i've been working with john huerta for a while on a project to understand how the simple lie algebra "g2" is related to the geometry of a rolling ball. this idea has probably been pretty much completely worked out by other people already, but as usual that doesn't stop me from trying to understand it in my own way, sometimes consulting what other people have done and sometimes not. also this research isn't very narrowly focused and instead digresses off in other directions; the benefit of learning all sorts of things during such digressions is perhaps one of the excuses for taking such a long time working on what should probably be a simple project not especially close to the research frontier.
i forget exactly how it happened, but john and i were reading some paper that talked about an "invariant 2,3,5-distribution" on one of the "grassmanian" homogeneous spaces of g2 , and also about invariant distributions associated with graded nilpotent lie algebras, and for some reason i decided to stare at the "root system" of g2:
[still working on finding a good way to post hand-drawn pictures here]
and see whether i could see anything in this picture that meshed with what the paper was talking about.
attaching maps for bruhat cells...
ReplyDeleteso is the case of a hermitian symmetric space precisely just the case where the canonical invariant distribution is "trivial"? which seems to sort of make sense because the symmetric space structure should make the action of the "rotations" on the "transvections" projectively transitive, right? try to clarify this though...
ReplyDeletebaez is trying to convince me that besides the a-series and c-series examples, there's also a b-series example of invariant contact distribution. so far this possible example seems more mysterious because i don't see an obvious systematic connection of the b-series geometries to symplectic geometry.
ReplyDeleteso for example, baez suggests that the 7-dimensional grassmanian of so(3,4) (given by the rank 2 partial linear isometries from euclidean 3-space to euclidean 4-space) has an invariant contact distribution. which might be interesting because the other grassmanian of g2, namely the one with an invariant contact distribution, is naturally seen as embedded in this grassmanian of so(3,4); thus we can ask how the contact distributions get along with the embedding.
ReplyDeletei suppose that we can consider these invariant contact distributions on grassmanians as in some sense "the next best thing" after hermitian symmetric spaces. the simplest of these graded nilpotent lie algebras are the abelian ones, and the heisenberg ones which give the invariant contact distributions seem in some sense "as close to abelian as possible". well, or something like that.
ReplyDeletein the case of a partial flag manifold as opposed to just a grassmanian, the nilpotent lie algebra has a special grading for each component of the flag. to get a single canonical grading i seem to be using the sum of these special gradings.
ReplyDelete