Trivial extensions of monomial algebras


  • 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



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.



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