Gradings on simple Jordan and Lie algebras
The simple classical Lie algebras over an algebraically closed field \(F\) of characteristic \(0\) consist of either the zero trace \(n\times n\) matrices for \(n\geq 2\), or of the skew symmetric elements in \(M_n(F)\) relative to an orthogonal or symplectic involution. \textit{Yu. A.~Bahturin} and \textit{M. V.~Zaicev} [Can. Math. Bull. 45, 499--508 (2002; Zbl 1022.16032)] determined the gradings on \(M_n(F)\). Their results are instrumental here. In the paper under review, the gradings on matrix algebras with involutions (that is, gradings where the homogeneous components are invariant under the involution) are determined first. Then this is used to determine all the gradings by finite abelian groups of the simple classical Lie algebras other than \(D_4\), and of the simple special Jordan algebras other than the Jordan algebras of symmetric nondegenerate bilinear forms. Gradings by torsion free groups are studied too.
scientific article; zbMATH DE number 2136612
scientific article