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Solution of a Problem in Monoidal Categorification by Additive Categorification

Alessandro Contu

Abstract

In 2021, Kashiwara-Kim-Oh-Park constructed cluster algebra structures on the Grothendieck rings of certain monoidal subcategories of the category of finite-dimensional representations of a quantum loop algebra, generalizing Hernandez-Leclerc's pioneering work from 2010. 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. Thus, 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.

Solution of a Problem in Monoidal Categorification by Additive Categorification

Abstract

In 2021, Kashiwara-Kim-Oh-Park constructed cluster algebra structures on the Grothendieck rings of certain monoidal subcategories of the category of finite-dimensional representations of a quantum loop algebra, generalizing Hernandez-Leclerc's pioneering work from 2010. 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. Thus, 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.
Paper Structure (15 sections, 44 theorems, 150 equations, 1 figure, 1 table)

This paper contains 15 sections, 44 theorems, 150 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Module categories of quantum loop algebras
  3. The category $\mathscr{C}_\mathfrak{g}^\mathbb{Z}$
  4. Duality datum and PBW pairs.
  5. Combinatorics of $i$-boxes
  6. Combinatorics of $Q$-data
  7. Monoidal Categorification
  8. Reminder on monoidal categorification
  9. Admissible seeds associated to chains of $i$-boxes
  10. Reminder on cluster-tilting theory
  11. Cluster categories associated to $Q(\underline{w}_0)$
  12. A Jacobi algebra associated to the initial seed
  13. An algebra of global dimension $\leq 2$ associated to the initial seed
  14. An additive avatar of affine determinant modules
  15. An additive avatar of monoidal seeds

Key Result

Lemma 2.8

Let $\mathfrak{C}=(\mathfrak{c}_k)$ be a chain of $i$-boxes of length $l\leq \infty$ and $1\leq s<l$ an integer such that $\mathfrak{c}_s$ is movable. Then the $i$-box $\mathfrak{c'}_s$ of $\nu_s\mathfrak{C}$ is movable and $\nu_s\nu_s\mathfrak{C}=\mathfrak{C}$.

Figures (1)

  • Figure 1: Unfoldings for non-simply-laced $\mathfrak{g}$

Theorems & Definitions (105)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: KKOP_mon_cat_quant_aff_II
  • Definition 2.4: KKOP_mon_cat_quant_aff_II
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Example 2.9
  • ...and 95 more