Models for Free Nilpotent Lie Algebras Matthew Grayson ∗ University of California, San Diego Robert Grossman ∗ University of Illinois, Chicago March, 1988 This is a draft of a paper which later appeared in the Journal Algebra, Volume 35, 1990, pp. 177-191. If a Lie algebra g can be generated by M of its elements E 1 ,...,E M , and if any other Lie algebra generated by M other elements F 1 ,...,F M is a homomorphic image of g under the map E i → F i , we say that it is the free Lie algebra on M generators. The free nilpotent Lie algebra g M,r on M generators of rank r is the quotient of the free Lie algebra by the ideal g r+1 generated as follows: g 1 = g, and g k = [g k−1 , g]. Let N denote dimension of g M,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 alge- bras g M,r . It is well known [19] that that there is a representation of g M,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 E 1 , ... ,E M defined in R N with the property that they generate a Lie algebra isomorphic to g M,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 ˙ x(t)=(E 1 + u(t)E 2 )(x(t)) may be computed explicitly * Research supported in part by Postdoctoral Research Fellowships from the National Science Foundation. 1