Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T18:31:13.665Z Has data issue: false hasContentIssue false
  • English
  • Français

Tannakian Duality for Affine Homogeneous Spaces

Published online by Cambridge University Press:  20 November 2018

Teodor Banica
Teodor Banica*
Affiliation:
Department of Mathematics, University of Cergy-Pontoise, F-95000 Cergy-Pontoise, France, e-mail : teo.banica@gmail.com
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.

Associated with any closed quantum subgroup $G\,\subset \,U_{N}^{+}$ and any index set $I\,\subset \,\{1,\,.\,.\,.\,,\,N\}$ is a certain homogeneous space ${{X}_{G,I}}\subset S_{\mathbb{C},+}^{N-1},$ called an affine homogeneous space. Using Tannakian duality methods, we discuss the abstract axiomatization of the algebraic manifolds $X\subset S_{\mathbb{C},+}^{N-1}$ that can appear in this way.

Keywords

46L6546L89quantum isometrynoncommutative manifold
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 3 , 01 September 2018 , pp. 483 - 494
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Banica, T., Weingarten Integration over noncommutative homogeneous Spaces. Ann. Math. Blaise Pascal 24(2017), no. 2, 195–224. http://dx.doi.org/10.5802/ambp.368Google Scholar
[2] Banica, T. and Collins, B., Integration over compact quantum groups. Publ. Res. Inst. Math. Sei. 43(2007), 277–302. http://dx.doi.org/10.2977/prims/1201011782Google Scholar
[3] Banica, T. and Goswami, D., Quantum isometries and noncommutative spheres. Comm. Math. Phys. 298(2010), 343–356. http://dx.doi.org/10.1007/s00220-010-1060-5Google Scholar
[4] Banica, T. and Meszäros, S., Uniqueness resultsfor noncommutative spheres and projeetive Spaces. Illinois J. Math. 59(2015), 219–233.Google Scholar
[5] Collins, B. and Matsumoto, S., Weingarten calculus via orthogonality relations: new appücations. ALEA Lat. Am. J. Probab. Math. Stat. 14(2017), no. 1, 631–656.Google Scholar
[6] Collins, B. and Sniady, P., Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group. Comm. Math. Phys. 264(2006), 773–795. http://dx.doi.org/10.1007/s00220-006-1554-3Google Scholar
[7] Goswami, D., Quantum group oj isometries in classical and noncommutative geometry. Comm. Math. Phys. 285(2009), 141–160. http://dx.doi.org/10.1007/s00220-008-0461-1Google Scholar
[8] Goswami, D., Existence and examples of quantum isometry groups for a class of compact metric Spaces. Adv. Math. 280(2015), 340–359. http://dx.doi.Org/10.1016/j.aim.2015.03.024Google Scholar
[9] Malacarne, S., Woronowicz's Tannaka-Krein duality andfree orthogonal quantum groups. Math. Scand. 122(2018), 151–160. http://dx.doi.org/10.7146/math.scand.a-97320Google Scholar
[10] Raum, S., Isomorphisms andfusion rules of orthogonal free quantum groups and their complexifications. Proc. Amer. Math. Soc. 140(2012), 3207–3218. http://dx.doi.org/10.1090/S0002-9939-2012-11264-XGoogle Scholar
[11] Wang, S., Free produets of compact quantum groups. Comm. Math. Phys. 167(1995), 671–692. http://dx.doi.org/10.1007/BF02101540Google Scholar
[12] Weingarten, D., Asymptotic behavior of group Integrals in the limit of infinite rank. J. Mathematical Phys. 19(1978), 999–1001. http://dx.doi.Org/10.1063/1.523807Google Scholar
[13] Woronowicz, S. L., Compact matrixpseudogroups. Comm. Math. Phys. 111(1987), 613–665. http://dx.doi.Org/10.1007/BF01219077Google Scholar
[14] Woronowicz, S. L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups. Invent. Math. 93(1988), 35–76. http://dx.doi.org/10.1007/BF01393687Google Scholar