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Trivial Extensions of Gentle Algebras and Brauer Graph Algebras

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journal contribution
posted on 2015-03-09, 10:05 authored by Sibylle Schroll
We show that two well-studied classes of tame algebras coincide: namely, the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. We then explore the connection between gentle algebras and symmetric special biserial algebras by explicitly determining the trivial extension of a gentle algebra by its minimal injective co-generator. This is a symmetric special biserial algebra and hence a Brauer graph algebra of which we explicitly give the Brauer graph. We further show that a Brauer graph algebra gives rise, via admissible cuts, to many gentle algebras and that the trivial extension of a gentle algebra obtained via an admissible cut is the original Brauer graph algebra. As a consequence we prove that the trivial extension of a Jacobian algebra of an ideal triangulation of a Riemann surface with marked points in the boundary is isomorphic to the Brauer graph algebra with Brauer graph given by the arcs of the triangulation.

Funding

Engineering and Physical Sciences Research Council, grant number EP/K026364/1

History

Citation

Journal of Algebra 444 (2015) 183–200

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

  • AO (Author's Original)

Published in

Journal of Algebra 444 (2015) 183–200

Available date

2015-03-09

Publisher version

http://www.sciencedirect.com/science/article/pii/S0021869315004093

Notes

Added an example. 2010 Mathematics Subject Classification. Primary 16G10, 16G20; Secondary 16S99, 13F60

Language

en

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