Lleonard Rubio y Degrassi "On the first (relative) Hochschild cohomology and (relative) fundamental groups"

Hochschild cohomology is a fascinating invariant of an associative algebra which possesses a rich structure. In particular, the first Hochschild cohomology group HH¹(A) of an algebra A is a Lie algebra, which is a derived invariant.

In the first half of the talk, I will explain how maximal tori in HH¹(A), together with fundamental groups associated to presentations of A, can be used to deduce information about the shape of the Gabriel quiver of A. In particular, I will show that every maximal torus in HH¹(A) arises as the dual of some fundamental group of A. By combining this, with known invariance results for Hochschild cohomology, I will deduce that the largest rank of a fundamental group of A is a derived invariant.

In the second half, I will introduce a relative version of the fundamental group of a bound quiver. Similarly to the classical case, I will show how these groups are related with relative Hochschild cohomology.

This talk is based on joint works with Benjamin Briggs and Jonathan Lindell.