Models for free nilpotent Lie algebras

Abstract

If a Lie algebra g can be generated by M of its elements E1, . . . , EM , and if any other Lie algebra generated by M other elements F1, . . . , FM is a homomorphic image of g under the map Ei → Fi, we say that it is the free Lie algebra on M generators. The free nilpotent Lie algebra gM,r on M generators of rank r is the quotient of the free Lie algebra by the ideal gr+1 generated as follows: g1 = g, and gk = [gk−1, g]. Let N denote dimension of gM,r. Free Lie algebras are those which have as few relations as possible: only those which are a conseqence of the anti-commutativity of the bracket and of the the Jacobi identity. Free nilpotent Lie algebras add the relations that any iterated Lie bracket of more than r elements vanishes. For further details, see [18] or [13]. In this paper we are interested in explicit computations of the Lie algebras gM,r. It is well known [19] that that there is a representation of gM,r on upper triangular N by N matrices. The problem is that many computations are difficult using this representation. In this paper we present an algorithm that yields vector fields E1, ... , EM defined in R with the property that they generate a Lie algebra isomorphic to gM,r. See [6] for another approach to this problem. In Section 3, we restrict to two generators and give three consequences of this algorithm. First, the form of the vector fields is such that flows of the control system ẋ(t) = (E1 + u(t)E2)(x(t)) may be computed explicitly ∗Research supported in part by Postdoctoral Research Fellowships from the National Science Foundation.