On Depth-Three Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component

Abstract

We consider the finite-dimensional transitive irreducible graded Lie algebras of depth three over an algebraically closed field of characteristic three with classical reductive component L ₀, and show that the L ₀-module L ₋₁ in such algebras must be restricted. This result contrasts with the depth-one case, in which non-restricted modules L ₋₁ can occur, but is similar to the depth-two case, where L ₋₁ is necessarily restricted.

Key Words:

• Lie algebras
• Modular Lie algebras
• Graded Lie algebras

Mathematics Subject Classification:

• Primary 17B50, 17B70
• Secondary 17B20

Acknowledgment

The second author gratefully acknowledges partial support from the Russian Foundation of Basic Research #02-01-00725. He would also like to express his appreciation for the hospitality of The Ohio State University, both at Columbus and at Mansfield, and for the support of The Ohio State University at Mansfield.

Notes

^#Communicated by K. Misra.