| Title:
|
Braided coproduct, antipode and adjoint action for $U_q(sl_2)$ (English) |
| Author:
|
Pandžić, Pavle |
| Author:
|
Somberg, Petr |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
60 |
| Issue:
|
5 |
| Year:
|
2024 |
| Pages:
|
365-376 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q(\mathfrak{sl}_n)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q(\mathfrak{sl}_2)$. The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid. (English) |
| Keyword:
|
quantum group |
| Keyword:
|
quantum $\mathfrak{sl}_2$ |
| Keyword:
|
quantum adjoint action |
| Keyword:
|
tensor categories |
| Keyword:
|
braided tensor product |
| Keyword:
|
braided adjoint action |
| MSC:
|
16T20 |
| MSC:
|
20G42 |
| DOI:
|
10.5817/AM2024-5-365 |
| . |
| Date available:
|
2024-12-13T18:51:00Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152657 |
| . |
| Reference:
|
[1] Burdik, C., Navratil, O., Posta, S.: The adjoint representation of quantum algebra $U_q(sl_2)$.J. Nonlinear Math. Phys. 16 (1) (2009), 63–75. MR 2571814, 10.1142/S1402925109000066 |
| Reference:
|
[2] Huang, J.-S., Pandžić, P.: Dirac cohomology, unitary representations and a proof of a conjecture of Vogan.J. Amer. Math. Soc. 15 (1) (2002), 185–202. MR 1862801, 10.1090/S0894-0347-01-00383-6 |
| Reference:
|
[3] Huang, J.-S., Pandžić, P.: Dirac Operators in Representation Theory.Mathematics: Theory and Applications, Birkhauser, 2006. MR 2244116 |
| Reference:
|
[4] Klimyk, A., Schmüdgen, K.: Quantum groups and their representations.Texts and Monographs in Physics, pp. xx+552, Springer-Verlag, Berlin, 1997. MR 1492989 |
| Reference:
|
[5] Majid, S.: Quantum and braided linear algebra.J. Math. Phys. 34 (1993), 1176–1196, https://doi.org/10.1063/1.530193. MR 1207978, 10.1063/1.530193 |
| Reference:
|
[6] Majid, S.: Transmutation theory and rank for quantum braided groups.Mathematical Proceedings of the Cambridge Philosophical Society, vol. 113, 1993, pp. 45–70. MR 1188817, 10.1017/S0305004100075769 |
| Reference:
|
[7] Majid, S.: Algebras and Hopf algebras in braided categories.Advances in Hopf algebras, vol. 158, Marcel Dekker, Lec. Notes Pure Appl. Math. ed., 1994. MR 1289422 |
| Reference:
|
[8] Majid, S.: Foundations of Quantum Group Theory.Cambridge University Press, 1995. MR 1381692 |
| Reference:
|
[9] Majid, S.: Quantum and braided ZX calculus.J. Phys. A: Math. Theor. 55 (2022), 34 pp., paper No. 254007. MR 4438638, 10.1088/1751-8121/ac631f |
| Reference:
|
[10] Pandžić, P., Somberg, P.: Dirac operator for the quantum group $U_q(\mathfrak{sl}_3)$.in preparation. |
| Reference:
|
[11] Pandžić, P., Somberg, P.: Dirac operator and its cohomology for the quantum group $U_q(\mathfrak{sl}_2)$.J. Math. Phys. 58 (4) (2017), 13 pp., Paper No. 041702. MR 3632540 |
| Reference:
|
[12] Parthasarathy, R.: Dirac operator and the discrete series.Ann. of Math. 96 (1972), 1–30. MR 0318398, 10.2307/1970892 |
| Reference:
|
[13] Vogan, D.: Dirac operators and unitary representations.3 talks at MIT Lie groups seminar, Fall 1997. |
| . |
