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

Abstract

We consider the problem of classifying gradings by groups on a finite-dimensional algebra $\mathcal{A}$ (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 $\mathcal{A}$ is obtained from an almost fine grading on $\mathcal{A}$ in an essentially unique way, which is not the case with fine gradings. For abelian $G$, 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 $\Phi$ and, in the simple case, construct an adapted $\Phi$-grading.