Semilinear clannish algebras arising from surfaces with orbifold points

Abstract

Semilinear clannish algebras have been recently introduced by the first author and Crawley-Boevey as a generalization of Crawley-Boevey's clannish algebras. In the present paper, we associate semilinear clannish algebras to the (colored) triangulations of a surface with marked points and orbifold points, and exhibit a Morita equivalence between these algebras and the Jacobian algebras constructed a few years ago by Geuenich and the second author.

Quivers with potentials associated with triangulations of Riemann surfaces

“…The main aim of this talk is to present [2]. We construct semilinear clannish algebras for the colored triangulations of a surface with marked points and orbifold points, and prove that they are Morita-equivalent to the Jacobian algebras of the species with potential constructed by Geuenich and myself a few years ago in [3,4].…”

“…We show in [2] that K σ Q(τ )/I(τ, ξ), where I(τ, ξ) := Z ∪ {q s j | s j ∈ S(τ )} , is a semilinear clannish algebra.…”

“…As in [2], we associate a semilinear clannish algebra to each colored triangulation (τ, ξ) as follows. Fix one of the following two choices of field K:…”

“…This way, e.g., ab and aib are distinct paths if t Theorem 1. [2] If Σ is a surface with marked points and orbifold points satisfying (1), then for any colored triangulation (τ, ξ) of Σ, the Jacobian algebra P(A(τ, ξ), W (τ, ξ)) and the semilinear clannish algebra K σ Q(τ, ξ)/I(τ, ξ) are Morita-equivalent (K = C in the B-like situation, K = R in the C-like situation).…”