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

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.