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Exceptional Lie Algebras and Related Algebraic and Geometric Structures

Exceptional Lie Algebras and Related Algebraic and Geometric Structures EXCEPTIONAL LIE ALGEBRAS AND RELATED ALGEBRAIC AND GEOMETRIC STRUCTURES J. R. FAULKNERf AND J. C. FERRARJ 1. Introduction 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 non- Desarguian projective planes) are just different manifestations of the same pheno- menon. 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. In §2, we briefly survey the relevant areas in the theory of Lie algebras (§2.1) and the related Chevalley groups (§2.2), for these provide the principal motivation for our study. §§3, 4, 5 are devoted to some well known (alternative algebras in §3, Jordan algebras in §4) and not so well known (3-ternary algebras in §5) classes http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Exceptional Lie Algebras and Related Algebraic and Geometric Structures

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References (27)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/9.1.1
Publisher site
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Abstract

EXCEPTIONAL LIE ALGEBRAS AND RELATED ALGEBRAIC AND GEOMETRIC STRUCTURES J. R. FAULKNERf AND J. C. FERRARJ 1. Introduction 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 non- Desarguian projective planes) are just different manifestations of the same pheno- menon. 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. In §2, we briefly survey the relevant areas in the theory of Lie algebras (§2.1) and the related Chevalley groups (§2.2), for these provide the principal motivation for our study. §§3, 4, 5 are devoted to some well known (alternative algebras in §3, Jordan algebras in §4) and not so well known (3-ternary algebras in §5) classes

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1977

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