Shuffle algebras, lattice paths and quantum toroidal \mathfrakgl_n|m

2025/07/01 by Alexandr Garbali, Garbali, Alexandr, Andrei Neguţ +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2507.00538

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe and compute various families of commuting elements of the matrix shuffle algebra of type \mathfrakgl_n|m, which is expected to be isomorphic to quantum toroidal \mathfrakgl_n|m. Our formulas are given in terms of partial traces of products of R-matrices of the quantum affine algebra U_t(\mathfrakgl_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.

Citations

• Shuffle algebras, lattice paths and Macdonald functions
• Shuffle algebras, lattice paths and the commuting scheme
• A tale of two shuffle algebras
• Affine Laumon spaces and a conjecture of Kuznetsov
• Quantum toroidal and shuffle algebras
• Drinfeld realization of the elliptic Hall algebra
• Kernel function and quantum algebras
• On correlation functions of Drinfeld currents and shuffle algebras
• A reproducing kernel for nonsymmetric Macdonald polynomials

Related

• Quantum toroidal and shuffle algebras
• Commutative subalgebra of a shuffle algebra associated with quantum toroidal \mathfrakgl_m|n
• The R-matrix of the quantum toroidal algebra
• A tale of two shuffle algebras
• An integral form of quantum toroidal \mathfrakgl₁