Graded subalgebras associated with a subalgebra of a graded free Lie algebra
If \(L=\oplus_{n\geq 1}L^ n\) is a graded free Lie algebra over a field K and A is a subalgebra of L, let gr A (resp. Gr A) denote the subalgebra of L generated by the homogeneous components of minimum (resp. maximum) degree of elements of A. It is shown that for any positive integer r, gr \(A_ r=(gr A)_ r\) and Gr \(A_ r=(Gr A)_ r\) where \(B_ r\), for a Lie algebra B denotes the r-th term of the lower central series of B. As an application of this result it is shown that if B is a free subalgebra of a free nilpotent Lie algebra N and Y is a free generating set of N, then \(gr_ Y B\) is also free where the gradation of N is the one induced by Y. It is known [\textit{M. Boral}, Proc. Am. Math. Soc. 94, 23-28 (1985; Zbl 0532.17004)] that the only free non-abelian ideal in a free nilpotent Lie algebra is the algebra itself. A simple proof of this result is given as another application.
scientific article; zbMATH DE number 3899119