Almost fine gradings on algebras and classification of gradings up to isomorphism

TL;DR

This paper introduces almost fine gradings on finite-dimensional algebras, providing a classification framework that simplifies understanding all gradings up to isomorphism, especially for semisimple Lie algebras.

Contribution

It defines almost fine gradings, establishes their relation to all gradings, and applies the theory to classify gradings on semisimple Lie algebras over algebraically closed fields.

Findings

Every G-grading is derived from an almost fine grading

Method to obtain all almost fine gradings from known fine gradings for abelian groups

Constructed root system-based gradings for semisimple Lie algebras

Abstract

We consider the problem of classifying gradings by groups on a finite-dimensional algebra $A$ (with any number of multilinear operations) over an algebraically closed field. We introduce a class of gradings, which we call almost fine, such that every $G$-grading on $A$ is obtained from an almost fine grading on $A$ in an essentially unique way, which is not the case with fine gradings. For abelian groups, we give a method of obtaining all almost fine gradings if fine gradings are known. We apply these ideas to the case of semisimple Lie algebras in characteristic $0$: to any abelian group grading with nonzero identity component, we attach a (possibly nonreduced) root system and, in the simple case, construct an adapted grading by this root system.