# Differentiable Manifolds/Lie algebras and the vector field Lie bracket

 Differentiable Manifolds ← Diffeomorphisms and related vector fields Lie algebras and the vector field Lie bracket Integral curves and Lie derivatives →

## Lie algebras

Definition 7.1:

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

${\displaystyle [\cdot ,\cdot ]:L\times L\to L}$

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

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

hold.

Definition 7.2:

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

## The vector field Lie bracket

Definition 7.3:

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

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

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

Proof:

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

1.1 We prove linearity:

{\displaystyle {\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:

{\displaystyle {\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 ${\displaystyle [\mathbf {V} ,\mathbf {W} ]}$  is differentiable of class ${\displaystyle {\mathcal {C}}^{n}}$ .

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

Theorem 6.5:

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

Proof:

1. First we note that ${\displaystyle {\mathfrak {X}}(M)}$  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 ${\displaystyle \mathbf {V} ,\mathbf {W} ,\mathbf {U} \in {\mathfrak {X}}(M)}$  and ${\displaystyle c\in \mathbb {R} }$ .

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

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

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

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

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

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

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

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

{\displaystyle {\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 ${\displaystyle \mathbf {V} (p),\mathbf {W} (p)}$  and ${\displaystyle \mathbf {U} (p)}$ .${\displaystyle \Box }$

 Differentiable Manifolds ← Diffeomorphisms and related vector fields Lie algebras and the vector field Lie bracket Integral curves and Lie derivatives →