Partial derivatives and endomorphisms of some relatively free Lie algebras

We define partial derivatives for free Lie algebras and extensions of nilpotent Lie algebras. The derivatives are analogs of the Fox derivatives in group theory. We give a matrix criterion for invertibility of an endomorphism of a free Lie algebra or a free Lie algebra in $\mathfrak{N}_c\mathfrak{M}$, $(c\ge1)$ , where $\mathfrak{M}$ stands for an arbitrary homogeneous variety of Lie algebras and $\mathfrak{N}_c$ is the variety of nilpotent Lie algebras of class $\le c+1$. A criterion of primitiveness is presented for a system of elements in a free Lie algebra of $\mathfrak{N}_c\mathfrak{A}$, where $\mathfrak{A}$ is the variety of abelian Lie algebras. We also prove algorithmic recognizability of automorphisms of an arbitrary free polynilpotent algebra of finite rank among all the endomorphisms of the algebra.