Non-Commutative Integration, Zeta Functions and the Haar State for SU q (2)

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• 作者：Marco Matassa (1)
  
  1. SISSA ; Via Bonomea 265 ; I-34136 ; Trieste ; Italy
  
• 关键词：Spectral triple ; Non ; commutative integral ; Zeta function ; Spectral dimension ; Primary 58B32 ; Secondary 58B34 ; 33D80
• 刊名：Mathematical Physics, Analysis and Geometry
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：18
• 期：1
• 全文大小：960 KB
• 参考文献：1. Carey, AL, Gayral, V, Rennie, A, Sukochev, FA (2012) Integration on locally compact noncommutative spaces. J. Funct. Anal. 263: pp. 383-414 CrossRef
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  3. Carey, AL, Phillips, J, Rennie, A (2010) Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra On. Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology 6: pp. 339-380 CrossRef
  4. Carey, AL, Phillips, J, Rennie, A, Sukochev, FA (2006) The local index formula in semifinite von Neumann algebras I: Spectral flow. Adv. Math. 202: pp. 451-516 CrossRef
  5. Carey, AL, Rennie, A, Sedaev, A, Sukochev, FA (2007) The Dixmier trace and asymptotics of zeta functions. J. Funct. Anal. 249: pp. 253-283 CrossRef
  6. Carey, AL, Rennie, A, Tong, K (2009) Spectral flow invariants and twisted cyclic theory for the Haar state on SUq(2). J. Geom. Phys. 59: pp. 1431-1452 CrossRef
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  15. Kaad, J: On modular semifinite index theory, arXiv preprint arXiv:1111.6546 (2011)
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  19. Kr盲hmer, U., Wagner, E., 2011: Twisted spectral triples and covariant differential calculi. Banach Center Publ. 93, 177鈥?88
  20. Kr谩hmer, U, Wagner, E (2013) A residue formula for the fundamental Hochschild class of the Podle艣 sphere. Journal of K- theory: K-theory and its Applications to Algebra, Geometry, and Topology 12: pp. 257-271 CrossRef
  21. Matassa, M.: On the spectral and homological dimension of / 魏-Minkowski space. arXiv preprint arXiv:1309.1054 (2013)
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• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Analysis
  Geometry
  Group Theory and Generalizations
  Applications of Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9656

文摘

We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group SU q (2). In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of SU q (2). We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of SU q (2). Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.