1989/01/01 by Nathan Jacobson · 2 citations
Mathematics · #Advanced Topics in Algebra #Mathematics #Automorphism #Lie algebra #Locally nilpotent #Nilpotent #Pure mathematics #Vector space #Lie conformal algebra #Universal enveloping algebra #Closure (psychology) #Space (punctuation) #Discrete mathematics #Combinatorics #Nilpotent group
paper · doi:10.1007/978-1-4612-3694-8_16
openalex publication_date 1989/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In a recent paper Borei and Serre proved the theorem: If \mathfrakL is a Lie algebra of characteristic 0 and \mathfrakL has an automorphism of prime period without fixed points ≠ 0, then \mathfrakL is nilpotent.1 In this note we give a proof valid also for characteristic p ≠ 0. By the same method we can prove several other similar results on automorphisms and derivations. Our method is based on decompositions of the Lie algebra which determine weakly closed sets of linear transformations. Such a set \mathfrakW has, by definition, the closure property that if A, B ∈ \mathfrakW then there exists a γ(A, B) in the base field such that A B + γ BA ∈ \mathfrakW The main result we shall need is the generalized Engel theorem that if \mathfrakW is a weakly closed set of nilpotent linear transformations in a finite-dimensional vector space, then the enveloping associative algebra \mathfrakW* of \mathfrakW is nilpotent [3].