JORDAN–SCHWINGER-TYPE REALIZATIONS OF THREE-DIMENSIONAL POLYNOMIAL ALGEBRAS

A three-dimensional polynomial algebra of order m is defined by the commutation relations [P0,P±] = ± P±, [P+,P-] = ϕ(m)(P0) where ϕ(m) (P0) is an mth order polynomial in P0 with the coefficients being constants or central elements of the algebra. It is shown that two given mutually commuting polynomial algebras of orders l and m can be combined to give two distinct (l+m+1)th order polynomial algebras. This procedure follows from a generalization of the well-known Jordan–Schwinger method of construction of su(2) and su(1,1) algebras from two mutually commuting boson algebras.