$A(\infty)$ -代数的结构与Hochschild和Harrison上同调
Structure of $A(\infty)$-algebra and Hochschild and Harrison cohomology
摘要：
Stasheff的$A(\infty)$ -代数$(M,\{m_i:\otimes^iM\to M, i=1,2,3,...\})$实际上是一个DG-代数$(M,m_1,m_2)$，不一定有结合积$m_2$，但这种非结合性是由更高的同伦$m_{i>2}$来度量的。然而，这样的结构出现在严格结合的情况下，也就是说，在同源代数$H(C)$的DG-代数$C$与自由$H_i(C)$ -s，特别是在上同调代数$H^*(X,Λ)$的拓扑空间$X$。显然，$A(\infty)$ -代数$(H^*(X,Λ),\{m_i\})$比上同调代数$H^*(B,Λ)$携带更多的信息。自然会出现一个问题时，这种结构是退化的，也就是说，当一个$A(\infty)$ -代数$(M, \{m_i\})$同构于一个更高的操作$m_i, i\geq 3$平凡？在本文中，我们介绍了这种退化的障碍。也就是说，操作$\{m_i\}$我们解释为Hochschild扭上链$m=m_3+m_4+..., m_i\in C^n(M,M)$满足$δm=m\smile_1m$其中$\smile_1$是$C^*(M,M)$的Gerstenhabers产品。使用广义产品$f\smile_1(g_1,...,g_k)$，我们定义扰动的Hochschild扭曲上链（即$A(\infty)$结构），并特别证明，如果一个分次代数$(M,μ)$所有的Hochschild上同调$Hoch^{n,2-n}(M,M)=0$的$n\geq3$，然后任何$A(\infty)$ -代数结构$\{m_i\}$上的$M$与$m_1=0, m_2=μ$，是退化的。
英文摘要：
Stasheff's $A(\infty)$-algebra $(M,\{m_i:\otimes^iM\to M, i=1,2,3,...\})$ in fact is a DG-algebra $(M,m_1,m_2)$ with not necessarily associative product $m_2$ but this nonassociativity is measured by higher homotopies $m_{i>2}$. Nevertheless such structure arises in the strictly associative situation too, namely in the homology algebra $H(C)$ of a DG-algebra $C$ with free $H_i(C)$-s, particularly in the cohomology algebra $H^*(X,Λ)$ of a topological space $X$. It is clear that the $A(\infty)$-algebra $(H^*(X,Λ),\{m_i\})$ carries more information than the cohomology algebra $H^*(B,Λ)$. Naturally arises a question when this structure is degenerate, that is when an $A(\infty)$-algebra $(M, \{m_i\})$ is isomorphic to one with higher operations $m_i, i\geq 3$ trivial? In this paper we introduce the obstructions for such degeneracy. Namely, operations $\{m_i\}$ we interpret as Hochschild twisting cochain $m=m_3+m_4+..., m_i\in C^n(M,M)$ satisfying $δm=m\smile_1m$ where $\smile_1$ is Gerstenhabers product in $C^*(M,M)$. Using the generalized product $f\smile_1(g_1,...,g_k)$ we define perturbations of Hochschild twisting cochains (i.e. of $A(\infty)$ structures) and in particular prove that if for a graded algebra $(M,μ)$ all Hochschild cohomologies $Hoch^{n,2-n}(M,M)=0$ for $n\geq3$ then any $A(\infty)$-algebra structure $\{m_i\}$ on $M$ with $m_1=0, m_2=μ$, is degenerate.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
