Metabelian thin Lie algebras (Q5946411)

A thin Lie algebra is a graded Lie algebra \(L=\bigoplus_{i=1}^{\infty}L_i\) over a field, generated by its two-dimensional component \(L_1\), and such that \(L_{i+1}=[x,L_1]\) for all nonzero \(x\in L_i\) and for all \(i\geq 1\). This notion is a natural analogue for graded Lie algebras of one introduced for (pro-) \(p\)-groups by \textit{R. R. A. Brandl} [Arch. Math. 50, No. 6, 502--510 (1998; Zbl 0652.20035)]. In current terminology, the thin pro-\(p\) groups are the pro-\(p\) groups of width two and obliquity zero, see the book by \textit{G. Klaas}, \textit{C. R. Leedham-Green} and \textit{W. Plesken} [Linear pro-\(p\)-groups of finite width, Lect. Notes Math. 1674 (1997; Zbl 0901.20013)].NEWLINENEWLINEIn the paper under review the authors study the metabelian thin Lie algebras, taking advantage of a standard description of the free two-generated metabelian Lie algebra in terms of a polynomial ring in two variables. This allows them to give a complete classification of the metabelian thin Lie algebras over an arbitrary field. In particular, it turns out that all metabelian thin Lie algebras of dimension greater than six are quotients of infinite-dimensional ones. The latter are in a bijective correspondence with the quadratic field extensions of the underlying field. The authors also compute their graded automorphism groups.