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.
1
Universidad de Zaragoza, Zaragoza, Spain
2
Memorial University of Newfoundland, St. John’s, Canada
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
@article{10_4171_dm_1006,
author = {Alberto Elduque and Mikhail Kochetov},
title = {Almost fine gradings on algebras and classification of~gradings up to isomorphism},
journal = {Documenta mathematica},
pages = {887--908},
year = {2025},
volume = {30},
number = {4},
doi = {10.4171/dm/1006},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/1006/}
}
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AU - Alberto Elduque
AU - Mikhail Kochetov
TI - Almost fine gradings on algebras and classification of gradings up to isomorphism
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