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On the structure of the cohomology of nilpotent Lie algebras

The exterior algebra over the centre of a Lie algebra acts on the cohomology of the Lie algebra in a natural way. Focusing on nilpotent Lie algebras, we explore the module structure afforded by this action. We show that for all two-step nilpotent Lie algebras, this module structure is non-trivial, which partially answers a conjecture of Cairns and Jessup [4]. The presence of free submodules indicates that the Lie algebra satisfies Halperin's Toral rank conjecture [11]. We prove that two specific classes of two-step nilpotent Lie algebras enjoy cohomology spaces with free submodules.