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

Elduque, Alberto (Universidad de Zaragoza) ; Kochetov, Mikhail

Resumen: 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.

Idioma: Inglés

DOI: 10.4171/DM/1006

Año: 2025

Publicado en: Documenta Mathematica 30, 4 (2025), 887-908
ISSN: 1431-0635

Financiación: info:eu-repo/grantAgreement/ES/DGA/E22-23R
Financiación: info:eu-repo/grantAgreement/EUR/ISCII-ERDF/A way to make Europe
Financiación: info:eu-repo/grantAgreement/ES/MCINN/PID2021-123461NB-C21

Tipo y forma: Article (Published version)

Área (Departamento): Área Algebra (Dpto. Matemáticas)