Affine cohomology classes for filiform Lie algebras

TL;DR

This paper classifies second cohomology groups of filiform nilpotent Lie algebras up to dimension 11 and certain higher dimensions, linking affine cohomology classes to affine structures on associated Lie groups.

Contribution

It provides a comprehensive classification of cohomology spaces for filiform Lie algebras and establishes a connection between affine cohomology classes and affine structures.

Findings

Classified $H^2(rak{g},K)$ for all filiform nilpotent Lie algebras up to dimension 11.

Proved that the existence of an affine cohomology class implies an affine structure on the Lie algebra.

Identified cases where the absence of affine cohomology classes indicates no affine structure exists.

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 $[\omega]\in H^2(\mathfrak{g},K)$. 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 $b_1(\mathfrak{g})=b_2(\mathfrak{g})=2$.