Some properties of Fox’s derivations for Lie algebras

Abstract

Let F be a free sum of Lie algebras A_i(i ∈ I) and a free Lie algebra G with basis {g_j|j ∈ J{ and its ideal N has trivial intersection with each summand A_i. Let U(F) be the universal enveloping algebra of F, NU the ideal in U(F) which is generated by N. In this paper we describe an elements v of the algebra F, such that D_l(v) ≡0 mod N_U, where l belongs to a subset of I ∪ J and D_k: U(F) → U(F)(k ∈ I ∪ J) are the Fox derivations of the universal enveloping algebra U(F). Using obtained description, we prove a theorems on freedom.