Almost fine gradings on algebras and classification of gradings up to isomorphism
Alberto Elduque, Mikhail Kochetov

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 (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 -grading on is obtained from an almost fine grading on 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 : 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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Sphingolipid Metabolism and Signaling · Algebraic structures and combinatorial models
