Structure of $A(\infty)$-algebra and Hochschild and Harrison cohomology
| dc.creator | Kadeishvili, Tornike | |
| dc.date | 2002-10-21 | |
| dc.date.accessioned | 2026-07-07T04:52:12Z | |
| dc.date.available | 2026-07-07T04:52:12Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/math/0210331 | |
| dc.identifier | http://arxiv.org/abs/math/0210331 | |
| dc.identifier | Proc. of A. Razmadze Math. Inst., 91 (1988), 20-27 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65382 | |
| dc.subject | Algebraic Topology | |
| dc.subject | 55S30, 55S35, 16E40 | |
| dc.title | Structure of $A(\infty)$-algebra and Hochschild and Harrison cohomology | |
| dc.type | text |