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Finite dimensional algebras and quantum groups. (English) Zbl 1154.17003

Mathematical Surveys and Monographs 150. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4186-0/hbk). xxv, 759 p. (2008).
The book presents an introduction to the structure and realization of quantum enveloping algebras defined by their Drinfeld-Jimbo presentation. The birth of quantum groups in the 1980s created a good framework for applying and extending the methods and ideas that connect representation theory of finite dimensional algebras to representations of quivers and the root space decomposition of some associated Lie algebras. A fundamental object of the theory is the Ringel-Hall algebra, introduced by C. M. Ringel [Invent. Math. 101, No. 3, 583–592 (1990; Zbl 0735.16009)], associated with the representation category of a finite dimensional algebra over the finite field \(\mathbb F_q\). Ringel proved that in the finite type case, the structure constants of the Ringel-Hall algebra are polynomials in \(q\), and the resulting generic Ringel-Hall algebra is isomorphic to the positive part of the corresponding quantum enveloping algebra.
A geometric approach was initiated by G. Lusztig [J. Am. Math. Soc. 3, No. 2, 447–498 (1990; Zbl 0703.17008)], who obtained a geometric realization of the \(\pm\)-parts of quantum enveloping algebras associated with symmetrizable Cartan matrices, canonical bases for these algebras and also their representations. J. A. Green [Invent. Math. 120, No. 2, 361–377 (1995; Zbl 0836.16021)] defined a comultiplication for Ringel-Hall algebras of hereditary algebras and extended Ringel’s algebraic realization to arbitrary types. A different approach to quantum enveloping algebras was initiated by A. A. Beilinson, G. Lusztig and R. MacPherson [Duke Math. J. 61, No. 2, 655–677 (1990; Zbl 0713.17012)], who constructed a realization for the entire quantum \(gl_n\) via quantum Schur algebras. An explicit basis for the entire quantum enveloping algebra and multiplication formulas for any basis element by a generator was thus obtained.
The book presents all these approaches in detail. Two other related topics are discussed: the connection between representations of quivers and representations of species, and the Kazhdan-Lusztig theory for Hecke algebras and cells. The book consists of an introductory Chapter 0, followed by 14 chapters: 1. Representations of quivers; 2. Algebras with Frobenius morphisms; 3. Quivers with automorphisms; 4. Coxeter groups and Hecke algebras; 5. Hopf algebras and universal enveloping algebras; 6. Quantum enveloping algebras; 7. Kazhdan-Lusztig combinatorics for Hecke algebras; 8. Cells and representations of symmetric groups; 9. The integral theory of quantum Schur algebras; 10. Ringel-Hall algebras; 11. Bases of quantum enveloping algebras of finite type; 12. Green’s theorem; 13. Serre relations in quantum Schur algebras; 14. Constructing quantum \(gl_n\) via quantum Schur algebras. There are also three appendices: A. Varieties and affine algebraic groups; B. Quantum linear groups through coordinate algebras; C. Quasi-hereditary and cellular algebras.
The chapters are arranged in five parts. Part 1 (Chapters 1–3) presents the theory of finite dimensional algebras and the connection to representations of quivers with automorphisms. Part 2 (Chapters 4–6) introduces algebras associated with Cartan matrices, which are presented by generators and relations. Some of these are showed to have Hopf algebra structures. Part 3 (Chapters 7–9) presents an approach to representation theory of symmetric groups by viewing the symmetric group algebra as a specialization of the Hecke algebra associated with a symmetric group. Part 4 (Chapters 10–12) presents the approach to quantum enveloping algebras by using Ringel-Hall algebras. Part 5 (Chapters 13–14) presents the Beilinson-Lusztig-MacPherson approach to quantum enveloping algebras by using quantum Schur algebras.
With few exceptions, the book is self-contained. It is accessible to graduate students and mathematicians that are not experts in the field. There are some useful exercises and interesting notes at the end of each chapter.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16T05 Hopf algebras and their applications
17B20 Simple, semisimple, reductive (super)algebras
16G20 Representations of quivers and partially ordered sets
20C30 Representations of finite symmetric groups
20C08 Hecke algebras and their representations
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