Three-dimensional quadratic algebras: Some realizations and representations

Abstract

Four classes of three-dimensional quadratic algebra of the type [Q0, Q±] = ±Q±, [Q+, Q-] = aQ20 + bQ0 + c, where (a, b, c) are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of su(2) and su(1, 1). The resulting matrix representations and single variable differential operator realizations are obtained. Some remarks on the mathematical and physical relevance of such algebras are given.