Some Theorems on Leibniz Algebras

Abstract

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.

Key Words:

• Engel subalgebras
• Leibniz algebras
• Nilpotent
• Subnormal

2000 Mathematics Subject Classification:

• 17A32

Notes

Communicated by K. Misra.