Multiloop Realization of Extended Affine Lie Algebras and Lie Tori

An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from centreless Lie tori, so the realization problem for EALAs reduces to the realization problem for centreless Lie tori. We show that all but one family of centreless Lie tori can be realized using multiloop algebras (in place of loop algebras). We also obtain necessary and sufficient for two centreless Lie tori realized in this way to be isotopic, a relation that corresponds to isomorphism of the corresponding families of EALAs. An extended affine Lie algebra (EALA) over a field of characteristic zero consists of a Lie algebra E, together with an nondegenerate invariant symmetric bilinear form ( , ) on E, and a nonzero finite dimensional ad-diagonalizable subalgebra H of E, such that a list of natural axioms are imposed (see [N2] and the references therein). (Although, by definition, an EALA consists of a triple (E,H, ( , )), we usually abbreviate it as E.) One of the axioms states that the group generated by the isotropic roots of E is a free abelian group Λ of finite rank, and the rank of Λ is called the nullity of E. As the term EALA suggests, the defining axioms for an EALA are modeled after the properties of affine Kac-Moody Lie algebras; and in fact affine Kac-Moody Lie algebras are precisely the extended affine Lie algebras of nullity 1. So it is natural to look for a realization theorem for EALAs of arbitrary nullity ≥ 1. (Nullity 0 EALAs are finite dimensional simple Lie algebras and we do not consider them in this context.) The classical procedure for realizing affine Lie algebras using loop algebras proceeds in two steps [K, Chap. 7 and 8]. In the first step, the derived algebra modulo its centre of the affine algebra is constructed as the loop algebra of a diagram automorphism of a finite dimensional simple Lie algebra. This loop algebra is naturally graded by Q×Z, where Q is the root lattice of a finite irreducible (but not necessarily reduced) root system. In the second step, the affine algebra itself, together with a Cartan subalgebra and a nondegenerate invariant bilinear form for the affine algebra, is built from the graded loop algebra by forming a central extension (with one dimensional centre) and adding a (one dimensional) graded algebra of derivations. The replacement for the derived algebra modulo its centre in EALA theory is the centreless core of the EALA, and centreless cores of EALAs have been characterized Date: September 6, 2007. 2000 Mathematics Subject Classification. Primary: 17B65; Secondary: 17B67, 17B70 . The authors Allison and Pianzola gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.