# Moscow Mathematical Journal

Volume 13, Issue 1, January–March 2013 pp. 1–18.

Post-Lie Algebra Structures and Generalized Derivations of Semisimple Lie Algebras**Authors**: Dietrich Burde (1) and Karel Dekimpe (2)

**Author institution:**(1) Fakultät für Mathematik, Universität Wien, Nordbergstr. 15, 1090 Wien, Austria

(2) Katholieke Universiteit Leuven, Campus Kortrijk, 8500 Kortrijk, Belgium

**Summary:**

We study post-Lie algebra structures on pairs of Lie algebras (𝔤,𝔫), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple 𝔤 and solvable 𝔫 we show that there exist no post-Lie algebra structures on (𝔤,𝔫). For semisimple 𝔫 and certain solvable 𝔤 we construct natural post-Lie algebra structures. On the other hand we prove that there are no post-Lie algebra structures for semisimple 𝔫 and solvable, unimodular 𝔤. We also determine the generalized (α,β,γ)-derivations of 𝔫 in the semisimple case. As an application we classify certain post-Lie algebra structures related to generalized derivations.

2010 Mathematics Subject Classification. 17B30, 17D25.

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