On Jacobian algebras associated with the once-punctured torus

TL;DR

This paper studies two different Jacobian algebras from the once-punctured torus, analyzing their component graphs and relating g-vectors to geometric coefficients, revealing structural connectivity and parametrization.

Contribution

It determines the connectedness of the strongly reduced component graphs for both finite and infinite dimensional Jacobian algebras and links g-vectors to universal geometric coefficients.

Findings

The component graph is connected in both cases.

g-vectors of indecomposable components match universal geometric coefficients.

Finite-dimensional algebra's g-vectors are precisely the universal geometric coefficients.

Abstract

We consider two non-degenerate potentials for the quiver arising from the once-punctured torus, which are a natural choice to study and compare: the first is the Labardini-potential, yielding a finite-dimensional Jacobian algebra, whereas the second potential gives rise to an infinite dimensional Jacobian algebra. In this paper we determine the graph of strongly reduced components for both Jacobian algebras. Our main result is that the graph is connected in both cases. Plamondon parametrized the strongly reduced components for finite-dimensional algebras using generic g-vectors. We prove that the generic g-vectors of indecomposable strongly reduced components of the finite-dimensional Jacobian algebra are precisely the universal geometric coefficients for the once-punctured torus, which were determined by Reading.