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Title: On squares of representations of compact Lie algebras

Abstract

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.

Authors:
 [1];  [2]
  1. Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching (Germany)
  2. Department of Computer Science, University College London, Gower St., London WC1E 6BT (United Kingdom)
Publication Date:
OSTI Identifier:
22479583
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 56; Journal Issue: 8; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; IRREDUCIBLE REPRESENTATIONS; LIE GROUPS; MULTIPLICITY; QUANTUM SYSTEMS; TENSORS

Citation Formats

Zeier, Robert, and Zimborás, Zoltán. On squares of representations of compact Lie algebras. United States: N. p., 2015. Web. doi:10.1063/1.4928410.
Zeier, Robert, & Zimborás, Zoltán. On squares of representations of compact Lie algebras. United States. doi:10.1063/1.4928410.
Zeier, Robert, and Zimborás, Zoltán. Sat . "On squares of representations of compact Lie algebras". United States. doi:10.1063/1.4928410.
@article{osti_22479583,
title = {On squares of representations of compact Lie algebras},
author = {Zeier, Robert and Zimborás, Zoltán},
abstractNote = {We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.},
doi = {10.1063/1.4928410},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 8,
volume = 56,
place = {United States},
year = {2015},
month = {8}
}