The coordinate algebra of extended affine Lie algebras of type A(1).

Abstract

In this thesis we classify the coordinate algebra of extended affine Lie algebras, EALA's for short, of type A1. The coordinate algebras of EALA's of the other reduced types were already described in [6], [7] and [2]. In Chapter 1 we show that the coordinate algebra of EALA's of type A1 is a certain Z n-graded Jordan algebra called a Jordan torus. The main result in this thesis is the classification of Jordan tori. For this purpose we study more general objects, division graded alternative or Jordan algebras in Chapter 2. In Chapter 3 we classify Jordan tori. They fall into five classes namely three types of Hermitian tori, Clifford tori and the Albert torus . Our second goal is the classification of division Z n-graded alternative algebras in Chapter 4, which generalizes the classification of alternative tori in [7]. In the associative case, we get a generalization of quantum tori, and in the nonassociative case, we obtain the Cayley torus or 3 closely related octonion rings . In the final chapter we classify graded forms and derivations of Jordan tori. These results serve as a preparation of classifying tame EALA's of type A 1.