Post-Lie algebra structures appear in various areas of mathematics, such as homology of generalized partition posets and the study of Koszul operads, affine actions on Lie groups, Rota-Baxter operators and the classical Yang-Baxter equation, étale and prehomogenous modules or decompositions of Lie algebras. The existence and classification of these structures often involves the study of decompositions of Lie algebras.  In this talk we will focus on the link between post-Lie algebra structures, decompositions of Lie algebras and étale and prehomogenous modules. In particular, we will show results on post-Lie algebra structures on semisimple Lie algebras and strongly disemisimple Lie algebras, i.e. Lie algebras that decompose as a direct vector space sum into the sum of two semisimple subalgebras.

This talk is based on joint work with Dietrich Burde and Karel Dekimpe.

(D. Burde, K. Dekimpe and M.Monadjem: Rigidity results for Lie algebras admitting a post-Lie algebra structure. International Journal of Algebra and Computation, Vol. 32, Issue 08, 1495-1511 (2022))