Lie Groups of Jacobi polynomials and Wigner d-matrices

TL;DR

This paper explores the symmetry group $SU(2,2)$ related to Jacobi polynomials and Wigner d-matrices, constructing representations and subgroup structures to unify integer and half-integer spin cases.

Contribution

It introduces a novel $SU(2,2)$ symmetry framework for Jacobi polynomials and Wigner matrices, including the construction of unitary irreducible representations and subgroup analysis.

Findings

Constructed a unitary irreducible representation of $SU(2,2)$.

Identified subgroup structures like $SU(1,1)$, $SO(3,2)$, and $Spin(3,2)$.

Separated integer and half-integer spin representations within the framework.

Abstract

A symmetry $SU(2,2)$ group in terms of ladder operators is presented for the Jacobi polynomials, $J_{n}^{(\alpha,\beta)}(x)$, and the Wigner $d_j$-matrices where the spins $j=n+(\alpha+\beta)/2$ integer and half-integer are considered together. A unitary irreducible representation of $SU(2,2)$ is constructed and subgroups of physical interest are discussed. The Universal Enveloping Algebra of $su(2,2)$ also allows to construct group structures $(SU(1,1), SO(3,2), Spin(3,2))$ whose representations separate integers and half-integers values of the spin $j$. Appropriate $L^2$--functions spaces are realized inside the support spaces of all these representations. Operators acting on these $L^2$-functions spaces belong thus to the corresponding Universal Enveloping Algebra.