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Unbounded Operators, Lie Algebras, and Local Representations
Book chapter

Unbounded Operators, Lie Algebras, and Local Representations

Palle E. T Jorgensen and Feng Tian
Operator Theory, pp.1221-1243
Springer Basel
06/20/2015
DOI: 10.1007/978-3-0348-0667-1_47

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Abstract

A numberUnbounded operators of results on integrability and extendability of Lie algebras of unbounded skew-symmetric operators with common dense domain in Hilbert space are proved. By integrability for a Lie algebra 𝔤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{g}$$\end{document}, it means that there is an associated unitary representation 𝒰\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{U}$$\end{document} of the corresponding simply connected Lie group such that 𝔤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{g}$$\end{document} is the differential of 𝒰\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{U}$$\end{document}. The results extend earlier integrability results in the literature and are new even in the case of a single operator. Applications include a new invariant for certain Riemann surfaces.
Local Representation Riemann Surface Single Operator Unbounded Operator Unitary Representation

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