Lie bracketLieBracket2004-02-16 15:43:322006-10-14 16:17:22Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand*{\abs}[1]{| #1 |}The Lie bracket is an antisymmetric, bilinear, first order differential operator on vector fields. It may be defined either in terms of local coordinates or in a global, coordinate-free fashion. Though both defintions are prevalent, it is perhaps easier to formulate the Lie Bracket without the use of coordinates at all, as a commutator:
{\bf Definition} (Global, coordinate-free) Suppose $X$ and $Y$ are vector fields on a smooth manifold $M$. Regarding these vector fields as operators on functions, the Lie bracket is their commutator:
\begin{align*}
[X,Y](f)=X(Y(f))-Y(X(f)).
\end {align*}
{\bf Definition} (Local coordinates) Suppose $X$ and $Y$ are vector fields on a smooth $n$-dimensional manifold $M$,
suppose $(x^1,\ldots, x^n)$ are local coordinates around some point $x\in M$,
and suppose that in these local coordinates
\begin{eqnarray*}
X(x)&=&X^i(x) \frac{\partial}{\partial x^i}\Big|_x, \\
Y(x)&=&Y^i(x) \frac{\partial}{\partial x^i}\Big|_x.
\end{eqnarray*}
Then the \emph{Lie bracket} of the above vector fields is the locally defined vector field
$$[X,Y](x) = X^i \frac{\partial Y^j}{\partial x^i} \frac{\partial}{\partial x^j}\Big|_x-Y^i \frac{\partial X^j}{\partial x^i} \frac{\partial}{\partial x^j}\Big|_x.$$
(The Einstein summation convention employed in the above equations ---
repeated indices are to be summed from the range 1 to $n$.)
\subsubsection*{Properties}
Suppose $X,Y,Z$ are smooth vector fields on a smooth manifold $M$.
\begin{itemize}
\item $[X,Y]=\mathcal{L}_XY$ where $\mathcal{L}_XY$ is the Lie derivative.
\item $[\cdot,\cdot]$ is anti-symmetric and bi-linear.
\item Vector fields on $M$ with the Lie bracket is a Lie algebra. That is to say, the Lie bracket satisfies the Jacobi identity:
\[ [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]]=0. \]
\item The Lie bracket is covariant with respect to changes of coordinates.
\end{itemize}