Irreducible quantum group modules with finite dimensional weight spaces

By Dennis Hasselstrøm Pedersen
PhD Dissertations
October 2015
Abstract:

We classify all irreducible weight modules for a quantized enveloping algebra $U_q(\mathfrak{g})$ at most $q\in\mathbb{C}^*$ when the simple Lie algebra $\mathfrak{g}$ is not of type $G_2$. More precisely, our classificiation is carried out when $q$ is either an odd root of unity or transcendental over $\mathbb{Q}$.

By a weight module we mean a finitely generated $U_q$-module which has finite dimensional weight spaces and is a sum of those. Our approach follows the procedures used by S. Fernando and O. Mathieu to solve the corresponding problem for semisimple complex Lie algebra modules. To achieve this we have to overcome a number of obstacles not present in the classical case.

In the process we also construct twisting functors rigerously for quantum group modules, study twisted Verma modules and show that these admit a Jantzen filtration with corresponding Jantzen sum formula.

Thesis advisor: Henning Haahr Andersen
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