A new approach to the representation theory of semisimple Lie algebras and quantum algebras (Q1592123)

A new approach, as an alternative to the very well known Gelfand Tsetlin one, is proposed for explicitly constructing the infinitesimal generators in an arbitrary finite-dimensional representation of semisimple algebras and is extended to the case of related quantum algebras. As starting point, the famous Weyl formulas for the known dimensions \((r)\) and characters of irreducible representations of semisimple groups are used, introducing the explicit expressions for the simple root generators of both Lie and quantum semisimple algebras. These generators are found here as solutions of a system of finite-difference equations and are presented in the form of \(N_l\times N_l\)-matrices, where \(N_l\) is the dimension of the corresponding representation. The defining relations of a quantum algebra are rewritten in terms of only \(2r\) generators, instead of the \(3r\) generators used in the traditional approach. The proposed general construction is applied to the rank-two algebras, and as examples the explicit calculations for the algebras \(A_2\), \(B_2=C_2\) and \(G_2\) are realized in detail.