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Transactions of the American Mathematical Society

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


Authors: Florian Eisele and Theo Raedschelders
Journal: Trans. Amer. Math. Soc. 373 (2020), 7607-7638
MSC (2010): Primary 16E40, 16G10, 16G60
DOI: https://doi.org/10.1090/tran/8064
Published electronically: September 9, 2020
MathSciNet review: 4169669
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Abstract: For an arbitrary finite-dimensional algebra $ A$, we introduce a general approach to determining when its first Hochschild cohomology $ \mathrm {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 $ \mathrm {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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Additional Information

Florian Eisele
Affiliation: Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom
MR Author ID: 971499
Email: florian.eisele@city.ac.uk

Theo Raedschelders
Affiliation: Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Elsene, Belgium
MR Author ID: 1186588
Email: theo.raedschelders@vub.be

DOI: https://doi.org/10.1090/tran/8064
Keywords: Hochschild cohomology, finite-dimensional algebras, Lie algebras, representation type
Received by editor(s): April 26, 2019
Received by editor(s) in revised form: October 6, 2019
Published electronically: September 9, 2020
Additional Notes: The second author was supported by an EPSRC postdoctoral fellowship EP/R005214/1.
Article copyright: © Copyright 2020 American Mathematical Society