Cup product on $A_\infty$-cohomology and deformations

Alexey A Sharapov,Evgeny D Skvortsov

doi:10.4171/jncg/411

Alexey A Sharapov, Evgeny D Skvortsov

Open Access

https://doi.org/10.4171/jncg/411

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Abstract

We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal $A_\infty$-algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on the $A_\infty$-cohomology, which we define in terms of the brace operations. As an example, we construct a minimal $A_\infty$-algebra from the Weyl-Moyal $\ast$-product algebra of polynomial functions.

Highlights

The concept of homotopy associative algebras, which first appeared in the context of algebraic topology [21], has evolved into a mature algebraic theory with numerous applications in theoretical and mathematical physics [13, 22]
We propose a simple formula for the deformation of families of differential graded algebras (DGA)’s in the category of minimal A∞-algebras
The basis for our construction is provided by a cup product on A∞-cohomology

Summary

Introduction

The concept of homotopy associative algebras (or A∞-algebras), which first appeared in the context of algebraic topology [21], has evolved into a mature algebraic theory with numerous applications in theoretical and mathematical physics [13, 22]. Given a one-parameter family A = An of DGA’s with differential ∂ : An → An−1, one can define a minimal A∞-algebra deforming the associative product in A in the direction of an (inhomogeneous) Hochschild cocycle ∆ given by any linear combination of One can define the family of graded algebras A = A0 A1, where A0 = A, A1 = M, and the product is given by

Results

Conclusion