Abstract
We show that each irreducible, transitive finite-dimensional graded Lie algebra over a field of prime characteristic p contains an initial subalgebra in which the pth power of the adjoint transformation associated with any element in the lowest gradation space is zero.
Highlights
In the classification of the simple finite-dimensional Lie algebras over fields of prime characteristic, irreducible transitive finite dimensional graded Lie algebras play a fundamental role [1]
It is well known that in Lie algebras of Cartan type, there is a subalgebra, the “initial piece,” which contains the sum of the negative gradations spaces of the Lie algebra, and in which the pth power of the adjoint representation associated with any element of the lowest gradation space is zero
We prove that any irreducible, transitive finite-dimensional graded Lie algebra contains such an initial subalgebra
Summary
In the classification of the simple finite-dimensional Lie algebras over fields of prime characteristic, irreducible transitive finite dimensional graded Lie algebras play a fundamental role [1]. The simple finite dimensional Lie algebras over algebraically closed fields of characteristic greater than three have been classified [2]. It is well known that in Lie algebras of Cartan type, there is a (not necessarily proper) subalgebra, the “initial piece,” which contains the sum of the negative gradations spaces of the Lie algebra, and in which the pth power of the adjoint representation associated with any element of the lowest gradation space is zero. We prove that any irreducible, transitive finite-dimensional graded Lie algebra contains such an initial subalgebra.
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