Revista de la
Unión Matemática Argentina
Trivial extensions of monomial algebras
María Andrea Gatica, María Valeria Hernández, and María Inés Platzeck

Volume 67, no. 1 (2024), pp. 173–196    

Published online: April 24, 2024

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

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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.

References

  1. I. Assem and A. Skowroński, Iterated tilted algebras of type $\tilde{\mathbf{A}}_n$, Math. Z. 195 no. 2 (1987), 269–290.  DOI  MR  Zbl
  2. 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
  3. 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
  4. S. Schroll, Trivial extensions of gentle algebras and Brauer graph algebras, J. Algebra 444 (2015), 183–200.  DOI  MR  Zbl