Upper and Lower Central Series in a Pair of Lie Algebras

The 𝐵aer’s theorem in the termes of the 𝐿ie algebras states that for a 𝐿ie algebra 𝐿 the finiteness of dim(𝐿/𝑍i(𝐿)) implies the finiteness of dim(γi+1(𝐿)) for all non negative integers i. 𝐿et (𝑁, 𝐿) denote a pair of 𝐿ie algebras, where 𝑁 is an ideal of 𝐿, and di = di(𝐿) denote the minimal number of generators of 𝐿/𝑍i(𝑁, 𝐿) for all non negative integers i. In this paper, we consider the pair (𝑁, 𝐿) and show that if dn is finite, then the converse of 𝐵aer’s theorem is true. In fact, we shall show that if for all i ≥ n, dn and dim(γi+1(𝑁, 𝐿)) are finite, then 𝑁/𝑍i(𝑁, 𝐿)) is finite. In particular, we give an upper bound as following, dim( 𝑁 𝑍i(𝑁, 𝐿) ) ≤ ((dn) n dndn+1 . . . di−1)dim(γi+1(𝑁, 𝐿)) ≤ (dn) i (dimγi+1(𝑁, 𝐿)) for all non negative integers i.