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Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models

David Hernandez

TL;DR

This work surveys the representation theory of $\mathcal{U}_q(\hat{\mathfrak{g}})$ and its crossroads with cluster categorification and quantum integrable models. It develops foundational tools such as the category $\mathcal{C}$ of finite-dimensional modules, the $q$-character map, and the Grothendieck ring $K(\mathcal{C})$, then connects these to monoidal categorifications of cluster algebras, Baxter polynomiality, and $QQ$-systems via prefundamental representations and $Q$-operators. It further integrates shifted quantum affine algebras $\mathcal{U}_q^{\mu}(\hat{\mathfrak{g}})$, their Coulomb-branch interpretations, and the associated cluster structures, extending to truncated and finite-level variants with detailed conjectures and examples. Overall, the article provides a unifying framework linking representation theory, cluster algebras, and integrable models, offering concrete constructions, canonical bases, and deep conjectures that guide future work in simply- and non-simply-laced types.

Abstract

In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster categorification and quantum integrable models. We will also give new examples and conjectures.

Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models

TL;DR

This work surveys the representation theory of and its crossroads with cluster categorification and quantum integrable models. It develops foundational tools such as the category of finite-dimensional modules, the -character map, and the Grothendieck ring , then connects these to monoidal categorifications of cluster algebras, Baxter polynomiality, and -systems via prefundamental representations and -operators. It further integrates shifted quantum affine algebras , their Coulomb-branch interpretations, and the associated cluster structures, extending to truncated and finite-level variants with detailed conjectures and examples. Overall, the article provides a unifying framework linking representation theory, cluster algebras, and integrable models, offering concrete constructions, canonical bases, and deep conjectures that guide future work in simply- and non-simply-laced types.

Abstract

In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster categorification and quantum integrable models. We will also give new examples and conjectures.

Paper Structure

This paper contains 11 sections, 23 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Quantum affine algebras and their representations
  3. Quantum integrable models
  4. Quantum affine Borel
  5. ODE/IM and Bethe Ansatz
  6. Cluster structures
  7. Quantum Grothendieck rings
  8. Shifted quantum groups
  9. Shifted quantum groups and cluster algebras
  10. Truncated shifted quantum groups
  11. Representations of truncations

Key Result

Theorem 2.2

There is a bijection between simple representationsIn this lecture, we will only consider type I representations of quantum enveloping algebras, that is, the eigenvalues of the $k_i$ belong to $q^{\mathbb{Z}}$. Any simple finite-dimensional representation can be obtained from a type I representation

Theorems & Definitions (71)

  • Example 2.1
  • Theorem 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 61 more