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)).