Geodesic
A new approach to the representation theory of semisimple Lie algebras and quantum algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 264-284

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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. 
@article{TMF_2000_123_2_a7,
     author = {A. N. Leznov},
     title = {A new approach to the representation theory of semisimple {Lie} algebras and quantum algebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {264--284},
     publisher = {mathdoc},
     volume = {123},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a7/}
}
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A. N. Leznov. A new approach to the representation theory of semisimple Lie algebras and quantum algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 264-284. http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a7/