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Shuffle algebras, lattice paths and quantum toroidal $\mathfrak{gl}_{n|m}$ (2507.00538v1)

Published 1 Jul 2025 in math.QA, math-ph, and math.MP

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.

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