**Mathematical Surveys and Monographs**

Volume: 150;
2008;
759 pp;
Hardcover

MSC: Primary 05; 16; 17; 20;

Print ISBN: 978-0-8218-4186-0

Product Code: SURV/150

List Price: $123.00

Individual Member Price: $98.40

**Electronic ISBN: 978-1-4704-1377-4
Product Code: SURV/150.E**

List Price: $123.00

Individual Member Price: $98.40

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#### Supplemental Materials

# Finite Dimensional Algebras and Quantum Groups

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*Bangming Deng; Jie Du; Brian Parshall; Jianpan Wang*

The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel–Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson–Lusztig–MacPherson realization for the entire quantum \(\mathfrak {gl}_n\).

The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac–Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel–Hall algebras. Cartan matrices can also be used to define an important class of groups—Coxeter groups—and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum \(\mathfrak{gl}_n\) through a completion process of a limit algebra (the Beilinson–Lusztig–MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature.

#### Table of Contents

# Table of Contents

## Finite Dimensional Algebras and Quantum Groups

- Contents vii8 free
- Preface xiii14 free
- Notational conventions xxiii24 free
- Leitfaden xxv26 free
- Chapter 0. Getting started 128 free
- §0.1. Cartan matrices and their two realizations 128
- §0.2. Free algebras and presentations with generators and relations 633
- §0.3. Examples: the realization problem 1239
- §0.4. Counting over finite fields: Gaussian polynomials 1744
- §0.5. Canonical bases: the matrix construction 2249
- §0.6. Finite dimensional semisimple Lie algebras 2552
- Exercises and notes 3461

- Part 1. Quivers and Their Representations 4168
- Chapter 1. Representations of quivers 4370
- §1.1. Quivers and their representations 4471
- §1.2. Euler forms, Cartan matrices, and the classification of quivers 4976
- §1.3. Weyl groups and root systems 5582
- §1.4. Bernstein–Gelfand–Ponomarev reflection functors 6087
- §1.5. Gabriel's theorem 6592
- §1.6. Representation varieties and generic extensions 7097
- Exercises and notes 74101

- Chapter 2. Algebras with Probenius morphisms 83110
- §2.1. F[sub(q)]-structures on vector spaces 84111
- §2.2. Algebras with Probenius morphisms and Frobenius twists 86113
- §2.3. F-stable A-modules 91118
- §2.4. A construction of indecomposable F-stable modules 94121
- §2.5. A functorial approach to the representation theory 98125
- §2.6. Almost split sequences 105132
- §2.7. Irreducible morphisms 112139
- §2.8. Frobenius folding of almost split sequences 117144
- Exercises and notes 121148

- Chapter 3. Quivers with automorphisms 127154
- §3.1. Quivers with automorphisms and valued quivers 128155
- §3.2. Automorphisms of Dynkin and tame quivers 135162
- §3.3. Modulated quivers and Auslander–Reiten quivers 140167
- §3.4. Preprojective and preinjective components 145172
- §3.5. Modulated quivers attached to quivers with automorphisms 150177
- §3.6. Frobenius folding of Auslander–Reiten quivers 156183
- §3.7. Finite dimensional algebras over a finite field 164191
- §3.8. Representations of tame quivers with automorphisms 170197
- Exercises and notes 174201

- Part 2. Some Quantized Algebras 181208
- Chapter 4. Coxeter groups and Hecke algebras 183210
- §4.1. Coxeter groups 184211
- §4.2. An example: symmetric groups 193220
- §4.3. Parabolic subgroups and affine Weyl groups 197224
- §4.4. Hecke algebras 203230
- §4.5. Hecke monoids 208235
- §4.6. Counting with finite general linear groups 212239
- §4.7. Integral Hecke algebras associated with GL[sub(n)](q) 218245
- Exercises and notes 223250

- Chapter 5. Hopf algebras and universal enveloping algebras 229256
- §5.1. Coalgebras, bialgebras, and Hopf algebras 230257
- §5.2. Universal enveloping algebras and PBW bases 239266
- §5.3. Universal enveloping algebras of Kac–Moody Lie algebras 244271
- §5.4. Symmetry structures of Kac–Moody Lie algebras 247274
- §5.5. Braid group actions 252279
- §5.6. Quantum sl[sub(2)] 256283
- Exercises and notes 263290

- Chapter 6. Quantum enveloping algebras 271298
- §6.1. Quantum enveloping algebras 271298
- §6.2. The elementary structure of U 275302
- §6.3. The Hopf algebra structure of U 278305
- §6.4. The adjoint action and triangular decomposition 283310
- §6.5. Annihilators of integrable U-modules 289316
- §6.6. Integrable U[sub(v)](sl[sub(2)])-modules and their symmetries 295322
- §6.7. Symmetries of integrable U-modules 302329
- §6.8. Symmetry of U and braid group actions 305332
- §6.9. An integral structure 308335
- §6.10. A PBW theorem for finite type 315342
- Exercises and notes 318345

