The variety of Lie algebras from a riemannian viewpoint.
by
Jorge Lauret
FaMAF, Univ. Nac. Cordoba, Argentina.

Oral Communication

Let as consider as parameter space for the real Lie algebras of a given dimension n, the set L of all Lie brackets on a fixed real vector space g. Since the Jacobi identity is determined by polynomial conditions, we have that L is actually an algebraic subset of \Lambda² g^*\otimesg.

Endowing g with a fixed inner product <·, ·>, we identify each element \mu in L with the riemannian manifold (G_\mu, <·, ·>), that is, the corresponding simply connected Lie group endowed with the left invariant riemannian metric determined by <·, ·>. In this way, the `change of basis' action of Gl(n) on L given by j.\mu = j\mu(j^-1·, j^-1·) has the following riemannian interpretation: each j in Gl(n) defines by exponentiation a riemannian isometry between (G_\mu, <j·, j·>) and (G_j.\mu, <·, ·>), so that we can say that the orbit O(\mu)=Gl(n).\mu covers all the left invariant riemannian metrics on G_\mu.

This somewhat strange interpretation of the points of the algebraic variety L, will allow as to consider and study functionals and flows on L, inspired on some well known curvature functionals and flows on the space of riemannian metrics. Furthermore, we will use repeatedly in this paper the isometry given above, to translate algebraic properties of the orbit O(\mu) to geometric properties of left invariant riemannian metrics on G_\mu.

Consider for instance the functional on L given by F(\mu)=tr(R_\mu²), where R_\mu is a symmetric transformation of g defined via the Ricci curvature operator of \mu. The riemannian functional F can be formally defined on the whole space \Lambda² g^*\otimesg (as a 4-degree homogeneous polynomial) since R_\mu is given just in terms of <·, ·> and \mu. We will see that the gradient of F at \mu in L is always tangent to the Gl(n)-orbit O(\mu). This interesting property implies that the positive and negative gradient flows of the restriction F|_S to the sphere S of \Lambda² g^*\otimesg, starting from a point \mu in L \cap S, will lies in the orbit O(\mu) for every time, and thus the flows will (semi) converges to a point in the clousure [`(O(\mu))]. We therefore obtain that the gradient flows of F define degenerations, a concept introduced and studied by theoretic physists and also by mathematicians.

We say that \mu degenerates to \lambda (denoted by \mu --> \lambda) if \lambda in [`(O(\mu))], the clousure of O(\mu) with respect to the the usual (metric) topology of \Lambda² g^*\otimesg. Every degeneration will be assumed nontrivial, that is, \lambda (and thus the entire orbit O(\lambda)) lies in the boundary of O(\mu). The maximum length k of a sequence

\mu = \mu0 --> \mu1 --> ... --> \muk-1 --> \muk=0

of degenerations is called the level of \mu. It will be usually denote by level(\mu).

A problem particulary ineresting from the physical point of view, is to find all the Lie algebras that degenerates to a given one. We use some results on the behavior of the critical points of F|_S, to study the Lie algebras \mu with level(\mu)=1. This means that the only possible degeneration of \mu is \mu --> 0, and they have the bigger sets of Lie algebras degenerating on them. We prove a geometric characterization of such a Lie brackets, obtaining in particular that G_\mu admits only one left invariant riemannian metric up to isometry and scaling. We then show that only two Lie algebras satisfies this pretty strong condition: the direct sum of the 3-dimensional Heisenberg Lie algebra with an abelian factor and the solvable Lie algebra R H\oplusR^n-1, where [H, X]=X for any X in R^n-1 and R^n-1 is abelian.

In a second part of the paper, we apply some results on the minimal vectors of representations of real reductive Lie groups, to the action of a reductive subgroup G subset Gl(n) on L subset \Lambda² g^*\otimesg, finding some relationships with the geometric interpretation of L discussed in the first part. Indeed, we prove that \mu in L is a minimal vector for G, that is, ||\mu|| <= ||j.\mu|| for all j in G, if and only if p(R_\mu)=0. Here p:sym( g) --> p is the orthogonal projection and L(G)=k\oplusp is a Cartan decomposition of the Lie algebra L(G) of G with k subset so( g) and p subset sym( g).

In particular, for G=Gl(n) we obtain that certain special flow (along orbits) introduced by A. Neeman coincides with the gradient flow of F.

A Gl(n)-orbit on L can never be closed, unless it is { 0}. Indeed, if j_t=t^-1I then lim_{t --> 0}j_t.\mu = 0 and so 0 in [`(O(\mu))] for every \mu in L. A natural question is then what happens if we do not allow the multiplication by a scalar, considering for instance Sl(n)-orbits instead of Gl(n)-orbits. Under what conditions on \mu we would have that 0 in [`(Sl(n).\mu)]?, when Sl(n).\mu will be closed?.

We use the appearance of the riemannian tensor R_\mu in the algebraic context of minimal vectors, to give the following geometric characterization: Sl(n).\mu is closed if and only if G_\mu admits a left invariant riemannian metric (·, ·) such that the curvature operator R_{(·, ·)} is a multiple of the identity. As an inmediate consequence, we obtain that if \mu in L is nilpotent (and \mu =/= 0) then the orbit Sl(n).\mu is not closed, since in this case R_\mu coincides with the Ricci curvature operator and so the above condition is nothing but the Einstein condition. By a result due to J. Milnor, a non-abelian nilpotent Lie group can never admit an Einstein left invariant riemannian metric.

The riemannian characterization of the closed Sl(n)-orbits obtained, allows as to apply some ideas and techniques due to I. Dotti, used in the study of the Ricci curvature of unimodular Lie groups, in order to prove that an orbit Sl(n).\mu is closed if and only if \mu is semisimple. Moreover, 0 in [`(O(\mu))] for every non-semisimple Lie bracket \mu in L.

Date received: June 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-19.