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Quantum quaternion spheres

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Abstract

For the quantum symplectic group SP q (2n), we describe the C ∗-algebra of continuous functions on the quotient space S P q (2n)/S P q (2n−2) as an universal C ∗-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ∗-algebra in terms of generators of the C ∗-algebra.

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References

  1. Blackadar Bruce, K-theory for operator algebras, volume 5 of mathematical sciences research institute publications ( 1998) ( Cambridge: Cambridge University Press) second edition

  2. Chakraborty Partha Sarathi and Pal Arupkumar, Characterization of spectral triples: A combinatorial approach, arXiv:math.OA/0305157 (2003)

  3. Chakraborty Partha Sarathi and Pal Arupkumar, Equivariant spectral triples on the quantum SU(2) group, K-Theory 28 (2) (2003) 107–126

    Article  MathSciNet  MATH  Google Scholar 

  4. Chakraborty Partha Sarathi and Pal Arupkumar, Characterization of SU q (ℓ+1)-equivariant spectral triples for the odd dimensional quantum spheres, J. Reine Angew. Math. 623 (2008) 25–42

    MathSciNet  MATH  Google Scholar 

  5. Chakraborty Partha Sarathi and Sundar S, K-groups of the quantum homogeneous space SU q (n)/SU q (n−2), Pacific J. Math. 252 (2) (2011) 275–292

    Article  MathSciNet  MATH  Google Scholar 

  6. Connes Alain, Cyclic cohomology, quantum group symmetries and the local index formula for SU q (2), J. Inst. Math. Jussieu 3 (1) (2004) 17–68

    Article  MathSciNet  MATH  Google Scholar 

  7. Dȧbrowski Ludwik, Landi Giovanni, Sitarz Andrzej, Suijlekom Walter van and Várilly Joseph C., The Dirac operator on SU q (2), Comm. Math. Phys. 259 (3) (2005) 729–759

    Article  MathSciNet  MATH  Google Scholar 

  8. Fulton William and Harri Joe, Representation theory, volume 129 of Graduate Texts in Mathematics ( 1991) ( New York: Springer-Verlag) A first course, Readings in Mathematics

    Google Scholar 

  9. Hong Jeong Hee and Szymański Wojciech, Quantum spheres and projective spaces as graph algebras, Comm. Math. Phys. 232 (1) (2002) 157–188

    Article  MathSciNet  MATH  Google Scholar 

  10. Klimyk Anatoli and Schmüdgen Konrad, Quantum groups and their representations, Texts and Monographs in Physics ( 1997) ( Berlin: Springer-Verlag)

    Book  MATH  Google Scholar 

  11. Korogodski Leonid I and Soibelman Yan S, Algebras of functions on quantum groups, Part I, volume 56 of Mathematical Surveys and Monographs ( 1998) ( Providence, RI: American Mathematical Society)

    Book  MATH  Google Scholar 

  12. Neshveyev Sergey and Tuset Lars, Quantized algebras of functions on homogeneous spaces with Poisson stabilizers, Comm. Math. Phys. 312 (1) (2012) 223–250

    Article  MathSciNet  MATH  Google Scholar 

  13. Pal Arupkumar and Sundar S, Regularity and dimension spectrum of the equivariant spectral triple for the odd-dimensional quantum spheres, J. Noncommut. Geom. 4 (3) (2010) 389–439

    MathSciNet  MATH  Google Scholar 

  14. Podkolzin G B and Vainerman L I, Quantum Stiefel manifold and double cosets of quantum unitary group, Pacific J. Math. 188 (1) (1999) 179–199

    Article  MathSciNet  MATH  Google Scholar 

  15. Piotr Podleṡ, Symmetries of quantum spaces: Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1) (1995) 1–20

    Article  MathSciNet  MATH  Google Scholar 

  16. Sheu Albert J L, Compact quantum groups and groupoid C ∗-algebras, J. Funct. Anal. 144 (2) (1997) 371–393

    Article  MathSciNet  MATH  Google Scholar 

  17. Vaksman L L and Soibelman Ya S, An algebra of functions on the quantum group SU(2), Funktsional. Anal. i Prilozhen. 22 (3) (1988) 1–14, 96

    MathSciNet  Google Scholar 

  18. Vaksman L L and Soibelman Ya S, Algebra of functions on the quantum group SU(n+1), and odd-dimensional quantum spheres, Algebra i Analiz 2 (5) (1990) 101–120

    MathSciNet  Google Scholar 

  19. Woronowicz S L, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1) (1987) 117–181

    Article  MathSciNet  Google Scholar 

  20. Woronowicz S L, Tannaka-Kren duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1) (1988) 35–76

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhelobenko D P, Classical groups. Spectral analysis of finite-dimensional representations, Uspehi Mat. Nauk 1 (103) (1962) 27–120, 17

    MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank Prof. Arup Kumar Pal, for his constant support. He would also like to thank S. Sundar for useful discussions on various topics.

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  1. Indian Statistical Institute, 7, SJSS Marg, New Delhi, 110 016, India

    BIPUL SAURABH

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  1. BIPUL SAURABH
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Correspondence to BIPUL SAURABH.

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Communicating Editor: B V Rajarama Bhat

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SAURABH, B. Quantum quaternion spheres. Proc Math Sci 127, 133–164 (2017). https://doi.org/10.1007/s12044-016-0318-z

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  • DOI: https://doi.org/10.1007/s12044-016-0318-z

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