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Title: Universal T-matrix, representations of OSp{sub q}(1/2) and little Q-Jacobi polynomials

Abstract

We obtain a closed form expression of the universal T-matrix encapsulating the duality between the quantum superalgebra U{sub q}[osp(1/2)] and the corresponding supergroup OSp{sub q}(1/2). The classical q{yields}1 limit of this universal T-matrix yields the group element of the undeformed OSp{sub q}(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp{sub q}(1/2) are readily constructed employing the above-mentioned universal T-matrix and the known finite dimensional representations of the dually related deformed U{sub q}[osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations, and the orthogonality of the representations of the quantum supergroup OSp{sub q}(1/2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q=-q. Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.

Authors:
; ; ;  [1];  [2];  [2]
  1. Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Daisen Campus, Sakai, Osaka 590-0035 (Japan)
  2. (India)
Publication Date:
OSTI Identifier:
20861568
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 12; Other Information: DOI: 10.1063/1.2399360; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; DUALITY; GROUP THEORY; JACOBIAN FUNCTION; POLYNOMIALS; QUANTUM MECHANICS; R MATRIX; RECURSION RELATIONS; S MATRIX; SUPERSYMMETRY

Citation Formats

Aizawa, N., Chakrabarti, R., Mohammed, S. S. Naina, Segar, J., Department of Theoretical Physics, University of Madras, Guindy Campus, Chennai 600 025, and Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600 004. Universal T-matrix, representations of OSp{sub q}(1/2) and little Q-Jacobi polynomials. United States: N. p., 2006. Web. doi:10.1063/1.2399360.
Aizawa, N., Chakrabarti, R., Mohammed, S. S. Naina, Segar, J., Department of Theoretical Physics, University of Madras, Guindy Campus, Chennai 600 025, & Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600 004. Universal T-matrix, representations of OSp{sub q}(1/2) and little Q-Jacobi polynomials. United States. doi:10.1063/1.2399360.
Aizawa, N., Chakrabarti, R., Mohammed, S. S. Naina, Segar, J., Department of Theoretical Physics, University of Madras, Guindy Campus, Chennai 600 025, and Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600 004. Fri . "Universal T-matrix, representations of OSp{sub q}(1/2) and little Q-Jacobi polynomials". United States. doi:10.1063/1.2399360.
@article{osti_20861568,
title = {Universal T-matrix, representations of OSp{sub q}(1/2) and little Q-Jacobi polynomials},
author = {Aizawa, N. and Chakrabarti, R. and Mohammed, S. S. Naina and Segar, J. and Department of Theoretical Physics, University of Madras, Guindy Campus, Chennai 600 025 and Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600 004},
abstractNote = {We obtain a closed form expression of the universal T-matrix encapsulating the duality between the quantum superalgebra U{sub q}[osp(1/2)] and the corresponding supergroup OSp{sub q}(1/2). The classical q{yields}1 limit of this universal T-matrix yields the group element of the undeformed OSp{sub q}(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp{sub q}(1/2) are readily constructed employing the above-mentioned universal T-matrix and the known finite dimensional representations of the dually related deformed U{sub q}[osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations, and the orthogonality of the representations of the quantum supergroup OSp{sub q}(1/2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q=-q. Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.},
doi = {10.1063/1.2399360},
journal = {Journal of Mathematical Physics},
number = 12,
volume = 47,
place = {United States},
year = {Fri Dec 15 00:00:00 EST 2006},
month = {Fri Dec 15 00:00:00 EST 2006}
}
  • The infinite dimensional highest weight representations of U{sub q}osp(1{vert_bar}2) for the deformation parameter q being a root of unity are investigated. As in the cases of q-deformed nongraded Lie algebras, we find that every irreducible representation is isomorphic to the tensor product of a highest weight representation of sl{sub 2}(R) and a finite dimensional one of U{sub q}osp(1{vert_bar}2). The structure is investigated in detail. {copyright} {ital 1997 American Institute of Physics.}
  • We show that our construction of realizations for algebras and quantum algebras can be generalized to quantum superalgebras too. We study an example of quantum superalgebra U{sub q}(osp(1/2)) and give the boson-fermion realization with respect to one pair of q-boson operators and one pair of fermions.
  • Using the method of projection operators, analytical formulas for Racah coefficients and 6-j symbols of the quantum superalgebra U{sub q}(osp(1{vert_bar}2)) are derived. The formulas obtained by this method are transformed by means of algebraic identities into symmetrical analytical formulas, the form of which are very similar to the classical formulas obtained by Racah and Regge for su(2) Racah coefficients and 6-j symbols. Symmetry properties of U{sub q}(osp(1{vert_bar}2)) Racah coefficients and 6-j symbols following from these analytical formulas are studied. In particular, it is shown that, similarly to the su(2) classical case, in addition to the usual tetrahedral symmetry, 6-j symbolsmore » of the quantum superalgebra U{sub q}(osp(1{vert_bar}2)) satisfy a Regge type symmetry. {copyright} {ital 1997 American Institute of Physics.}« less
  • It is shown that the universal R matrix in the tensor product of two irreducible representation spaces of the quantum superalgebra U{sub q}{bold (}osp(1{vert_bar}2){bold )} can be expressed by Clebsch{endash}Gordan coefficients. The Racah sum rule satisfied by U{sub q}{bold (}osp(1{vert_bar}2){bold )} Racah coefficients and 6{minus}j symbols is derived from the properties of the universal R matrix in the tensor product of three representation spaces. Considering the tensor product of four irreducible representations, it is shown that Biedenharn{endash}Elliott identity holds for U{sub q}{bold (}osp(1{vert_bar}2){bold )} Racah coefficients and 6{minus}j symbols. A recursion relation for U{sub q}{bold (}osp(1{vert_bar}2){bold )} 6{minus}j symbols ismore » derived from the Biedenharn{endash}Elliott identity. {copyright} {ital 1998 American Institute of Physics.}« less