is a graded, connected, Hopf algebra with a distinguished basis that is declared orthonormal, such that each component of the grading is of ﬁnite rank, and such that multiplication and comultiplication are adjoint to each other and positive. If then, moreover, there is only one distinguished basis element that is primitive, the Hopf algebra is isomorphic to Symm, the Hopf (and more) algebra of symmetric functions with the Schur functions as distinguished basis. Quite surprisingly the second (co)multiplication on Symm, which makes each graded summand a ring in its own right and which is distributive over the ﬁrst one (in the Hopf algebra sense), turns up during the proof of the theorem, the Bernstein morphism. This certainly calls for more investigations. The enormously rich structure of Symm is discussed extensively (various lambda ring structures, Frobenius, and Verschiebung morphims, Adams operations, ... etc. Correspondingly a fair amount of space is given to the big Witt vectors.

1As a rule a professional mathematician is well advised to stay away from ’primers’; we hope and believe that these are an exception.

ix x LIE ALGEBRAS AND HOPF ALGEBRAS

Chapter ﬁve is on the representations of the symmetric groups; more precisely it is on the direct sum of the RSn where Sn is the symmetric group on n letters andasanAbeliangroupRSn is the Grothendieck group of (virtual) complex representations. The direct sum, with RS0 = Z by decree, is given a multiplication and a comultiplication by using induction up from two factor Young subgroups and restriction down to such subgroups. The result is a PSH algebra. That is not at all diﬃcult to see: basically Frobenius reciprocity and the Mackey double coset formula do the job. The distinguished basis is formed by the irreducible representations and there is only one primitive among these, viz the trivial repre- ∞ sentation of the symmetric group on one letter. It follows that RS = RSn is a n=0 PSH algebra and that it is isomorphic to Symm. It also follows that Symm itself is PSH (with one distinguished primitive). So far that was not clear. At the end of chapter four the situation was that if there were a PSH algebra with one distinguished primitive it would be isomorphic to Symm, making Symm itself also PSH. The stumbling block in proving directly that Symm is PSH is positivity. There seems to be no direct proof, without going through the representation theory of the symmetric groups, of the fact that the product of two Schur functions is a positive linear combination of (other) Schur functions. The question of positivity of multiplication and comultiplication, such a vexing matter in the case of Symm, becomes a triviality in the case of RS. Indeed, the product is describable as follows. Take a representation ρ of Sp and a representation σ of Sq and form the tensor product ρ ⊗ σ which is a representation of the Young subgroup Sp × Sq of Sp+q and induce this representation up to a representation of all of Sp+q. If the two initial representations are real (as opposed to virtual), so is their tensor product and the induced representation of Sp+q. Thus multiplication is positive. Comultiplication can be treated in a similar way, or handled via duality. The isomorphism between Symm and RS is far more than just an isomorphism of Hopf algebras; it also says things regarding the second (co)multiplications, the various lambda ring structures and plethysms, ... ; see [7]. It won’t be argued here, but it may well turn out to be the case that among the many Hopf algebras isomorphic to Symm the incarnation RS is the central one; see loc. cit. Of course the fact that Symm and RS are isomorphic (as algebras) is much older than the Zelevinsky theorem; see [8], section I.7, for a classical treatment.

Chapter six is about two generalizations of Symm that have become important in the last 25 years or so: the Hopf algebra QSymm of quasi-symmetric functions and the Hopf algebra NSymm of non commutative symmetric functions. They are dual to each other and this duality extends the autoduality of Symm via a natural imbedding Symm ⊂ QSymm and a natural projection NSymm −→ Symm. Both Hopf algebras carry a good deal more structure. The seminal paper which started all this is [5], but there, and in a slew of subsequent papers, things were done over a ﬁeld of characteristic zero and not over the integers as here. It is somewhat startling to discover how many concepts, proofs, constructions, ... have natural non commutative and quasi analogs; and not rarely these analogs are more elegant than their counterparts in the world of symmetric functions. PREFACE xi

Things can be generalized still further to the Hopf algebra MPR of permutations of Malvenuto, Poirier, and Reutenauer, [9]; [10]. This one is twisted autodual and this duality extends the duality between NSymm and QSymm via a natural imbedding NSymm −→ MPR and a natural projection MPR −→ QSymm. This is the subject matter of chapter seven. There are still further generalizations, complete with duality, [6], but the investigation of these has only just started. Like MPR theyareHopfalgebrasofendomorphismsofHopfalgebras.

