September 2022 Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces
Karl-Hermann Neeb
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Kyoto J. Math. 62(3): 577-613 (September 2022). DOI: 10.1215/21562261-2022-0017


Let V be a standard subspace in the complex Hilbert space H, and let U:GU(H) be a unitary representation of a finite-dimensional Lie group. We assume the existence of an element hg such that U(expth)=ΔVit is the modular group of V and that the modular involution JV normalizes U(G). We want to determine the semigroup SV={gG:U(g)VV}. In previous work, we have seen that its infinitesimal generators span a Lie algebra on which adh defines a 3-grading, and here we completely determine the semigroup SV under the assumption that adh defines a 3-grading on g. Concretely, we show that the adh-eigenspaces g±1 contain closed convex cones C±, such that


where GV={gG:U(g)V=V} is the stabilizer of V. To obtain this result, we compare several subsemigroups of G specified by the grading and the positive cone CU of U. In particular, we show that the orbit OV=U(G)V with the inclusion order is an ordered symmetric space covering the adjoint orbit Oh=Ad(G)h, endowed with the partial order defined by CU.


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Karl-Hermann Neeb. "Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces." Kyoto J. Math. 62 (3) 577 - 613, September 2022.


Received: 9 January 2020; Accepted: 20 July 2020; Published: September 2022
First available in Project Euclid: 19 July 2022

MathSciNet: MR4517997
zbMATH: 1521.22011
Digital Object Identifier: 10.1215/21562261-2022-0017

Primary: 22E45
Secondary: 81R05 , 81T05

Keywords: endomorphism semigroup , graded Lie group , ordered symmetric space , Quantum field theory , standard subspace

Rights: Copyright © 2022 by Kyoto University


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Vol.62 • No. 3 • September 2022
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