On matrix representations of deformed Lie algebras for quantized Hamiltonians

Abstract

Consider the Lie algebras L : [ K 1 , K 2 ] = F ( K 3 ) + G ( K 4 ) , [ K 3 , K 1 ] = uK 1 , [ K 3 , K 2 ] = - uK 2 , [ K 4 , K 1 ] = vK 1 , [ K 4 , K 2 ] = - vK 2 , [ K 3 , K 4 ] = 0 , subject to the physical conditions, K 3 and K 4 are real diagonal operators and K 2 = K 1 † († 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 ( K 3 ) and G ( K 4 ) . Representations satisfying K 1 + K 2 to be real are separately considered.