In this talk I will explain how to determine the first Hochschild homology and cohomology spaces
of a gentle algebra in terms of the associated quiver and the relations. This leads to an explicit
description of the Lie algebra structure of the first Hochschild cohomology space of special biserial
algebras (and hence of Brauer graph algebras with multiplicity one in each vertex) based on the
Hochschild cohomology of trivial extensions of gentle algebras.
This talk contains joint work with Sibylle Schroll and Cristian Chaparro Acosta.