A note on species realizations and nondegeneracy of potentials

Abstract

In this note we show that a mutation theory of species with potential can be defined so that a certain class of skew-symmetrizable integer matrices have a species realization admitting a non-degenerate potential. This gives a partial affirmative answer to a question raised by Jan Geuenich and Daniel Labardini-Fragoso. We also provide an example of a class of skew-symmetrizable 4 × 4 integer matrices, which are not globally unfoldable nor strongly primitive, and that have a species realization admitting a non-degenerate potential.

Cyclic derivations, species realizations and potentials

“…In [5], it is shown that if the underlying base field F is uncountable then a nondegenerate quiver with potential exists for every underlying quiver. Motivated by this result, the following theorem is proved in [9].…”

“…In [9,Corollary 3.6] a partially affirmative answer to Question 2.23 is given by proving the following: let B = (b ij ) ∈ Z n×n be a skew-symmetrizable matrix with skew-symmetrizer D = diag(d 1 , . .…”

“…This paper is organized as follows. In Section 2, we review the preliminaries taken from [1] and [9]. Instead of working with an usual quiver, we consider the completion of the tensor algebra of M over S, where M is an S-bimodule and S is a finite dimensional semisimple F -algebra.…”

“…([9, Theorem 3.5]) Suppose that the underlying field F is uncountable, then F S (M ) admits a nondegenerate potential.…”