On representation theory of quantum groups at roots of unity

Piotr Kondratowicz; Piotr Podleś

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 223-248
  • ISSN: 0137-6934

Abstract

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Irreducible representations of quantum groups (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.

How to cite

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Kondratowicz, Piotr, and Podleś, Piotr. "On representation theory of quantum $SL_{q}(2)$ groups at roots of unity." Banach Center Publications 40.1 (1997): 223-248. <http://eudml.org/doc/252182>.

@article{Kondratowicz1997,
abstract = {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.},
author = {Kondratowicz, Piotr, Podleś, Piotr},
journal = {Banach Center Publications},
keywords = {quantum group; quantum enveloping algebra; irreducible representation; Hopf algebra; Haar functional},
language = {eng},
number = {1},
pages = {223-248},
title = {On representation theory of quantum $SL_\{q\}(2)$ groups at roots of unity},
url = {http://eudml.org/doc/252182},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Kondratowicz, Piotr
AU - Podleś, Piotr
TI - On representation theory of quantum $SL_{q}(2)$ groups at roots of unity
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 223
EP - 248
AB - 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.
LA - eng
KW - quantum group; quantum enveloping algebra; irreducible representation; Hopf algebra; Haar functional
UR - http://eudml.org/doc/252182
ER -

References

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  1. [1] H. H. Andersen, J. Paradowski, Fusion categories arising from semisimple Lie algebras, Commun. Math. Phys. 169, (1995), 563-588. Zbl0827.17010
  2. [2] G. Cliff, A tensor product theorem for quantum linear groups at even roots of unity, J. Algebra 165, (1994), 566-575. Zbl0812.17012
  3. [3] C. De Concini, V. Lyubashenko, Quantum function algebra at roots of 1, Preprints di Matematica 5, Scuola Normale Superiore Pisa, February 1993. Zbl0846.17008
  4. [4] D. V. Gluschenkov, A. V. Lyakhovskaya, Regular Representation of the Quantum Heisenberg Double (q is a root of unity), UUITP - 27/1993, hep-th/9311075. 
  5. [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. Zbl0602.17005
  6. [6] G. Lusztig, Modular representations and quantum groups, Contemporary Mathematics 82, (1989), 59-77. Zbl0665.20022
  7. [7] P. Podleś, Complex Quantum Groups and Their Real Representations, Publ. RIMS, Kyoto University 28, (1992), 709-745. Zbl0809.17003
  8. [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 . Zbl0715.17015
  9. [9] P. Roche, D. Arnaudon, Irreducible Representations of the Quantum Analogue of SU(2), Lett. Math. Phys. 17, (1989), 295-300. Zbl0694.17005
  10. [10] M. Takeuchi, Some topics on , J. Algebra 147, (1992), 379-410. Zbl0760.16015
  11. [11] B. Parshall, J. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 439, Providence, 1991. 
  12. [12] S. L. Woronowicz, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. RIMS, Kyoto University 23, (1987), 117-181. Zbl0676.46050
  13. [13] S. L. Woronowicz, Compact Matrix Pseudogroups, Commun. Math. Phys. 111, (1987), 613-665. Zbl0627.58034
  14. [14] S. L. Woronowicz, Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Commun. Math. Phys. 122, (1989), 125-170. Zbl0751.58042
  15. [15] S. L. Woronowicz, The lecture 'Quantum groups' at Faculty of Physics, University of Warsaw (1990/91) 
  16. [16] S. L. Woronowicz, New quantum deformation of SL(2,𝐂). Hopf algebra level, Rep. Math. Phys. 30, (1991), 259-269. Zbl0759.17010
  17. [17] S. L. Woronowicz, S. Zakrzewski, Quantum deformations of the Lorentz group. The Hopf *-algebra level, Comp. Math. 90, (1994), 211-243. Zbl0798.16026

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