The remarks about verbal ideals of a free finite-generated Lie algebra

In the article we consider verbal ideals of the free finite-generated Lie algebras over field of zero characteristic. This ideals are connected with an action of the universal envelopment algebra of the Lie algebra of derivations of a free finite-generated Lie algebra. We estimate the multiplicities of the irreducible components in the modules of the concequences of the identities. For the homogeneous identities which generate the irreducible module of degree $n$, we study the decomposition to irreducible components his module of consequences degree of $n+1$. In the case two-generated of the free Lie algebra we also find the vectors oldest weight in these irreducible components.