Note on Malcev algebras. (English) Zbl 0166.04203

A Malcev algebra is a vector spare over a field whose multiplication satisfies \[ x^2 = 0, \quad (xy)(zx) + (xy \cdot z)x + (yz\cdot x)x + (zx \cdot x) y = 0. \] Relative to any bilinear product, define a ternary composition by \[ [xyz] = x(yz) - y(xz) + (xy)z. \tag{*} \] The following theorems are proved:
I. If \(M\) is an algebra over the field \(\Phi\), \(\operatorname{char} \Phi \ne 2, 3\), whose multiplication satisfies \(x^2 = 0\) for all \(x\in M\), then \(M\) is a Malcev algebra if and only if every transformation of the form \(z \to [xyz]\) \((x, y\in M)\) is a derivation.
II. Let \(M\) be a Malcev algebra over the field \(\Phi\), \(\operatorname{char} \Phi \ne 2\). Then there is a Lie algebra \(\mathfrak L\supseteq M\), and a subalgebra \(\mathfrak D\subseteq \mathfrak L\) such that (1) \(\mathfrak L = M \oplus \mathfrak D\), (2) \(M\) is stable under the adjoint action of \(\mathfrak D\), (3) for \(x, y \in M\), the projection of \([x, y]\) on \(M\) along is the original (Malcev) product of \(x\) and \(y\). The author concludes from the above that a Malcev algebra is a Lie algebra if and only if it is a Lie triple system relative to the ternary product defined by (*).


17D10 Mal’tsev rings and algebras