Ana Garcia Elsener "Monomial Jacobian algebras"

Our starting point is a finite dimensional algebra kQ/I, where Q is a connected quiver (oriented graph) and I an ideal over the path algebra kQ.

There is a way to define a subclass of these algebras, called Jacobian, with good homological properties. One has to consider the sum W over all possible cycles over Q, this is a potential for Q. Then we need to define a cyclic derivative 𝜕_𝛼 with respect to each arrow 𝛼 in Q. The ideal I is obtained from these derivatives applied to W . It is known from cluster theory, based on works by C. Amiot, that Jacobian algebras are 1-Gorenstein and stably 3-Calabi-Yau.

In this talk we will show that when we consider monomial algebras, that is when I is a set of monomials in kQ, then the converse statement holds:

Let kQ/I be finite dimensional monomial algebra that is 1-Gorenstein and stably 3-Calabi-Yau, then kQ/I is Jacobian.

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