Jordan-Schwinger realizations of three-dimensional polynomial algebras

Abstract

A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0,P_{\pm }]$ $=$ $\pm P_{\pm }$, $[P_{+},P_{-}]$ $=$ ${\varphi }^{(m)}(P_0)$ where ${\varphi }^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ 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.

10 pages, LaTeX2e

References in corpus (2)

• Nonlinear supersymmetry on the plane in magnetic field and quasi-exactly solvable systems
• Three dimensional quadratic algebras: Some realizations and representations

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