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## Banach Center Publications

1997 | 40 | 1 | 223-248
Tytuł artykułu

### On representation theory of quantum $SL_{q}(2)$ groups at roots of unity

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Irreducible representations of quantum groups $SL_q(2)$ (in Woronowicz' approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the case of q being an odd root of unity. Here we find the irreducible representations for all roots of unity (also of an even degree), as well as describe "the diagonal part" of the tensor product of any two irreducible representations. An example of a not completely reducible representation is given. Non-existence of Haar functional is proved. The corresponding representations of universal enveloping algebras of Jimbo and Lusztig are provided. We also recall the case of general q. Our computations are done in explicit way.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
223-248
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
• Department of Mathematical Methods in Physics, University of Warsaw , Hoża 74, 00-682 Warszawa, Poland
autor
• Department of Mathematical Methods in Physics, University of Warsaw , Hoża 74, 00-682 Warszawa, Poland
Bibliografia
• [1] H. H. Andersen, J. Paradowski, Fusion categories arising from semisimple Lie algebras, Commun. Math. Phys. 169, (1995), 563-588.
• [2] G. Cliff, A tensor product theorem for quantum linear groups at even roots of unity, J. Algebra 165, (1994), 566-575.
• [3] C. De Concini, V. Lyubashenko, Quantum function algebra at roots of 1, Preprints di Matematica 5, Scuola Normale Superiore Pisa, February 1993.
• [4] D. V. Gluschenkov, A. V. Lyakhovskaya, Regular Representation of the Quantum Heisenberg Double ${U_{q}sl(2),Fun_{q}(SL(2))}$ (q is a root of unity), UUITP - 27/1993, hep-th/9311075.
• [5] M. Jimbo, A q-analogue of U(gl(N+1)), Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys. 11, (1986), 247-252.
• [6] G. Lusztig, Modular representations and quantum groups, Contemporary Mathematics 82, (1989), 59-77.
• [7] P. Podleś, Complex Quantum Groups and Their Real Representations, Publ. RIMS, Kyoto University 28, (1992), 709-745.
• [8] N. Yu. Reshetikhin, L. A. Takhtadzyan, L. D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J., Vol. 1, No. 1, (1990), 193-225 .
• [9] P. Roche, D. Arnaudon, Irreducible Representations of the Quantum Analogue of SU(2), Lett. Math. Phys. 17, (1989), 295-300.
• [10] M. Takeuchi, Some topics on $GL_q(n)$, J. Algebra 147, (1992), 379-410.
• [11] B. Parshall, J. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 439, Providence, 1991.
• [12] S. L. Woronowicz, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. RIMS, Kyoto University 23, (1987), 117-181.
• [13] S. L. Woronowicz, Compact Matrix Pseudogroups, Commun. Math. Phys. 111, (1987), 613-665.
• [14] S. L. Woronowicz, Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Commun. Math. Phys. 122, (1989), 125-170.
• [15] S. L. Woronowicz, The lecture 'Quantum groups' at Faculty of Physics, University of Warsaw (1990/91)
• [16] S. L. Woronowicz, New quantum deformation of SL(2,𝐂). Hopf algebra level, Rep. Math. Phys. 30, (1991), 259-269.
• [17] S. L. Woronowicz, S. Zakrzewski, Quantum deformations of the Lorentz group. The Hopf *-algebra level, Comp. Math. 90, (1994), 211-243.
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