On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras (Q2182346)

Gentle algebras are special biserial algebras and one of the most important classes of tame algebras. Algebras of tame representation type are of great interest as they have infinitely many isomorphism classes of indecomposable modules, yet they do usually exhibit discernible patterns in their representation theory. In this paper the authors have determined the first Hochschild homology and cohomology with different coefficients for gentle algebras and have given a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra. An explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras with trivial multiplicities (multiplicity one) based on trivial extensions of gentle algebras was presented and it was shown how the Hochschild cohomology is encoded in the Brauer graph. In particular, it was shown that except in one low-dimensional case, the resulting Lie algebras are all solvable. A nice example of a Brauer graph algebra which can be obtained both as a trivial extension of a gentle algebra with finite global dimension and also as a trivial extension of a gentle algebra with infinite global dimension was exhibited, and in this way also it was shown that the global dimension is an invariant which in fact plays no role in the Lie algebra structure of the first Hochschild cohomology space of the trivial extension.