# Algebra Seminar: On the first (relative) Hochschild cohomology and the (relative) fundamental groups

- Date: 6 February 2024, 15:15–17:00
- Location: Ångström Laboratory, Å64119 + Zoom
- Type: Seminar
- Lecturer: Lleonard Rubio y Degrassi (Uppsala)
- Organiser: Matematiska institutionen
- Contact person: Volodymyr Mazorchuk

Lleonard Rubio y Degrassi from Uppsala University holds a seminar with the title "On the first (relative) Hochschild cohomology and the (relative) fundamental groups". Welcome to join!

Welcome to join on site or via Zoom link (Meeting ID: 645 5572 6999, please contact the organizer for passcode)

**Abstract: ** Hochschild cohomology is a fascinating invariant of an associative algebra which possesses a rich structure. In particular, the first Hochschild cohomology group HH^1(A) of an algebra A is a Lie algebra, which is a derived invariant. In the half part of talk, I will explain how maximal tori in HH^1(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 HH^1(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 assume that A is a monomial algebra. For some families of subalgebras of A, I will introduce the notion of relative fundamental group and I will show how these relative fundamental groups are related with the relative Hochschild cohomology of A. This talk is based on joint works with Benjamin Briggs and Jonathan Lindell.

This is a seminar in our algebra seminar series.