Differentiable Manifolds/Lie algebras and the vector field Lie bracket

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Lie algebras

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Definition 7.1:

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

 

such that for all   and   the three rules

  1.   and   (bilinearity)
  2.   (skew-symmetry)
  3.   (Jacobi's identity)

hold.

Definition 7.2:

Let   with   be a Lie algebra. A subset of   which is a Lie algebra with the restriction of   on that subset is called a Lie subalgebra.

The vector field Lie bracket

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Definition 7.3:

Let   be a manifold of class  . We define the vector field Lie bracket, denoted by  , as follows:

 

Theorem 6.4: If   are vector fields of class   on  , then   is a vector field of class   on   (i. e.   really maps to  )

Proof:

1. We show that for each  ,  . Let   and  .

1.1 We prove linearity:

 

1.2 We prove the product rule:

 

2. We show that   is differentiable of class  .

Let   be arbitrary. As   are vector fields of class  ,   and   are contained in  . But since   are vector fields of class  ,   and   are contained in  . But the sum of two differentiable functions is again differentiable (this is what theorem 2.? says), and thus   is in  , and since   was arbitrary,   is differentiable of class  . 

Theorem 6.5:

If   is a manifold, and   is the vector field Lie bracket, then   and   form a Lie algebra together.

Proof:

1. First we note that   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   and  .

2.1 We prove bilinearity. For all   and  , we have

 

and hence, since   and   were arbitrary,

 

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

 

2.2 We prove skew-symmetry. We have for all   and  :

 

2.3 We prove Jacobi's identity. We have for all   and  :

 

, where the last equality follows from the linearity of   and  . 

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