论文标题
范德·伯格（Van den Bergh
Around Van den Bergh's double brackets for different bimodule structures
论文作者
论文摘要
从M. van den Bergh的意义上讲，双泊松支架是对关联代数$ a $的操作，该操作以明确的方式诱导了每个表示空间上的泊松支架$ \ operatoratorname {rep}（ape}（a，n）$。在本说明中，我们研究了更改双括号基础的莱布尼兹规则的影响。此更改是在$ a \ otimes a $上的$ a-bimodule结构的合适选择。在最重要的情况下，我们描述了$ a-bimodule结构的选择如何固定与雅各比身份的类似物，并在表示空间上获得诱导的泊松支架。目前的理论还编码用于编写数学物理学矩阵的泊松托符的广泛张量表示法的形式化。
A double Poisson bracket, in the sense of M. Van den Bergh, is an operation on an associative algebra $A$ which induces a Poisson bracket on each representation space $\operatorname{Rep}(A,n)$ in an explicit way. In this note, we study the impact of changing the Leibniz rules underlying a double bracket. This change amounts to make a suitable choice of $A$-bimodule structure on $A\otimes A$. In the most important cases, we describe how the choice of $A$-bimodule structure fixes an analogue to Jacobi identity, and we obtain induced Poisson brackets on representation spaces. The present theory also encodes a formalisation of the widespread tensor notation used to write Poisson brackets of matrices in mathematical physics.
