Arkiv för Matematik

  • Ark. Mat.
  • Volume 53, Number 2 (2015), 271-301.

Un scindage du morphisme de Frobenius quantique

Michel Gros and Masaharu Kaneda

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Résumé

Nous montrons que le morphisme de Frobenius quantique construit par Lusztig dans le cadre des algèbres enveloppantes quantiques $U_{\mathcal{B}}$ spécialisées en une racine de l’unité admet un scindage multiplicatif (non unitaire). Nous utilisons pour ce faire une base de la partie torique de la petite algèbre quantique constituée d’idempotents orthogonaux deux à deux et de somme 1 et faisons de même dans le cas «modulaire» pour l’algèbre des distributions d’un groupe algébrique semi-simple.

Abstract

We show that the quantum Frobenius morphism constructed by Lusztig in the setting of the quantum enveloping algebra $U_{\mathcal{B}}$ specialized at a root of unity admits a (nonunital) multiplicative splitting. We construct the splitting using a basis of the toral part of the small quantum algebra consisting of pairwise orthogonal idempotents summing up to 1, and likewise in the modular case of the algebra of distributions for a semisimple algebraic group.

Note

M.K. a bénéficié d’un soutien partiel JSPS Grants in Aid for Scientific Research 235-40023.

Article information

Source
Ark. Mat., Volume 53, Number 2 (2015), 271-301.

Dates
Received: 19 August 2014
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485802712

Digital Object Identifier
doi:10.1007/s11512-014-0205-8

Mathematical Reviews number (MathSciNet)
MR3391172

Zentralblatt MATH identifier
1331.17010

Rights
2014 © Institut Mittag-Leffler

Citation

Gros, Michel; Kaneda, Masaharu. Un scindage du morphisme de Frobenius quantique. Ark. Mat. 53 (2015), no. 2, 271--301. doi:10.1007/s11512-014-0205-8. https://projecteuclid.org/euclid.afm/1485802712


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Bibliographie

  • Andersen, H. H. et Kaneda, M., Loewy series of modules for the first Frobenius kernel in a reductive algebraic group, Proc. Lond. Math. Soc. 59 (1989), 74–98.
  • Andersen, H. H., Polo, P. et Wen, K., Injective modules for quantum algebras, Amer. J. Math. 114 (1992), 571–604.
  • Bourbaki, N., Éléments de mathématique, Algèbre, Chapitres 1–3, Hermann, Paris, 1970.
  • Gros, M., A Splitting of the Frobenius morphism on the whole algebra of distributions of SL2, Algebr. Represent. Theory 15 (2012), 109–118.
  • Gros, M. et Kaneda, M., Contraction par Frobenius de G-modules, Ann. Inst. Fourier (Grenoble) 61 (2011), 2507–2542.
  • Humphreys, J. E., Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math. 9, Springer, Berlin, 1972.
  • Jantzen, J. C., Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs 107, Amer. Math. Soc., Providence, RI, 2003.
  • Kaneda, M., On the Frobenius morphism of flag schemes, Pacific J. Math. 163 (1994), 315–336.
  • Kumar, S. et Littelmann, P., Algebraization of Frobenius splitting via quantum groups, Ann. of Math. 155 (2002), 491–551.
  • Littelmann, P., Contracting modules and standard monomial theory for symmetrizable Kac–Moody algebras, J. Appl. Math. Stoch. Anal. 11 (1998), 551–567.
  • Lusztig, G., Modular representations and quantum groups , dans Classical Groups and Related Topics (Beijing, 1987), Contemp. Math. 82, pp. 59–77, Amer. Math. Soc., Providence, RI, 1989.
  • Lusztig, G., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257–296.
  • Lusztig, G., Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89–113.
  • Lusztig, G., Introduction to Quantum Groups, Progress in Math. 110, Birkhäuser, Boston, MA, 1993.
  • McGerty, K., Generalized q-Schur algebras and quantum Frobenius, Adv. Math. 214 (2007), 116–131.
  • Parshall, B. et Scott, L., On $p\mspace {1mu}$-filtrations of Weyl modules, Preprint, 2013.