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Integration of Lie Algebra Representations

  • Karl-Hermann NeebEmail author
  • Gestur Ólafsson
Chapter
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 32)

Abstract

A central problem in the context of reflection positive representations of a symmetric Lie group \((G,\tau )\) on a reflection positive Hilbert space \((\mathscr {E},\mathscr {E}_+,\theta )\) is to construct on the associated Hilbert space \(\widehat{\mathscr {E}}\) a unitary representations of the 1-connected Lie group \(G^c\) with Lie algebra \({\mathfrak {g}}^c = {\mathfrak {h}}+ i {\mathfrak {q}}\). As we have seen in Remark 3.3.9, the main point is to “integrate” a unitary representation of the Lie algebra \({\mathfrak {g}}^c\) on a pre-Hilbert space. In general this problem need not have a solution, but we shall see below that in the reflection positive contexts, where the Hilbert spaces are mostly constructed from G-invariant positive definite kernels or positive definite G-invariant distributions, there are natural assumptions that apply in all cases that we consider.

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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