Crystalline bases of quantum groups. Edited by Charles Cochet. (Bases cristallines des groupes quantiques.) (French) Zbl 1066.17007
Cours Spécialisés (Paris) 9. Paris: Société Mathématique de France (ISBN 2-85629-126-0/pbk). viii, 115 p. (2002).
The author studies the crystal bases of quantum groups and their applications for representation theory.
The enveloping algebras were introduced by Drinfeld and Jimbo in 1985. The notion of crystal bases was inaugurated later (in 1990) by the author. In an independent way, Lusztig defined at the same time the notion of canonical bases arising from quantized enveloping algebras. These notions together with the \(q\)-analogue quantum enveloping algebras have been extensively studied in the literature.
In this course, the author describes the crystal bases. After studying in detail the \(sl_2\) case, she introduces crystal bases for \(U_q(sl_2)\)-modules. Then, the theory is generalized for the quantum enveloping algebra \(U_q(g)\) for any symmetrizable Kac-Moody algebra.
The consequences for representation theory are then investigated. In particular, the author shows how the calculus of highest weight modules and the computation of coefficients for the tensor product of two representations can be reduced to combinatorial calculus using crystal bases.
The enveloping algebras were introduced by Drinfeld and Jimbo in 1985. The notion of crystal bases was inaugurated later (in 1990) by the author. In an independent way, Lusztig defined at the same time the notion of canonical bases arising from quantized enveloping algebras. These notions together with the \(q\)-analogue quantum enveloping algebras have been extensively studied in the literature.
In this course, the author describes the crystal bases. After studying in detail the \(sl_2\) case, she introduces crystal bases for \(U_q(sl_2)\)-modules. Then, the theory is generalized for the quantum enveloping algebra \(U_q(g)\) for any symmetrizable Kac-Moody algebra.
The consequences for representation theory are then investigated. In particular, the author shows how the calculus of highest weight modules and the computation of coefficients for the tensor product of two representations can be reduced to combinatorial calculus using crystal bases.
Reviewer: Angela Gammella (Creil)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
