New bounds on the betti numbers of nilpotent lie algebras

Cohomology operations for Lie algebras

“…The toral rank r of a manifold is the largest rank of a torus acting freely on it, and the TRC states that dim H * (M ; R) ≥ 2 r . By analogy with nilmanifolds and their corresponding nilpotent Lie algebras over Q, the TRC for a Lie algebra L is dim H * L ≥ 2 dim Z [4]. As far as we are aware, this conjecture is open for Lie algebras over an arbitrary field k, although we will assume k = R throughout this paper.…”

“…The TRC for Lie algebras is known to hold for a large class of nilpotent Lie algebras; it is true for algebras L possessing a grading [7] (see also [19]). In particular, it is true for nilpotent Lie algebras of dimension at most 14 [4], for free n-step nilpotent Lie algebras, and for all 2-step nilpotent Lie algebras (see [4] and [18] for other proofs of this fact).…”

Free submodules for the central representation in the cohomology of Lie algebras

“…The TRC is known to hold for nilpotent Lie algebras of dimension at most 14 [1]. It holds for two-step nilpotent Lie algebras (see [6] and [1]) and more generally for positively graded Lie algebras where the centre is the summand of highest grading (see [3] and [7]).…”

“…The TRC is known to hold for nilpotent Lie algebras of dimension at most 14 [1]. It holds for two-step nilpotent Lie algebras (see [6] and [1]) and more generally for positively graded Lie algebras where the centre is the summand of highest grading (see [3] and [7]). Recently Hannes Pouseele and Paulo Tirao gave a remarkably simple result, which establishes the TRC for a class of Lie algebras that includes algebras of large nilpotency class that are not positively graded [5].…”

A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras

“…Besides this class, a few special cases have been added recently. For example, it was shown in [2] that the TRC holds for g if its center has dimension ≤ 5 or has codimension ≤ 7.…”