Gelfand-Tsetlin Bases for Elliptic Quantum Groups

2024/08/19 by Hitoshi Konno, Konno, Hitoshi, Kohei Motegi +1
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Geometric and Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2408.09712

openalex publication_date 2024/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the level-0 representations of the elliptic quantum group U_q,p(\widehat\mathfrakgl_N). We give a classification theorem of the finite-dimensional irreducible representations of U_q,p(\widehat\mathfrakgl_N) in terms of the theta function analogue of the Drinfeld polynomial for the quantum affine algebra U_q(\widehat\mathfrakgl_N). We also construct the Gelfand-Tsetlin bases for the level-0 U_q,p(\widehat\mathfrakgl_N)-modules following the work by Nazarov-Tarasov for the Yangian Y(\mathfrakgl_N)-modules. This is a construction in terms of the Drinfeld generators. For the case of tensor product of the vector representations, we give another construction of the Gelfand-Tsetlin bases in terms of the L-operators and make a connection between the two constructions. We also compare them with those obtained by the first author by using the \mathfrakS_n-action realized by the elliptic dynamical R-matrix on the standard bases. As a byproduct, we obtain an explicit formula for the partition functions of the corresponding 2-dimensional square lattice model in terms of the elliptic weight functions of type A_N-1.

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