Differential geometry of Grassmannian embeddings of based loop groups
type: Item
instance of: scholarly article
title: Differential geometry of Grassmannian embeddings of based loop groups
zbMATH Open document ID: 0967.58005
DOI: 10.1016/S0926-2245(00)00020-6
author: Mauro Spera
author: Tilmann Wurzbacher
published in: Differential Geometry and its Applications
publication date: 2001-08-28T00:00:00Z
review text: Let \(F\) be a complex separable Hilbert space and a closed, infinite dimensional complex subspace \(F_{+} \) whose orthocomplement \(F_{-} \) is infinite dimensional as well. The restricted Grassmannian of the polarized Hilbert space \(F= F_{-}\oplus F_{+}\) is \(\text{Gr}(F,F_{+}) = { W \subset F |W}\) and is a closed complex subspace of \(F\) such that \(p_{+} : W \rightarrow F_{+}\) is Fredholm and \(p_{-}\) : \(W \rightarrow F_{-}\) is Hilbert-Schmidt. This is transitively acted upon by a ``restricted'' unitary group \(G\), and the isotropy group of the point \(F_{+}\) is the contractible group \(H= U(F_{+})\times U(F_{-})\). The authors prove that the reductive homogeneous space \(\text{Gr}(F,F_{+})=G/H\) is an isotropy irreducible Hermitian symmetric space. Then they compute the Riemannian curvature tensor of \(\text{Gr}(F,F_{+})\) as well as the ``linear divergence'' of its Ricci curvature tensor. For a compact Lie group \(K\) they consider the group \(LK\) of smooth loops in \(K\). There is an embedding \(LK/K \rightarrow \text{Gr}\) which was studied by means of the Gauss equations, and it is shown that the Ricci curvature of \(LK/K\) exists and is proportional to the metric induced from the Grassmannian. An Appendix on linear and logarithmic divergence is included.
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 22E67
zbMATH DE Number: 1526472
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: Grassmannians
zbMATH Keywords: based Lie groups
zbMATH Keywords: curvatures
reviewed by: Mihai Anastasiei
MaRDI profile type: Publication
Wikidata QID: Q115338403
full work available at URL: s0926-2245(00)00020-6
OpenAlex ID: W1997115800
description: scientific article; zbMATH DE number 1526472