1

Let $V=\{V_i\}$, $i=1,\dots,m<n$ be a set of linearly independent vector fields on some $n$-dim manifold $M$. Let, furthermore, $l(V)$ be a nilpotent and hence finite-dimensional Lie algebra generated by $V$. By Lie's third theorem $l(V)$ gives rise to a finite-dimensional Lie group $L(V)$ acting on $M$.

How should I proceed to determine a (finite-dimensional) representation of $L(V)$?

Dmitry
  • 1,337
  • A caveat: Lie's theorem is only local, so, technically speaking, $L(V)$ is only a local Lie group, not a Lie group. You then have to decide what you mean by a linear representation of a local group. Do you mean a representation of its Lie algebra? In any case, you need more information about $l(V)$ to determine a linear representation of this Lie algebra. – Moishe Kohan Feb 27 '17 at 16:10
  • Dear @Moishe, what would be the obstacle to extending $l(V)$ to a global Lie group? Will it change if I assume that $V_i$ are complete vector fields? By a finite-dimensional representation I understand a map $\phi:L(V)\rightarrow GL(X)$, where $X$ is a vector space. It is said that this allows for passing to a matrix representation by a proper choice of basis. I wonder if a matrix representation could give some more insight into the structure of the underlying Lie group. – Dmitry Feb 28 '17 at 09:00
  • 1
    You need the entire l(V) to be complete, see http://math.stackexchange.com/questions/788744/example-of-a-sum-of-complete-vector-fields. – Moishe Kohan Feb 28 '17 at 09:58

0 Answers0