Deformation of algebras

Summary

A deformation of an associative algebra $A$ is a family of associative products on $A$ depending on a parameter $t$. At $t = 0$ it gives back the original product. In the formal theory of Murray Gerstenhaber, the parameter is a formal variable and the products are power series in $t$. The first-order terms are classified by the second Hochschild cohomology of $A$, and the obstructions to extending them lie in the third. Deformations of algebras are the algebraic core of deformation quantization.