Geodesic
Almost fine gradings on algebras and classification of gradings up to isomorphism
Documenta mathematica, Tome 30 (2025) no. 4, pp. 887-908

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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 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 Φ and, in the simple case, construct an adapted Φ-grading.
DOI : 10.4171/dm/1006
Classification : 17A01, 17A36, 17B70, 16W50, 20G07
Mots-clés : grading, classification, automorphism group, Lie algebra, root grading 
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Alberto Elduque; Mikhail Kochetov. Almost fine gradings on algebras and classification of gradings up to isomorphism. Documenta mathematica, Tome 30 (2025) no. 4, pp. 887-908. doi: 10.4171/dm/1006

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