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Title: Algebraic special functions and SO(3,2)

Abstract

A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to getmore » a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.« less

Authors:
 [1]
  1. Departamento de Física Teórica and IMUVA, Universidad de Valladolid, E-47011, Valladolid (Spain)
Publication Date:
OSTI Identifier:
22220727
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 333; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; HARMONICS; IRREDUCIBLE REPRESENTATIONS; LEGENDRE POLYNOMIALS; LIE GROUPS; SPACE; SPHERICAL HARMONICS

Citation Formats

Celeghini, E., E-mail: celeghini@fi.infn.it, and Olmo, M.A. del,. Algebraic special functions and SO(3,2). United States: N. p., 2013. Web. doi:10.1016/J.AOP.2013.02.010.
Celeghini, E., E-mail: celeghini@fi.infn.it, & Olmo, M.A. del,. Algebraic special functions and SO(3,2). United States. https://doi.org/10.1016/J.AOP.2013.02.010
Celeghini, E., E-mail: celeghini@fi.infn.it, and Olmo, M.A. del,. 2013. "Algebraic special functions and SO(3,2)". United States. https://doi.org/10.1016/J.AOP.2013.02.010.
@article{osti_22220727,
title = {Algebraic special functions and SO(3,2)},
author = {Celeghini, E., E-mail: celeghini@fi.infn.it and Olmo, M.A. del,},
abstractNote = {A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining in this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.},
doi = {10.1016/J.AOP.2013.02.010},
url = {https://www.osti.gov/biblio/22220727}, journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = ,
volume = 333,
place = {United States},
year = {Sat Jun 15 00:00:00 EDT 2013},
month = {Sat Jun 15 00:00:00 EDT 2013}
}