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Quiver moduli and quantum loop algebras

Andrei Neguţ

TL;DR

The paper develops a comprehensive geometric and K-theoretic realization of simple modules in the Borel category O for quantum loop algebras U_q(L{\mathfrak{g}}), extending known HL-type Euler-characteristic connections from reachable to all highest ell-weights {\boldsymbol{\psi}}. It constructs and analyzes shuffle algebra models (big and small) to describe q-characters, and couples them with framed moduli of quiver representations and their critical K-theory. The central result is a categorification: coefficients of χ_q(L^{≠0}({\boldsymbol{\psi}})) correspond to dimensions of spaces from 𝒮_{\boldsymbol{n}}/(𝒮_{\boldsymbol{n}} ∩ Θ(\boldsymbol{ψ})_{\boldsymbol{x}}), realized geometrically via framed stable critical K-theory of quivers with potential; for strongly symmetrizable {\mathfrak{g}}, this yields an explicit, geometry-driven construction of all simple modules in Borel U_q(L{\mathfrak{g}}) and their q-characters. The approach integrates shuffle algebra techniques, quiver moduli, and critical K-theory (with stable envelopes) to generalize HL results, providing a robust framework with potential extensions to quantum toroidal algebras. This advances understanding of the interface between representation theory, algebraic geometry, and mathematical physics, offering concrete computational tools through K-theoretic and combinatorial structures.

Abstract

A classic result of Hernandez-Leclerc and Kashiwara-Kim-Oh-Park relates the q-characters of so-called reachable simple modules of quantum affine algebras to the Euler characteristics of certain quiver moduli spaces. We categorify and generalize this relation to all simple modules in the Hernandez-Jimbo category O using critical K-theory (our results hold for quantum toroidal as well as quantum affine algebras)

Quiver moduli and quantum loop algebras

TL;DR

The paper develops a comprehensive geometric and K-theoretic realization of simple modules in the Borel category O for quantum loop algebras U_q(L{\mathfrak{g}}), extending known HL-type Euler-characteristic connections from reachable to all highest ell-weights {\boldsymbol{\psi}}. It constructs and analyzes shuffle algebra models (big and small) to describe q-characters, and couples them with framed moduli of quiver representations and their critical K-theory. The central result is a categorification: coefficients of χ_q(L^{≠0}({\boldsymbol{\psi}})) correspond to dimensions of spaces from 𝒮_{\boldsymbol{n}}/(𝒮_{\boldsymbol{n}} ∩ Θ(\boldsymbol{ψ})_{\boldsymbol{x}}), realized geometrically via framed stable critical K-theory of quivers with potential; for strongly symmetrizable {\mathfrak{g}}, this yields an explicit, geometry-driven construction of all simple modules in Borel U_q(L{\mathfrak{g}}) and their q-characters. The approach integrates shuffle algebra techniques, quiver moduli, and critical K-theory (with stable envelopes) to generalize HL results, providing a robust framework with potential extensions to quantum toroidal algebras. This advances understanding of the interface between representation theory, algebraic geometry, and mathematical physics, offering concrete computational tools through K-theoretic and combinatorial structures.

Abstract

A classic result of Hernandez-Leclerc and Kashiwara-Kim-Oh-Park relates the q-characters of so-called reachable simple modules of quantum affine algebras to the Euler characteristics of certain quiver moduli spaces. We categorify and generalize this relation to all simple modules in the Hernandez-Jimbo category O using critical K-theory (our results hold for quantum toroidal as well as quantum affine algebras)

Paper Structure

This paper contains 26 sections, 13 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Shuffle algebras
  4. Euler characteristics
  5. General highest $\ell$-weights ${\boldsymbol{\psi}}$
  6. Critical $K$-theory
  7. Conclusion
  8. Acknowledgements
  9. Moduli of quiver representations
  10. Framed quiver representations
  11. Paths
  12. Cell decompositions
  13. Interlude on quiver Grassmannians
  14. Residues
  15. $q$-characters via shuffle algebras
  16. ...and 11 more sections

Key Result

Theorem 1.4

(general conjecture in HL Cluster, proof for Kirillov-Reshetikhin ${\boldsymbol{\psi}}$ in loc. cit., general proof in KKOP) For ${\mathfrak{g}}$ of finite type and "reachable" highest $\ell$-weights ${\boldsymbol{\psi}}$, where the algebraic variety $N^{\emph{stab}}_{\boldsymbol{x},\bm,{\mathbf{l}}}$ is defined as follows: The potential $W^{\emph{gr}}$ in eqn:critical intro is the sum over all

Theorems & Definitions (28)

  • Theorem 1.4
  • Theorem 1.6
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Definition 2.3
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.8
  • ...and 18 more