Three dimensional quadratic algebras: Some realizations and representations

TL;DR

This paper constructs and analyzes three-dimensional quadratic algebras using generalized bosonic realizations, providing matrix and differential operator representations, and discusses their mathematical and physical significance.

Contribution

It introduces new classes of quadratic algebras and derives their explicit matrix and differential operator representations.

Findings

Constructed four classes of quadratic algebras.

Derived matrix and differential operator realizations.

Discussed relevance in mathematical physics.

Abstract

Four classes of three dimensional quadratic algebras of the type $\lsb Q_0 , Q_\pm \rsb$ $=$ $\pm Q_\pm$, $\lsb Q_+ , Q_- \rsb$ $=$ $aQ_0^2 + bQ_0 + c$, 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.