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Characters of quantum loop algebras

Andrei Neguţ

TL;DR

This work ties q-characters of quantum loop algebras to a refined, graded character formula conditioned on a conjectural dimension conjecture. It develops a comprehensive shuffle-algebra framework, introduces slope subalgebras, and proves factorization results that decompose the shuffle algebra along catty-corner curves. The main achievement is a conditional, explicit expression for the refined character χ_ref^r, capturing the graded dimensions of slope-graded quotients and connecting to a product over positive roots; this advances understanding of how q-characters split into a nonzero-pole part and an ord-part across arbitrary Kac-Moody algebras. The results hinge on a conjectured horocycle-type generating-function formula and extend known finite-type cases to broader KM settings, with implications for category O representations and the structure of q-characters.

Abstract

The q-characters of quantum loop algebras are very important objects in representation theory. In [20], we showed that q-characters factor as a power series of the form studied in [9] times a character, an important phenomenon which had already been known in finite types. In the present paper, we prove a conjectural formula for the aforementioned character factor.

Characters of quantum loop algebras

TL;DR

This work ties q-characters of quantum loop algebras to a refined, graded character formula conditioned on a conjectural dimension conjecture. It develops a comprehensive shuffle-algebra framework, introduces slope subalgebras, and proves factorization results that decompose the shuffle algebra along catty-corner curves. The main achievement is a conditional, explicit expression for the refined character χ_ref^r, capturing the graded dimensions of slope-graded quotients and connecting to a product over positive roots; this advances understanding of how q-characters split into a nonzero-pole part and an ord-part across arbitrary Kac-Moody algebras. The results hinge on a conjectured horocycle-type generating-function formula and extend known finite-type cases to broader KM settings, with implications for category O representations and the structure of q-characters.

Abstract

The q-characters of quantum loop algebras are very important objects in representation theory. In [20], we showed that q-characters factor as a power series of the form studied in [9] times a character, an important phenomenon which had already been known in finite types. In the present paper, we prove a conjectural formula for the aforementioned character factor.

Paper Structure

This paper contains 29 sections, 10 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. $q$-characters
  3. Characters
  4. Kac-Moody Lie algebras
  5. Acknowledgements
  6. Shuffle algebras
  7. Basic definitions
  8. The quantum loop algebra
  9. The big shuffle algebra
  10. The shuffle algebra
  11. The double shuffle algebra
  12. The Hopf algebra structure
  13. The pairing
  14. Slopes
  15. Slope subalgebras
  16. ...and 14 more sections

Key Result

Theorem 1.3

For any semisimple complex Lie algebra ${\mathfrak{g}}$, we have for all ${\mathbf{r}} \in {\mathbb{Z}^I}$.

Theorems & Definitions (30)

  • Theorem 1.3
  • Definition 2.3
  • Remark 2.6
  • Proposition 2.7
  • Proposition 2.11
  • proof
  • Definition 2.13
  • Proposition 2.15
  • Proposition 2.17
  • proof
  • ...and 20 more