Trivial extensions of monomial algebras

Authors

  • María Andrea Gatica Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacional delSur (UNS)-CONICET, Bahía Blanca, Argentina
  • María Valeria Hernández Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, Santa Rosa, Argentina
  • María Inés Platzeck Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, Argentina

DOI:

https://doi.org/10.33044/revuma.2070

Abstract

We describe the ideal of relations for the trivial extension $T(\Lambda)$ of a finite-dimensional monomial algebra $\Lambda$. When $\Lambda$ is, moreover, a gentle algebra, we solve the converse problem: given an algebra $B$, determine whether $B$ is the trivial extension of a gentle algebra. We characterize such algebras $B$ through properties of the cycles of their quiver, and show how to obtain all gentle algebras $\Lambda$ such that $T(\Lambda) \cong B$. We prove that indecomposable trivial extensions of gentle algebras coincide with Brauer graph algebras with multiplicity one in all vertices in the associated Brauer graph, result proven by S. Schroll.

 

Downloads

Download data is not yet available.

References

I. Assem and A. Skowroński, Iterated tilted algebras of type Ã$_n$, Math. Z. 195 no. 2 (1987), 269–290.  DOI  MR  Zbl

D. J. Benson, Representations and cohomology. I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, Cambridge, 1991.  MR  Zbl

E. A. Fernández and M. I. Platzeck, Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner, J. Algebra 249 no. 2 (2002), 326–344.  DOI  MR  Zbl

S. Schroll, Trivial extensions of gentle algebras and Brauer graph algebras, J. Algebra 444 (2015), 183–200.  DOI  MR  Zbl

Downloads

Published

2024-04-24

Issue

Vol. 67 No. 1 (2024)

Section

Article

License

Copyright (c) 2024 María Andrea Gatica, María Valeria Hernández, María Inés Platzeck

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.