Pre-Lie algebra structures on reductive Lie algebras and etale affine representations

Abstract: Etale affine representations of Lie algebras and algebraic groups 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 Lie algebra g of G. For a Lie group G, a pre-Lie algebra structure on g corresponds to a left-invariant affine structure on G. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups.

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.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar