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Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups. (English) Zbl 0951.33001

Advanced Series in Mathematical Physics. 21. Singapore: World Scientific. ix, 371 p. (1995).
The aim of the book is to give different connections of hypergeometric functions of many variables with Kac-Moody algebras, their quantum deformations and their representations. The hypergeometric functions under consideration are given as integrals and are of the form \[ F(t_1,\cdots ,t_n; P; \{ a_{ij}\} ,k)= \int _{\gamma} \prod _{1\leq i<j\leq N} (t_i-t_j)^{a_{ij}/k} P(t_1,\cdots ,t_N) dt_{n+1}\wedge \cdots \wedge dt_N, \] where \(a_{ij}\) and \(k\) are complex parameters, \(P\) belongs to a suitable space of rational functions, and \(\gamma\) belongs to a suitable space of cycles. The differential-geometrical side of the function \(\prod (t_i-t_j)^{a_{ij}/k}\) is connected with the theory of Kac-Moody algebras. Complexes of multivalued differential forms associated with this function have a natural description in terms of Kac-Moody algebras. The differential equation for associated hypergeometric functions has a description in terms of Kac-Moody algebras and this differential equation is the Knizhnik-Zamolodchikov differential equation very well known for physisists. The topological side of the function \(\prod (t_i-t_j)^{a_{ij}/k}\) is connected with the theory of quantum groups. Univalued branches of this function define a local system of coefficients over the complement to the union of diagonal hyperplanes. Complexes of chains with coefficients in this local system have a natural description in terms of quantum groups. It is well-known that the representation theory of Kac-Moody algebras and their quantum deformations are similar and have striking differences at the same time. The hypergeometric functions provide a new approach to this problem. Namely, integration of multivalued differential forms over chains gives a correspondence between the corresponding objects of the representation theory of Kac-Moody algebras and the representation theory of quantum groups.
The book contains 15 chapters. The first chapters (chapters 1-5) are devoted to discussion of the correspondence between the representation theory of quantum groups and geometry of configurations of hyperplanes. In chapters 6-9, the author describes the correspondence between the universal \(R\)-matrix in the theory of quantum groups and the monodromy of configurations of hyperplanes depending on parameters. Chapters 10, 11 and the first part of chapter 12 are on the connections between the representation theory of Kac-Moody Lie algebras and the geometry of configurations of hyperplanes. The second part of chapter 12 and chapters 13 and 14 are devoted to discussion of the interrelations between the representation theory of Kac-Moody algebras and the representation theory of quantum groups coming from hypergeometric functions. In chapter 15 the author discuss how the constructions of the previous chapters could be applied to studying homology groups of configurations with coefficients more general than complex one-dimensional ones and to studying homology groups of braid groups. Chapters 2, 3, 6, 8 and the first part of chapter 13 are mainly geometric. Chapters 4, 5, 7, 9, 11, 15 and the first part of chapter 10 are algebraic. Chapter 12 and the second parts of chapters 10 and 13 are mainly analytic. The book is written clearly. But a reader has to be prepared for reading the book at least in size of the university courses.
Reviewer: A.Klimyk (Kyïv)

MSC:

33-02 Research exposition (monographs, survey articles) pertaining to special functions
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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