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 (complex) reductive groups, every étale 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 \(\mathfrak{g}\) of \(G\). For a Lie group \(G\), pre-Lie algebra structures on \(\mathfrak{g}\)
correspond to left-invariant affine structures 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 étale affine representations of reductive groups,
and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable
subgroups of the affine Cremona group.