Fusion Quivers

TL;DR

This paper introduces a categorical framework for quivers, establishing monoidal actions on their modules, leading to new algebraic structures like fusion products and preprojective algebras.

Contribution

It develops a novel categorical approach to quivers, defining monoidal actions and fusion products on modules, and explores their implications for algebraic structures.

Findings

Constructed rigid monoidal structures on quiver module categories.

Defined a fusion product inducing a graded ring with duality and trace.

Established a class of preprojective algebras with fusion modules.

Abstract

We develop a categorical approach to quivers and their modules. Naturally this leads to a notion of an action of a monoidal category on quivers. Using this, we construct for a large class of quivers rigid monoidal structures on their categories of modules. This fusion product on the quiver modules induces a graded ring structure with duality and trace on the moduli spaces of semisimple quiver modules. Our approach allows to consider a class of relations on such fusion quivers that are compatible with the rigid monoidal structure. In particular we obtain a class of preprojective algebras with fusion product on their modules.