# Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer:

Let $(A,\mu)$ be a commutative associative algebra over a field $\mathbb{F}$. The first order deformations of $\mu$ are classified up to equivalence by the second Hochschild cohomology group $HH_{\mu}^{2}(A)$. I'm wondering if this can be generalized in the following sense:

Given a $k^{th}$ order deformation $\mu_{(k)} = \mu + \mu_{1} h + ... + \mu_{k} h^k$ of $\mu$, in some cases it is possible to extend it to a $(k+1)^{th}$ order deformation $\mu_{(k+1)} = \mu_{(k)} + \mu_{k+1}h^{k+1}$, where $\mu_{k+1}$ is a bilinear map $A \times A \to A$. In this case it is natural to ask what are all the possible extensions up to equivalence. It turns out that the set of all possible deformations consists of the affine space $\mu_{k+1} + \ker(\delta_{\mu}^2),$ where $\delta_{\mu}^2 : HC^{2}(A) \to HC^{3}(A)$ is the Hochschild coboundary operator. Furthermore, if two extensions correspond to cohomologous elements of $HC^2(A)$, it turns out that the extensions are equivalent (in the sense that the resulting deformed algebras are isomorphic). My question is whether the converse holds, which is to say do equivalent extensions correspond to cohomologous elements? (And hence $HH_{\mu}^2(A)$ would classify the $(k+1)^{th}$ order deformations extending a particular $k^{th}$ order deformation of $\mu$.)

Here is what I have tried so far (in the case of $k = 2$): consider two equivalent $2^{nd}$ order deformations of $\mu$:

$\mu_{(2)} = \mu + \mu_{1} h + \eta h^2, \ \ \mu_{(2)}' = \mu + \mu_{1} h + \eta' h^2,$ with isomorphism $\Phi = 1 + \phi h + \psi h^2$. Then from the requirement that $\Phi(\mu_{(2)}(F,G)) = \mu_{(2)}'(\Phi(F), \Phi(G))$ for $F, G \in A$ we get that

$\delta_{\mu}^{1}(\phi) = 0,$ and

$\eta - \eta' = \delta_{\mu}^{1}(\psi) + \delta_{\mu_{1}}^{1}(\phi) + \mu \circ \phi \otimes \phi.$

Note that I use the notation $\delta_{\mu_{1}}^{1}(\phi)(F,G) = \mu_{1}(F, \phi(G)) + \mu_{1}(\phi(F),G) - \phi(\mu_{1}(F,G)).$

I need to end up with a formula like $\eta - \eta' = \delta_{\mu}^1 ( \gamma)$ for some $\gamma \in Hom_{\mathbb{F}}(A,A)$. It doesn't seem like its going to work, but I haven't been able to think of a counter example, and I have been assured that the result should be true.

P.S. I'd like to know if the analogous statement holds for deformations of a Poisson algebra.

• You're really trying to reinvent the wheel here. This is all in Hochschild's papers "The deformation theory of rings and algebras I-IV" from the 60's. – Ben Webster Mar 26 '15 at 8:28
• Correction to the previous comment: Hochschild -> Gerstenhaber. – Pasha Zusmanovich Mar 31 '15 at 18:25