Differentiable Manifolds/Lie algebras and the vector field Lie bracket

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Lie algebras[edit]

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[edit]

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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