On the Lie algebra structure of $H\!H^1(A)$ of a finite-dimensional algebra $A$
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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 $H\!H^1(A)\cong$ $\operatorname {\mathfrak {sl}}_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.References
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Additional Information
- Markus Linckelmann
- Affiliation: Department of Mathematics, City, University of London, London EC1V 0HB, United Kingdom
- MR Author ID: 240411
- Email: markus.linckelmann.1@city.ac.uk
- Lleonard Rubio y Degrassi
- Affiliation: Department of Mathematics, University of Murcia, 30100 Murcia, Spain
- MR Author ID: 1183180
- Email: lleonard.rubio@um.es
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1879-1890
- MSC (2010): Primary 16E40, 16G30, 16D90, 17B50
- DOI: https://doi.org/10.1090/proc/14875
- MathSciNet review: 4078074