Geodesic
PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES
Glasgow mathematical journal, Tome 49 (2007) no. 3, pp. 479-488

Voir la notice de l'article provenant de la source Cambridge University Press

Let , let be the quantum function algebra – over – associated to G, and let be the specialisation of the latter at a root of unity ε, whose order l is odd. There is a quantum Frobenius morphism that embeds the function algebra of G, in as a central Hopf subalgebra, so that is a module over . When , it is known by [3], [4] that (the complexification of) such a module is free, with rank ldim(G). In this note we prove a PBW-like theorem for , and we show that – when G is Matn or GLn – it yields explicit bases of over . As a direct application, we prove that and are free Frobenius extensions over and , thus extending some results of [5].
DOI : 10.1017/S0017089507003813
Mots-clés : Primary 20G42, Secondary 81R50 
GAVARINI, FABIO. PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES. Glasgow mathematical journal, Tome 49 (2007) no. 3, pp. 479-488. doi: 10.1017/S0017089507003813
@article{10_1017_S0017089507003813,
     author = {GAVARINI, FABIO},
     title = {PBW {THEOREMS} {AND} {FROBENIUS} {STRUCTURES} {FOR} {QUANTUM} {MATRICES}},
     journal = {Glasgow mathematical journal},
     pages = {479--488},
     year = {2007},
     volume = {49},
     number = {3},
     doi = {10.1017/S0017089507003813},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089507003813/}
}
TY  - JOUR
AU  - GAVARINI, FABIO
TI  - PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES
JO  - Glasgow mathematical journal
PY  - 2007
SP  - 479
EP  - 488
VL  - 49
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089507003813/
DO  - 10.1017/S0017089507003813
ID  - 10_1017_S0017089507003813
ER  - 
%0 Journal Article
%A GAVARINI, FABIO
%T PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES
%J Glasgow mathematical journal
%D 2007
%P 479-488
%V 49
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089507003813/
%R 10.1017/S0017089507003813
%F 10_1017_S0017089507003813

[1] 1.Andersen, H. H., Kexin, W. and Polo, P., Representations of quantum algebras, Invent. Math. 104 (1991), 1–59. Google Scholar | DOI

[2] 2.Bergman, G. M., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218. Google Scholar | DOI

[3] 3.Brown, K. A. and Gordon, I., The ramifications of the centres: quantised function algebras at roots of unity, Proc. London Math. Soc. (3) 84 (2002), 147–178. Google Scholar | DOI

[4] 4.Brown, K. A., Gordon, I. and Stafford, J. T., is a free module over preprint (2000), 3 pages. Google Scholar | arXiv

[5] 5.Brown, K. A., Gordon, I. and Stroppel, C., Cherednik, Hecke and quantum algebras as free modules and Calabi-Yau extensions preprint (2006), 31 pages. Google Scholar | arXiv

[6] 6.Chari, V. and Pressley, A., A guide to quantum groups (Cambridge University Press 1994). Google Scholar

[7] 7.De Concini, C. and Lyubashenko, V., Quantum function algebra at roots of 1, Adv. Math. 108 (1994), 205–262. Google Scholar | DOI

[8] 8.Dabrowski, L., Reina, C. and Zampa, A., A(SL(2)) at roots of unity is a free module over A(SL(2)) Lett. Math. Phys. 52 (2000), 339–342. Google Scholar | DOI

[9] 9.Gavarini, F., Quantum function algebras as quantum enveloping algebras, Comm. Algebra 26 (1998), 1795–1818. Google Scholar | DOI

[10] 10.Koelink, H. T., On *-representations of the Hopf *-algebra associated with the quantum group U (n), Compositio Math. 77 (1992), 199–231. Google Scholar

[11] 11.Levasseur, T. and Stafford, J. T., The quantum coordinate ring of the special linear group, J. Pure Appl. Algebra 86 (1993), 181–186. Google Scholar | DOI

[12] 12.Parshall, B. and Wang, J., Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439. Google Scholar

Cité par Sources :