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.
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M.K. a bénéficié d’un soutien partiel JSPS Grants in Aid for Scientific Research 235-40023.
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Gros, M., Kaneda, M. Un scindage du morphisme de Frobenius quantique. Ark Mat 53, 271–301 (2015). https://doi.org/10.1007/s11512-014-0205-8
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DOI: https://doi.org/10.1007/s11512-014-0205-8