On deformations of associative algebras

Abstract

In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra a are controlled by the second Hochschild cohomology group of a. More generally, any n-parameter first order deformation of a gives, due to commutativity of the cup-product on Hochschild cohomology, a graded algebra morphism Sym•(kn) → Exta a-bimod2•(a, a). We prove that any extension of the n-parameter first order deformation of a to an infinite order formal deformation provides a canonical 'lift' of the graded algebra morphism above to a dg-algebra morphism Sym•(kn) → RHom•(a,a), where the symmetric algebra Sym •(kn) is viewed as a dg-algebra (generated by the vector space kn placed in degree 2) equipped with zero differential.