Abstract:
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.
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