# Integration of Lie Algebra Representations

• Karl-Hermann Neeb
• 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.

## Keywords

Positive Definite Distribution Unitary Representation Problem Need Essential Selfadjointness Continuous Anti-linear Functionals
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