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Title: Quantization and harmonic analysis on nilpotent Lie groups

Abstract

Weyl Quantization is a procedure for associating a function on which the canonical commutation relations are realized. If G is a simply-connected, connected nilpotent Lie group with Lie algebra g and dual g/sup */, it is shown how to inductively construct symplectic isomorphisms between every co-adjoint orbit O and the bundle in Hilbert Space for some m. Weyl Quantization can then be used to associate to each orbit O a unitary representation rho/sub 0/ of G, recovering the classification of the unitary dual by Kirillov. It is used to define a geometric Fourier transform, F : L/sup 1/(G) ..-->.. functions on g/sup */, and it is shown that the usual operator-valued Fourier transform can be recovered from F, characters are inverse Fourier transforms of invariant measures on orbits, and matrix coefficients are inverse Fourier transforms of non-invariant measures supported on orbits. Realizations of the representations rho/sub 0/ in subspaces of L/sup 2/(O) are obtained.. Finally, the kernel function is computed for the upper triangular unipotent group and one other example.

Authors:
Publication Date:
Research Org.:
Yale Univ., New Haven, CT (USA)
OSTI Identifier:
5464732
Resource Type:
Thesis/Dissertation
Resource Relation:
Other Information: Thesis (Ph. D.)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LIE GROUPS; FOURIER TRANSFORMATION; QUANTIZATION; ALGEBRA; HILBERT SPACE; UNITARITY; BANACH SPACE; INTEGRAL TRANSFORMATIONS; MATHEMATICAL SPACE; MATHEMATICS; SPACE; SYMMETRY GROUPS; TRANSFORMATIONS; 658000* - Mathematical Physics- (-1987)

Citation Formats

Wildberger, N J. Quantization and harmonic analysis on nilpotent Lie groups. United States: N. p., 1983. Web.
Wildberger, N J. Quantization and harmonic analysis on nilpotent Lie groups. United States.
Wildberger, N J. Sat . "Quantization and harmonic analysis on nilpotent Lie groups". United States.
@article{osti_5464732,
title = {Quantization and harmonic analysis on nilpotent Lie groups},
author = {Wildberger, N J},
abstractNote = {Weyl Quantization is a procedure for associating a function on which the canonical commutation relations are realized. If G is a simply-connected, connected nilpotent Lie group with Lie algebra g and dual g/sup */, it is shown how to inductively construct symplectic isomorphisms between every co-adjoint orbit O and the bundle in Hilbert Space for some m. Weyl Quantization can then be used to associate to each orbit O a unitary representation rho/sub 0/ of G, recovering the classification of the unitary dual by Kirillov. It is used to define a geometric Fourier transform, F : L/sup 1/(G) ..-->.. functions on g/sup */, and it is shown that the usual operator-valued Fourier transform can be recovered from F, characters are inverse Fourier transforms of invariant measures on orbits, and matrix coefficients are inverse Fourier transforms of non-invariant measures supported on orbits. Realizations of the representations rho/sub 0/ in subspaces of L/sup 2/(O) are obtained.. Finally, the kernel function is computed for the upper triangular unipotent group and one other example.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {1983},
month = {1}
}

Thesis/Dissertation:
Other availability
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