3-Graded Lie Algebras with Jordan Finiteness Conditions

Abstract

A notion of socle is introduced for 3-graded Lie algebras (over a ring of scalars Φ containing ) whose associated Jordan pairs are non-degenerate. The socle turns out to be a 3-graded ideal and is the sum of minimal 3-graded inner ideals each of which is a central extension of the TKK-algebra of a division Jordan pair. Non-degenerate 3-graded Lie algebras having a large socle are essentially determined by TKK-algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras, which are also 3-graded. Classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space provide a source of examples of infinite dimensional strongly prime 3-graded Lie algebras with non-zero socle. Other examples can be found within the class of finitary simple Lie algebras

Key Words:

• 3-Graded Lie algebra
• Jordan pair
• Socle

Mathematics Subject Classification:

• 17B60
• 17B65
• 16W10
• 17C50

Acknowledgments

The authors are indebted to Alberto Elduque for his interesting remarks about a preliminary version of the paper. They also wish to thank the referee for the careful reading of the manuscript and his valuable comments and suggestions. The first and third authors were partially supported by the MCYT, BFM2001-1938-C02-01 and Fondos FEDER, and the Junta de Andalucía FQM264, while the second author was partially supported by the MCYT, BFM2001-1938-C02-02 and Fondos FEDER.

Notes

^#Communicated by B. Allison.