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arXiv · 1903.07380

On solvability of the first Hochschild cohomology of a finite-dimensional algebra

Florian Eisele·Theo Raedschelders
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Abstract

For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.

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Florian Eisele, Theo Raedschelders. 2019-04-26. On solvability of the first Hochschild cohomology of a finite-dimensional algebra. https://arxiv.org/abs/1903.07380

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