Generic, almost primitive and test elements of free Lie algebras (Q2781287)

Let \(\mathfrak V\) be a variety of Lie algebras over a field \(K\) defined by the set of identities \(\Sigma\), \(L=L(X)\) the free Lie algebra with the set \(X\) of free generators and \(\mathfrak V(L)\) the verbal ideal of \(L\) defined by \(\mathfrak V\). An element \(f\) of \(L\) is called \(\mathfrak V\)-generic if \(f\in\mathfrak V(L)\) but \(f\notin\mathfrak V(H)\) for every proper subalgebra \(H\) of \(L\). An element \(u\) of \(L(X)\) is said to be primitive if it is an element of some set of free generators of \(L(X)\). An almost primitive element of \(L\) is an element which is not primitive in \(L\) but is primitive in any proper subalgebra of \(L\) containing it. An element \(u\) of \(L(X)\) is called a test element if any endomorphism \(\varphi\) of \(L(X)\) with \(\varphi(u)=u\) is an automorphism of \(L(X)\) The authors construct a series of generic elements of free Lie algebras and find new almost primitive and test elements. An example of an almost primitive but not generic element is presented.