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VOL. 61 | 2011 Quantum groups and quantization of Weyl group symmetries of Painlevé systems
Gen Kuroki

Editor(s) Koji Hasegawa, Takahiro Hayashi, Shinobu Hosono, Yasuhiko Yamada

## Abstract

We shall construct the quantized $q$-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra for any symmetrizable generalized Cartan matrix. Then non-integral powers of the image of the Chevalley generators generate the quantized $q$-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Bäcklund transformations of $q$-Painlevé equations constructed by Hasegawa. We shall also prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.

## Information

Published: 1 January 2011
First available in Project Euclid: 24 November 2018

zbMATH: 1247.81213
MathSciNet: MR2867150

Digital Object Identifier: 10.2969/aspm/06110289