Enveloping algebras of anticommutative triple systems and 3-Lie algebras
In [Algebra Logic 29, No. 1, 48--59 (1990); translation from Algebra Logika 29, No. 1, 67--81 (1990; Zbl 0726.17001)] an arbitrary anticommutative triple system \(A\) (with characteristic \(p\neq 3)\) has been characterized by the author as a subspace of some anticommutative algebra \(B\) such that the trilinear operation \([xyz]\) of the system \(A\) coincides with the Jacobian \(J(x,y,z)=(xy)z+(zx)y+(yz)x\) which is defined in \(B\), and \(B=A\dot +{\mathfrak R}(A)\), where \({\mathfrak R}(A)\) is the Lie algebra of right multiplications of the system \(A\). \(B\) is called a standard \(\beta\)-enveloping algebra of \(A\). The paper under review continues the study of standard \(\beta\)-enveloping algebras of anticommutative triple systems. It is proved that if \(B\) is \(L\)-solvable then \(A\) is solvable, further, if \(B\) is solvable and \(A\) is a 3-Lie algebra then \(A\) is solvable. For a 3-Lie algebra \(A\), it is also proved that its standard \(\beta\)-enveloping algebra \(B\) is solvable if \(A\) is so-called 2-solvable.
scientific article; zbMATH DE number 4198250