Let $(A,\mu_0)$ be an associative unital $k$-algebra over a field $k$. A formal deformation of $(A,\mu_0)$ is sometimes defined as a $k[[t]]$-bilinear map $\mu: A[[t]]\times A[[t]]\rightarrow A[[t]]$ such that on $A\subset A[[t]]$ the map $\mu$ is of the form
$$\mu(a,b)=\mu_0(a,b)+\sum_{i\geq 1}\mu_i(a,b)t^i,$$ where for $i\geq 1$, the $\mu_i:A\times A\rightarrow A$ are $k$-bilinear maps.
So far and at first sight, this definition of a deformation of an algebra seems natural to me. We have different multiplications $\mu_i$ on the same underlying vector space $A$, and record these multiplications as a power series. At second sight, I would have expected that one requires that all $\mu_i$'s are associative and unital.
1.Question
- Why is this usually not done?
Furthermore, I see that this definition is too broad: So far, we just have different multiplications on the same underlying vector space, but we want to somehow capture that they are near to each other, i.e. deformations of $\mu_0$.
It seems that in the literarure on the algebraic deformation theory of associative algebras, this notion of "nearness" is implemented by requiring that the map $\mu$ itself is associative. I am wondering:
2. Question
- Why is this condition natural? Does it implement a notion of the $\mu_i$ being close to each other?