Lie Tori and Structurable Tori

Yoji Yoshii

Abstract

We report some recent progress on Lie tori. Lie n-tori are certain Z^{n}-graded Lie algebras which coincide with the cores of extended affine Lie algebras. Lie 1-tori of type A_{1} and BC_{1} are exactly affine Kac-Moody Lie algebras of type A_{1}^{(1)} and A_{2}^{(2)} respectively. We discuss Lie 2-tori of type A_{1} and BC_{1} which are coordinatized by analogues of the algebra of Laurent polynomials in two variables. It turns out that the coordinate algebras for type A_{1} include one infinite family containing the algebra of Laurent polynomials in two variables and precisely one other algebra. For type BC_{1} the situation is rather different. There are precisely five different coordinate algebras in that case.