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Finite Dimensional

Algebras and

Quantum Groups

For all their help, encouragement, and infinite patience,we dedicate this book to our wives and children:

Wenlian Guo and Zhuoran Deng

Chunli Yu and Andy Du

Karen Parshall

Huiqing Xu, Xin Wang, and Yun Wang

Finite Dimensional Algebras and

Quantum Groups

Bangming Deng1

Jie Du2

Brian Parshall3

Jianpan Wang4

1School of Mathematical Sciences, Beijing Normal Univer-

sity, Beijing 100875, China

E-mail address: dengbm@bnu.edu.cn

2School of Mathematics and Statistics, University of New

South Wales, Sydney NSW 2052, Australia

E-mail address: j.du@unsw.edu.au

3Department of Mathematics, University of Virginia, Char-

lottesville, VA 22904-4137, USA

E-mail address: bjp8w@virginia.edu

4Department of Mathematics, East China Normal University,

Shanghai 200062, China

E-mail address: jpwang@ecnu.edu.cn

2000 Mathematics Subject Classification. Primary 05E10, 16G20, 16W35,17B37, 20C08;

Secondary 16G70, 20C30, 20G05, 20G42

Key words and phrases. Algebra with Frobenius morphism, Cartan matrix,Hecke algebra, Kazhdan–Lusztig cell, quantum enveloping algebra,

quantum gln, quantum Schur algebra, quasi-hereditary algebra, quiverwith automorphism, representation, Ringel–Hall algebra

.

Abstract. This research/advanced graduate-level text combines, forthe first time in book form, the two theories given in its title. As a mainfeature, the book emphasizes the finite dimensional algebra approachto quantum groups. More precisely, it investigates the Ringel–Hall al-gebra realization for the positive part of a quantum enveloping algebraassociated with a symmetrizable Cartan matrix and looks closely at theBeilinson–Lusztig–MacPherson realization for the entire quantum gln.

The book begins with the two realizations of generalized Cartanmatrices, namely, the graph realization and the root datum realization.From there, it develops the representation theory of quivers with auto-morphisms and the theory of quantum enveloping algebras associatedwith Kac-Moody Lie algebras. These two independent theories eventu-ally meet, under the umbrella of Ringel–Hall algebras.

Cartan matrices can also be used to define an important class ofgroups — Coxeter groups — and their associated Hecke algebras. Heckealgebras associated with symmetric groups give rise to an interestingclass of quasi-hereditary algebras, the quantum Schur algebras. Thestructure of these finite dimensional algebras is used to build the en-tire quantum gln through a completion process of a limit algebra (theBeilinson–Lusztig–MacPherson algebra).

