Homological properties of metabelian restricted Lie algebras (Q6146415)

This article focuses on homological properties and presentability of metabelian restricted Lie algebras over fields. Let \(k\) be a field. A Lie algebra \(\mathcal L\) over \(k\) is said to be \textit{metabelian} when \([[\mathcal L, \mathcal L], \mathcal L] = 0\). Further, if one denotes by \(\mathcal U(\mathcal L)\) the universal enveloping algebra of \(\mathcal L\), then \(\mathcal L\) is said to be of \emph{type \(FP_m\)} if there exists a projective resolution of the trivial \(\mathcal U(\mathcal L)\)-module \(k\) such that every projective module is finitely generated up to degree \(m\). Moreover, when the field \(k\) has positive characteristic \(p\), a Lie algebra \(\mathcal L\) over \(k\) is said to be \textit{restricted} when it is endowed with a map \(-^{[p]} : \mathcal L \to \mathcal L\) that satisfies the following conditions: \begin{itemize} \item[(i)] \((\lambda x)^{[p]} = \lambda^p x^{[p]}\) for all \(\lambda \in k\) and \(x \in \mathcal L\). \item[(ii)] \((x + y)^{[p]} = x^{[p]} + y^{[p]} + \sum_{i = 1}^{p-1} s_i(x, y)\), where \(s_i(x, y\)) are elements in \(\mathcal L^p\) for all \(x, y \in \mathcal L\). \item[(iii)] \(\text{ad}\left(x^{[p]}\right) = \left( \text{ad}(x) \right)^p\) for all \(x \in \mathcal L\). \end{itemize} In the main results of this article, the authors classify type \(FP_m\) metabelian restricted Lie algebras that are split extensions of abelian-by-abelian restricted Lie algebras.