Schreier varieties of \(n\)-Lie algebras (Q1177570)

An \(n\)-Lie algebra is a vector space equipped with an \(n\)-ary anticommutative multiplication satisfying the generalized Jacobi identity \[ x_ 1...x_{n-1}(y_ 1...y_ n)=\sum_{k=1}^ ny_ 1...(x_ 1...x_{n-1}y_ k)...y_ n. \] The paper under review studies \(n\)-Lie algebras over infinite fields of characteristic more than \(n+1\) or of characteristic 0. The author obtains some polynomial identities for \(n\)- Lie algebras which generalize the standard identity. The main result is that the only Schreier variety of \(n\)-Lie algebras is the variety of all algebras with trivial multiplication \(x_ 1...x_ n=0\). Recall that a variety of universal algebras \(U\) is Schreier if every subalgebra of a \(U\)-free algebra is free in \(U\).