On matrix representations of deformed Lie algebras for quantized Hamiltonians

Consider the Lie algebras L:[K1,K2]=F(K3)+G(K4),[K3,K1]=uK1,[K3,K2]=-uK2,[K4,K1]=vK1,[K4,K2]=-vK2,[K3,K4]=0, subject to the physical conditions, K3 and K4 are real diagonal operators and († is for hermitian conjugation). Matrix representations are discussed and faithful representations of least degree for L satisfying the physical requirements are given for appropriate values of u,v∈R and certain conditions for the polynomials F(K3) and G(K4). Representations satisfying K1+K2 to be real are separately considered.