Differentiable Manifolds/Lie algebras and the vector field Lie bracket

Lie algebras

Definition 7.1:

Let $L$ be a $d$-dimensional real vector space. $L$ is called a Lie algebra iff it has a function

$[\cdot ,\cdot ]:L\times L\to L$

such that for all $\mathbf {u} ,\mathbf {v} ,\mathbf {w} \in L$ and $b\in \mathbb {R}$ the three rules

1. $[\mathbf {u} ,\mathbf {v} +b\mathbf {w} ]=[\mathbf {u} ,\mathbf {v} ]+b[\mathbf {u} ,\mathbf {w} ]$ and $[\mathbf {u} +b\mathbf {v} ,\mathbf {w} ]=[\mathbf {u} ,\mathbf {w} ]+b[\mathbf {v} ,\mathbf {w} ]$ (bilinearity)
2. $[\mathbf {u} ,\mathbf {v} ]=-[\mathbf {v} ,\mathbf {u} ]$ (skew-symmetry)
3. $[\mathbf {u} ,[\mathbf {v} ,\mathbf {w} ]]+[\mathbf {w} ,[\mathbf {u} ,\mathbf {v} ]]+[\mathbf {v} ,[\mathbf {w} ,\mathbf {u} ]]=0$ (Jacobi's identity)

hold.

Definition 7.2:

Let $L$ with $[\cdot ,\cdot ]$ be a Lie algebra. A subset of $L$ which is a Lie algebra with the restriction of $[\cdot ,\cdot ]$ on that subset is called a Lie subalgebra.

The vector field Lie bracket

Definition 7.3:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$. We define the vector field Lie bracket, denoted by $[\cdot ,\cdot ]$, as follows:

$[\cdot ,\cdot ]:{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M),[\mathbf {V} ,\mathbf {W} ](p)(\varphi ):=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )$

Theorem 6.4: If $\mathbf {V} ,\mathbf {W}$ are vector fields of class ${\mathcal {C}}^{n}$ on $M$, then $[\mathbf {V} ,\mathbf {W} ]$ is a vector field of class ${\mathcal {C}}^{n}$ on $M$ (i. e. $[\cdot ,\cdot ]$ really maps to ${\mathfrak {X}}(M)$)

Proof:

1. We show that for each $p\in M$, $[\mathbf {V} ,\mathbf {W} ](p)\in T(p)M$. Let $\varphi ,\vartheta \in {\mathcal {C}}^{\infty }(M)$ and $c\in \mathbb {R}$.

1.1 We prove linearity:

${\begin{aligned}{[\mathbf {V} ,\mathbf {W} ]}(p)(\varphi +c\vartheta )&=\mathbf {V} (p)(\mathbf {W} (\varphi +c\vartheta ))-\mathbf {W} (p)(\mathbf {V} (\varphi +c\vartheta ))\\&=\mathbf {V} (p)(\mathbf {W} \varphi +c\mathbf {W} \vartheta )-\mathbf {W} (p)(\mathbf {V} \varphi +c\mathbf {V} \vartheta )\\&=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )+c(\mathbf {V} (p)(\mathbf {W} \vartheta )-\mathbf {W} (p)(\mathbf {V} \vartheta ))\\&=[\mathbf {V} ,\mathbf {W} ](p)(\varphi )+c[\mathbf {V} ,\mathbf {W} ](p)(\vartheta )\end{aligned}}$

