Quantum groups and quantization of Weyl group symmetries of Painlevé systems

Details Quantum groups and quantization of Weyl group symmetries of Painlevé systems Quantum groups and quantization of Weyl group symmetries of Painlevé systems

2008-08-19

arxiv.org/abs/0808.2604v4

Advanced Studies in Pure Mathematics 61, 2011, pp. 289-325

Mathematics

Quantum Algebra

30 pages, LaTeX; v2: minor corrections; v3: minor corrections; v4: minor corrections (\otimes problem fixed)

Scientific paper

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 B\"acklund transformations for Painlev\'e equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. 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 B\"acklund transformations of q-Painlev\'e 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.

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