Parabolic Positive Representations of U_q(\mathfrakg_ℝ)

2020/08/19 by Ip, Ivan Chi-Ho
#13F60 #17B37 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2008.08589

Abstract

We construct a new family of irreducible representations of U_q(\mathfrakg_ℝ) and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of U_q(\mathfrakg_ℝ) act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations which corresponds to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type A_n acting on L²(ℝⁿ) with minimal functional dimension, and establish the properties of its central characters and universal R operator. We construct a positive version of the evaluation module of the affine quantum group U_q(\widehat\mathfraksl_n+1) modeled over this minimal positive representation of type A_n.

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