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Quantum loop groups and shuffle algebras via Lyndon words

Andrei Neguţ, Alexander Tsymbaliuk

Abstract

We study PBW bases of the untwisted quantum loop group $U_q(L\mathfrak{g})$ (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment of [29,30,43] in the finite type case. As an application, we prove that Enriquez' homomorphism [11] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [15] associated to $\mathfrak{g}$ is an isomorphism.

Quantum loop groups and shuffle algebras via Lyndon words

Abstract

We study PBW bases of the untwisted quantum loop group (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment of [29,30,43] in the finite type case. As an application, we prove that Enriquez' homomorphism [11] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [15] associated to is an isomorphism.

Paper Structure

This paper contains 7 sections, 49 theorems, 495 equations.

Table of Contents

  1. Introduction
  2. Lie algebras and Lyndon words
  3. Lyndon words and Weyl groups
  4. Quantum groups and shuffle algebras
  5. Quantum affine and quantum loop: two presentations
  6. Shuffle algebras of Feigin-Odesskii and Enriquez
  7. Appendix

Key Result

Theorem 1.5

There exists an injective algebra homomorphism: Fix a total order of $I$, which induces the following total order on the set $\{i^{(d)}\}_{i\in I}^{d\in {\mathbb{Z}}}$: This induces the lexicographic order on the words $[i_1^{(d_1)} \dots\, i_k^{(d_k)}]$ with respect to which we may define the notion of standard Lyndon loop words by analogy with LR (see Subsections sub:affine standard--sub:stand

Theorems & Definitions (120)

  • Theorem 1.5
  • Theorem 1.7
  • Definition 2.2
  • Definition 2.4
  • Claim 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Definition 2.9
  • Remark 2.10
  • Definition 2.12
  • ...and 110 more