Finally, chapter eight contains over ﬁfteen relatively short and shorter outlines, heavily bibliographical, on the roles of Hopf algebras in other parts of mathematics and theoretical physics. Much of this material has not previously appeared in the monographic literature.

In closing, we would like to express our cordial thanks to our friends and col- leagues (among them, we especially thank S.A.Ovsienko) who have read portions of preliminary versions of this book and oﬀered corrections, suggestions and other constructive comments improving the entire text. We also give special thanks to A. Katkow and V. V. Sergeichuk who have helped with the preparation of this manuscript. And last, but not least, we would like to thank a number of anonymous referees; several of whom made some very valuable suggestions (both for corrections and small additions). Needless to say any remaining inaccuracies (and there are always some) are the responsibility of the authors.

References [1] L. N. Bernstein, I. M. Gel’fand and V. A. Ponomarev, Coxeter functors and Gabriel’s theorem, Russiam math. Surveys 28 (1973), 17 - 32. [2] Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Amer. math. Soc., 1976. [3] Peter Gabriel, Unzerlegbare Darstellungen I, Manuscripta mathematica 6 (1972), 71 - 103. [4] Peter Gabriel, Indecomposable representations II, Symp. Math. INDAM 11 (Rome. 1971), Acad. Press, 1973, 81 - 104. [5] I. M. Gel’fand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh and Jean- Yves-Thibon, Noncommutative symmetric functions,Adv.Math.112 (1995), 218-348. [6] Michiel Hazewinkel, Hopf algebras of endomorphisms of Hopf algebras, Arabian J. for Science and Engineering 33(2C) (Dec. 2008), 239 - 272; arXiv: math.QA/0410364. [7] Michiel Hazewinkel, Witt vectors. Part 1, in: M. Hazewinkel, ed., Handbook of algebra. Volume 6, Elsevier, 2009, 319 - 472. [8]I.G.Macdonald,Symmetric functions and Hall polynomials, Clarendon Press, 1995. [9] C. Malvenuto, and Chr. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. of Algebra 177 (1994), 967-982. [10] Stéphane Poirier and Christophe Reutenauer, Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec 19 (1995), 79-90.

Michiel Hazewinkel, Bussum, Nov. 2009 Nadiya Gubareni, Cz¸estochowa, Nov. 2009 Vladimir V. Kirichenko, Kiev, Nov. 2009

Index

A braid, 313 Abelian Lie algebra, 3 braid closure, 315 (abstract) Cartan matrix, 89 braid on n strands, 313 abstract root system, 2, 47 braid equation, 292 action, 9 (Brauer-)Schur Weyl duality, 361 ad-nilpotent endomorphism, 13 Adams operations, 207 C adjoint action, 162 canonical curves, 255 adjoint Lie algebra, 10 Capelli identities, 350 adjoint Lie module, 3 Cartan decomposition, 68 adjoint representation, 10 Cartan matrix, 57, 89 admissible sequence of sinks, 85 Cartan number, 71 Ado theorem, 4, 28 Cartan subalgebra, 67 affine Lie algebra, 5, 91 Cartan-Killing classification theorem, 73 affine line, 144 Cartan’s first criterion (for solvability), 32 algebra, 102 Cartan’s semisimplicity criterion, 21 algebraic Bethe Ansatz, 309 Cartier theorem, 159 antipode, 138 Casimir element, 24 arrow of a quiver, 75 Casimir operator, 24 Alexander theorem, 316 Cauchy-Binet formulas, 351 Artin braid group, 314 Cauchy-Binet theorem, 351 Artin-Hasse exponential, 205 Cayley-Hamilton-Newton formulae, 351 associative algebra, 103 Cayley-Hamilton-Newton identities, 353 automorphism (of a Lie algebra), 8 Cayley-Hamilton relations, 351 center of a Lie algebra, 7 B centralizer, 4 base of a root system, 54 character formulas, 361 Baxter algebra, 376 Chen-Fox-Lyndon factorization, 242 Bergman centralizer theorem, 239 Chevalley generators, 89 Bernstein morphism, 192 Clarence Wilkerson theorem, 208 Bernstein-Gel’fand-Ponomarev theorem, 85 classical Capelli identity, 350 Bethe Ansatz, 310 classical Hall-Steinitz algebra of partitions, Bethe Ansatz equations, 311 333 bialgebra, 133 classical root system, 60 bicovariant calculi on quantum groups, 375 Clifford algebra, 361 bi-Hopf-module, 327 coalgebra, 103 bilinear form, 75 coassociative comultiplication, 101 bilinear form of a Coxeter diagram, 66 coassociativity, 103 binomial coalgebra, 107 coassociative coalgebra, 104 Birman-Murakami-Wenzl algebra, 366, 367 cocommutive coalgebra, 104 BMW algebra, 366 cocommutative bialgebra, 137 Borel-Weyl construction, 360 cofree k-coalgebra, 110 Boson-Fermion correspondence, 354 cofree graded coalgebra, 124 bracket, 4 coideal, 116

