Unbounded Operators, Lie Algebras, and Local Representations

Palle E. T. Jorgensen () and James Tian ()
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Palle E. T. Jorgensen: The University of Iowa, Department of Mathematics
James Tian: Mathematical Reviews

Chapter 48 in Operator Theory, 2026, pp 1533-1554 from Springer

Abstract: Abstract Several results are presented on the integrability and extendability of Lie algebras composed of unbounded skew-symmetric operators with a common dense domain in a Hilbert space. Integrability of a Lie algebra 𝔤 $$\mathfrak {g}$$ refers to the existence of an associated unitary representation U $$\mathcal {U}$$ of the corresponding simply connected Lie group, such that U $$\mathcal {U}$$ of the corresponding simply connected Lie group, such that 𝔤 $$\mathfrak {g}$$ is the differential of U $$\mathcal {U}$$ . The findings extend previous integrability results in the literature and introduce new insights even for the case of a single operator. Applications include the discovery of a new invariant for certain Riemann surfaces.

Keywords: Unbounded operators; Deficiency indices; Hilbert space; Lie algebras; Lie groups; Unitary representations; Local invariance; Spectral resolution; Riemann surfaces; Noncommutative geometry; Boundary value problems; Harmonic analysis; Skew-symmetric operators; Integrability; Extendability; 47L60; 47B25; 22E65; 81S05 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_47

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