Alessandro Contu, Université Paris
Cité
Solution of a problem in monoidal categorification
by additive categorification
Abstract: Categorification, broadly speaking, is the process of
replacing set-theoretic or algebraic structures with
category-theoretic analogues. This often allows, working with
categories, to deduce highly non-trivial properties of the original
structures. In 2021, Kashiwara--Kim--Oh--Park showed that certain
monoidal subcategories of the category of finite-dimensional
representations of a quantum loop algebra categorify a quantum
cluster algebra structure of their Grothendieck rings. They stated
the problem of finding explicit quivers for the seeds they used. We
provide a solution by using Palu’s generalized mutation rule applied
to the cluster categories associated with certain algebras of global
dimension at most 2, for example tensor products of path algebras of
representation-finite quivers. Our method is based on (and
contributes to) the bridge, provided by cluster combinatorics,
between the representation theory of quantum groups and that of
quivers with relations.