On deformations of associative algebras

Abstract

In a classic paper, Gerstenhaber showed that first order deformations of an associative $\mathbf{k}$-algebra $\mathsf{a}$ are controlled by the second Hochschild cohomology group of $\mathsf{a}$. More generally, any $n$-parameter first order deformation of $\mathsf{a}$ gives, due to commutativity of the cup-product on Hochschild cohomology, a graded algebra morphism $\mathrm{Sym}^\bullet(\mathbf{k}^n) \to \mathrm{Ext}^{2\bullet}_{\mathsf{a} – \mathrm{bimod}}(\mathsf{a},\mathsf{a})$. We prove that any extension of the $n$-parameter first order deformation of $\mathsf{a}$ to an infinite order formal deformation provides a canonical ‘lift’ of the graded algebra morphism above to a $dg$-algebra morphism $\mathrm{Sym}^\bullet(\mathbf{k}^n) \to \mathrm{RHom}^{\bullet}_{}(\mathsf{a},\mathsf{a})$, where the symmetric algebra $\mathrm{Sym}^\bullet(\mathbf{k}^n)$ is viewed as a $dg$-algebra (generated by the vector space $\mathbf{k}^n$ placed in degree 2) equipped with zero differential.