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Log-concavity of the cohomology of nilpotent Lie algebras in characteristic two | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Log-concavity of the cohomology of nilpotent Lie algebras in characteristic two

Grant CAIRNS

Department of Mathematics, La Trobe University, Melbourne, VIC 3086, Australia. E-mail: [email protected]

Received Date: April 18, 2009; Revised Date: July 24, 2009

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Abstract

It is known that the Betti numbers of the Heisenberg Lie algebras are unimodal over elds of characteristic two. This note observes that they are log-concave. An example is given of a nilpotent Lie algebra in characteristic two for which the Betti numbers are unimodal but not log-concave.

The Heisenberg Lie algebra of dimension 2m + 1 is the Lie algebra Equation having the basis Equation and nonzero relations Equation. For the cohomology with trivial coecients, the Betti numbers Equation have been explicitly computed in all characteristics [1,3,4]. Recall that the Betti numbers are unimodal if bibj for all 0 ≤ ijm and bibj for all m ≤ ij ≤ 2m + 1, and they are concave (resp., log-concave) if bi is at least as great as the arithmetic (resp., geometric) mean of the pair bi–1, bi+1 for all 1 ≤ i ≤ 2m. So concave implies log-concave which implies unimodal. In characteristic zero, unimodality is quite common. The Heisenberg Lie algebras play a key role in the construction of all known examples of Lie algebras in characteristic zero where the Betti numbers are not unimodal [2]. In fact, the Betti numbers of Equation are unimodal only in characteristic two [1]. On the other hand, we know of no nilpotent Lie algebra in characteristic two whose Betti numbers fail to be unimodal. In [1], the question was posed: in characteristic two, do all nilpotent Lie algebras have unimodal Betti numbers? Since logconcavity is a common route taken to prove unimodality, it is natural to ask whether the Betti numbers of the Heisenberg algebras are unimodal in characteristic two. We record the following observation as a theorem, though it is really just a corollary of the works [1,4].

Theorem 1. Over elds of characteristic two, the Betti numbers of Equation are log-concave; i.e.,Equation for all n.

Proof. For the rest of this note we x the characteristic to be two. Emil Skoldberg showed that the Poincare polynomial Equation is [4]

Equation     (1)

Though we will not need them, we mention that the individual Betti numbers are given in [1]; for all im,

Equation

To establish the log-concavity, we observe that the Betti numbers of Equation are essentially determined by those of Equation , with a curious correction for the middle two terms. Explicitly,

Equation    (2)

This relation is easily deduced from (1). Using induction, we assume that Sm is log-concave. Since (1+t)2 is log-concave, (1+t)2Sm(t) is thus also log-concave (see [5]). So in view of (2), to establish the log-concavity of Sm+1, it remains to verify it for the middle terms; that is, for Equation we require that Equation. But by Poincare duality, bm+1 = bm+2, and so we only require bm+1bm, and this is given by the unimodality of the Betti numbers, which was shown in [1]. This completes the proof.

The following example shows that, despite the above result, log-concavity is not a route for establishing unimodality in the general setting of nilpotent Lie algebras in characteristic two.

Example 2. Let Equation denote the 7-dimensional Lie algebra with basis x1, . . . , x7 and de ning relations:

Equation

Clearly Equation is nilpotent (and actually graded and liform). Direct calculations using Mathematica show that in characteristic two, the Betti numbers are

Equation

As Equation, the Betti numbers are not log-concave.

Acknowledgement

The author thanks Peter Cameron for asking whether the Betti numbers of the Heisenberg Lie algebras are log-concave.

References

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