Contents

Preface xiii

Notational conventions xxiii

Leitfaden xxv

Chapter 0. Getting started 1

§0.1. Cartan matrices and their two realizations 1

§0.2. Free algebras and presentations with generators and relations 6

§0.3. Examples: the realization problem 12

§0.4. Counting over finite fields: Gaussian polynomials 17

§0.5. Canonical bases: the matrix construction 22

§0.6. Finite dimensional semisimple Lie algebras 25

Exercises and notes 34

Part 1. Quivers and Their Representations

Chapter 1. Representations of quivers 43

§1.1. Quivers and their representations 44

§1.2. Euler forms, Cartan matrices, and the classification of quivers 49

§1.3. Weyl groups and root systems 55

§1.4. Bernstein–Gelfand–Ponomarev reflection functors 60

§1.5. Gabriel’s theorem 65

§1.6. Representation varieties and generic extensions 70

Exercises and notes 74

vii

viii Contents

Chapter 2. Algebras with Frobenius morphisms 83

§2.1. Fq-structures on vector spaces 84

§2.2. Algebras with Frobenius morphisms and Frobenius twists 86

§2.3. F -stable A-modules 91

§2.4. A construction of indecomposable F -stable modules 94

§2.5. A functorial approach to the representation theory 98

§2.6. Almost split sequences 105

§2.7. Irreducible morphisms 112

§2.8. Frobenius folding of almost split sequences 117

Exercises and notes 121

Chapter 3. Quivers with automorphisms 127

§3.1. Quivers with automorphisms and valued quivers 128

§3.2. Automorphisms of Dynkin and tame quivers 135

§3.3. Modulated quivers and Auslander–Reiten quivers 140

§3.4. Preprojective and preinjective components 145

§3.5. Modulated quivers attached to quivers with automorphisms 150

§3.6. Frobenius folding of Auslander–Reiten quivers 156

§3.7. Finite dimensional algebras over a finite field 164

§3.8. Representations of tame quivers with automorphisms 170

Exercises and notes 174

Part 2. Some Quantized Algebras

Chapter 4. Coxeter groups and Hecke algebras 183

§4.1. Coxeter groups 184

§4.2. An example: symmetric groups 193

§4.3. Parabolic subgroups and affine Weyl groups 197

§4.4. Hecke algebras 203

§4.5. Hecke monoids 208

§4.6. Counting with finite general linear groups 212

§4.7. Integral Hecke algebras associated with GLn(q) 218

Exercises and notes 223

Chapter 5. Hopf algebras and universal enveloping algebras 229

§5.1. Coalgebras, bialgebras, and Hopf algebras 230

§5.2. Universal enveloping algebras and PBW bases 239

Contents ix

§5.3. Universal enveloping algebras of Kac–Moody Lie algebras 244

§5.4. Symmetry structures of Kac–Moody Lie algebras 247

§5.5. Braid group actions 252

§5.6. Quantum sl2 256

Exercises and notes 263

Chapter 6. Quantum enveloping algebras 271

§6.1. Quantum enveloping algebras 271

§6.2. The elementary structure of U 275

§6.3. The Hopf algebra structure of U 278

§6.4. The adjoint action and triangular decomposition 283

§6.5. Annihilators of integrable U-modules 289

§6.6. Integrable Uv(sl2)-modules and their symmetries 295

§6.7. Symmetries of integrable U-modules 302

§6.8. Symmetry of U and braid group actions 305

§6.9. An integral structure 308

§6.10. A PBW theorem for finite type 315

Exercises and notes 318

Part 3. Representations of Symmetric Groups

Chapter 7. Kazhdan–Lusztig combinatorics for Hecke algebras 325

§7.1. R-polynomials and Kazhdan–Lusztig bases 326

§7.2. Multiplication formulas and Kazhdan–Lusztig polynomials 328

§7.3. Inverse Kazhdan–Lusztig polynomials and dual bases 332

§7.4. Cells 335

§7.5. Knuth and Vogan classes 338

§7.6. q-permutation modules and their canonical bases 342

§7.7. Cell modules and the Ext1-vanishing property 349

§7.8. The positivity property 353

Exercises and notes 361

Chapter 8. Cells and representations of symmetric groups 367

§8.1. The row-insertion algorithm 368

§8.2. The RSK correspondence 370

§8.3. The symmetry of the RSK correspondence 375

§8.4. Knuth equivalence classes in Sr 379

x Contents

§8.5. Left cells in symmetric groups 382

§8.6. The irreducibility of cell modules 388

§8.7. An Artin–Wedderburn decomposition for H(Sr)Q(v) 392

§8.8. A poset isomorphism 395

Exercises and notes 399

Chapter 9. The integral theory of quantum Schur algebras 405

§9.1. The quantum Schur algebra 406

§9.2. Specht modules and Specht data 412

§9.3. Canonical bases for quantum Schur algebras 415

§9.4. The cellular property of quantum Schur algebras 418

§9.5. Standard modules: canonical bases, duality, and beyond 423

§9.6. The integral double centralizer property 427

Exercises and notes 431

Part 4. Ringel–Hall Algebras: A Realization for the ±-Parts

Chapter 10. Ringel–Hall algebras 437

§10.1. Integral Hall algebras 438

§10.2. Ringel’s fundamental relations 441

§10.3. Ringel–Hall algebras 444

§10.4. Hall polynomials 449

§10.5. Generic Ringel–Hall algebras of finite type 456

Exercises and notes 461

Chapter 11. Bases of quantum enveloping algebras of finite type 467

§11.1. Generic extension monoids 468

§11.2. Reduced filtrations and distinguished words 472

§11.3. Monomial bases 478

§11.4. Reflection functors and subalgebras of Ringel–Hall algebras 483

§11.5. The Lusztig symmetries and PBW-type bases 488

§11.6. An elementary algebraic construction of canonical bases 494

§11.7. An example: canonical basis of U+v (sl3) 497

Exercises and notes 500

Chapter 12. Green’s theorem 505

§12.1. Comultiplication on Ringel–Hall algebras 505

§12.2. Some counting lemmas 511

Contents xi

§12.3. Proof of Green’s formula 516

§12.4. Green algebras and Lusztig’s theorem 523

§12.5. Green’s theorem 527

Exercises and notes 532

Part 5. The BLM Algebra: A Realization for Quantum gln

Chapter 13. Serre relations in quantum Schur algebras 537

§13.1. n-step flags and the orbit–matrix correspondence 538

§13.2. Dimensions of orbits 541

§13.3. Orbits corresponding to almost diagonal matrices 544

§13.4. A quantumization for quantum Schur algebras 546

§13.5. The fundamental multiplication formulas 550

§13.6. Some partial orderings on Ξ(n) and Ξ̃(n) 558

§13.7. The BLM triangular relations 560

§13.8. Extending the fundamental multiplication formulas 567

§13.9. Generators and relations 572

§13.10. Presentations for quantum Schur algebras 577

Exercises and notes 587

Chapter 14. Constructing quantum gln via quantum Schur algebras 591

§14.1. A stabilization property 592

§14.2. The BLM algebra K and its canonical basis 595

§14.3. The completion K̂ of K and multiplication formulas 598

§14.4. Embedding Uv(gln) into K̂ 602

§14.5. Z -forms of Uv(gln) 606

§14.6. Integral quantum Schur–Weyl reciprocity 609

§14.7. A connection with Ringel–Hall algebras 614

Exercises and notes 617

Appendices

Appendix A. Varieties and affine algebraic groups 623

§A.1. Affine varieties 624

§A.2. Varieties 630

§A.3. Affine algebraic groups 633

§A.4. Parabolic subgroups and the Chevalley–Bruhat ordering 643

§A.5. Representation theory: a first view 645

xii Contents

§A.6. Representations in positive characteristic; Frobeniusmorphisms 649

§A.7. Induced representations and the Weyl character formula 654

§A.8. Higher Ext functors; ∆- and ∇-filtrations 658

Exercises and notes 660

Appendix B. Quantum linear groups through coordinate algebras 669

§B.1. Quantum linear algebra 670

§B.2. Quantum linear groups 677

§B.3. Multiparameter quantum matrix spaces 683

§B.4. An application: quantum Schur algebras 691

Exercises and notes 695

Appendix C. Quasi-hereditary and cellular algebras 699

§C.1. Heredity ideals 700

§C.2. Quasi-hereditary algebras and highest weight categories 704

§C.3. Regular rings of Krull dimension at most 2 709

§C.4. Integral quasi-hereditary algebras 715

§C.5. Algebras with a Specht datum 719

§C.6. Cellular algebras 720

Exercises and notes 726

Bibliography 733

Index of notation 749

Index of terminology 755