Yadira Valdivieso-Díaz

arXiv2016

A note on (co)homologies of algebras from unpunctured surfaces

In a previous paper, the author compute the dimension of Hochschild cohomology groups of Jacobian algebras from (unpunctured) triangulated surfaces, and gave a geometric interpretation of those numbers in terms of the number of internal triangles, the number of vertices and the existence of certain kind of boundaries. The aim of this note is computing the cyclic (co)homology and the Hochschild homology of the same family of algebras and giving an interpretation of those dimensions through elements of the triangulated surface.

math.RT↗

arXiv2016

Jacobian algebras with periodic module category and exponential growth

The Jacobian algebra associated to a triangulation of a closed surface $S$ with a collection of marked points $M$ is (weakly) symmetric and tame. We show that for these algebras the Auslander-Reiten translate acts 2-periodical on objects. Moreover, we show that excluding only the case of a sphere with $4$ (or less) punctures, these algebras are of exponential growth. These four properties implies that there is a new family of algebras symmetric, tame and with periodic module category. As a consequence of the 2-periodical actions of the Auslander-Reiten translate on objects, we have that the Auslander-Reiten quiver of the generalized cluster category $\cC_{(S,M)}$ consists only of stable tubes of rank $1$ or $2$.

math.RT↗