Affine cohomology classes for filiform Lie algebras

Abstract

We classify the cohomology spaces $H^2(\mathfrak{g},K)$ for all filiform nilpotent Lie algebras of dimension $n\le 11$ over $K$ and for certain classes of algebras of dimension $n\ge 12$. The result is applied to the determination of affine cohomology classes $[ω]\in H^2(\mathfrak{g},K)$. We prove the general result that the existence of an affine cohomology class implies an affine structure of canonical type on $\mathfrak{g}$, hence a canonical left-invariant affine structure on the corresponding nilpotent Lie group. For certain filiform algebras the absence of an affine cohomology class implies the nonexistence of any affine structure. Of particular interest are algebras $\mathfrak{g}$ with minimal Betti numbers $b_1(\mathfrak{g})=b_2(\mathfrak{g})=2$.