Some results on finite dimensional modules of current and loop algebras

In the field of finite dimensional modules of current and loop algebras a lot of research was done and progress was made in the last two decades. Resuming the discussions we showed in the present thesis that Demazure modules are fusion products of ``smaller`` Demazure modules and calculated a decomposition of the fundamental Demazure modules as g-modules. Combining both results we obtained new dimension and character formulas for Demazure modules. As an application we constructed certain affine highest weight modules as the direct limit of fusion products of Demazure modules. We proved that the fundamental Demazure modules are isomorphic to Kirillov-Reshetikhin modules for the current algebra. Furthermore we proved as an analogon in the combinatorial representation theory that the Kirillov-Reshetikhin crystal contains the Demazure crystal. We give a new and elementary proof of the dimension formula of Weyl modules for the loop algebra in the simply laced case. For twisted loop algebras we provide a classification of Weyl modules and proved the analog theorems from the untwisted case, e.g. tensor product structure, dimension and character formulas