Patrick Eberlein (University of North Carolina)

Structure of 2-step nilpotent Lie algebras of type (p,q)

Abstract. For pairs of integers $p \ge 1$, $q \ge 2$ let N(p,q)denote the set of 2-step nilpotent Lie algebra structures on $\mathbb{R} ^{p+q}$ that have a derived algebra $[\mathbb{R} ^{p+q}, \mathbb{R} ^{p+q}]$ of dimension p and codimension q. The space N(p,q) is a smooth connected manifold of dimension pq + pD, where D =(1/2)q(q-1). Moreover, N(p,q) is also a fiber bundle over the Grassmann manifold G(p,p+q) of p-dimensional subspaces of $\mathbb{R} ^{p+q}$. There is a natural action of $GL(p+q,\mathbb{R} )$ on N(p,q) whose orbits are the isomorphism classes of N(p,q).

Let $so(q,\mathbb{R} )$ denote the real $q \times q$ skew symmetric matrices, and let $GL(q,\mathbb{R} )$ act on $so(q,\mathbb{R} )$ by g(A) = gAg^*, where g^* denotes the transpose of g. Let G(p,so(q,R)) denote the Grassmann manifold of p-dimensional subspaces of $so(q,\mathbb{R} )$. The action of $GL(q,\mathbb{R} )$ on $so(q,\mathbb{R} )$ extends naturally to an action of $GL(q,\mathbb{R} )$ on $G(p,so(q,\mathbb{R} ))$.

Theorem. The quotient space $X(p,q) = N(p,q) / GL(p+q,\mathbb{R} )$ of isomorphism classes in N(p,q) is homeomorphic to the compact quotient space $G(p,so(q,\mathbb{R} )) / GL(q,\mathbb{R} )$.

Corollary. (Duality) X(p,q) is homeomorphic to X(D-p,q) for all integers $p \ge 1$, $q \ge 2$, where D = (1/2)q(q-1).

Generic local rigidity. We define the dimension of X(p,q) to be the smallest codimension of an orbit of $GL(p+q,\mathbb{R} )$ in N(p,q), or equivalently, the smallest codimension of an orbit of $GL(q,\mathbb{R} )$ in $G(p,so(q,\mathbb{R} ))$. The cases where X(p,q) has dimension zero, or equivalently where $GL(p+q,\mathbb{R} )$ has open orbits in N(p,q), is of special interest. Every element $\{R^{p+q}, [\; ,\;]\}$ in an open orbit of $GL(p+q,\mathbb{R} )$ is locally rigid; that is, $\{\mathbb{R} ^{p+q}, [\; ,\; ]\}$is isomorphic to all elements $\{\mathbb{R} ^{p+q}, [\; , \;]'\}$ in some open set of N(p,q)that contains $\{R^{p+q}, [\; ,\;]\}$. The union of all open GL(p+q,R) orbits in N(p,q) is a dense open (possibly disconnected) subset of N(p,q).

Spaces of dimension zero. Up to the duality $(p,q)\rightarrow (D-p,q)$ the following is a complete list of pairs for which X(p,q) has dimension zero: [1] (1, 2k) (Heisenberg algebras); [2] (D, q), $q \ge 2$ (free 2-step nilpotent Lie algebras); [3] (2, 2k+1); [4] (2, 4); [5] (2, 6); [6] (3,4); [7] (3, 5); [8] (4, 5). Examples [1] through [5] were previously well known.

Dimension of X(p,q). The dimension of X(2,2k) is k-3 for $k \ge 4$. Except for this case and the cases of dimension zero listed above X(p,q)has dimension 1 + p(D-p) - q².

Lattices. In another direction we describe some highly nongeneric examples in X(p,q) that arise from p-dimensional subalgebras and Lie triple systems in $so(q,\mathbb{R} )$. All of these examples of 2-step nilpotent Lie algebras admit a rational structure, and hence the corresponding simply connected, 2-step nilpotent Lie groups admit a lattice by the criterion of Mal'cev.