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Mathematics > Representation Theory

arXiv:1804.02644 (math)
[Submitted on 8 Apr 2018 (v1), last revised 12 Aug 2019 (this version, v3)]

Title:Quantized Vershik-Kerov Theory and Quantized Central Measures on Branching Graphs

Authors:Ryosuke Sato
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Abstract:We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila-Voiculescu's work (Stratila-Voiculescu, 1975) (or (Enomoto-Izumi, 2016)), and give its serious investigation when the system consists of quantum unitary groups $U_q(N)$ with $q$ in $(0,1)$. The key features of this work are: The "quantized trace" of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a KMS state with respect to a certain one-parameter automorphism group related to the so-called scaling group. In this paper we provide a Vershik-Kerov type approximation theorem for extremal quantized characters (called the ergodic method) and also compare our quantized character theory for the inductive system of $U_q(N)$ with Gorin's theory on $q$-Gelfand-Tsetlin graphs (Gorin, 2012).
Comments: 21 pages
Subjects: Representation Theory (math.RT); Operator Algebras (math.OA); Probability (math.PR)
Cite as: arXiv:1804.02644 [math.RT]
  (or arXiv:1804.02644v3 [math.RT] for this version)
  https://doi.org/10.48550/arXiv.1804.02644
Journal reference: J. Funct. Anal. 277 (2019) 2522-2557
Related DOI: https://doi.org/10.1016/j.jfa.2018.11.016

Submission history

From: Ryosuke Sato [view email]
[v1] Sun, 8 Apr 2018 07:48:47 UTC (29 KB)
[v2] Mon, 10 Sep 2018 18:00:48 UTC (28 KB)
[v3] Mon, 12 Aug 2019 02:57:55 UTC (29 KB)
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