This talk is in two parts, both focusing on the structure of the first Hochschild cohomology group.
If two self-injective finite dimensional algebras are connected by a stable equivalence of Morita type then their Hochschild Cohomology algebras are isomorphic in positive degrees. Over a field of positive characteristic Hochschild cohomology possesses the structure of a restricted graded Lie algebra, and Linckelmann asked whether this too passes across a stable equivalence of Morita type. In the first part I will discuss some work with Rubio y Degrassi where we answer this question and give a few applications.
In the second part I will talk about the fundamental group(s) of a finite dimensional algebra, and how it (they!) can be seen inside the first Hochschild cohomology group. This part is ongoing joint work with Rubio y Degrassi and Saorín.