2022/03/28 by Ivan Chi-Ho Ip, Ip, Ivan Chi-Ho, Ryuichi Man +1
Mathematics · Physics and Astronomy · #13F60 #17B37 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2203.14828
openalex publication_date 2022/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we construct a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces. It is motivated by a new observation arising from modifying the representation in the simplest case of Uq(\mathfraksl(2,ℝ)) compatible with Faddeev's modular double, while having a surprising tensor product decomposition. For higher rank, the representations are obtained by the polarization of Chevalley generators of Uq(\mathfrakg) in a new realization as universally Laurent polynomials of a certain skew-symmetrizable quantum cluster algebra. We also calculate explicitly the Casimir actions of the maximal An-1 degenerate representations of Uq(\mathfrakg_ℝ) for general Lie types based on the complexification of the central parameters.