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 A₂, B₂=C₂, D₂, and G₂ 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×N_l matrices, where N_l is the dimension of the representation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 264–284, May, 2000.