Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module? - MathOverflow

Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?

2009-11-04T18:58:15Z

Let be a finite dimensional Lie algebra over k, let U be its enveloping algebra, and let M be a -module (not necessarily finite dimensional). Call the invariant dimension of M the largest i such that . This is the same as the degree of the largest non-vanishing Lie algebra cohomology group.

Here are two equivalent statements of my question.

If M is a cyclic -module (that is, its generated by a single element as a U-module), then is the number of relations of M always greater than the invariant dimension minus 1?
If I is a left ideal in U, then is the number of generators of I always greater than the invariant dimension?

The reason why I would suspect such a thing is that it is true in the case of an abelian Lie algebra; that is, when U is a polynomial ring. In this case, the invariant dimension of an ideal I coincides with the height of I, and so (2) becomes Krull's height theorem (and (1) follows immediately from (2)).

Answer by Simon Wadsley for Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?

2009-11-19T20:55:58Z

This looks like the kind of thing that one might be able to prove by filtering $U$ by word-length over $g$ and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that $g$ is abelian, since the associated graded ring is a polynomial ring in $\dim g$ variables and the number of generators of $I$ as a left ideal will be at least as big as the number of generators of $\mathrm{gr}(I)$ its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as a similar invariant, the grade of the module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)

Edit: Expansion of strategy:

Proposition 7.1 of Auslander-Gorenstein Rings by Ajitabh, Smith and Zhang tells us that $\mathrm{Ext}^j_U(M,k)$ is isomorphic to the $k$-vectorspace dual of $\mathrm{Ext}^{d-j}_U(k,M)$ where $d=\dim g$. Thus invariant dimension is $d-min(j|\mathrm{Ext}^j_U(M,k)\neq 0)$. If one can relate the non-vanishing of $\mathrm{Ext}^j_U(M,U)$ and $\mathrm{Ext}^j(M,k)$ then it might be possible to reduce the problem to something relating the grade of $M$ to the number of relations of $M$. This can be done by passing to the associated graded ring.

Answer by Victor Protsak for Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?

2010-05-01T12:10:28Z

This is FALSE as stated, even for the augmentation ideal. For example, consider the 3-dimensional Heisenberg Lie algebra g and the trivial module k. Then it's easy to see that the third (=top) cohomology is non-zero, but due to the noncommutativity of g, the left ideal I=annihilator of k may be generated by only two elements. The inequality proposed by Simon, in fact, goes the other way: the number of generators for I is at most the number of generators of gr I.

Admittedly, this is a cheap shot, since the right notion of the number of generators for a left ideal I of U(g) is the minimal dimension of an ad(g)-invariant subspace of U(g) generating I - in particular, this number survives the passage to the associated graded. I don't know whether the modified statement is true.