Jordan systems, bounded symmetric domains and associated group orbits with holomorphic and CR extension theory

Abstract

The first chapter will deal with the one to one correspondence between the positive hermitian Jordan triple systems and the bounded symmetric domains. We start by defining the various Jordan systems. Then we continue by giving classification results leading to the classification of the Jordan pairs under finiteness conditions. This is followed by defining bounded symmetric domains, and some motivation on why they became a topic of interest. Up to this point we will be dealing with the Jordan systems over a general ring of scalars, we now restrict ourselves to the complex number field. We show bounded symmetric domains are connected to the positive hermitian Jordan triple systems. A look at Peirce decompositions of the Jordan systems will prove important. With this completed, we deal with the one to one correspondence mentioned. This will involve some results relating to Riemannian manifolds and Lie theory, we can then present the result as presented by Ottmar Loos in [Loos1]. A table of the classification is also included to conclude the chapter. We note here that throughout this thesis when we say we are dealing with bounded symmetric domains we mean circled bounded symmetric domains. In the second chapter we look at some holomorphic extension theory. This is started by looking at and defining manifolds and varieties. We then look at normal spaces and normalisations of spaces with some extension theory. After that we look briefly at Stein manifolds and domains of holomorphy, explaining their importance in relation to (maximal) holomorphic and CR extension theory. We will also look at a CR extension theorem due to Albert Boggess and John Polking. To complete this section we will present a application to the extension theorem from the paper [KaupZait] by Dimitri Zaitsev and Wilhelm Kaup. Our third chapter is based on [KaupZait] giving an overview of results contained within. We will look at orbits associated with bounded symmetric domains with respect to (the connected identity component of) the automorphism group of the domain. We look at various hulls and domains that are relevant to the extension theory results of the paper. Then we state the results, and sketch the extension theorem proof. In the final chapter we will generalise these results. In [KaupZait] , the results are for irreducible bounded symmetric domains, we will look at what happens in the reducible case. We will introduce a generalised notation and generalise the results from the irreducible case when needed.