Partial derivatives and endomorphisms of some relatively free Lie algebras
Let \(L\) be a free Lie algebra of \(x_ 1,\dots, x_ n\) over a field \(k\) of characteristic zero. Then the universal envelope \(U(L)\) is a free associative algebra on \(x_ 1,\dots, x_ n\). If \(f\in U(L)\) has zero constant term then \(f\) has a unique representation \(f=\sum_{i=1}^ n x_ i u_ i\), where \(u_ i\in U(L)\). Put \(u_ i= {{\partial f} \over {\partial x_ i}}\). Assume that \(\varphi\) is an endomorphism of the Lie algebra \(L\) and \(f_ j= \varphi(x_ j)\), \(j=1,\dots, n\). Denote by \(J(\varphi)\) the Jacobi matrix \(J(\varphi)= {{\partial f_ j} \over {\partial x_ i}}\). The following are equivalent: (i) \(\varphi\) is an automorphism of \(L\); (ii) the matrix \(J(\varphi)\) is invertible in \(\text{Mat} (n,U(L))\). A similar result is proved for a relatively free Lie algebra \(\widetilde{L}\) of a variety \(N_ c V\) where \(N_ c\) is a variety of all nilpotent Lie algebras of a class \(\leq c+1\) and \(V\) is an arbitrary proper variety of Lie algebras. Let \(\widetilde{L}\) be a free algebra of rank \(n\) in a variety \(N_ c A\) where \(A\) is the variety of all abelian Lie algebras. If \(f_ 1,\dots, f_ m\in \widetilde{L}\) then the following are equivalent: (iii) \(f_ 1,\dots, f_ m\) can be extended to a base of \(\widetilde{L}\); (iv) the matrix \(J(\varphi)\) of size \(n\times m\) is left invertible. Analogous results for groups have been obtained by \textit{V. A. Roman'kov} [Ukr. Math. J. 43, No. 7/8, 930-935 (1991); translation from Ukr. Mat. Zh. 43, No. 7/8, 996-1002 (1991; Zbl 0743.20033)] and by \textit{E. I. Timoshenko}. The last theorem of the paper states that automorphisms of free polynilpotent Lie algebras of finiterank are algorithmically recognizable among all its endomorphisms.
scientific article; zbMATH DE number 652228