On the quantum groups and semigroups of maps between noncommutative spaces
Czechoslovak Mathematical Journal (2017)
- Volume: 67, Issue: 1, page 97-121
- ISSN: 0011-4642
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topSadr, Maysam Maysami. "On the quantum groups and semigroups of maps between noncommutative spaces." Czechoslovak Mathematical Journal 67.1 (2017): 97-121. <http://eudml.org/doc/287902>.
@article{Sadr2017,
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.},
author = {Sadr, Maysam Maysami},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hopf-algebra; bialgebra; quantum group; noncommutative geometry; quantum semigroup structure},
language = {eng},
number = {1},
pages = {97-121},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the quantum groups and semigroups of maps between noncommutative spaces},
url = {http://eudml.org/doc/287902},
volume = {67},
year = {2017},
}
TY - JOUR
AU - Sadr, Maysam Maysami
TI - On the quantum groups and semigroups of maps between noncommutative spaces
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 97
EP - 121
AB - 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.
LA - eng
KW - Hopf-algebra; bialgebra; quantum group; noncommutative geometry; quantum semigroup structure
UR - http://eudml.org/doc/287902
ER -
References
top- Banica, T., 10.2140/pjm.2005.219.27, Pac. J. Math. 219 (2005), 27-51. (2005) Zbl1104.46039MR2174219DOI10.2140/pjm.2005.219.27
- Banica, T., Bichon, J., Collins, B., Quantum permutation groups: a survey, Noncommutative Harmonic Analysis with Applications to Probability Papers presented at the 9th Workshop, Będlewo, Poland, 2006, Banach Center Publications 78, Polish Academy of Sciences, Institute of Mathematics, Warsaw M. Bożejko et al. (2008), 13-34. (2008) Zbl1140.46329MR2402345
- Baues, H. J., Algebraic Homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, Cambridge (1989). (1989) Zbl0688.55001MR0985099
- Brzeziński, T., Majid, S., 10.1007/BF02096884, Commun. Math. Phys. 157 (1993), 591-638. (1993) Zbl0817.58003MR1243712DOI10.1007/BF02096884
- Gersten, S. M., 10.1016/0021-8693(71)90098-6, J. Algebra 19 (1971), 396-415. (1971) Zbl0264.18009MR0291253DOI10.1016/0021-8693(71)90098-6
- Hovey, M., Palmieri, J. H., Strickland, N. P., 10.1090/memo/0610, Mem. Am. Math. Soc. Vol. 128 (1997), 114 pages. (1997) Zbl0881.55001MR1388895DOI10.1090/memo/0610
- Jardine, J. F., 10.4153/CJM-1981-025-9, Can. J. Math. 33 (1981), 302-319. (1981) Zbl0444.55018MR0617621DOI10.4153/CJM-1981-025-9
- Lam, T. Y., 10.1007/978-1-4612-0525-8, Graduate Texts in Mathematics 189, Springer, New York (1999). (1999) Zbl0911.16001MR1653294DOI10.1007/978-1-4612-0525-8
- Majid, S., Foundations of Quantum Group Theory, Cambridge Univ. Press, Cambridge (1995). (1995) Zbl0857.17009MR1381692
- May, J. P., 10.1006/aima.2001.1996, Adv. Math. 163 (2001), 1-16. (2001) Zbl0994.18004MR1867201DOI10.1006/aima.2001.1996
- Milne, J. S., Basic Theory of Affine Group Schemes, Available online: www.jmilne.org /math/CourseNotes/AGS.pdf (2012). (2012)
- Podleś, P., Quantum spaces and their symmetry groups, PhD Thesis, Department of Mathematical Methods in Physics Faculty of Physics, Warsaw University (1989). (1989)
- Sadr, M. M., 10.1155/2012/725270, Int. J. Math. Math. Sci. 2012 (2012), Article ID 725270, 10 pages. (2012) Zbl1267.46079MR3009563DOI10.1155/2012/725270
- Skalski, A., Sołtan, P. M., 10.4153/CJM-2015-037-9, Can. J. Math. 68 (2016), 698-720. (2016) Zbl06589338MR3492633DOI10.4153/CJM-2015-037-9
- Sołtan, P. M., 10.1016/j.geomphys.2008.11.007, J. Geom. Phys. 59 (2009), 354-368. (2009) Zbl1160.58007MR2501746DOI10.1016/j.geomphys.2008.11.007
- Sołtan, P. M., 10.4171/JNCG/48, J. Noncommut. Geom. 4 (2010), 1-28. (2010) Zbl1194.46108MR2575388DOI10.4171/JNCG/48
- Sołtan, P. M., On quantum maps into quantum semigroups, Houston J. Math. 40 (2014), 779-790. (2014) Zbl1318.46051MR3275623
- Sweedler, M. E., Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, New York (1969). (1969) Zbl0194.32901MR0252485
- Wang, S., 10.1007/BF02101540, Commun. Math. Phys. 167 (1995), 671-692. (1995) Zbl0838.46057MR1316765DOI10.1007/BF02101540
- Wang, S., 10.1007/s002200050385, Commun. Math. Phys. 195 (1998), 195-211. (1998) Zbl1013.17008MR1637425DOI10.1007/s002200050385
- Woronowicz, S. L., 10.1007/3-540-09964-6_354, Mathematical Problems in Theoretical Physics Proc. Int. Conf. on Mathematical Physics, Lausanne, 1979, Lect. Notes Phys. Vol. 116, Springer, Berlin 407-412 (1980). (1980) Zbl03810280MR0582650DOI10.1007/3-540-09964-6_354
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