Projective representations of compact quantum groups and quantizations of non-compact Lie groups (FWOAL763)

A group is an abstraction of the notion of symmetry. A representation of a group is a concrete realization of that group as symmetries of some mathematical object. For example, for real compact simple Lie groups such as the group of rotations of the sphere, the linear representations have been known for a long time. These are important in many geometrical applications, as well as for the theory of special functions. More recently, quantum groups have been invented, offering a vast generalization of group theory. In particular, it turns out that any compact simple Lie group fits naturally into a one-parameter family of compact quantum groups. Although the linear representation theory of these quantum groups is similar to that of their classical counterparts, this is not true for more general representations such as projective representations. The first aim of this project is to study the projective representations of the above compact quantum groups from an analytical viewpoint, and to examine the structure of their associated crossed products. A second aim is to use these projective representations to construct, in an explicit fashion, one-parameter quantizations of classes of non-compact Lie groups. A third aim is to study the representation theory of the constructed non-compact quantum groups, and to elicit its connections to the theory of q-special functions.

Keywords:AATO, WISK, DWIS, CAMP, TWIS, ALG, Mathematics

Disciplines:Other mathematical sciences and statistics not elsewhere classified