Is it possible to generate geometric properties which are invariant under affine transformations?
I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine transformations. How would you find invariants of this Lie group? Is there a way to generate invariants? How can Lie algebras be used to find invariants of Lie groups?
An interesting geometric property for Lie groups is the existence of left-invariant affine structures on Lie groups, e.g., see here. There is a bijection to so-called left-symmetric algebra structures on Lie algebras. The geometric problem thus can be reduced to an algebraic question on Lie algebras. In this case, working on the Lie algebra level has many advantages.