Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On squares of representations of compact Lie algebras   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.

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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom