Gram-Schmidt process, iwasawa decomposition, and reduction of structure in principal bundles

Abstract

Using the classical Gram--Schmidt process from the beginning linear algebra, we are able to derive group theory results about the linear groups and reduction of structure group results for vector bundles. By starting with these very elementary considerations,we see that there are applications to both group representation theory as well as to the topology of groups, their classifying spaces, and principal bundles. This theory also has applications to modular forms which play an important role in systems with an SL(2,ℤ)-symmetry. One example is the Verlinde algebra which arises later in a twisted K-theory calculation. © Springer-Verlag Berlin Heidelberg 2008.