Let be a matrix Lie group with associated Lie algebra . denotes the group of Lie algebra automorphisms of under composition. Recall we have the adjoint representation
of , given by
Of course, is always in the Lie algebra, essentially due to the fact that conjugation “passes through” the exponential unmolested.
It was remarked in the PMATH 763 lectures that if denotes the centre of , then . However, after doing some reading I found this was not quite true and that indeed , where denotes the connected component of the identity, and denotes the centralizer of in . I couldn’t find a proof anywhere, but luckily it wasn’t too difficult to solve.
The proof of this gives us a bit of insight into the imperfect but palatable relationship between commutativity in and commutativity in .
If , to show the idea is that in a sufficiently small neighbourhood of the identity. Using this we can prove (by scaling) that for any , which is what we wanted. The details are a quick exercise.
Conversely, if , then to show the idea is to use the fact that is generated by . Again I leave the details for you.
Of course, if is connected, then and so
the centre of .