Applications of Crystal Bases to Current Problems in Representation Theory

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

We study finite dimensional representations of current algebras, loop algebras and their quantized versions. For the current algebra of a simple Lie algebra of type {\tt ADE}, we show that Kirillov-Reshetikhin modules and Weyl modules are in fact all Demazure modules. As a consequence one obtains an elementary proof of the dimension formula for Weyl modules for the current and the loop algebra. Further, we show that the crystals of the Weyl and the Demazure module are the same up to some additional label zero arrows for the Weyl module. For the current algebra $\Lgc$ of an arbitrary simple Lie…