No abstract

If ~' is a Jordan algebra, then Str(s¢) = L~O Der(s~¢) forms a Lie algebra called the structure algebra of d, where L, denotes the left multiplication operator by a for ae ~4 and Der(.~) is the algebra of derivations of J. The Tits-Koecher construction gives the vector space ~(~Str(~c)Gd the structure of a Lie algebra, where s] is a second copy of ~¢ (see Section 8.5 of [6]). In this paper, we define and study a cla~s of non-associative algebras containing the class of Jordan algebras and allowing the construction of generalizations of the structure algebra and the Tits-Koecher algebra. The algebras in this class, called structurable algebras, are unital algebras with involution. The class is defined by an identity of degree 4 and includes associative algebras, Jordan algebras (with the identity map as involution), tensor products of two composition algebras, the 56-dimensional Freudenthal module for E; with a natural binary product, and algebras constructed from hermitian forms in a manner generalizing the usual construction of Jordan algebras from quadratic forms (see Section 8). Throughout the paper, motivation for definitions and proofs is provided by the examples of Jordan algebras and composition algebras.The present paper is devoted primarily to the study of finite dimensional central simple structurable algebras over a field 4} of characteristic zero. The main result (Theorem 25) gives a complete description of these algebras. The primary toot used to obtain this description is the form defined by (x,y)= tr(Lx~+~.~), where -denotes the involution (see Section 6). A subsequent paper will deal with the generalization of the Tits-Koecher construction. The Lie algebras obtained from central simple structurable algebras using this construction include all finite dimensional central simple Lie algebras over 4} that contain at least one non-zero ad-nilpotent element.Constructions analogous to the Tits-Koecher construction have been studied in [1,3,4,8] and [16]. The algebras used to construct Lie algebras in these papers are ternary algebras. The simple Lie algebras obtained from these ternary algebras can be obtained from structurable algebras [although the graded structure obtained may differ in the two cases]. The formulation of the constructions in terms of binary algebras rather than ternary algebras results in simplification not only of

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.

No abstract

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-