Affine cohomology classes for filiform Lie algebras
Dietrich Burde
TL;DR
The paper analyzes when filiform nilpotent Lie algebras admit canonical affine structures by studying affine cohomology classes in $H^2(\mathfrak{g},K)$. It develops an adapted-basis framework to parameterize filiform algebras, connects affine cohomology to central extensions, and proves that the existence of an affine class yields a canonical affine structure, while minimal second Betti number $b_2(\mathfrak{g})=2$ often obstructs such classes. The authors compute $H^2(\mathfrak{g},K)$ for all $n\le 11$ and for select $n\ge 12$ cases, presenting explicit cocycles and a dichotomy between algebras that do and do not admit affine cohomology classes. They further classify higher-dimensional cases, showing universal affine structure for certain classes $\mathfrak{A}_n^1(K)$ and polynomial-dependent behavior for $\mathfrak{A}^2_n(K)$, with no affine structure for $\mathfrak{A}^2_{13}(K)$ and implications for the minimal faithful module dimension.
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 $b_1(\mathfrak{g})=b_2(\mathfrak{g})=2$.