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

Andrea Solotar,Cristian Chaparro,Sibylle Schroll

doi:10.1016/j.jalgebra.2020.02.003

Andrea Solotar, Cristian Chaparro + Show 1 more

Open Access

https://doi.org/10.1016/j.jalgebra.2020.02.003

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Abstract

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 [32]. 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.

Highlights

Gentle algebras have gained much traction in recent years due to their pivotal role in several different areas of mathematics. They appear in the guise of Jacobian algebras of quivers with potential from unpunctured marked surfaces in cluster theory [16, 4, 9, 1, 2], they play a role in certain gauge model theories in theoretical physics in the work of Cecotti, where they appear as Jacobian algebras of quivers and superpotentials corresponding to a Gaiotto A1-theory with only irregular punctures and at least one such puncture [5] and they are an important element in homological mirror symmetry of 2-manifolds, in that the derived category of a differential graded smooth gentle algebra is equivalent to the partially wrapped Fukaya category of a surface with stops [13, 19]
In this article we will give a description of the Lie algebra structure of the first Hochschild cohomology space of gentle algebras and any Brauer graph algebra with multiplicity 1, using the fact that the latter algebras are trivial extensions of gentle algebras
In this article we give a description of these spaces and provide bases for the four summands that are adapted to make a link with the ribbon graph of the gentle algebra

Summary

Introduction

Gentle algebras have gained much traction in recent years due to their pivotal role in several different areas of mathematics. Hochschild cohomology, Gerstenhaber brackets, Brauer graph algebras, trivial extensions, Lie algebras. In this article we will give a description of the Lie algebra structure of the first Hochschild cohomology space of gentle algebras and any Brauer graph algebra with multiplicity 1, using the fact that the latter algebras are trivial extensions of gentle algebras. It is already known that this happens for other tame algebras, such as the special biserial algebras considered in [20] and the toupie algebras with no branches of length one –that is, exactly the tame ones–, see [3] In view of these results, in an earlier version of the paper we posed the following question: Question: Is it true that, except in some low dimensional cases, the first Hochschild cohomology space of any finite dimensional algebra of tame representation type is a solvable Lie algebra?. We will assume that all algebras are indecomposable and finite dimensional

Background

Nilpotent and solvable Lie algebras