Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n geqq 2n geqq 2, let H(n) be the group of all n x n matrices having determinant ±1pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) in { H(n), {rm SL}(n)} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?langle {rm Sem}(n) rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?langle {rm Sem}(n) rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].