One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry

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• 作者：Biswarup Das ; Uwe Franz ; Anna Kula ; Adam Skalski
• 关键词：Hopf $$^*$$ ?algebra ; Cocycle ; Generating functional ; Quantum Lévy process ; Quantum group ; Haagerup property ; Primary 16T20 ; Secondary 16T05 ; 46L65
• 刊名：Mathematische Zeitschrift
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：281
• 期：3-4
• 页码：949-965
• 全文大小：497 KB
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  7.Franz, U., Skalski, A., Tomatsu, R.: Idempotent states on compact quantum groups and their classification on \(U_q(2)\) , \(SU_q(2)\) , and \(SO_q(3)\) . J. Noncommun. Geom. 7(1), 221-54 (2013)MATH MathSciNet CrossRef
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• 作者单位：Biswarup Das (1)
  Uwe Franz (2)
  Anna Kula (3) (4)
  Adam Skalski (1) (5) (6)
  
  1. Institute of Mathematics of the Polish Academy of Sciences, ul. ?niadeckich 8, 00-56, Warszawa, Poland
  2. Département de mathématiques de Besan?on, Université de Franche-Comté, 16 route de Gray, 25 030, Besan?on cedex, France
  3. Instytut Matematyczny, Uniwersytet Wroc?awski, pl. Grunwaldzki 2/4, 50-384, Wroc?aw, Poland
  4. Instytut Matematyki, Uniwersytet Jagielloński, ul.?ojasiewicza 6, 30-48, Kraków, Poland
  5. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland
  6. CNRS, Département de mathématiques de Besan?on, Université de Franche-Comté, 16 route de Gray, 25 030, Besan?on cedex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

We prove that under a symmetry assumption all cocycles on Hopf \(^*\)-algebras arise from generating functionals. This extends earlier results of R. Vergnioux and D. Kyed and has two quantum group applications: all quantum Lévy processes with symmetric generating functionals decompose into a maximal Gaussian and purely non-Gaussian part and the Haagerup property for discrete quantum groups is characterized by the existence of an arbitrary proper cocycle. Keywords Hopf \(^*\)-algebra Cocycle Generating functional Quantum Lévy process Quantum group Haagerup property