John R. Faulkner

A construction of Lie algebras from a class of ternary algebras

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 E8 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 E8 from a 56-dimensional space which is a module for a Lie algebra of type E7. A similar construction was given by H. Freudenthal in [2]. 1. A class of ternary algebras. We shall be interested in a module TM over an arbitrary commutative associative ring D with 1 which possesses an alternating bilinear form and a ternary product which satisfy (TI) = + z for x,y, ze M; (T2) = + x for x, y, z e 9; (T3) , w> = , z> + for x, y, z, w e 9; Received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 subject classifications. Primary 1730.

On the geometry of inner ideals

The notion of an inner ideal, which has arisen in the study of Jordan algebras, is extended here to an arbitrary finite dimensional module M for a finite dimensional Lie algebra with nondegenerate symmetric associative bilinear form. The extension is made by first defining a product xy∗z for x, z ϵ M, y∗ ϵ M∗ (the contragredient module). With suitable identification of M∗ with M, this product is, in various cases, that of a Jordan triple system, a Lie triple system, a symplectic ternary algebra, and other ternary algebras. The inner ideals of M are used to describe several special geometries previously defined by ad hoc methods on certain Lie modules. Finally, for a split semisimple Lie algebra over a field of characteristic zero, the inner ideals are shown to correspond to the objects of a geometry defined by Tits from the corresponding Chevalley group.

Realization of graded-simple algebras as loop algebras

Abstract Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient 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 sufficient 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. 2000 Mathematics Subject Classification: 16W50, 17B70; 17B65, 17B67.

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 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.

Structurable triples, Lie triples, and symmetric spaces

Structurable triple Systems are introduced and shown to be coordinates for certain Lie triple Systems. Rotational Symmetrie spaces are characterized using real Structurable algebras. 1991 Mathematics Subject Classification: 17A40, 17A30, 53C35.

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.

Lie Tori of Type BC 2 and Structurable Quasitori

Structurable quasitori are introduced and shown to give all Lie tori of type BC 2 via the Kantor construction. Structurable quasitori, including Jordan quasitori, are classified. The methods include a use of generalized quadrangles of order (2,t).

GENERALIZED QUADRANGLES AND CUBIC FORMS

A Jordan pair is constructed from a pair of cubic forms satisfying the adjoint identities. Given some parameters and an incidence structure S having three points on each line and no more than one line through two points, a pair of cubic forms are constructed. These forms satisfy the adjoint identities if and only if S is either a star or a generalized quadrangle and the parameters are precisely determined.

Projective remoteness planes

A new axiomatization involving incidence and remoteness of planes with nondivision coordinate rings is introduced and a coordinatization theorem is obtained. A geometric process of splitting points and lines to obtain another plane with the same coordinates is described. It is also shown that a group of Steinberg type is parametrized by a nonassociative ring. The notion of elementary basis sets for an associative ring is introduced and constructions of projective and affine planes are given. A plane with reflections determining a system of rotations is shown to have commutative, associative coordinates.

Weyl Images of Kantor Pairs

Kantor pairs arise naturally in the study of 5-graded Lie algebras. In this article, we begin the study of simple Kantor pairs of arbitrary dimension. We introduce Weyl images of Kantor pairs and use them to construct examples of Kantor pairs including a new class of central simple Kantor pairs.