Title：Pre-Lie algebra structures and etale affine representations

Reporter：Dietrich Burde

Work Unit：Universität Wien, Austria

Time：2023/09/14 20:00-22:00

Address：Zoom id：904 645 6677 Password:2023

**Summary of the report: **

Pre-Lie algebras and Post-Lie algebras arise in many areas of mathematics and physics. They are also related to etale affine representations of Lie algebras and algebraic groups. They also arise in the context of affine geometry on Lie groups, operad theory, deformation theory and Young-Baxter equations. For reductive groups, every etale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the associated Lie algebra. Pre-Lie algebra structures also correspond to left-invariant affine structures on Lie groups. In this talk we present results on the existence of etale affine representations of reductive groups and Lie algebras and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable subgroups of the affine Cremona group.

**Introduction of the Reporter: **

Dietrich Burde is a professor at the Department of Mathematics at the University of Vienna. He obtained his Ph.D. in 1992 at the University of Bonn in Germany. His research interests lie in algebra and geometry, in the area of Lie groups, Lie algebras, algebraic groups and representation theory. He has received research grants by the Austrian Science Foundation on the topic "Affine Geometry on Lie Groups and Lie-algebraic Structures".