Lie algebras with almost constant-free derivations
Let \(L\) be a finite-dimensional Lie algebra of characteristic 0 admitting a nilpotent Lie algebra of derivations \(D\). By a classical result of Jacobson, if \(D\) is constant-free (that is, without non-zero constants, \(x^\delta=0\) for all \(\delta\in D\) only if \(x=0\)), then \(L\) is nilpotent. We prove that if \(D\) is almost constant-free, then \(L\) is almost nilpotent in the following precise sense: if \(m\) is the dimension of the Fitting null-component with respect to \(D\), then \(L\) contains a nilpotent subalgebra \(N\) of codimension bounded by a function of \(m\) and \(n\), and the nilpotency class of \(N\) is bounded in terms of \(n\) only. This includes strengthening Jacobson's theorem by giving in the case \(m=0\) a bound for the nilpotency class of \(L\) in terms of the number of weights of \(D\). The main result can also be stated as a theorem about a locally finite Lie algebra over an algebraically closed field of characteristic 0 containing a nilpotent subalgebra with finitely many weights such that its Fitting null-component is of finite dimension -- then the algebra contains a nilpotent subalgebra of finite codimension (with similar bounds for the codimension and the nilpotency class). The results on derivations are based on results on \((\mathbb Z/p\mathbb Z)\)-graded Lie algebras, where \(p\) is a prime, with zero component of dimension \(m\) (these results in turn generalize the Higman-Kreknin-Kostrikin theorem on the nilpotency of Lie algebras with a regular automorphism of prime order).
scientific article; zbMATH DE number 5115356
scientific article