控制分次李代数与双李代数的上同调
Controlling graded Lie algebras and cohomologies of double Lie algebras
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中文总结 AI 辅助
本文构造控制双李代数结构的分次李代数，并证明其扭转后与Fairon-Valeri复形同构，进而将双李代数上同调等同于矩阵代数上斜对称Rota-Baxter算子派生结合代数的循环上同调。
中文摘要 AI 辅助
我们在循环斜对称上链空间上构造了一个分次李代数结构，其Maurer-Cartan元素刻画了双李代数结构。用固定的双李括号进行扭转后，得到一个上链复形，该复形在重新索引和逐度符号调整后同构于Fairon和Valeri引入的复形的正arity部分。利用双李properad的Koszul消解，我们将所得微分分次李代数等同于固定双李代数形变复形上的卷积微分分次李代数。我们还构造了一个控制有限维对称Frobenius代数上斜对称Rota-Baxter算子的分次李代数。对于有限维向量空间$V$和$A=\en(V)$，我们将$V$上双李代数结构与$A$上斜对称Rota-Baxter算子之间的已知对应提升为其控制分次李代数之间的同构。在由相应Maurer-Cartan元素扭转后，该同构变为微分分次李代数之间的同构，从而诱导相关上同调群之间的同构。我们进一步证明了对称Frobenius代数上斜对称Rota-Baxter算子的上同调可视为其派生结合代数的循环上同调。因此，双李代数的上同调被等同于与矩阵代数上斜对称Rota-Baxter算子相关的派生结合代数的循环上同调。
英文摘要
We construct a graded Lie algebra structure on the space of cyclically skew-symmetric cochains whose Maurer-Cartan elements characterize double Lie algebra structures. Twisting by a fixed double Lie bracket yields a cochain complex that is isomorphic to the positive arity part of the complex introduced by Fairon and Valeri after reindexing and degreewise sign adjustment. Using the Koszul resolution of the double Lie properad, we identify the resulting differential graded Lie algebra with the convolution differential graded Lie algebra on the deformation complex of the fixed double Lie algebra. We also construct a graded Lie algebra governing skew-symmetric Rota-Baxter operators on a finite-dimensional symmetric Frobenius algebra. For a finite-dimensional vector space $V$ and $A=\en(V)$, we lift the known correspondence between double Lie algebra structures on $V$ and skew-symmetric Rota-Baxter operators on $A$ to an isomorphism of their governing graded Lie algebras. After twisting by corresponding Maurer-Cartan elements, this isomorphism becomes an isomorphism of differential graded Lie algebras and hence induces an isomorphism between the associated cohomology groups. We further prove that the cohomology of a skew-symmetric Rota-Baxter operator on a symmetric Frobenius algebra can be seen as the cyclic cohomology of its descendent associative algebra. Thus the cohomology of a double Lie algebra is identified with the cyclic cohomology of the descendent associative algebra associated with the skew-symmetric Rota-Baxter operator on the matrix algebra.
发表机构
- Chern Institute of Mathematics & LPMC, Nankai University（南开大学陈省身数学研究所及LPMC）
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