New approach to representation theory of semisimple Lie algebras and quantum algebras

摘要

A method to construct in explicit form the generators of the simple roots ofan arbitrary finite-dimensional representation of a quantum or standardsemisimple algebra is found. The method is based on general results from theglobal theory of representations of semisimple groups. The rank two algebras$A_2$, $B_2=C_2$, $D_2$ and $G_2$ are considered as examples. The generators ofthe simple roots are presented as solutions of a system of finite differenceequations and given in the form of $N_limes N_l$ matrices, where $N_l$ is thedimension of the representation.