Subalgebras of free leibniz algebras

Abstract

Leibniz algebras are in some sense non-(anti)commutative analogs of Lie algebras. The variety of all Lie algebras has the Schreier property (i.e. any subalgebra of the free algebra is free). This is not the case in the variety of all Leibniz algebras. Nevertheless we prove that: 1) the variety of all Leibniz algebras has the property of differential separability for subalgebras; 2) the Jacobian conjecture is true for free Leibniz algebras; 3) the free Leibniz algebras are finitely separable (in particular, it follows that the occurrence problem for free Leibniz algebras is solvable).