Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property
Let \(k\) be a field of characteristic different from 2 and \(F\) a free algebra in a Schreier variety \(V\) of nonassociative \(k\)-algebras defined by a set of homogeneous identities. An element \(v\in F\) is primitive if \(v\) can be included into some set of generators of \(F\). Let \(F\) have two free generators and \(u\in F\setminus 0\). Suppose that \(\phi\) is an endomorphism of \(F\) such that the automorphic orbit of \(u\) is stable under the action of \(\phi\). Then \(\phi\) is an automorphism of \(F\). A non-primitive element \(u\in F\) is almost primitive if it is primitive in any proper subalgebra of \(F\) containing \(u\). It is shown that \(x_1x_2+x_3x_4+\cdots +x_{2m-1}x_{2m}\) is almost primitive in \(F\) provided \(\{x_1,x_2,\ldots, x_{2m}\}\) is the set of free generators in \(F\). Similarly \(x_1x_2+x_3x_4+\cdots +x_{2m-1}x_{2m}+x_{2m}x_{2m+1}\) is almost primitive in the free nonassociative algebra \(F\) provided \(\{x_1,x_2,\ldots, x_{2m+1}\}\) is the set of free generators in \(F\). There are given various examples of almost primitive elements in a free Lie algebra, free Lie \(p\)-algebra, free Lie superalgebra, free Lie \(p\)-superalgebra. An element \(a\neq 0\) in an algebra \(A\) is a test element if for any endomorphism \(\phi\) of \(A\) the equality \(\phi(a)=a\) means that \(\phi\) is an automorphism. An almost primitive element \(a\in F\setminus 0\) is a test one if its constituent of degree 1 vanishes. It means that all examples of almost primitive elements listed above are in fact test elements. Let \(F\) be a free Lie algebra with a free set of generators \(X=\{x_1,\ldots,x_n\}\). Then the universal envelope \(U\) is a free associative algebra on \(X\). If \(u\in U\), then \(u=\alpha+x_1a_1+\cdots +x_na_n\), where \(a_i\in U,\alpha\in k\). Put \(a_i=\frac{\partial u}{\partial x_i}\). An element \(a\in F\) is \(\Delta\)-primitive if the elements \(\frac{\partial a}{\partial x_1},\ldots \frac{\partial u}{\partial x_n}\) generate the left ideal \(\Delta=Ux_1+\cdots +Ux_n\) in \(U\). Suppose that \(n\) is even. Then any \(\Delta\)-primitive element is an automorphic image of \([x_1,x_2]+[x_3,x_4]+\cdots +[x_{n-1},x_n]\). If \(n\) is odd then \(F\) has no \(\Delta\)-primitive elements. The automorphism group acts transitively on the set of all \(\Delta\)-primitive elements.
scientific article; zbMATH DE number 1471831
scientific article