Lie Groups of Jacobi polynomials and Wigner d-matrices

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.

21 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1307.7380

References in corpus (2)

• d-Orthogonal polynomials and su(2)
• Jacobi polynomials and SU(2,2)