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Shuffle algebras, lattice paths and quantum toroidal $\mathfrak{gl}_{n|m}$

Alexandr Garbali, Andrei Neguţ

Abstract

We describe and compute various families of commuting elements of the matrix shuffle algebra of type $\mathfrak{gl}_{n|m}$, which is expected to be isomorphic to quantum toroidal $\mathfrak{gl}_{n|m}$. Our formulas are given in terms of partial traces of products of $R$-matrices of the quantum affine algebra $U_t(\dot{\mathfrak{gl}}_{n|m})$, and have a lattice path interpretation. Our calculations are based on the machinery of the quantum toroidal algebras and a new anti-homomorphism between matrix shuffle algebras.

Shuffle algebras, lattice paths and quantum toroidal $\mathfrak{gl}_{n|m}$

Abstract

We describe and compute various families of commuting elements of the matrix shuffle algebra of type , which is expected to be isomorphic to quantum toroidal . Our formulas are given in terms of partial traces of products of -matrices of the quantum affine algebra , and have a lattice path interpretation. Our calculations are based on the machinery of the quantum toroidal algebras and a new anti-homomorphism between matrix shuffle algebras.

Paper Structure

This paper contains 31 sections, 19 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Conic partition function
  3. An anti-isomorphism of shuffle algebras
  4. Formulas for elements of commuting matrix shuffle algebra of the quantum toroidal $\mathfrak{gl}_{n|m}$
  5. Overview of the paper
  6. The matrix shuffle algebra of quantum toroidal $\mathfrak{gl}_{n|m}$
  7. The $R$-matrix
  8. Symmetric tensors and wheel conditions
  9. The shuffle product
  10. The double shuffle algebra
  11. The quantum toroidal algebra
  12. The super case
  13. Diagrammatic representation
  14. An anti-isomorphism of shuffle algebras
  15. The shuffle algebra $\mathcal{A}'$
  16. ...and 16 more sections

Key Result

Theorem 1

For any $n,m \geqslant 0$, the generating function $Z(v)$ can be expressed as: where $y_i:=t^{-\epsilon_i/2}\epsilon_i u_i$, $u_{n+m+1}:=q u_1$, $\epsilon_i=(-1)^{\delta_{i>n}}$, and $S_k^{(1)}\ldots S_k^{(n+m)} \in \emph{End}(V^{\otimes k})(z_1\ldots z_k)$ is a collection of matrix-valued rational functions which can be computed recursively: where $[a,b]=a*b-b*a$. Theorem thm:Z_intro is conting

Theorems & Definitions (49)

  • Theorem 1
  • Remark 1
  • Definition 1
  • Example 2
  • Proposition 1: [Prop. 2.6, 4.6, 4.10]
  • Proposition 2
  • Definition 2
  • Definition 3
  • Theorem 3
  • Definition 4
  • ...and 39 more