The derived non-commutative Poisson bracket on Koszul Calabi-Yau algebras
The derived non-commutative Poisson bracket on Koszul Calabi-Yau algebras
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DOI：
10.4171/jncg/11-1-4
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发表时间：
2015-04
期刊：
影响因子：
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通讯作者：
Xiaojun Chen;A. Eshmatov;F. Eshmatov;Song Yang
中科院分区：
文献类型：
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作者：
Xiaojun Chen;A. Eshmatov;F. Eshmatov;Song Yang
设$A$是一个Koszul（或更一般地，$N$-Koszul）Calabi-Yau代数。受Kontsevich，Ginzburg和货车den Bergh工作的启发，我们证明了$A$上存在一个导出的非交换Poisson结构，它导出了$A$上循环同调的分次李代数结构;此外，我们证明了$A$的Hochschild同调是循环同调上的李模，Connes长正合列实际上是李模序列.最后，我们表明，Leibniz-Loday括号与派生的非交换泊松结构上$A$自然映射到Gerstenhaber括号的Hochschild上同调的Koszul对偶代数，因此对$A$本身。文中还讨论了它与文献中其它括号的关系，并给出了具体的例子。
Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie algebra structure on the cyclic homology of $A$; moreover, we show that the Hochschild homology of $A$ is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz-Loday bracket associated to the derived non-commutative Poisson structure on $A$ is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of $A$ itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.