Group gradings on restricted Cartan type Lie algebras

TL;DR

This paper classifies G-gradings on certain simple restricted Lie algebras of types W, S, and H over algebraically closed fields of characteristic p>3, using automorphism group schemes to understand their structure.

Contribution

It provides a complete classification of G-gradings on these Lie algebras, introducing methods involving automorphism group schemes for the first time in this context.

Findings

Classified G-gradings on W(m;1) and S(m;1) Lie algebras.

Determined automorphism group schemes for S(m;1) and H(m;1).

Connected gradings to numerical and group-theoretical invariants.

Abstract

For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism group schemes, which we determine for the simple restricted Lie algebras of types S(m;1) and H(m;1). The ground field is assumed to be algebraically closed of characteristic p>3.