# Speciality problem for Malcev algebras

A Malcev algebra is an algebra that satisfies the identities

\[ xx=0, \ \ \ J(xy,z,x)=J(x,y,z)x, \]

where $J(x,y,z)=(xy)z+(yz)x+(zx)y$. Clearly, any Lie algebra is a Malcev algebra. If $A$ is an alternative algebra then it forms a Malcev algebra $A^{(-)}$ with respect to the commutator multiplication $[a,b]=ab-ba$. The most known examples of non-Lie Malcev algebras are the algebra $O^{(-)}$ for an octonion algebra $O$ and its subalgebra $sl(O)$ consisting of octonions with zero trace. Every simple non-Lie Malcev algebra is isomorphic to $sl(O)$.

The problem of speciality, formulated by A.I.Malcev in 1955, asks whether any Malcev algebra is isomorphic to a subalgebra of $A^{(-)}$ for certain alternative algebra $A$. In other words, it asks whether an analogue of the celebrated Poincare-Bikhoff-Witt theorem is true for Malcev algebras. We show that the answer to this problem is negative, by constructing a Malcev algebra which is not embeddable into an algebra $A^{(-)}$ for any alternative algebra $A$.

It is a joint work with A.Buchnev, V.Filippov, and S.Sverchkov.