Spécialisation des groupes quantiques à une racine de l’unité et représentations de dimension finie

Abstract

In this master thesis, we study the restricted specialization of quantum enveloping algebras defined by Lusztig, as well as their representation theory.

First, we review briefly the class of quantum groups $U_{q}$ introduced by Drinfel’d and Jimbo as deformations of the classical universal enveloping algebra of a simple Lie algebra and Lusztig’s construction of an integral form on it.

Then, we consider the restricted specialization ${U_{\epsilon}^{{\mathrm{res}}}}$ at a root of the unity $\epsilon$. Contrary to most authors, we do not assume that the order $l$ of the root is odd, that it is prime with the extra-diagonal coefficients of the Cartan matrix, nor that it is greater than the absolute values of these coefficients. We deal with the finite dimensional representations of the restricted specialization, define a character for these representations and classify the irreducible ones.

When $l$ is greater than the absolute values of the extra-diagonal coefficients of the Cartan matrix, we propose a way to generalize Lusztig’s factorization of irreducible ${U_{\epsilon}^{{\mathrm{res}}}}$-modules as a tensor product of a module over a finite dimensional subalgebra of ${U_{\epsilon}^{{\mathrm{res}}}}$ and the pullback of a module over the classical enveloping algebra. In general, it seems necessary to consider the pullback of a $U_{{\epsilon^{{\prime}}}}^{{\mathrm{res}}}(\mathfrak{g})$-module, for some root of unity $\epsilon^{{\prime}}$ whose order is, in most cases, not too large.

Finally we come back to the usual restrictions on $l$ and give an overview of the framework of tilting modules. We mention that they form a family of finite dimensional ${U_{\epsilon}^{{\mathrm{res}}}}$-modules stable by various operations and explain how we can deduce a semisimple category from them.