Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-08T00:03:04.940Z Has data issue: false hasContentIssue false
  • English
  • Français

HALF-LIBERATED MANIFOLDS AND THEIR QUANTUM ISOMETRIES

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  10 June 2016

TEODOR BANICA
TEODOR BANICA*
Affiliation:
Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France E-mail: teodor.banica@u-cergy.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss the half-liberation operation X → X*, for the algebraic submanifolds of the unit sphere, $X\subset S^{N-1}_\mathbb C$. There are several ways of constructing this correspondence, and we take them into account. Our main results concern the computation of the affine quantum isometry group G+(X*), for the sphere itself.

MSC classification

Primary: 46L65: Quantizations, deformations
Secondary: 46L54: Free probability and free operator algebras
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 463 - 492
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Banica, T., A note on free quantum groups, Ann. Math. Blaise Pascal 15 (2008), 135–146.Google Scholar
2. Banica, T., Liberations and twists of real and complex spheres, J. Geom. Phys. 96 (2015), 1–25.Google Scholar
3. Banica, T. and Collins, B., Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. 43 (2007), 277–302.CrossRefGoogle Scholar
4. Banica, T. and Goswami, D., Quantum isometries and noncommutative spheres, Comm. Math. Phys. 298 (2010), 343–356.Google Scholar
5. Banica, T. and Speicher, R., Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461–1501.Google Scholar
6. Banica, T. and Vergnioux, R., Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60 (2010), 2137–2164.Google Scholar
7. Bercovici, H. and Pata, V., Stable laws and domains of attraction in free probability theory, Ann. of Math. 149 (1999), 1023–1060.Google Scholar
8. Bhowmick, J., D'Andrea, F. and Dabrowski, L., Quantum isometries of the finite noncommutative geometry of the standard model, Comm. Math. Phys. 307 (2011), 101–131.Google Scholar
9. Bhowmick, J. and Goswami, D., Quantum isometry groups: examples and computations, Comm. Math. Phys. 285 (2009), 421–444.Google Scholar
10. Bichon, J. and Dubois-Violette, M., Half-commutative orthogonal Hopf algebras, Pacific J. Math. 263 (2013), 13–28.CrossRefGoogle Scholar
11. Chirvasitu, A., On quantum symmetries of compact metric spaces, J. Geom. Phys. 94 (2015), 141–157.CrossRefGoogle Scholar
12. Collins, B. and Śniady, P., Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group, Comm. Math. Phys. 264 (2006), 773–795.Google Scholar
13. Goswami, D., Existence and examples of quantum isometry groups for a class of compact metric spaces, Adv. Math. 280 (2015), 340–359.Google Scholar
14. Goswami, D. and Joardar, S., Rigidity of action of compact quantum groups on compact, connected manifolds, preprint 2013.Google Scholar
15. Huang, H., Faithful compact quantum group actions on connected compact metrizable spaces, J. Geom. Phys. 70 (2013), 232–236.CrossRefGoogle Scholar
16. Quaegebeur, J. and Sabbe, M., Isometric coactions of compact quantum groups on compact quantum metric spaces, Proc. Indian Acad. Sci. Math. Sci. 122 (2012), 351–373.Google Scholar
17. Raum, S., Isomorphisms and fusion rules of orthogonal free quantum groups and their complexifications, Proc. Amer. Math. Soc. 140 (2012), 3207–3218.CrossRefGoogle Scholar
18. Speicher, R., Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994), 611–628.CrossRefGoogle Scholar
19. Voiculescu, D. V., Dykema, K. J. and Nica, A., Free random variables, CRM Monograph Series. 1 (American Mathematical Society, Providence, RI, 1992).Google Scholar
20. Wang, S., Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671–692.Google Scholar
21. Wang, S., Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211.CrossRefGoogle Scholar
22. Weingarten, D., Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys. 19 (1978), 999–1001.CrossRefGoogle Scholar
23. Woronowicz, S. L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.Google Scholar
24. Woronowicz, S. L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35–76.CrossRefGoogle Scholar