Quantum groups and quantization of Weyl group symmetries of Painlevé systems. (English) Zbl 1247.81213

Hasegawa, Koji (ed.) et al., Exploring new structures and natural constructions in mathematical physics. Collected papers of the conference upon the occasion of the retirement of Professor Akihiro Tsuchiya, Nagoya, Japan, March 5–8, 2007. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-64-8/hbk). Advanced Studies in Pure Mathematics 61, 289-325 (2011).
The author constructs quantized \(q\)-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras proposed by M. Noumi and Y. Yamada [in: Physics and combinatorics. Proceedings of the international workshop, Nagoya, Japan, August 23–27, 1999. Singapore: World Scientific. 287–319 (2001; Zbl 0991.37047)] as conceptual generalizations of the Bäcklund transformations for Painlevé equations.
The author considers a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra for any symmetrizable generalized Cartan matrix. Then he shows that non-integral powers of the image of the Chevalley generators generate the quantized \(q\)-analogue of the birational Weyl group action. In the last part of the paper, using the same method, the author reconstructs the quantized birational Weyl group actions obtained in [K. Hasegawa, Advanced Studies in Pure Mathematics 61, 275–288 (2011; Zbl 1241.81114)]. The author also proves that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.
For the entire collection see [Zbl 1231.81003].


81R12 Groups and algebras in quantum theory and relations with integrable systems
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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