Classification of group gradings on simple Lie algebras of types A, B, C and D

TL;DR

This paper classifies all possible gradings by abelian groups on simple Lie algebras of types A, B, C, and D over algebraically closed fields, using invariants to distinguish isomorphism classes.

Contribution

It provides a comprehensive classification of G-gradings on these Lie algebras, extending previous work to all classical types with explicit invariants.

Findings

Complete classification of G-gradings for types A, B, C, D

Use of numerical and group-theoretical invariants

Results applicable over algebraically closed fields of characteristic not 2

Abstract

For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple Lie algebras of types A_n (n >= 1), B_n (n >= 2), C_n (n >= 3) and D_n (n > 4), in terms of numerical and group-theoretical invariants. The ground field is assumed to be algebraically closed of characteristic different from 2.