Lie algebra representations, nilpotent matrices, and the \(C\)-numerical range
type: Item
instance of: scholarly article
title: Lie algebra representations, nilpotent matrices, and the \(C\)-numerical range
zbMATH Open document ID: 1089.15027
published in: Linear Algebra and its Applications
publication date: 2006-03-22T00:00:00Z
review text: This is a survey paper on some applications of the representation theory of Lie algebras to linear algebra. The authors illustrate the power of representation theoretic methods by discussing their role in three different areas: Jordan canonical forms for tensor products of matrices; Geometry and parametrization of unitary orbits of nilpotent matrices; The calculation of the \(C\)-numerical range of nilpotent matrices. For example, using the Clebsch-Gordan decomposition for Lie group representation of \(SL_2({\mathbb C})\), the authors obtain an explicit formula for the Jordan form of the Kronecker product of two invertible Jordan blocks. They discuss the relations of classification of Lie algebra representations of \( \mathfrak{gl}_2({\mathbb C})\) to the task of parametrizing nilpotent similarity orbits. Further, applications to quantum mechanics are discussed, i.e.\ the maximization of the transfer function of \(N+1\) weakly coupled spin-\(\frac 12\)-systems arising in NMR-spectroscopy and quantum computing. Their approach consists in reformulating the task as a least squares matching problem for two unitary representations of \(\mathfrak{su}_2({\mathbb C})\). This then leads us to explicit formulas for the conjectured maxima inn terms of the Clebsch--Gordan decomposition. It is shown that unitary transformations exist that achieve these values.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 15A90
Mathematics Subject Classification ID: 81P68
zbMATH DE Number: 5014299
zbMATH Keywords: Lie algebra
zbMATH Keywords: representations
zbMATH Keywords: nilpotent matrices
zbMATH Keywords: \(C\)-numerical range
zbMATH Keywords: Clebsch-Gordan decomposition
zbMATH Keywords: Jordan canonical forms
zbMATH Keywords: tensor products
zbMATH Keywords: Kronecker product
zbMATH Keywords: NMR-spectroscopy
zbMATH Keywords: quantum computing
author: G. Dirr
author: Uwe Helmke
author: Martin Kleinsteuber
reviewed by: A. R. Alimov
MaRDI profile type: Publication
full work available at URL: j.laa.2005.05.019
OpenAlex ID: W2091405020
Wikidata QID: Q115344939
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DOI: 10.1016/J.LAA.2005.05.019
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description: scientific article; zbMATH DE number 5014299