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We describe the ideal of relations for the trivial extension T (Λ) of a finite-dimensional monomial algebra Λ.When Λ 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 Λ such that T (Λ) ∼ = 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.
Gatica et al. (Wed,) studied this question.
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