On some algebraic and geometric aspects of the quantum unitary group

2024/04/27 by Debabrata Jana, Jana, Debabrata
Mathematics · #16T20 #46L89 #58B32 #58B34 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2404.17863

openalex publication_date 2024/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the compact quantum group U_q(2), where q is a non-zero complex deformation parameter such that |q|≠ 1. Let C(U_q(2)) denote the underlying C^*-algebra of the compact quantum group U_q(2). We prove that if q is a non-real complex number and q^′ is real, then the underlying C^*-algebras C(U_q(2)) and C(U_q^′(2)) are non-isomorphic. This is in sharp contrast with the case of braided SU_q(2), introduced earlier by Woronowicz et al., where q is a non-zero complex deformation parameter. In another direction, on a geometric aspect of U_q(2), we introduce torus action on the C^*-algebra C(U_q(2)) and obtain a C^*-dynamical system (C(U_q(2)),\mathbbT³,α). We construct a \mathbbT³-equivariant spectral triple for U_q(2) that is even and 3⁺-summable. It is shown that the Dirac operator is K-homologically nontrivial.

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