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Abstract <p> 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 also describe all double Poisson brackets on the algebra of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2\times 2\)</EquationSource> </InlineEquation> upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(3\)</EquationSource> </InlineEquation>. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra. </p>

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Double Poisson Brackets on Low-Dimensional Algebras

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • G. I. Sharygin,
  • A. Hernández Rodríguez

摘要

Abstract

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 also describe all double Poisson brackets on the algebra of \(2\times 2\) upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank \(2\) and \(3\) . In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.