Upper and Lower Central Series in a Pair of Lie Algebras

Document Type : Research Paper


1 School of Mathematics and Computer sciences, Damghan University, Damghan, Iran

2 Damghan University


The Baer's theorem in the termes of the Lie algebras states that for a Lie algebra $L$ the finiteness of $\mathrm{dim}(L/Z_i(L))$ implies the finiteness of $\mathrm{dim}(\gamma_{i+1}(L))$. Let $(N,L)$ denote a pair of Lie algebras, where $N$ is an ideal of $L$, and $d_i=d_i(L)$ denote the minimal number of generators of $L/Z_i(N, L)$. In this paper we shall consider the pair $(N, L)$ and show that if $d_n$ is finite then the converse of Baer's theorem is true. In fact we shall show that if $d_n$ and $\mathrm{dim}(\gamma_{i+1}(N, L))$ are finite, where $i\geq n$, then $N/Z_i(N, L))$ is finite. In particular, we shall provide an upper bound as following,
$$\mathrm{dim}(\frac{N}{Z_i(N, L)}) \leq ((d_n)^nd_nd_{n+1}\ldots d_{i-1})\mathrm{dim}(\gamma_{i+1}(N, L))$$$$\leq (d_n)^i(\mathrm{dim}\gamma_{i+1}(N, L)).$$
for all non negative integers i.


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