407 408 LIE ALGEBRAS AND HOPF ALGEBRAS coinvariants, 169 dual partition, 177 column determinant, 349 Dynkin diagram, 58 commutative algebra, 104 Dynkin quiver, 86 commutative bialgebra, 137 commutative Lie algebra, 3 E commutator, 2, 4 eigenvector, 29 comodule, 122 end vertex, 75 comodule-module duality, 122 Engel theorem, 13, 15 completely reducible L-module, 23 endomorphism algebra representation, 361 completely reducible representation, 23 enveloping algebra, 36 composition, 176, 267 epimorphism of Lie algebras, 8 composition algebra, 345 equivalent representation, 10 comultiplication, 101 exceptional root system, 61 concatenation, 241 Exp algebra, 210 conilpotent comultiplication, 159 exponent sequence, 175 conservation laws, 361 exponentiation functor, 211 convolution, 126 extended Coxeter-Dynkin diagram, 91 convolution product, 138 extended twisted generic composition coradical, 128 algebra, 345 coradical filtration, 128 extended twisted Hall algebra, 342 coroot, 70 extended Yang-Baxter cotensor coalgebra, 325 operator, 320 covariant differential calculi, 374 F Coxeter diagram, 61 faithful representation, 9, 131 Coxeter element, 85, 86 finite dimensional Lie algebra, 3 Coxeter functor, 85, 86 finite dual coalgebra of an algebra, 113 Coxeter graph, 61 finiteness lemma, 48 Coxeter group, 61 first Adams=Frobenius theorem, 208 Coxeter matrix, 61 forgotten symmetric function, 180 Coxeter transformation, 85 formal group, 167, 168 crossed product, 161 free Baxter algebra, 378 free k-algebra, 110 D free Lie algebra, 43 decomposable L-module, 23 Friedrichs theorem, 148 decomposable representation, 10, 23 Frobenius axiom, 221 defect of a monomial, 39 Frobenius endomorphism, 200 derivation, 6 Frobenius reciprocity, 217, 363 derived algebra, 9 FRT construction, 296 derived series, 11 FRT theorem, 298 descending central series, 12 functor Curve, 256 descent set, 267 fundamental commutation relations, 290, diagonalizable endomorphism, 16 291 difference, 4 differential calculus on an algebra, 374 G dimension vector, 81 Gabriel quiver, 338 divided power algebra, 246 Gabriel theorem, 87 divided power coalgebra, 105 general linear Lie algebra, 4 divided power Hopf algebra, 143, 377 generalized Cartan matrix, 89 divided power sequence, 150, 239 generating function, 183 divided powers, 245 generic algebra of polynomials, 357 double commutant theorem, 361 generic composition algebra, 345 double dual morphism, 115 ghost component, 197, 198 double composition algebra, 345 graded algebra, 124, 153 double Ringel-Hall (Hopf) algebra, 344 graded coalgebra, 124, 153 Drinfel’d double, 344 graded-cofree coalgebra, 325 dual algebra of a coalgebra, 112 graded dual, 124, 156 dual bialgebra, 152 graded Hopf algebra, 154 dual Hopf algebra, 152 graded module, 123, 153 INDEX 409 graded morphism, 134 Lax operator, 308 Grothendieck group, 223 lattice statistical mechanics models, 301, group graded algebra, 284 364 group-like element, 149 left coideal, 116 left comodule, 122 H left integral of a bialgebra, 163 Hall-Steinitz algebra, 336 left integral in a Hopf algebra, 166 Hecke algebra, 367 left module, 122 height, 77 left reflection functor, 80 Heisenberg algebra, 363 length of a monomial, 39 Heisenberg model, 307 length of a solvable Lie algebra, 11 hereditary algebra, 338 length of a Weyl group, 57 highest root, 53 Leray theorem, 157 highest weight, 34 Levi decomposition, 26 homogenous space, 363 Levi factor, 26 Hopf algebra, 138 Levi subalgebra, 26 Hopf algebra structure, 184 Levi theorem, 26 Hopf algebra of an algebra, 138 lexicographic order, 183, 241 Hopf module, 169 Lie algebra, 2 Hopf module algebra, 159 Lie algebra morphism, 8 Hopf pairing, 343 Lie bracket, 2 Hopf quiver, 330 Lie homomorphism, 8 Lie Hopf algebra, 141 I LieHopf, 241 ideal of a Lie algebra, 7 Lie module, 23 indecomposable module, 23 Lie subalgebra, 3 indecomposable root system, 50 Lie theorem, 29, 30 induction, 217 linear Lie algebra, 4 infinite dimensional Lie algebra, 3 Lielevicius theorem, 193 initial vertex, 75 Lorentz group, 365 inner derivation, 11 Lorentz inner product, 5 integral for a Hopf algebra, 153 Lorentz Lie algebra, 5 invariance lemma, 29 lower central series, 12 invariants, 169 LSD permutation, 271 invariant subspace, 9 Lyndon word, 241 inversion, 39 irreducible L-module, 23 M irreducible representation, 10, 23 Mackey decomposition, 219 irreducible root system, 50 Mackey double coset theorem, 219 isomorphic L-modules, 23 Mackey tensor product theorem, 220 isomorphic root systems, 52 MacMachon master theorem, 354 isomorphism of Lie algebras, 8 magnetic spin chain, 307 isomorphism of Lie modules, 23 Manin matrix, 347 Markov move, 317 J Jacobi identity, 3 Markov theorem, 318 Jordan decomposition, 17 Maschke theorem, 167 Jordan-Chevalley decomposition, 16, 17 maximal root, 53 maximal vector, 33 K measuring, 127 knot, 311 metric tensor, 20 knot invariants, 312 Milnor-Moore theorem, 150 Kac-Moody Lie algebra, 88, 90 Minkowski space, 365 Killing form, 19 module of coinvariants, 169 monodromy matrix, 305 L monoid bialgebra of a finite monoid, 135 L-module, 23 monoid bialgebra of an arbitrary monoid, lambda algebra, 203 136 lambda ring, 205 monomial quasisymmetric function, 235 410 LIE ALGEBRAS AND HOPF ALGEBRAS monomial symmetric function, 177 Q monomorphism of Lie algebras, 8 q-derivative, 373 morphism of L-modules, 23 q-determinant, 349 MPR Hopf algebra, 263 q-Manin matrix, 348 multiparameter deformation, 289 q-number, 373 q-special function, 380 N q-Schur algebra, 368 negative root, 52, 77 QSymm, 235 Newton primitives, 146 quadratic algebra, 296 Nichols-Zoeller theorem, 151 quantization, 289 nil-algebra, 11 quantum calculi, 372 nilpotency length, 12 quantum determinant, 349 nilpotent endomorphism, 13 quantum enveloping algebra, 294 nilpotent Lie algebra, 12 quantum field theory, 365 nilpotent part, 17 quantum group, 285 Noether theorem, 361 quantum group SLq(2; k), 145 nondegenerate form, 20 quantum group matrix, 348 nondecreasing sequence, 39 quantum matrix algebra, 290 nonnegative definite quadratic quantum universal enveloping algebra form, 76 Uqsl(2; k), 294 normalizer, 14 quantum Yang-Baxter equation, 291 NSymm, 232 quasi bialgebra, 155 quasisymmetric function, 176 O quasisymmetric polynomial, 175 ordered monomial, 39 quasisymmetric power series, 176 orientation on a valued graph, 342 quasitriangular Hopf algebra, 365 outer derivation, 11 quiver, 75 overlapping shuffle product, 236, 379 quotient coalgebra, 108 P quotient Lie algebra, 8 Pareigis Hopf algebra, 144 R partial Coxeter functor, 78 radical of a Lie algebra, 15 partition, 176 radical of a form, 20 path, 324 rank (of a free Lie algebra), 44 path algebra, 324 rank (of a root system), 48 path coalgebra, 324 refinement, 270 PBW basis, 41 reflection, 47, 66 PBW algebra, 289 reflection functor, 78 PSH algebra, 185, 186 reflexive module, 125 Peter-Weyl theory, 365 regular representation, 10 plethysm, 213 renormalization, 372 Poincaré group, 365 representation of a quiver, 79 Poincaré-Birkhoff-Witt theorem, 38, 41, 42 representation of a Lie algebra, 9 point (of a quiver), 75 restriction, 217 Poisson bracket, 6 Ribbon Schur function, 272 positive root, 52, 77 right coideal, 116 positive definite quadratic form, 76 right comodule, 279 positive semi-definite quadratic form, 76 right Hopf module, 169 power sum, 183 right integral of a bialgebra, 163 primitive element, 146 right reflection functor, 80 projection formula, 221 Ringel-Hall (Hopf) algebra 339 projective limit description, 177 root, 47, 68 product of representations of associative root system, 48 algebras, 132 root space decomposition, 68 product of representations of groups, 131 root subspace, 68 product of representations of Lie algebras, Rota-Baxter algebra, 378 132 row determinant, 349 proper algebra, 131 proper tail, 241 INDEX 411

