Séminaires : Séminaire d'Algèbre

Equipe(s) : gr,
Responsables :J. Alev, D. Hernandez, B. Keller, Th. Levasseur, et S. Morier-Genoud.
Email des responsables : Jacques Alev <jacques.alev@univ-reims.fr>, David Hernandez <david.hernandez@imj-prg.fr>, Bernhard Keller <bernhard.keller@imj-prg.fr>, Thierry Levasseur <Thierry.Levasseur@univ-brest.fr>, Sophie Morier-Genoud <sophie.morier-genoud@imj-prg.fr>
Salle : 001
Adresse :IHP


Orateur(s) Naihong HU - Shanghai,
Titre Loewy Filtration and Quantum de Rham Cohomology
Horaire14:00 à 15:00
RésumeThis talk is about : (1) the indecomposable submodule structures of quantum divided power algebra $\mathcal{A}_q(n)$ introduced in my earlier work (2000) and its truncated objects $\mathcal{A}_q(n, m)$, where an ``intertwinedly-lifting`` method is established to prove the indecomposability of a module when its socle is non-simple. (2) The Loewy filtrations are described for all homogeneous subspaces $\mathcal{A}^{(s)}_q(n)$ or $\mathcal{A}_q^{(s)}(n, m)$, the Loewy layers and dimensions are determined. The rigidity of these indecomposable modules is proved. An interesting combinatorial identity is derived from our realization model for a class of indecomposable $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules. (3) Meanwhile, the quantum Grassmann algebra $\Omega_q(n)$ over $\mathcal{A}_q(n)$ is defined and constructed, together with the quantum de Rham complex $(\Omega_q(n), d^\bullet)$ via defining the appropriate $q$-differentials, and its subcomplex $(\Omega_q(n, m), d^\bullet)$. For the latter, the corresponding quantum de Rham cohomology modules are decomposed into the direct sum of some sign-trivial modules. This is a joint work with H.X. Gu.