Representations of Lie Algebras by non-Skewselfadjoint Operators in Hilbert Space

We study non-selfadjoint representations of a finite dimensional real Lie algebra \(\fg\). To this end we embed a non-selfadjoint representation of \(\fg\) into a more complicated structure, that we call a \(\fg\)-operator vessel and that is associated to an overdetermined linear conservative input/...

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Bibliographic Details
Published in:arXiv.org 2016-08
Main Authors: Shamovich, Eli, Vinnikov, Victor
Format: Article
Language:English
Subjects:
Affine transformations
Algebra
Characteristic functions
Group theory
Hilbert space
Lie groups
Operators (mathematics)
Quantum theory
Representations
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Summary:We study non-selfadjoint representations of a finite dimensional real Lie algebra \(\fg\). To this end we embed a non-selfadjoint representation of \(\fg\) into a more complicated structure, that we call a \(\fg\)-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group \(\fG\). We develop the frequency domain theory of the system in terms of representations of \(\fG\), and introduce the joint characteristic function of a \(\fg\)-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the \(ax+b\) group, the group of affine transformations of the line.
ISSN:2331-8422
DOI:10.48550/arxiv.1209.4224