On the Lie algebra structure of the first Hochschild cohomology of\n gentle algebras and Brauer graph algebras

2018/11/06 by Cristian Chaparro, Sibylle Schroll, Chaparro, Cristian +3 · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1811.02211

openalex publication_date 2018/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we determine the first Hochschild homology and cohomology with\ndifferent coefficients for gentle algebras and we give a geometrical\ninterpretation of these (co)homologies using the ribbon graph of a gentle\nalgebra as defined in earlier work by the second author. We give an explicit\ndescription of the Lie algebra structure of the first Hochschild cohomology of\ngentle and Brauer graph algebras (with multiplicity one) based on trivial\nextensions of gentle algebras and we show how the Hochschild cohomology is\nencoded in the Brauer graph. In particular, we show that except in one\nlow-dimensional case, the resulting Lie algebras are all solvable.\n

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