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
- and (bilinearity)
- (Jacobi's identity)
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
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 )
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 .
If is a manifold, and is the vector field Lie bracket, then and form a Lie algebra together.
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 .