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Lie Groups of Jacobi polynomials and Wigner d-matrices

A symmetry $SU(2,2)$ group in terms of ladder operators is presented for the Jacobi polynomials, $J_{n}^{(α,β)}(x)$, and the Wigner $d_j$-matrices where the spins $j=n+(α+β)/2$ 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.