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

2025/10/07 by Hernandez, David · 1 citation
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2510.06437

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.

Citations

• Quantum cluster algebras and representations of shifted quantum affine algebras
• Characters of quantum loop algebras
• Representations of shifted affine quantum groups and Coulomb branches
• Symmetries of Grothendieck rings in representation theory
• Monoidal Jantzen filtrations
• Isomorphisms among quantum Grothendieck rings and cluster algebras
• Weyl group symmetry of q-characters
• Monoidal categorification and quantum affine algebras II
• q-Opers, QQ-Systems, and Bethe Ansatz
• Coulomb branches of quiver gauge theories with symmetrizers
• Cluster algebra structures on module categories over quantum affine algebras
• Quantum affine algebras and cluster algebras
• Crystal bases and categorifications
• Simplicity of heads and socles of tensor products
• Algèbres amassées et applications
• Quantization of soliton systems and Langlands duality
• Level 0 Monomial crystals
• Algebraic Approach to q,t-Characters
• t-analogs of q-characters of quantum affine algebras of type A_n, D_n
• t-analogs of q-characters of Kirillov-Reshetikhin modules of quantm affine algebras
• Quiver varieties and t-analogs of q-characters of quantum affine algebras
• Cluster algebras I: Foundations
• Perverse sheaves and quantum Grothendieck rings
• Quiver varieties and finite dimensional representations of quantum affine algebras
• Deformations of W-algebras associated to simple Lie algebras

Cited by

• Braid group action and quasi-split affine iquantum groups III

Related

• Quantum affine algebras and cluster algebras
• A quantum cluster algebra approach to representations of simply-laced\n quantum affine algebras
• Quantum Affine Algebras and their Representations
• Quantum affine algebras and affine Hecke algebras
• Based cluster algebras of infinite rank