ar X iv : 0 70 9 . 11 81 v 3 [ m at h . R A ] 2 5 D ec 2 00 7 Isotopy for extended affine Lie algebras and Lie tori

Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study isotopy for centreless Lie tori, and show that Neher’s construction provides a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general nonassociative, and, for many types of centreless Lie tori, there are classical definitions of isotopy for the coordinate algebras. We show for those types that an isotope of a Lie torus is coordinatized by an isotope of its coordinate algebra, thereby connecting the two notions of isotopy. In writing the article, we have not assumed prior knowledge of the theories of EALAs, Lie tori or isotopy.