Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize a‰ne Kac-Moody Lie algebras. In this paper, we obtain necessary and su‰cient conditions for a Z n -graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and su‰cient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group.We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields.

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Abstract. 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 conditions 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 a 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 is 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; 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, Chaps. 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 ×ℤ, where 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 alge-

Abstract. A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type Ea are obtained.A construction of Lie algebras from Jordan algebras discovered independently by J. Tits [7] and M. Koecher [4] has been useful in the study of both kinds of algebras. In this paper, we give a similar construction of Lie algebras from a ternary algebra with a skew bilinear form satisfying certain axioms. These ternary algebras are a variation on the Freudenthal triple systems considered in [1]. Most of the results we obtain for our construction are parallel to those for the TitsKoecher construction (see [3, Chapter VIII]).In §1, we define the ternary algebras, derive some basic results about them, and give two examples of such algebras. In §2, the Lie algebras are constructed and shown to be simple if and only if the skew bilinear form is nondegenerate. In §3, we give a characterization, in the case the base ring contains 1/2, of the Lie algebras obtained by our construction in terms of their structure as modules for the threedimensional simple Lie algebra. Finally, in §4, we identify some of the simple Lie algebras obtained by our construction from the examples of §1. In particular, we show that we can construct a Lie algebra of type Pa from a 56-dimensional space which is a module for a Lie algebra of type P7. A similar construction was given by H. Freudenthal in [2].

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