S Tannaka-Klein duality, 368 Scheffer sequence, 377 target vertex, 75 Schur algebra, 368 terminal vertex, 75 Schur function, 180 trace form, 24 Schur lemma, 24, 217 transform matrix, 305 Schur-Weyl duality, 366 trivial representation, 9 Schurian Hopf quiver, 330 twisted module, 161 second Adams=Frobenius theorem, 209 twisted Ringel-Hall algebra, 339 selfadjoint bialgebra, 155 semidirect product of groups, 160 U semisimple endomorphism, 16 umbral calculus, 376 semisimple element, 67 universal enveloping algebra, 36 universal enveloping bialgebra, 136 semisimple Lie algebra, 15 universal measuring coalgebra, 172 semisimple part, 17 Serre relations, 89 V Serre theorem, 89 valued graph, 341 set of roots, 77 valued quiver, 342 Shuffle, 241 Verschiebung endomorphism, 200 shuffle Hopf algebra, 141 vertex (of a quiver), 75 shuffle product, 141 Virasoro algebra, 6 sigma ring, 205 simple Lie algebra, 9 W simple Lie module, 23 weight, 29, 33 simple-pointed Hopf algebra, 332 weight space, 29, 33 simple reflection, 74 Weyl algebra, 368 simple root, 52, 77 Weyl group, 56, 74 sink, 78 Weyl group of a quiver, 75 sink-admissible sequence, 85 Weyl relations, 89 skew group ring, 160 Weyl theorem, 25 skew Hopf pairing, 344 Wigner-Eckart theorem, 369 skew pairing, 343 Witt algebra, 7, 10 slender module, 125 wll-ordering, 252 smash product, 160 world-cut-coalgebra, 103 Solomon descent algebra, 274 Y solvable Lie algebra, 11 Yang-Baxter equation, 291 source, 78 Yangian, 369 source vertex, 75 Young subgroup, 221 source-admissible sequence, 85 special linear Lie algebra, 4 Z species, 342 Zelevinsky theorem, 188, 189, 191 standard basis, 32 zero dual coalgebra of an algebra, 113 standard monomial, 39 start vertex, 75 string, 49 string property, 49 structure constants, 19 sub coalgebra, 108, 116 submodule, 23 subrepresentation, 9 Sweedler sigma notation, 277 Sweedler Hopf algebra, 141 symmetric bilinear form, 75 symmetric polynomial, 175 symmetrizable matrix, 89

