SYMMETRIC SPACES WITH DISSECTING INVOLUTIONS

• Published in: Transformation groups Vol. 27; no. 2; pp. 635 - 649
• Main Authors: NEEB, K.-H., ÓLAFSSON, G.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2022; Springer Nature B.V
• Subjects: Algebra; Automorphisms; Hilbert space; Isomorphism; Lie Groups; Mathematics; Mathematics and Statistics; Quantum field theory; Quantum theory; Riemann manifold; Topological Groups
• ISSN: 1083-4362; 1531-586X
• DOI: 10.1007/s00031-020-09595-z

An involutive diffeomorphism
σ
of a connected smooth manifold
M
is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs (
M,
σ), where
M
is an irreducible connected symmetric space, not necessarily Riemannian, and
σ
is a dissecting involutive automorphism. In particular, we show that the only irreducible, connected and simply connected Riemannian symmetric spaces with dissecting isometric involutions are
S
n
and ℍ
n
, where the corresponding fixed point spaces are
S
n
−
1
and ℍ
n
 − 1
, respectively.