Abstract
Let \(\mathcal{N}\) denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field \(\mathbb{F}\), there exists a smallest ideal I of L such that L/I ∈ \(\mathcal{N}\). This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L\(\mathcal{N}\). In this paper, we define the subalgebra S(L) = ∩H≤LIL(H\(\mathcal{N}\)). Set S0(L) = 0. Define Si+1(L)/Si(L) = S(L/Si(L)) for i > 1. By S∞(L) denote the terminal term of the ascending series. It is proved that L = S∞(L) if and only if L\(\mathcal{N}\) is nilpotent. In addition, we investigate the basic properties of a Lie algebra L with S(L) = L.
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Wei Meng was supported by the National Natural Science Foundation of China (11761079). Hailou Yao was supported by the National Natural Science Foundation of China (11671126).
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Meng, W., Yao, H. On the nilpotent residuals of all subalgebras of Lie algebras. Czech Math J 68, 817–828 (2018). https://doi.org/10.21136/CMJ.2018.0006-17
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DOI: https://doi.org/10.21136/CMJ.2018.0006-17