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

TL;DR

This paper constructs and computes commuting elements in the shuffle algebra of type _{n|m}, revealing their structure via R-matrices and lattice paths, and relates it to quantum toroidal _{n|m}.

Contribution

It introduces formulas for commuting elements in the shuffle algebra of _{n|m} using R-matrices and lattice paths, and establishes a new anti-homomorphism linking these algebras.

Findings

Explicit formulas for commuting elements in the shuffle algebra.

A lattice path interpretation of the algebraic structures.

A new anti-homomorphism between matrix shuffle algebras.

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.