Abstract
We give a brief overview of generalized symmetry of classical spaces (manifolds/metric spaces/varieties etc.) in terms of (co)actions of Hopf algebras, both in the algebraic and the analytic set-up.
Partially supported by J C Bose Fellowship from D.S.T. (Govt. of India).
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References
Banica, T.: Quantum automorphism groups of small metric spaces. Pac. J. Math. 219(1), 27–51 (2005)
Banica, T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224(2), 243–280 (2005)
Banica, T., Bichon, J.: Quantum groups acting on 4 points. J. Reine Angew. Math. 626, 75–114 (2009)
Banica, T., Bichon, J., Collins, B.: Quantum permutation groups: a survey. noncommutative harmonic analysis with applications to probability. Polish Acad. Sci. Inst. Math. 78, 13–34. Banach Center Publications, Warsaw (2007)
Banica, T., Moroianu, S.: On the structure of quantum permutation groups. Proc. Am. Math. Soc. 135(1), 21–29 (2007)
Bichon, J.: Quantum automorphism groups of finite graphs. Proc. Am. Math. Soc. 131(3), 665–673 (2003)
Bichon, J.: Algebraic quantum permutation groups. Asian-Eur. J. Math. 1(1), 1–13 (2008)
Connes, A.: Noncommutative Geometry. Academic Press, London, New York (1994)
Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, Berkeley (1986)
Drinfeld, V.G.: Quasi-Hopf algebras. Leningr. Math. J. 1, 1419–1457 (1990)
Etingof, P., Walton, C.: Semisimple hopf actions on commutative domains. Adv. Math. 251, 47–61 (2014)
Goswami, D.: Quantum group of isometries in classical and non commutative geometry. Commun. Math. Phys. 285(1), 141–160 (2009)
Goswami, D.: Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds. preprint (2018)
Goswami, D., Bhowmick, J.: Quantum Isometry Groups. Springer, Infosys Series (2017)
Goswami, D., Joardar, S.: Non-existence of faithful isometric action of compact quantum groups on compact, connected Riemannian manifolds. Geom. Funct. Anal. 28(1), 146–178 (2018)
Huang, H.: Faithful compact quantum group actions on connected compact metrizable spaces. J. Geom. Phys. 70, 232–236 (2013)
Jimbo, M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 10, 63–69 (1985)
Jimbo, M.: A q-diff erence analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985)
Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. Ecole Norm. super. 33(6), 837–934 (2000)
Kustermans, J., Vaes, S.: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(1), 68–92 (2003)
Manin, Y: Some remarks on Koszul algebras and quantum groups. Ann. Inst. Fourier Grenoble 37(4), 191–205 (1987)
Manin, Y.: Quantum Groups and Non-commutative Geometry. CRM, Montreal (1988)
Masuda, T., Nakagami, Y., Woronowicz, S.L.: A \(C^{\ast }\)-algebraic framework for quantum groups. Int. J. Math. 14(9), 903–1001 (2003)
Reshetikhin, N.Y., Takhtajan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Algebra i analiz 1, 178 (1989) (Russian), English translation in Leningrad Math. J. 1
Soibelman, Y.S., Vaksman, L.L.: On some problems in the theory of quantum groups. representation theory and dynamical systems. Adv. Soviet Math. Am. Math. Soc. Providence, RI, 9, 3–55 (1992)
Walton, C., Wang, X.: On quantum groups associated to non-noetherian regular algebras of dimension 2. arXiv:1503.0918
Wang, S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195, 195–211 (1998)
Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987)
Woronowicz, S.L.: Compact quantum groups. In: Connes, A. (ed.) et al. Symétries Quantiques (Quantum symmetries) (Les Houches), p. 1998. Elsevier, Amsterdam (1995)
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Goswami, D. (2020). Quantum Symmetry of Classical Spaces. In: Roy, P., Cao, X., Li, XZ., Das, P., Deo, S. (eds) Mathematical Analysis and Applications in Modeling. ICMAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 302. Springer, Singapore. https://doi.org/10.1007/978-981-15-0422-8_8
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