Abstract. We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPAstructures, and classify all CPA-structures on parabolic subalgebras of simple Lie algebras.
Abstract. Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.
Nilpotent Lie algebras and invertible derivationsWhich information about the structure of a Lie algebra is contained in its group of automorphisms or in its Lie algebra of derivations? Several partial answers to this question, in terms of sufficient conditions for the nilpotency or solvability of the Lie algebra, are known -see for example the theorems by Borel, Serre and Mostow [BS, BM]. We will mainly be interested in the following well-known theorem by Jacobson on the invertibility of Lie algebra derivations. Once again, it was asked whether the converse were true; but counterexamples were found: nilpotent Lie algebras admitting only nilpotent pre-derivations. Such Lie algebras are called strongly nilpotent, [Bu]. By stating Jacobson's theorem even more generally, in terms of
We show that for a given nilpotent Lie algebra g with Z(g) ⊆ [g, g] all commutative post-Lie algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras F g,c and discover a strong relationship to solving systems of linear equations of type [x, u] + [y, v] = 0 for generator pairs x, y ∈ F g,c . We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra F g,c has only central CPA-structures.
We prove an explicit formula for the invariant µ(g) for finitedimensional semisimple, and reductive Lie algebras g over C.Here µ(g) is the minimal dimension of a faithful linear representation of g. The result can be used to study Dynkin's classification of maximal reductive subalgebras of semisimple Lie algebras.
Mathematics Subject Classification (2000). 17B20, 17B10.
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