A K-theoretic approach to the classification of symmetric spaces

TL;DR

This paper introduces a K-theoretic framework for classifying Hermitian symmetric spaces of non-compact type, linking algebraic invariants to classical root system classifications.

Contribution

It develops a novel K-theory approach for JB*-triples, extending C*-theory and incorporating grid-based invariants related to Lie algebra root systems.

Findings

K-theory can classify Hermitian symmetric spaces

Introduction of grid-based invariants linked to root systems

Extension of K-theory to JB*-triples

Abstract

We show that the classification of the symmetric spaces can be achieved by K-theoretical methods. We focus on Hermitian symmetric spaces of non-compact type, and define K-theory for JB*-triples along the lines of C*-theory. K-groups have to be provided with further invariants in order to classify. Among these are the cycles obtained from so called grids, intimately connected to the root systems of an underlying Lie-algebra and thus reminiscent of the classical classification scheme.