A new approach to the representation theory of semisimple Lie algebras and quantum algebras

We propose a method for explicitly constructing the simple-root generators in an arbitrary finite-dimensional representation of a semisimple quantum algebra or Lie algebra. The method is based on general results from the global 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 simple-root generators are represented as solutions of a system of finite-difference equations and are given in the form of $N_l\times N_l$ matrices, where $N_l$ is the dimension of the representation.