ar X iv : 0 70 9 . 11 81 v 1 [ m at h . R A ] 8 S ep 2 00 7 Isotopy for extended affine Lie algebras and Lie tori

If A is a unital associative algebra and u is invertible in A, one can define an algebra A, called the u-isotope of A, which is equal to A as a vector space but has a new product x ·u y = xuy. This isotope A (u) is again unital and associative but with a shifted identity element u. More generally there are definitions of isotope for several other classes of unital nonassociative algebras, notably Jordan algebras [Mc2, §I.3.2], alternative algebras [Mc1] and associative algebras with involution [Mc2, §I.3.4]. In each case, the isotope is obtained very roughly by shifting the identity element in the algebra, and two algebras are said to be isotopic if one is isomorphic to an isotope of the other. In the associative case, the u-isotope A is isomorphic to A under left multiplication by u, and therefore isotopy has not played a role in associative theory. That is not true in general though, and in particular isotopy plays an important role in Jordan theory (see for example [Mc2, §II.7]). In contrast, isotopes and isotopy have not been defined for Lie algebras, for the evident reason that Lie algebras are not unital. In this article we study notions of isotope and isotopy, which were recently introduced in [ABFP2], for a class of graded Lie algebras called Lie tori. The point to emphasize here is that forming an isotope of a Lie torus does not change the multiplication at all, but rather it shifts the grading. We are primarily interested in the case when the Lie torus is centreless, and there are two basic reasons why we are interested in isotopy in this case. First, centreless Lie tori arise naturally in the construction of families of extended affine Lie algebras (EALAs), and we see in this article that isotopes and