On the cohomology of a class of nilpotent Lie algebras

Abstract

Let g denote a finite dimensional nilpotent Lie algebra over C containing an Abelian ideal a of codimension 1, with z £ g\a. We give a combinatorial descript tion of the Betti numbers of 0 in terms of the Jordan decomposition 0 = (^ dj i=i induced by ad(z)\ a . As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.

Affine cohomology classes for filiform Lie algebras

“…The last case corresponds to filiform algebras. It is well known [1] that filiform algebras play also an interesting role in the study of Betti numbers b i (g) of nilpotent Lie algebras g. They produce lower bounds for the Betti numbers. More precisely computations have shown that for any small n, there exists a filiform algebra f n such that b i (f n ) ≤ b i (g) for all i and all nilpotent Lie algebras g of dimension n. Such filiform Lie algebras very often do not admit any affine structure.…”

“…For filiform Lie algebras however both conjectures are clear. Nevertheless the explicit determination of Betti numbers of filiform algebras leads to formidable combinatorial problems, see [1].…”

“…Definition. Let A be a vector space over a field K and let A × A → A, (x, y) → x · y be a K-bilinear product which satisfies (1) x · (y · z) − (x · y) · z = y · (x · z) − (y · x) · z for all x, y, z ∈ A. Then A together with the product is called left-symmetric algebra or LSA.…”

“…The term left-symmetric becomes evident, if we rewrite condition (1) as (x, y, z) = (y, x, z), where (x, y, z) = x · (y · z) − (x · y) · z.…”

The mod 4 behaviour of total Lie algebra cohomology

“…, n − 1 . According to [1], the cohomology of f n is given by a standard partition problem: let κ i denote the number of i-tuples (a 1 , . .…”