Affine Kac-Moody symmetric spaces

Abstract

Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody symmetric spaces are infinite dimensional symmetric spaces associated to affine Kac-Moody groups. They have the structure of tame Fréchet manifolds; the natural Ad-invariant scalar product on affine Kac-Moody algebras is Lorentzian, making affine Kac-Moody symmetric spaces into Lorentzian symmetric spaces. Similar to affine Kac-Moody groups sharing most of their structure properties with simple Lie group, also Kac-Moody symmetric spaces share most of their structure properties with their finite dimensional Riemannian counterparts. In particular the classification of Kac-Moody symmetric spaces follows the lines of the classification of finite dimensional Riemannian symmetric spaces: There are four types distinguished, which fall into the two classes of Kac-Moody symmetric spaces of the compact type and of the noncompact type. Symmetric spaces of the compact type and of the noncompact type are related by a duality relation. In addition the geometry of the rank 1-building blocks is the same for Kac-Moody symmetric spaces as for finite dimensional Riemannian symmetric spaces. Kac-Moody symmetric spaces appear in mathematics and theoretical physics: for example their isotropy representations are essentially equivalent to polar actions on Hilbert spaces; twin cities can be embedded equivariantly into the tangent space. In theoretical physics Kac-Moody symmetric spaces got recently a prominent place due to various conjectures relating them to certain formulations of supergravity theories and M-theory.