Let $V=\{V_i\}$, $i=1,\dots,m How should I proceed to determine a (finite-dimensional) representation of $L(V)$?
Representation of a Lie group
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differential-geometry
representation-theory
lie-groups
lie-algebras
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0A 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. – 2017-02-27
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0Dear @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. – 2017-02-28
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1You need the entire l(V) to be complete, see http://math.stackexchange.com/questions/788744/example-of-a-sum-of-complete-vector-fields. – 2017-02-28