Elementary zeros of lie algebras of vector fields

WE CONSIDER a finite dimensional Lie algebra ℒ of ϐk (k ≥ 1) differentiable vector fields on a finite dimensional manifold M. A point p ∈ M is a zero for ℒ if X(p) = 0 for very X∈ ℒ. For the classical special case of (the Lie algebra spanned by) a single vector field X, the zero p is said to be elementary (or simple) if the derivative at p of the principal part of X is invertible. The equivalent coordinate-free formulation of this definition is that the vector field, regarded as a map X:M→TM, be transverse to the zero section at p [1]. From this point of view it can be seen that elementary zeros are stable, i.e. they persist under perturbation of X. In addition, such zeros can bc assipncd an index of + I or - I and, when all zeros arc elementary and M is compact, the vector field satisfies the Poincare-Hopf index formula. It is reasonable to ask how these notions should be formulated for zeros of an arbitrary Lie algebra. The correct definition turns out to involve the first Lit algebra cohomology of the linear part of ℒ at p. That the first cohomology is relevant is suggested by results of Hirsch [5, 6] and Stowe [14], which relate stability of stationary points of group actions with the group cohomology of the linear action, and the relationship between the cohomologies of Lit groups and Lie algebras [15]. Section I reviews basic results concerning cohomology of Lie algebras. Section 2 applies these results to the definition and description of elementary zeros for a Lie algebra of vector fields on a manifold. Section 3 contains examples and a result, concerning commuting vector fields on a compact surface, which illustrate limitations to defining an index for a zero of a Lie algebra of vector fields