Strongly primitive species with potentials I: mutations

Abstract

Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.

Cyclic derivations, species realizations and potentials

“…We now give an example ([9, p.8]) 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 nondegenerate potential. This gives an example of a class of skew-symmetrizable 4×4 integer matrices which are not covered by [8].…”

“…Remark 1. By [8,Example 14.4] we know that the class of all matrices given by (3) does not admit a global unfolding. Moreover, since we are not assuming that a and b are coprime, then such matrices are not strongly primitive; hence they are not covered by [8].…”

“…The theory of quivers with potentials has proven useful in many subjects of mathematics such as cluster algebras, Teichmüller theory, KP solitons, mirror symmetry, Poisson geometry, among many others. There have been different generalizations of the notion of a quiver with potential and mutation where the underlying F -algebra, F being a field, is replaced by more general algebras, see [4,8,10]. This paper is organized as follows.…”

A note on species realizations and nondegeneracy of potentials

“…In [7], D. Labardini-Fragoso and A. Zelevinsky give a partially positive answer to Question 2.23 provided that the skew-symmetrizer has pairwise coprime diagonal entries. We remark that this is a stronger condition than the one we impose in [1].…”

“…Finally, in Section 5 we give an example of a class of skew-symmetrizable 4 × 4 integer matrices which are not globally unfoldable nor strongly primitive (in the sense of [7, Definition 14.1]), and that have a species realization admitting a non-degenerate potential. This gives an example of a class of skew-symmetrizable 4×4 integer matrices which are not covered by [7].…”