# Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces

Neeb KH (2022)

**Publication Type:** Journal article

**Publication year:** 2022

### Journal

**Book Volume:** 62

**Pages Range:** 577-613

**Journal Issue:** 3

**DOI:** 10.1215/21562261-2022-0017

### Abstract

Let V be a standard subspace in the complex Hilbert space H, and let U : G -> U(H) be a unitary representation of a finite-dimensional Lie group. We assume the existence of an element h is an element of g such that U(exp th) = delta(it)(V) is the modular group of V and that the modular involution J(V) normalizes U(G). We want to determine the semi-group S-V = {g is an element of G: U(g)V subset of V}. In previous work, we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup SV under the assumption that ad h defines a 3-grading on g. Concretely, we show that the ad h-eigenspaces g(+/- 1) contain closed convex cones C-+/- , such that S-V = exp(C+)G(V) exp(C-) , where G(V) = {g is an element of G : 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 C-U of U. In particular, we show that the orbit O-V = U(G)V with the inclusion order is an ordered symmetric space covering the adjoint orbit O-h = Ad(G)h , endowed with the partial order defined by C-U.

### Authors with CRIS profile

### How to cite

**APA:**

Neeb, K.H. (2022). Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces. *Kyoto Journal of Mathematics*, *62*(3), 577-613. https://doi.org/10.1215/21562261-2022-0017

**MLA:**

Neeb, Karl Hermann. "Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces." *Kyoto Journal of Mathematics* 62.3 (2022): 577-613.

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