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On the Lie algebra structure of $ HH^1(A)$ of a finite-dimensional algebra $ A$


Authors: Markus Linckelmann and Lleonard Rubio y Degrassi
Journal: Proc. Amer. Math. Soc. 148 (2020), 1879-1890
MSC (2010): Primary 16E40, 16G30, 16D90, 17B50
DOI: https://doi.org/10.1090/proc/14875
Published electronically: January 15, 2020
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Abstract: Let $ A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the $ \operatorname {Ext}$-quiver of $ A$ is a simple directed graph, then $ H\!H^1(A)$ is a solvable Lie algebra. The second main result shows that if the $ \operatorname {Ext}$-quiver of $ A$ has no loops and at most two parallel arrows in any direction, and if $ H\!H^1(A)$ is a simple Lie algebra, then $ \operatorname {char}(k)\neq $ $ 2$ and $ H\!H^1(A)\cong $ $ \sl _2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.


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Additional Information

Markus Linckelmann
Affiliation: Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom
Email: markus.linckelmann.1@city.ac.uk

Lleonard Rubio y Degrassi
Affiliation: Department of Mathematics, University of Murcia, 30100 Murcia, Spain
Email: lleonard.rubio@um.es

DOI: https://doi.org/10.1090/proc/14875
Received by editor(s): April 15, 2019
Received by editor(s) in revised form: September 10, 2019
Published electronically: January 15, 2020
Additional Notes: The second author was supported by the projects, “Oberwolfach Leibniz Fellows”, DAAD Short-Term Grant (57378443), and by Fundación \lq Séneca\rq of Murcia (19880/GERM/15). He also would like to thank the University of Leicester for its support
Communicated by: Sarah Witherspoon
Article copyright: © Copyright 2020 American Mathematical Society