THE GEOMETRY OF THE EXCEPTIONAL LIE GROUPS

Section 3.4 Vector Fields

We return to the example introduced in Section 1.2 of an elementary example of a Lie algebra that is not constructed using matrices. Consider the following derivative operators on Euclidean 3-space, \(\RR^3\text{:}\)

where the notation \(\partial_x\) is short for the operator \(\frac{\partial}{\partial x}\text{.}\) We can commute the action of these operators by acting on any (differentiable) function. For example,

and it is easy to see that the mixed second derivatives cancel. Thus, the commutator of (first-order) derivative operators is a (first-order) derivative operator! Working out the right-hand side yields

and we conclude that

and similarly for cyclic permutations of \(x\text{,}\) \(y\text{,}\) \(z\text{.}\) The astute reader may already have recognized these operators as angular momentum operators, corresponding to rotational symmetry; the Lie algebra they generate (by real linear combinations) is a copy of \(\so(3)\text{.}\)

More generally, a vector field on a manifold acts on functions on the manifold by taking directional derivatives. For example, if the manifold is \(\RR^3\text{,}\) the vector field

acts on real functions \(f\) of three variables as

However, in this context, we must check explicitly that the Jacobi identity (3.3.1) is satisfied. In this particular example, that computation is straightforward: Expand \(X\text{,}\) \(Y\text{,}\) \(Z\) in terms of \(\{\partial_x,\partial_y,\partial_z\}\text{.}\) Terms in the Jacobi identity involving all three of these elements are identically zero (Why?), and the remaining terms cancel using the antisymmetry of the commutator. This conclusion also holds for more general Lie algebras of vector fields, although the computation is somewhat messier, again making use of even and odd permutations.