Narrow Lie algebras: A coclass theory and a characterization of the Witt algebra
Guided by the accuracy of the coclass conjecture of \textit{C. R. Leedham-Green} and \textit{M. F. Newman} [Arch. Math. 35, 193-202 (1980; Zbl 0437.20016)] the authors have developed a coclass theory for Lie algebras. The analogue of the solvability of pro-\(p\) groups of finite coclass does not exist, as is shown by the positive part of the Witt algebra \(\text{Der} (F[x])\). Let \(F\) denote the collection of \(N\)-graded, finitely generated, residually nilpotent Lie algebras of characteristic 0 which are generated by \(L_1\). Also, let \(M\) be the Lie algebra with basis \(x,y_1,y_2,\dots\) with \([y_i,x]= y_i+1\) and \([y_i,y_j]=0\). Here \(L_1\) has basis \(x,y_1\) and \(L_j\) has basis \(y_j\), \(j>1\). Let \({\mathcal Z}_j(L)\) be the \(j\)th term in the upper central series of \(L\). The main result is that for each \(L\in F\) and \(S= S(L)= \{i\geq 1:\dim L^i/ L^{i+1}>1\}<\infty\), there exists a polynomial \(f(s)\) such that \(L/{\mathcal Z}_{f(s)} (L)\simeq M/M^d\) for some \(d\leq\infty\). \(f(s)\) is of degree 4 and if \(L\) is infinite, then \(f(s)=s\) will do. A number of consequences are noted. Lie algebras of finite coclass satisfy \(s\leq\infty\). If they are in \(F\), then they are solvable. If \(L\) is not generated by \(L_1\), the above counterexample shows \(L\) need not be solvable. But under certain conditions, it is the only counterexample.
scientific article; zbMATH DE number 1013987
scientific article