On finite presentability and the homological type \(FP_n\) of Lie algebras: a survey
The paper under review is a compact, but rather exhaustive survey of finite presentability and homological finiteness properties of Lie algebras, with an eye on corresponding results in geometric group theory. Among the covered topics, one finds the result of \textit{A. Wasserman} that in a finitely presented soluble Lie algebra every ideal of codimension one is finitely generated as a Lie algebra [Isr. J. Math. 106, 79--92 (1998; Zbl 0930.17006)], and the criterion of \textit{G. Baumslag} for a metabelian Lie algebra to be embeddable in a finitely presented one [J. Algebra 45, 295--305 (1977; Zbl 0364.17005)], with its generalisation by \textit{J. R. J. Groves} and the author to embeddings in a metabelian Lie algebra of homological type \(FP_{m}\) [J. Lond. Math. Soc., II. Ser. 73, No. 2, 475--492 (2006; Zbl 1155.20313)]. The author also answers a question of Petrogradsky, exhibiting a metabelian Lie algebra \(L\) over a field \(k\) of characteristic different from \(2\), such that \(L\) is not finitely presented, but \(H_{2}(L, k)\) is finite dimensional. For this example one relies on the treatment of \textit{R. M. Bryant} and \textit{J. R. J. Groves} of finite presentability for metabelian Lie algebras [J. Algebra 218, No. 1, 1--25 (1999; Zbl 0935.17005); J. Lond. Math. Soc., II. Ser. 60, No. 1, 45--57 (1999; Zbl 0940.17006)].
scientific article; zbMATH DE number 7074581
scientific article