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A tale of two shuffle algebras

Andrei Neguţ

Abstract

As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra.

A tale of two shuffle algebras

Abstract

As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra.

Paper Structure

This paper contains 5 sections, 26 theorems, 394 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Shuffle algebras and $R$--matrices
  3. Quantum toroidal ${\mathfrak{gl}}_n$
  4. The shuffle algebra with spectral parameters
  5. The extended shuffle algebra with spectral parameter

Key Result

Theorem 1.5

There exist injective algebra homomorphisms: Denoting the images of these maps by $U^\uparrow_{q,{\overline{q}}}(\ddot{{\mathfrak{gl}}}_n)$ and $U^\downarrow_{q,{\overline{q}}}(\ddot{{\mathfrak{gl}}}_n)$ yields the decompositions featuring in eqn:decomp 1, eqn:half plus, eqn:half minus. Moreover, there exist topological coproducts on the subal which extend the Drinfeld-Jimbo coproduct on the hori

Figures (20)

  • Figure 1: Various crossings
  • Figure 2: Reidemeister III move - version 1
  • Figure 3: Reidemeister III move - version 2
  • Figure 4: Reidemeister III move - version 3
  • Figure 5: $A*B$ as a braid
  • ...and 15 more figures

Theorems & Definitions (61)

  • Theorem 1.5
  • Proposition 2.4
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.8
  • Proposition 2.9
  • proof
  • Proposition 2.11
  • proof
  • ...and 51 more