Bruce Allison

Extended affine Lie algebras and their root systems

Covering extended affine Lie algebras and their root systems, this work is intended for graduate students, research mathematicians, and mathematical physicists interested in Lie theory.

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.

Structurable tori and extended affine Lie algebras of type BC1

Structurable n-tori are nonassociative algebras with involution that generalize the quantum n-tori with involution that occur as coordinate structures of extended affine Lie algebras. We show that the core of an extended affine Lie algebra of type BC1 and nullity n is a central extension of the Kantor Lie algebra obtained from a structurable n-torus over C. With this result as motivation, we prove general properties of structurable n-tori and show that they divide naturally into three classes. We classify tori in one of the three classes in general, and we classify tori in the other classes when n=2. It turns out that all structurable 2-tori are obtained from hermitian forms over quantum 2-tori with involution.

Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

Let M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n-tuples of commuting finite order automorphisms. It is a classical result that M_1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in M_1. In this paper, we classify the algebras in M_2, and further determine the relationship between M_2 and two other classes of Lie algebras: the class of all loop algebras of affine Lie algebras and the class of all extended affine Lie algebras of nullity 2.

Trace forms for structurable algebras

Structurable algebras were defined in [2] as a generalization of Jordan algebras, and finite-dimensional simple structurable algebras of characteristic 0 were classified in that paper. As a generalization of the Tits-Koecher construction for Jordan algebras, a Lie algebra Xx(&, ) has been associated in [3] with each structurable algebra (Ccp, ). By transferring properties of X(&, -) to (&, -), a structure theory for linite-dimensional structurable algebras of characteristic 0 was obtained in [9-111. In this paper we consider two trace forms on structurable algebras. The symmetric bilinear form

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.