Exceptional Lie Algebras and Related Algebraic and Geometric Structures

Abstract

Certain algebraic structures, most notably associative, alternative, and Jordan algebras are strongly linked via construction and classification to simple Lie algebras and to interesting geometries. These geometries are in turn linked to simple Lie algebras via their groups of collineations. These linkages serve to illustrate how various notions of exceptionality in algebra and geometry (e.g., non-classical Lie algebras, non-associative alternative algebras, non-special Jordan algebras, and nonDesarguian projective planes) are just different manifestations of the same phenomenon. It is the intent of this survey to discuss briefly the general classes of structures in which the exceptional objects occur, to describe the linkage between the exceptional objects, and to illustrate the utility of these linkages in understanding the nature of these diverse exceptional structures.