Double Poisson brackets on low dimensional algebras

Авторы: Шарыгин Г.И., Rodriguez A.Hernandez
Журнал: Труды Математического института им.В.А.Стеклова РАН
Том: 330
Год издания: 2025
Издательство: МИАН
Местоположение издательства: Москва
Аннотация: In this paper, we describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are “inner”, i.e. given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebras over algebraically closed fields are also given by inner derivations. We further give a description of all double Poisson brackets on the algebra of 2 ˆ 2 upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank two and three. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.
Добавил в систему: Шарыгин Георгий Игорьевич

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