Representations of s ⁢ l q ⁢ 3 at the roots of unity

Nicoletta Cantarini

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1996)

  • Volume: 7, Issue: 4, page 201-212
  • ISSN: 1120-6330

Abstract

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In this paper we study the irreducible finite dimensional representations of the quantized enveloping algebra U q ⁢ g associated to g = s ⁢ l ⁢ 3 , at the roots of unity. It is known that these representations are parametrized, up to isomorphisms, by the conjugacy classes of the group G = S ⁢ L ⁢ 3 . We get a complete classification of the representations corresponding to the submaximal unipotent conjugacy class and therefore a proof of the De Concini-Kac conjecture about the dimension of the U q ⁢ g -modules at the roots of 1 in the case of g = s ⁢ l ⁢ 3 .

How to cite

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Cantarini, Nicoletta. "Representations of \( sl_{q} (3) \) at the roots of unity." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 7.4 (1996): 201-212. <http://eudml.org/doc/244290>.

@article{Cantarini1996,
abstract = {In this paper we study the irreducible finite dimensional representations of the quantized enveloping algebra \( \mathcal\{U\}\_\{q\} (g) \) associated to \( g = sl (3) \), at the roots of unity. It is known that these representations are parametrized, up to isomorphisms, by the conjugacy classes of the group \( G = SL(3) \). We get a complete classification of the representations corresponding to the submaximal unipotent conjugacy class and therefore a proof of the De Concini-Kac conjecture about the dimension of the \( \mathcal\{U\}\_\{q\} (g) \)-modules at the roots of \( 1 \) in the case of \( g = sl (3) \).},
author = {Cantarini, Nicoletta},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Enveloping algebra; Representation; Cartan matrix; irreducible representation; quantized universal enveloping algebras; subregular unipotent conjugacy class; De Concini-Kac-Procesi conjecture},
language = {eng},
month = {12},
number = {4},
pages = {201-212},
publisher = {Accademia Nazionale dei Lincei},
title = {Representations of \( sl\_\{q\} (3) \) at the roots of unity},
url = {http://eudml.org/doc/244290},
volume = {7},
year = {1996},
}

TY - JOUR
AU - Cantarini, Nicoletta
TI - Representations of \( sl_{q} (3) \) at the roots of unity
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1996/12//
PB - Accademia Nazionale dei Lincei
VL - 7
IS - 4
SP - 201
EP - 212
AB - In this paper we study the irreducible finite dimensional representations of the quantized enveloping algebra \( \mathcal{U}_{q} (g) \) associated to \( g = sl (3) \), at the roots of unity. It is known that these representations are parametrized, up to isomorphisms, by the conjugacy classes of the group \( G = SL(3) \). We get a complete classification of the representations corresponding to the submaximal unipotent conjugacy class and therefore a proof of the De Concini-Kac conjecture about the dimension of the \( \mathcal{U}_{q} (g) \)-modules at the roots of \( 1 \) in the case of \( g = sl (3) \).
LA - eng
KW - Enveloping algebra; Representation; Cartan matrix; irreducible representation; quantized universal enveloping algebras; subregular unipotent conjugacy class; De Concini-Kac-Procesi conjecture
UR - http://eudml.org/doc/244290
ER -

References

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  1. DE CONCINI, C. - KAC, V. G., Representations of quantum groups at roots of 1. Progress in Math., 92, Birkhäuser, 1990, 471-506. Zbl0738.17008MR1103601
  2. DE CONCINI, C. - KAC, V. G., Representations of quantum groups at roots of 1: reduction to the exceptional case. International Journal of Modern Physics A, vol. 7, Suppl. 1A, 1992, 141-149. Zbl0867.17007MR1187546
  3. DE CONCINI, C. - KAC, V. G. - PROCESI, C., Quantum coadjoint action. Journal of the American Mathematical Society, 5, 1992, 151-190. Zbl0747.17018MR1124981DOI10.2307/2152754
  4. DE CONCINI, C. - KAC, V. G. - PROCESI, C., Some remarkable degenerations of quantum groups. Comm. Math. Phys., 157, 1993, 405-427. Zbl0795.17006MR1244875
  5. DRINFELD, V. G., Hopf algebras and quantum Yang-Baxter equation. Soviet. Math. Dokl., 32, 1985, 254-258. Zbl0588.17015MR802128
  6. DRINFELD, V. G., Quantum groups. Proc. ICM, Berkeley, 1, 1986, 798-820. MR934283
  7. HESSELINK, W., The normality of closures of orbits in a Lie algebra. Commentarii Math. Helvet., 54, 1979, 105-110. Zbl0395.14014MR522033DOI10.1007/BF02566258
  8. JIMBO, M., A q -difference analogue of U ⁢ g and the Yang-Baxter equation. Lett. Math. Phys., 10, 1985, 63-69. Zbl0587.17004MR797001DOI10.1007/BF00704588
  9. LEVENDORSKEI, S. Z. - SOIBELMAN, YA. S., Algebras of functions on compact quantum groups, Schubert cells and quantum tori. Comm. Math. Phys., 139, 1991, 141-170. Zbl0729.17011MR1116413
  10. LUSZTIG, G., Quantum groups at roots of 1. Geom. Ded., 35, 1990, 89-114. Zbl0714.17013MR1066560DOI10.1007/BF00147341
  11. TITS, J., Sur les constants de structure et le théorème d'existence des algebres de Lie semi-simple. Publ. Math. IHES, 31, 1966, 21-58. Zbl0145.25804MR214638

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