| Title:
|
Covariantization of quantized calculi over quantum groups (English) |
| Author:
|
Akrami, Seyed Ebrahim |
| Author:
|
Farzi, Shervin |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
145 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
415-433 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce a method for construction of a covariant differential calculus over a Hopf algebra $A$ from a quantized calculus $da=[D,a]$, $a\in A$, where $D$ is a candidate for a Dirac operator for $A$. We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra $A^\circ $. We apply this method to the Dirac operator for the quantum $\rm SL(2)$ given by S. Majid. We find that the differential calculus obtained by our method is the standard bicovariant 4D-calculus. We also apply this method to the Dirac operator for the quantum $\rm SL(2)$ given by P. N. Bibikov and P. P. Kulish and show that the resulted differential calculus is $8$-dimensional. (English) |
| Keyword:
|
Hopf algebra |
| Keyword:
|
quantum group |
| Keyword:
|
covariant first order differential calculus |
| Keyword:
|
quantized calculus |
| Keyword:
|
Dirac operator |
| MSC:
|
58B32 |
| MSC:
|
81Q30 |
| idZBL:
|
07286022 |
| idMR:
|
MR4221843 |
| DOI:
|
10.21136/MB.2019.0142-18 |
| . |
| Date available:
|
2020-11-18T09:57:50Z |
| Last updated:
|
2021-04-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148433 |
| . |
| Reference:
|
[1] Bibikov, P. N., Kulish, P. P.: Dirac operators on the quantum group ${\rm SU}(2)$ and the quantum sphere.J. Math. Sci., New York 100 (1997), 2039-2050. Zbl 0954.58004, MR 1627837, 10.1007/BF02675726 |
| Reference:
|
[2] Brzeziński, T., Majid, S.: A class of bicovariant differential calculi on Hopf algebras.Lett. Math. Phys. 26 (1992), 67-78. Zbl 0776.58005, MR 1193627, 10.1007/BF00420519 |
| Reference:
|
[3] Connes, A.: Noncommutative Geometry.Academic Press, San Diego (1994). Zbl 0818.46076, MR 1303779 |
| Reference:
|
[4] Klimyk, A., Schmüdgen, K.: Quantum Groups and Their Representations.Texts and Monographs in Physics. Springer, Berlin (1997). Zbl 0891.17010, MR 1492989, 10.1007/978-3-642-60896-4 |
| Reference:
|
[5] Majid, S.: Foundations of Quantum Group Theory.Cambridge Univ. Press, Cambridge (1995). Zbl 0857.17009, MR 1381692, 10.1017/CBO9780511613104 |
| Reference:
|
[6] Majid, S.: Riemannian geometry of quantum groups and finite groups with nonuniversal differentials.Commun. Math. Phys. 225 (2002), 131-170. Zbl 0999.58004, MR 1877313, 10.1007/s002201000564 |
| Reference:
|
[7] Woronowicz, S. L.: Differential calculus on compact matrix pseudogroups (quantum groups).Commun. Math. Phys. 122 (1989), 125-170. Zbl 0751.58042, MR 0994499, 10.1007/BF01221411 |
| . |
