📂Geometry

Lie Brackets of Vector Fields

Definition¹

On two differentiable vector fields $X, Y$ on a differentiable manifold $M$, $[X, Y]$ is defined as follows, and is called the (Lie-)bracket or Lie algebra.

$$ \begin{equation} [X, Y] := XY - YX \end{equation} $$

Explanation

Vector field $X, Y$ can be seen as an operator acting on $\mathcal{D}(M)$, and $XY$ although not a vector field, $[X, Y] = XY - YX$ becomes a vector field.

$(1)$ satisfying such equation is generally called a commutator.

The following theorem states that (a), (b), (c) are generally satisfied properties not only for Lie-brackets but also for commutators. Especially, (c) is known as the Jacobi identity.

Theorem

Let $X, Y, Z$ be a differentiable vector field on $M$. Let $a, b$ be a real number and $f, g$ be a differentiable function on $M$. Then, the following holds:

(a) $[X, Y] = -[Y, X]$

(b) $[aX + bY, Z] = a[X, Y] + b[Y, Z]$

(c) $[ [X, Y], Z] + [ [Y, Z], X] + [ [Z, X], Y] = 0$

(d) $[fX, gY] = fg[X, Y] + fX(g)Y - gY(f)X$

Proof

(d)

Since the differentiation of a product $X(gY) = X(g)Y + gXY$ holds,

$$ \begin{align*} [fX, gY] &= fX(gY) - gY(fX) \\ &= \left( fX(g)Y + fgXY \right) - \left( gY(f)X - gfYX \right) \\ &= fgXY - fgYX + fX(g)Y + gY(f)X\\ &= fg(XY - YX) + fX(g)Y + gY(f)X\\ &= fg[X, Y] + fX(g)Y + gY(f)X \end{align*} $$

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