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I am trying to understand the notion (and notation) of the Lie derivative on a general manifold by trying to convert the notation the concrete example of the Lie group O(n).

Let $X,Y$ be smooth vector fields on a smooth manifold $M$, $p \in M$ and the local flow $\psi_t: U \rightarrow M$ of X in a neighborhood $U$ of $p$. Then the $\textit{Lie derivative}$ and thus the Lie bracket is defined as:

$$ [X,Y]_p := \mathcal{L}_X(Y_p) = \lim_{t\rightarrow 0} \frac{(d\psi)_{-t}\; Y_{\psi_t(p)} - Y_p}{t}$$

In words: one uses the pullback of the vector field $Y$ along the flow of $X$ to define this derivative.

Now, for $M=O(n)$, with associated Lie algebra $\mathfrak{o}(n)=\{ X \in M(n,\mathbb{R}) : X = -X^T \}$, the Lie bracket for $A,B \in \mathfrak{o}(n)$ can be written as:

$$[A,B] = \lim_{t\rightarrow 0} \frac{\gamma(t) - B}{t}, $$

where the curve $\gamma$ is given by

$$ \gamma(t) = \exp(tA)B\exp(-tA). $$

Unfortunately, I got stuck in defining the abstract notation appropriately to arrive at the latter expression.

Does someone has an idea how to define $\psi$ and the vector fields in this situation? Or, does someone know a better $\textit{concrete}$ example where one can well understand the Lie derivative?

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    Vector fields are equivalent to derivations (http://en.wikipedia.org/wiki/Derivation_(abstract_algebra)) on the algebra $C^{\infty}(M)$ of smooth functions, and then the Lie derivative is just the commutator of the corresponding derivations.2012-08-08

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