On squares of representations of compact Lie algebras

TL;DR

This paper proves that tensor squares of representations become more complex when restricted from a compact semisimple Lie algebra to a proper subalgebra, providing a new method to identify subalgebras in quantum control.

Contribution

It establishes a classification-free criterion based on multiplicities for detecting proper subalgebras of compact semisimple Lie algebras using tensor squares.

Findings

Sum of multiplicities increases under restriction to subalgebras

Sum of squares of multiplicities can be computed via linear algebra

Provides a practical test for subalgebra identification in quantum systems

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.