On structure of graded restricted simple Lie algebras of Cartan type as modules over the Witt algebra

TL;DR

This paper investigates the structure of graded restricted simple Lie algebras of Cartan type, revealing their decomposition into modules over the Witt algebra and explicitly determining their composition factors.

Contribution

It provides a detailed decomposition of these Lie algebras into restricted baby Verma modules and simple modules, extending understanding of their module structure.

Findings

Decomposition into restricted baby Verma modules and simple modules

Explicit determination of composition factors

Identification of subalgebras isomorphic to the Witt algebra

Abstract

Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.