Class of Representations of the IU(n) and IO(n) Algebras and Respective Deformations to U(n, 1), O(n, 1)

Abstract

We define directly the matrix elements of the generators of the algebra of U(n) × I_2n on a chosen basis. This construction, though naturally infinite‐dimensional, has a very close formal resemblance (interpretable, if so desired, in terms of a suitably defined ``contraction'' procedure) to the Gel'fand‐Zetlin (GZ) representation for U(n + 1). The representations we obtain are characterized by (n ‐ 1) integers and one continuous parameter. We then exploit the formal analogy with the GZ pattern in order to prove the necessary commutation relations and to derive the explicit expressions for some invariants. However, direct alternative methods are indicated where they are useful. Our representation is easily enlarged to that of ${\text{(U(n)⊗U(1))×I}}_{\text{2n}}$ , which we use, together with a deformation formula to obtain a class of representations of the U(n, 1) algebra. The irreducible components are characterized by (n ‐ 1) integers and two continuous parameters. We compare our deformation formula with that of Rosen and Roman. We indicate briefly the typical difficulties that arise for the case IU(p,q) (q ≥ 1). Parallel constructions, finally, are given for the IO(n) and O(n, 1) algebras