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We shall construct the quantized q-analogues of the birational Weyl groupactions arising from nilpotent Poisson algebras, which are conceptualgeneralizations, proposed by Noumi and Yamada, of the B\"acklundtransformations for Painlev\'e equations. Consider a quotient Ore domain of thelower nilpotent part of a quantized universal enveloping algebra of arbitrarysymmetrizable Kac-Moody type. Then non-integral powers of the image of theChevalley generators generate the quantized q-analogue of the birational Weylgroup action. Using the same method, we shall reconstruct the quantizedB\"acklund transformations of q-Painlev\'e equations constructed by Hasegawa.We shall also prove that any subquotient integral domain of a quantizeduniversal enveloping algebra of finite or affine type is an Ore domain.

Quantum groups and quantization of Weyl group symmetries of Painlev\'e systems