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 is 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.
Observe that earlier a notion of a (non-alternative) universal enveloping
algebra U(M) for a Malcev algebra M was introduced by J.M.Perez-Izquierdo
and the speaker. The algebra U(M) has properties similar to those of the
universal enveloping algebras of Lie algebras.
It is a joint work with A.Buchnev, V.Filippov, and S.Sverchkov.
