Analogs of the Cartan criteria for \(n\)-Lie algebras
According to \textit{V. T. Filippov} [Sib. Mat. Zh. 26, No. 6, 126-140 (1985; Zbl 0585.17002)], an \(n\)-Lie algebra \((n\geq 2)\) over a field \(F\) of characteristic 0 is a linear space \(A\) over \(F\) with \(n\)-ary polylinear operation \([.,\dots,.]\) which is skewsymmetric by all arguments and satisfies to the generalized Jacobi identity \[ \bigl[ [x_1,\dots, x_n],y_2,\dots, y_n\bigr]= \sum^n_{i=1} \bigl[x_1,\dots, [x_i,y_2,\dots, y_n],\dots, x_n\bigr]. \] A subspace \(I\) of \(A\) is called an ideal of \(A\) if \([I,A,\dots, A]\subseteq I\). An ideal \(I\) of an \(n\)-Lie algebra \(A\) is called solvable in the sense of V. T. Filippov if \(I^{(r)}=0\) for some \(r\geq 0\) where \(I^{(0)}= I\), \(I^{(s+1)}= [I^{(s)}, I^{(s)},\dots, I^{(s)}]\). An ideal \(I\) of an \(n\)-Lie algebra \(A\) is called solvable in the sense of E. N. Kuz'min if \(I^{(r)}=0\) for some \(r\geq 0\) where \(I^{(0)}=I\), \(I^{(s+1)}= [I^{(s)}, I^{(s)}, A,\dots,A]\). It is shown that Cartan's criteria for finite-dimensional Lie algebras to be semisimple and solvable are fully adaptable to \(n\)-Lie algebras, provided that ideals of an \(n\)-Lie algebra are understood to be solvable in the sense of Kuz'min. Specifically, a characterization of the Kuz'min radical in terms of a trace form associated with some representation is presented. It is analogous to the characterization one has in the case of Lie algebras. Another analog of the Cartan theorem is proved for \(n\)-Lie algebras which are solvable in the sense of Filippov.
scientific article; zbMATH DE number 907268