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Finite Dimensional Algebras and Quantum Groups Mathematical Surveys and Monographs Volume 150 http://dx.doi.org/10.1090/surv/150 Finite Dimensional Algebras and Quantum Groups Mathematical Surveys and Monographs Volume 150 Finite Dimensional Algebras and Quantum Groups Bangming Deng JieDu Brian Parshall Jianpan Wang AHEM^ American Mathematical Society Providence, Rhode Island Editorial Committee Jerry L. Bona Michael G. Eastwood Ralph L. Cohen J. T. Stafford, Chair Benjamin Sudakov 2000 Mathematics Subject Classification. Primary 05E10, 16G20, 16W35, 17B37, 20C08; Secondary 16G70, 20C30, 20G05, 20G42. For additional information and updates on this book, visit www.ams.org/bookpages/surv-150 Library of Congress Cataloging-in-Publication Data Finite dimensional algebras and quantum groups / Bangming Deng... [et al.]. p. cm. — (Mathematical surveys and monographs ; v. 150) Includes bibliographical references and indexes. ISBN 978-0-8218-4186-0 (alk. paper) 1. Finite fields (Algebra) 2. Quantum groups. I. Deng, Bangming. QA247.3 .F52 2008 512/.3—dc22 2008028975 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08 for att their heCp, encouragement, and infinite patience, zue dedicate this boof^to our unves andchitdren: (Weniian Quo andZhuoran <Deng Chunti Ju and Andy <Du 'Karen Varshatl HuiaingXu, Xin 'Wang, andjun 'Wang 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 vn Vlll Contents Chapter 2. Algebras with Probenius morphisms 83 §2.1. Fg-structures on vector spaces 84 §2.2. Algebras with Probenius 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 s^ 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 U^s^-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. ^-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. ^-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 &r 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(&r)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. RingePs 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+^fe) 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 S(n) and H(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 K of K and multiplication formulas 598 §14.4. Embedding U„(gU into K 602 §14.5. Z-forms of Vv(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.l. 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 Xll Contents §A.6. Representations in positive characteristic; Frobenius morphisms 649 §A.7. Induced representations and the Weyl character formula 654 §A.8. Higher Ext functors; A- and V-filtrations 658 Exercises and notes 660 Appendix B. Quantum linear groups through coordinate algebras 669 §B.l. 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.l. Heredity ideals 700 §C2.
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