Mathematical Sciences Geometry/Topology Seminar

April 18:
Speaker: Patrick Eberlein, Univ. of North Carolina (Chapel Hill)
Title: Structure of the space of 2-step nilpotent Lie algebras

Abstract: Let n >=3 be an integer and let p be an integer with 1 <= p <= n-1. Let X(p,n) denote the space of 2-step nilpotent structures on R^n with center of dimension p. Let G(p,n) denote the Grassmann manifold of p-planes in R^n and let Z : X(p,n) --> G(p,n) denote the map that sends a 2-step nilpotent structure [ , ] on R^n to the center of {R^n, [ , ]}. One can show that Z : X(p,n) --> G(p,n) is a smooth fiber bundle and X(p,n) is a connected smooth manifold of dimension pq + pD, where q = n-p and D = (1/2)q(q-1).

The group G = GL(n,R) has a natural action on X(p,n), and the orbits of G are the isomorphism classes of 2-step nilpotent structures on R^n. The quotient space X(p,n) / G is therefore the set of isomorphism classes of 2-step nilpotent structures on R^n

If p >= 3 and q is sufficiently large relative to p, then the G-orbits in X(p,n) will have dimension smaller than that of X(p,n). We present some preliminary work on computing the maximal dimension of a G-orbit in X(p,n). This problem is equivalent to computing the dimension of the automorphism group of a generic 2-step structure{R^n, [ , ]). The answer is known in special cases and conjectured but not yet proved in general.

More generally, if X(p,n) / G has positive dimension, then it is an interesting problem to study the topology of this space. Computing the dimension of a maximal G-orbit in X(p,n) may be considered step zero in this program. We will present some partial results.

Elements of X(p,n) / G with rational structures
If X(p,n) / G has positive dimension,then a generic element of X(p,n) / G will have no rational structure; the elements with a rational structure form a countable subset of X(p,n) / G. If W is a p-dimensional subspace of the n x n skew symmetric matrices so(n,R), then there is natural 2-step nilpotent Lie algebra structure on N = R^n + W (direct sum). If W has a basis of matrices whose entries lie in the rationals Q, then N has a rational structure. The same is true if one replaces W by gWg^-1 for an element g of the orthogonal group O(n,R). More generally one may ask

1) Describe those p-dimensional subspaces W for which N = R^n + W has a rational structure.

2) Given a p-dimensional subspace W such that N = R^n + W admits a rational structure, describe all rational structures on N, up to equivalence by automorphisms of N.

We describe some partial answers to 1) and 2).