The coordinate algebra of extended affine Lie algebras of type $A_1$

Yoji Yoshii
(University of Ottawa)

The cores of extended affine Lie algebras of reduced types were classified except for type $A_1$. The main part of the classification of these Lie algebras is to determine their coordinate algebras. It turns out that in the $A_1$-case the coordinate algebra is a certain $\Bbb{Z}^n$-graded Jordan algebra, called a Jordan torus, which can be considered as a Jordan analogue of the algebra of Lauren polynomials in $n$ variables. In my talk I will start by defining extended affine Lie algebras and the cores. Then I will show how the Jordan tori appear as the coordinate algebra in the $A_1$-case. After this, I will explain the examples of Jordan tori, and how the classification works. It turns out that Jordan tori are strongly prime, and so Zelmanov's Prime Structure Theorem can be applied. I will briefly show that there are only three classes of Hermitian type, one class of Clifford type and a unique exceptional Jordan torus, called the Albert torus.