Dijana Jakelic
University of Virginia
Charlotsville, USA
On representations of quantum groups: crystal bases and completions.
Abstract: Quantum groups emerged from quantum statistical mechanics in the mid 1980's. Since then they have turned out to be the fundamental algebraic structure behind many branches of mathematics. Crystal bases, defined by Kashiwara, provide an extremely powerful tool in the study of certain representations of quantum groups. Completion was introduced by Enright on a category of representations of a complex semisimple Lie algebra as a process of obtaining new representations from a given one, the latter sitting as a subrepresentation. I will survey the aforementioned topics and discuss some results and certain issues in bringing the concepts of crystal bases and completions together.