Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 97-121, 2017


On the quantum groups and semigroups of maps between noncommutative spaces

Maysam Maysami Sadr

Received July 18, 2015.  First published February 24, 2017.

Abstract:  We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are introduced: quantum group of gauge transformations, Pontryagin dual of a quantum group, and Galois-Hopf-algebra of an algebra extension.
Keywords:  Hopf-algebra; bialgebra; quantum group; noncommutative geometry
Classification MSC:  16T05, 16T10, 16T20, 58B34
DOI:  10.21136/CMJ.2017.0393-15


References:
[1] T. Banica: Quantum automorphism groups of small metric spaces. Pac. J. Math. 219 (2005), 27-51. DOI 10.2140/pjm.2005.219.27 | MR 2174219 | Zbl 1104.46039
[2] T. Banica, J. Bichon, B. Collins: Quantum permutation groups: a survey. Noncommutative Harmonic Analysis with Applications to Probability Papers presented at the 9th Workshop, Bedlewo, Poland, 2006, Banach Center Publications 78, Polish Academy of Sciences, Institute of Mathematics, Warsaw M. Bozejko et al. (2008), 13-34. MR 2402345 | Zbl 1140.46329
[3] H. J. Baues: Algebraic Homotopy. Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge (1989). MR 0985099 | Zbl 0688.55001
[4] T. Brzeziński, S. Majid: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157 (1993), 591-638. DOI 10.1007/BF02096884 | MR 1243712 | Zbl 0817.58003
[5] S. M. Gersten: Homotopy theory of rings. J. Algebra 19 (1971), 396-415. DOI 10.1016/0021-8693(71)90098-6 | MR 0291253 | Zbl 0264.18009
[6] M. Hovey, J. H. Palmieri, N. P. Strickland: Axiomatic stable homotopy theory. Mem. Am. Math. Soc. Vol. 128 (1997), 114 pages. DOI 10.1090/memo/0610 | MR 1388895 | Zbl 0881.55001
[7] J. F. Jardine: Algebraic homotopy theory. Can. J. Math. 33 (1981), 302-319. DOI 10.4153/CJM-1981-025-9 | MR 0617621 | Zbl 0444.55018
[8] T. Y. Lam: Lectures on Modules and Rings. Graduate Texts in Mathematics 189, Springer, New York (1999). DOI 10.1007/978-1-4612-0525-8 | MR 1653294 | Zbl 0911.16001
[9] S. Majid: Foundations of Quantum Group Theory. Cambridge Univ. Press, Cambridge (1995). MR 1381692 | Zbl 0857.17009
[10] J. P. May: Picard groups, Grothendieck rings, and Burnside rings of categories. Adv. Math. 163 (2001), 1-16. DOI 10.1006/aima.2001.1996 | MR 1867201 | Zbl 0994.18004
[11] J. S. Milne: Basic Theory of Affine Group Schemes. Available online: www.jmilne.org /math/CourseNotes/AGS.pdf (2012).
[12] P. Podleś: Quantum spaces and their symmetry groups. PhD Thesis, Department of Mathematical Methods in Physics Faculty of Physics, Warsaw University (1989).
[13] M. M. Sadr: A kind of compact quantum semigroups. Int. J. Math. Math. Sci. 2012 (2012), Article ID 725270, 10 pages. DOI 10.1155/2012/725270 | MR 3009563 | Zbl 1267.46079
[14] A. Skalski, P. M. Sołtan: Quantum families of invertible maps and related problems. Can. J. Math. 68 (2016), 698-720. DOI 10.4153/CJM-2015-037-9 | MR 3492633 | Zbl 06589338
[15] P. M. Sołtan: Quantum families of maps and quantum semigroups on finite quantum spaces. J. Geom. Phys. 59 (2009), 354-368. DOI 10.1016/j.geomphys.2008.11.007 | MR 2501746 | Zbl 1160.58007
[16] P. M. Sołtan: Quantum $\rm SO(3)$ groups and quantum group actions on $M_2$. J. Noncommut. Geom. 4 (2010), 1-28. DOI 10.4171/JNCG/48 | MR 2575388 | Zbl 1194.46108
[17] P. M. Sołtan: On quantum maps into quantum semigroups. Houston J. Math. 40 (2014), 779-790. MR 3275623 | Zbl 1318.46051
[18] M. E. Sweedler: Hopf Algebras. Mathematics Lecture Note Series, W. A. Benjamin, New York (1969). MR 0252485 | Zbl 0194.32901
[19] S. Wang: Free products of compact quantum groups. Commun. Math. Phys. 167 (1995), 671-692. DOI 10.1007/BF02101540 | MR 1316765 | Zbl 0838.46057
[20] S. Wang: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195 (1998), 195-211. DOI 10.1007/s002200050385 | MR 1637425 | Zbl 1013.17008
[21] S. L. Woronowicz: Pseudospaces, pseudogroups and Pontrjagin duality. Mathematical Problems in Theoretical Physics Proc. Int. Conf. on Mathematical Physics, Lausanne, 1979, Lect. Notes Phys. Vol. 116, Springer, Berlin 407-412 (1980). DOI 10.1007/3-540-09964-6_354 | MR 0582650 | Zbl 03810280

Affiliations:   Maysam Maysami Sadr, Department of Mathematics, Institute for Advanced Studies in Basic Sciences, No. 444, Prof. Yousef Sobouti Blvd., P. O. Box 45195-1159, Zanjan 45137-66731, Zanjan, Iran, e-mail: sadr@iasbs.ac.ir

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