On squares of representations of compact Lie algebras

Journal Article · Sat Aug 15 04:00:00 EDT 2015 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4928410· OSTI ID:22479583

Zeier, Robert ^[1]; Zimborás, Zoltán ^[2]

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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)

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.

• OSTI ID: 22479583
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 8 Vol. 56; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
IRREDUCIBLE REPRESENTATIONS
LIE GROUPS
MULTIPLICITY
QUANTUM SYSTEMS
TENSORS