1.2 We prove the product rule:

${\begin{aligned}{[\mathbf {V} ,\mathbf {W} ]}(p)(\varphi \vartheta )&=\mathbf {V} (p)(\mathbf {W} (\varphi \vartheta ))-\mathbf {W} (p)(\mathbf {V} (\varphi \vartheta ))\\&=\mathbf {V} (p)(\varphi \mathbf {W} \vartheta +\vartheta \mathbf {W} \varphi )-\mathbf {W} (p)(\varphi \mathbf {V} \vartheta +\vartheta \mathbf {V} \varphi )\\&=\mathbf {V} (p)(\varphi \mathbf {W} \vartheta )+\mathbf {V} (p)(\vartheta \mathbf {W} \varphi )-\mathbf {W} (p)(\varphi \mathbf {V} \vartheta )-\mathbf {W} (p)(\vartheta \mathbf {V} \varphi )\\&=\varphi (p)\mathbf {V} (p)(\mathbf {W} \vartheta )+\overbrace {\mathbf {(} Y\vartheta )(p)} ^{=Y(p)(\vartheta )}\mathbf {V} (p)(\varphi )+\vartheta (p)\mathbf {V} (p)(\mathbf {W} \varphi )+\overbrace {\mathbf {(} Y\varphi )(p)} ^{=Y(p)(\varphi )}\mathbf {V} (p)(\vartheta )\\&~~~~-\varphi (p)\mathbf {W} (p)(\mathbf {V} \vartheta )-\overbrace {\mathbf {(} X\vartheta )(p)} ^{=X(p)(\vartheta )}\mathbf {W} (p)(\varphi )-\vartheta (p)\mathbf {W} (p)(\mathbf {V} \varphi )-\overbrace {\mathbf {(} X\varphi )(p)} ^{=X(p)(\varphi )}\mathbf {W} (p)(\vartheta )\\&=\varphi (p)\mathbf {V} (p)(\mathbf {W} \vartheta )-\varphi (p)\mathbf {W} (p)(\mathbf {V} \vartheta )+\vartheta (p)\mathbf {V} (p)(\mathbf {W} \varphi )-\vartheta (p)\mathbf {W} (p)(\mathbf {V} \varphi )\\&=\varphi (p)[\mathbf {V} ,\mathbf {W} ](p)(\vartheta )+\vartheta (p)[\mathbf {V} ,\mathbf {W} ](p)(\varphi )\end{aligned}}$

2. We show that $[\mathbf {V} ,\mathbf {W} ]$ is differentiable of class ${\mathcal {C}}^{n}$.

Let $\varphi \in {\mathcal {C}}^{n}(M)$ be arbitrary. As $\mathbf {V} ,\mathbf {W}$ are vector fields of class ${\mathcal {C}}^{n}$, $\mathbf {V} \varphi$ and $\mathbf {W} \varphi$ are contained in ${\mathcal {C}}^{n}(M)$. But since $\mathbf {V} ,\mathbf {W}$ are vector fields of class ${\mathcal {C}}^{n}$, $\mathbf {V} (\mathbf {W} \varphi )$ and $\mathbf {W} (\mathbf {V} \varphi )$ are contained in ${\mathcal {C}}^{n}(M)$. But the sum of two differentiable functions is again differentiable (this is what theorem 2.? says), and thus $[\mathbf {V} ,\mathbf {W} ]\varphi$ is in ${\mathcal {C}}^{n}(M)$, and since $\varphi$ was arbitrary, $[\mathbf {V} ,\mathbf {W} ]$ is differentiable of class ${\mathcal {C}}^{n}$.$\Box$

Theorem 6.5:

If $M$ is a manifold, and $[\cdot ,\cdot ]$ is the vector field Lie bracket, then ${\mathfrak {X}}(M)$ and $[\cdot ,\cdot ]$ form a Lie algebra together.

Proof:

1. First we note that ${\mathfrak {X}}(M)$ as defined in definition 5.? is a vector space (this was covered by exercise 5.?).

2. Second, we prove that for the vector Lie bracket, the three calculation rules of definition 6.1 are satisfied. Let $\mathbf {V} ,\mathbf {W} ,\mathbf {U} \in {\mathfrak {X}}(M)$ and $c\in \mathbb {R}$.

