Affine actions on Lie groups and post-Lie algebra structures

2011/09/01 by Burde, Dietrich, Dekimpe, Karel, Vercammen, Kim · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1109.0251

Abstract

We introduce post-Lie algebra structures on pairs of Lie algebras (\Lg,\Ln) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras \Lg and \Ln. One result is, for example, that if there exists a post-Lie algebra structure on (\Lg,\Ln), where \Lg is nilpotent, then \Ln must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.

Cited by

• An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence
• Structural and rigidity properties of Lie skew braces

Related

• Post-Lie algebra structures on pairs of Lie algebras
• On post-Lie algebras structures coming from simply transitive NIL-affine actions
• LR-algebras
• Post-Lie algebra structures for nilpotent Lie algebras
• Post-Lie algebra structures for perfect Lie algebras