T Taft Hopf algebra, 141 Tannaka reconstruction, 368 The main goal of this book is to present an introduction to and applications of the theory of Hopf algebras. The authors also discuss some important aspects of the theory of Lie algebras. The first chapter can be viewed as a primer on Lie algebras, with the main goal to explain and prove the Gabriel–Bernstein–Gelfand–Ponomarev theorem on the correspondence between the representations of Lie algebras and quivers; this material has not previously appeared in book form. The next two chapters are also “primers” on coalgebras and Hopf algebras, respec- tively; they aim specifically to give sufficient background on these topics for use in the main part of the book. Chapters 4–7 are devoted to four of the most beautiful Hopf algebras currently known: the Hopf algebra of symmetric functions, the Hopf algebra of representations of the symmetric groups (although these two are isomorphic, they are very different in the aspects they bring to the forefront), the Hopf algebras of the nonsymmetric and quasisymmetric functions (these two are dual and both generalize the previous two), and the Hopf algebra of permutations. The last chapter is a survey of applications of Hopf algebras in many varied parts of mathematics and physics. Unique features of the book include a new way to introduce Hopf algebras and coalgebras, an extensive discussion of the many universal properties of the functor of the Witt vectors, a thorough discussion of duality aspects of all the Hopf algebras mentioned, emphasis on the combinatorial aspects of Hopf algebras, and a survey of applications already mentioned. The book also contains an extensive (more than 700 entries) bibli- ography.

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