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- W3194592512 abstract "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 $igeq n$, then $N/Z_i(N, L))$ is finite. In particular, we shall provide an upper bound as following, $$mathrm{dim}(frac{N}{Z_i(N, L)}) leq ((d_n)^nd_nd_{n+1}ldots d_{i-1})mathrm{dim}(gamma_{i+1}(N, L))$$$$leq (d_n)^i(mathrm{dim}gamma_{i+1}(N, L)).$$" @default.
- W3194592512 created "2021-08-30" @default.
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- W3194592512 date "2021-08-01" @default.
- W3194592512 modified "2023-09-23" @default.
- W3194592512 title "Upper and lower central series in a pair of Lie algebras" @default.
- W3194592512 doi "https://doi.org/10.22128/gadm.2020.381.1034" @default.
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