- Part 3. Representations of Symmetric Groups 323350
- Chapter 7. Kazhdan–Lusztig combinatorics for Hecke algebras 325352
- §7.1.R-polynomials and Kazhdan–Lusztig bases 326353
- §7.2. Multiplication formulas and Kazhdan–Lusztig polynomials 328355
- §7.3. Inverse Kazhdan–Lusztig polynomials and dual bases 332359
- §7.4. Cells 335362
- §7.5. Knuth and Vogan classes 338365
- §7.6. q-permutation modules and their canonical bases 342369
- §7.7. Cell modules and the Ext[sup(1)]-vanishing property 349376
- §7.8. The positivity property 353380
- Exercises and notes 361388

- Chapter 8. Cells and representations of symmetric groups 367394
- §8.1. The row-insertion algorithm 368395
- §8.2. The RSK correspondence 370397
- §8.3. The symmetry of the RSK correspondence 375402
- §8.4. Knuth equivalence classes in [omitted][sub(r)] 379406
- §8.5. Left cells in symmetric groups 382409
- §8.6. The irreducibility of cell modules 388415
- §8.7. An Artin-Wedderburn decomposition for H([omitted][sub(r)])[sub(Q(v))] 392419
- §8.8. A poset isomorphism 395422
- Exercises and notes 399426

- Chapter 9. The integral theory of quantum Schur algebras 405432
- §9.1. The quantum Schur algebra 406433
- §9.2. Specht modules and Specht data 412439
- §9.3. Canonical bases for quantum Schur algebras 415442
- §9.4. The cellular property of quantum Schur algebras 418445
- §9.5. Standard modules: canonical bases, duality, and beyond 423450
- §9.6. The integral double centralizer property 427454
- Exercises and notes 431458

- Part 4. Ringel–Hall Algebras: A Realization for the ±-Parts 435462
- Chapter 10. Ringel–Hall algebras 437464
- Chapter 11. Bases of quantum enveloping algebras of finite type 467494
- §11.1. Generic extension monoids 468495
- §11.2. Reduced filtrations and distinguished words 472499
- §11.3. Monomial bases 478505
- §11.4. Reflection functors and subalgebras of Ringel–Hall algebras 483510
- §11.5. The Lusztig symmetries and PBW-type bases 488515
- §11.6. An elementary algebraic construction of canonical bases 494521
- §11.7. An example: canonical basis of U[sup(+)][sub(v)](sl[sub(3)]) 497524
- Exercises and notes 500527

- Chapter 12. Green's theorem 505532

- Part 5. The BLM Algebra: A Realization for Quantum gl[sub(n)] 535562
- Chapter 13. Serre relations in quantum Schur algebras 537564
- §13.1. n-step flags and the orbit–matrix correspondence 538565
- §13.2. Dimensions of orbits 541568
- §13.3. Orbits corresponding to almost diagonal matrices 544571
- §13.4. A quantumization for quantum Schur algebras 546573
- §13.5. The fundamental multiplication formulas 550577
- §13.6. Some partial orderings on Ξ(n) and Ξ(n) 558585
- §13.7. The BLM triangular relations 560587
- §13.8. Extending the fundamental multiplication formulas 567594
- §13.9. Generators and relations 572599
- §13.10. Presentations for quantum Schur algebras 577604
- Exercises and notes 587614

- Chapter 14. Constructing quantum gl[sub(n)] via quantum Schur algebras 591618
- §14.1. A stabilization property 592619
- §14.2. The BLM algebra K and its canonical basis 595622
- §14.3. The completion K of K and multiplication formulas 598625
- §14.4. Embedding U[sub(v)](gl[sub(n)]) into K 602629
- §14.5. Z-forms of U[sub(v)](gl[sub(n)]) 606633
- §14.6. Integral quantum Schur–Weyl reciprocity 609636
- §14.7. A connection with Ringel–Hall algebras 614641
- Exercises and notes 617644

- Appendices 621648
- Appendix A. Varieties and affine algebraic groups 623650
- §A.1. Affine varieties 624651
- §A.2. Varieties 630657
- §A.3. Affine algebraic groups 633660
- §A.4. Parabolic subgroups and the Chevalley–Bruhat ordering 643670
- §A.5. Representation theory: a first view 645672
- §A.6. Representations in positive characteristic; Frobenius morphisms 649676
- §A.7. Induced representations and the Weyl character formula 654681
- §A.8. Higher Ext functors; Δ- and ∇-filtrations 658685
- Exercises and notes 660687

- Appendix B. Quantum linear groups through coordinate algebras 669696
- Appendix C. Quasi-hereditary and cellular algebras 699726
- §C.1. Heredity ideals 700727
- §C.2. Quasi-hereditary algebras and highest weight categories 704731
- §C.3. Regular rings of Krull dimension at most 2 709736
- §C.4. Integral quasi-hereditary algebras 715742
- §C.5. Algebras with a Specht datum 719746
- §C.6. Cellular algebras 720747
- Exercises and notes 726753

- Bibliography 733760
- Index of notation 749776 free
- Index of terminology 755782

#### Readership

Graduate students and research mathematicians interested in quantum groups and finite-dimensional algebras.

#### Reviews

...prove[s] to be a valuable reference to researchers working in the field. It contains and collects many results which have not appeared before in book form.

-- Mathematical Reviews