2.1 We prove bilinearity. For all $p\in M$ and $\varphi \in {\mathcal {C}}^{n}(M)$, we have

${\begin{aligned}{[\mathbf {V} ,\mathbf {W} +c\mathbf {U} ]}(p)(\varphi )&=\mathbf {V} (p)((\mathbf {W} +c\mathbf {U} )\varphi )-(\mathbf {W} +c\mathbf {U} )(p)(\mathbf {V} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} \varphi +c\mathbf {U} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )-c\mathbf {U} (p)(\mathbf {V} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )+c\mathbf {V} (p)(\mathbf {U} \varphi )-c\mathbf {U} (p)(\mathbf {V} \varphi )\\&=[\mathbf {V} ,\mathbf {W} ](p)(\varphi )+c[\mathbf {V} ,\mathbf {U} ](p)(\varphi )\end{aligned}}$

and hence, since $p\in M$ and $\varphi \in {\mathcal {C}}^{n}(M)$ were arbitrary,

$[\mathbf {V} ,\mathbf {W} +c\mathbf {U} ]=[\mathbf {V} ,\mathbf {W} ]+c[\mathbf {V} ,\mathbf {U} ]$

Analogously (see exercise 1), it can be proven that

$[\mathbf {V} +c\mathbf {W} ,\mathbf {U} ]=[\mathbf {V} ,\mathbf {U} ]+c[\mathbf {W} ,\mathbf {U} ]$

2.2 We prove skew-symmetry. We have for all $p\in M$ and $\varphi \in {\mathcal {C}}^{n}(M)$:

$[\mathbf {V} ,\mathbf {W} ](p)(\varphi )=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )=-(\mathbf {W} (p)(\mathbf {V} \varphi )-\mathbf {V} (p)(\mathbf {W} \varphi ))=-[\mathbf {W} ,\mathbf {V} ](p)(\varphi )$

2.3 We prove Jacobi's identity. We have for all $p\in M$ and $\varphi \in {\mathcal {C}}^{n}(M)$:

${\begin{aligned}{[\mathbf {V} ,[\mathbf {W} ,\mathbf {U} ]]}(p)(\varphi )+[\mathbf {U} ,[\mathbf {V} ,\mathbf {W} ]](p)(\varphi )+[\mathbf {W} ,[\mathbf {U} ,\mathbf {V} ]](p)(\varphi )&=\mathbf {V} (p)([\mathbf {W} ,\mathbf {U} ]\varphi )-[\mathbf {W} ,\mathbf {U} ](p)(\mathbf {V} \varphi )\\&~~~~+\mathbf {U} (p)([\mathbf {V} ,\mathbf {W} ]\varphi )-[\mathbf {V} ,\mathbf {W} ](p)(\mathbf {U} \varphi )\\&~~~~+\mathbf {W} (p)([\mathbf {U} ,\mathbf {V} ]\varphi )-[\mathbf {U} ,\mathbf {V} ](p)(\mathbf {W} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} (\mathbf {U} \varphi )-\mathbf {U} (\mathbf {W} \varphi ))-\mathbf {W} (p)(\mathbf {U} (\mathbf {V} \varphi ))+\mathbf {U} (p)(\mathbf {W} (\mathbf {V} \varphi ))\\&~~~~+\mathbf {U} (p)(\mathbf {V} (\mathbf {W} \varphi )-\mathbf {W} (\mathbf {V} \varphi ))-\mathbf {V} (p)(\mathbf {W} (\mathbf {U} \varphi ))+\mathbf {W} (p)(\mathbf {V} (\mathbf {U} \varphi ))\\&~~~~+\mathbf {W} (p)(\mathbf {U} (\mathbf {V} \varphi )-\mathbf {V} (\mathbf {U} \varphi ))-\mathbf {U} (p)(\mathbf {V} (\mathbf {W} \varphi ))+\mathbf {V} (p)(\mathbf {U} (\mathbf {W} \varphi ))\\&=0\end{aligned}}$

, where the last equality follows from the linearity of $\mathbf {V} (p),\mathbf {W} (p)$ and $\mathbf {U} (p)$.$\Box$