Foundations of the theory of Kac-Moody Symmetric spaces
We investigate possiblities to develop a theory of infinite dimensional symmetric spaces. Those spaces are constructed as quotient spaces of loop groups (or loop group extensions) by fixed point groups of suitable involutions; they are furnished with a Hilbert-, Banach-, or Frechet structure. In this setting, Kac-Moody algebras appear as Lie algebras of Loop group extensions. Kac-Moody symmetric spaces share many properties of their finite dimensional counterparts. For example, one can complexify loop groups and derive a duality between spaces of compact and of noncompact type. A classification of Kac-Moody symmetric spaces was recently achieved by Ernst Heintze. We conjecture, that many more properties of finite dimensional symmetric spaces can be generalized to Kac-Moody symmetric spaces.