Семинар Международной лаборатории теории представлений и математической физики ВШЭ-Сколтех: Абнер Эрнандес Родригес

Abstract:  We will discuss the structure of double Poisson brackets in the sense of M. Van den Bergh on finite-dimensional algebras. In particular (using Hochschild homology and some results) we prove that all possible double Poisson brackets on matrix algebras are given by inner derivations of appropriate type. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebra are also given by inner derivations. We further give a description of all double Poisson brackets on a small non-semisimple algebra (namely, the algebra of 2×2 upper triangular matrices); in addition, we discuss Poisson structures induced from the double Poisson brackets in its representation spaces of dimensions two and three.

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