2381/11611356.v1 Cristian Chaparro Cristian Chaparro Sibylle Schroll Sibylle Schroll Andrea Solotar Andrea Solotar On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras University of Leicester 2020 math.RT Hochschild cohomology Gerstenhaber brackets Brauer graph algebras Trivial extensions Lie algebras 2020-04-01 07:34:42 Journal contribution https://figshare.le.ac.uk/articles/journal_contribution/On_the_Lie_algebra_structure_of_the_first_Hochschild_cohomology_of_gentle_algebras_and_Brauer_graph_algebras/11611356 In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in earlier work by the second author. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-dimensional case, the resulting Lie algebras are all solvable.