Let be a Lie group with Lie algebra . As is well known, the exponential map is a local homeomorphism near the identity. As such, the group law on can be locally pulled back to an operation defined on a neighbourhood of the identity in , defined as
where is the local inverse of the exponential map. One can view as the group law expressed in local exponential coordinates around the origin.
An asymptotic expansion for is provided by the Baker-Campbell-Hausdorff (BCH) formula
for all sufficiently small , where is the Lie bracket. More explicitly, one has the Baker-Campbell-Hausdorff-Dynkin formula
for all sufficiently small , where , is the adjoint representation , and is the function
which is real analytic near and can thus be applied to linear operators sufficiently close to the identity. One corollary of this is that the multiplication operation is real analytic in local coordinates, and so every smooth Lie group is in fact a real analytic Lie group.
It turns out that one does not need the full force of the smoothness hypothesis to obtain these conclusions. It is, for instance, a classical result that regularity of the group operations is already enough to obtain the Baker-Campbell-Hausdorff formula. Actually, it turns out that we can weaken this a bit, and show that even regularity (i.e. that the group operations are continuously differentiable, and the derivatives are locally Lipschitz) is enough to make the classical derivation of the Baker-Campbell-Hausdorff formula work. More precisely, we have
Theorem 1 ( Baker-Campbell-Hausdorff formula) Let be a finite-dimensional vector space, and suppose one has a continuous operation defined on a neighbourhood around the origin, which obeys the following three axioms:
- (Approximate additivity) For sufficiently close to the origin, one has
(In particular, for sufficiently close to the origin.)
- (Associativity) For sufficiently close to the origin, .
- (Radial homogeneity) For sufficiently close to the origin, one has
for all . (In particular, for all sufficiently close to the origin.)
Then is real analytic (and in particular, smooth) near the origin. (In particular, gives a neighbourhood of the origin the structure of a local Lie group.)
Indeed, we will recover the Baker-Campbell-Hausdorff-Dynkin formula (after defining appropriately) in this setting; see below the fold.
The reason that we call this a Baker-Campbell-Hausdorff formula is that if the group operation has regularity, and has as an identity element, then Taylor expansion already gives (2), and in exponential coordinates (which, as it turns out, can be defined without much difficulty in the category) one automatically has (3).
We will record the proof of Theorem 1 below the fold; it largely follows the classical derivation of the BCH formula, but due to the low regularity one will rely on tools such as telescoping series and Riemann sums rather than on the fundamental theorem of calculus. As an application of this theorem, we can give an alternate derivation of one of the components of the solution to Hilbert’s fifth problem, namely the construction of a Lie group structure from a Gleason metric, which was covered in the previous post; we discuss this at the end of this article. With this approach, one can avoid any appeal to von Neumann’s theorem and Cartan’s theorem (discussed in this post), or the Kuranishi-Gleason extension theorem (discussed in this post).
— 1. Proof of Baker-Campbell-Hausdorff formula —
We begin with some simple bounds of Lipschitz and type on the group law .
Proof: We begin with the first estimate. If , then is small, and (on multiplying by ) we have . By (2) we have
Lemma 3 (Adjoint representation) For all sufficiently close to the origin, there exists a linear transformation such that for all sufficiently close to the origin.
Proof: Fix . The map is continuous near the origin, so it will suffice to establish additivity, in the sense that
for sufficiently close to the origin.
Let be a large natural number. Then from (3) we have
where is the product of copies of . Conjugating this by , we see that
But from (2) we have
and thus (by Lemma 2)
But if we split as the product of and and use (2), we have
Putting all this together we see that
sending we obtain the claim.
From (2) we see that
for all sufficiently small. Combining these two properties (and using (4)) we conclude in particular that
for sufficiently small. Thus we see that is a continuous linear representation. In particular, is a continuous homomorphism into a linear group, and so we have the Hadamard lemma
where is the linear transformation
for sufficiently small, and so by the product rule we have
Also we clearly have for small. Thus we see that is linear in , and so we have
for some bilinear form .
One can show that this bilinear form in fact defines a Lie bracket, but for now, all we need is that it is manifestly real analytic (since all bilinear forms are polynomial and thus analytic), thus and depend analytically on .
We now give an important approximation to in the case when is small:
Proof: If we write , then (by (2)) and
we obtain the claim.
Using (3), the first summand can be expanded as
From (4) one has
Writing , and then letting , we conclude (from the convergence of the Riemann sum to the Riemann integral) that
and the claim follows.
We can then integrate this to obtain an exact formula for :
Corollary 5 (Baker-Campbell-Hausdorff-Dynkin formula) For sufficiently small, one has
The right-hand side is clearly real analytic in and , and Lemma 1 follows.
Proof: Let be a large natural number. We can express as the telescoping sum
We conclude that
Sending , so that the Riemann sum converges to a Riemann integral, we obtain the claim.
Remark 1 It seems likely that one can relax the type condition (2) in the above arguments to the weaker conditions
where is bounded by for some function that goes to zero at zero, and similarly for , as the effect of this is to replace various errors with errors that still go to zero as . However, type regularity is what is provided to us by Gleason metrics, so this type of regularity suffices for applications related to Hilbert’s fifth problem.
— 2. Building a Lie group from a Gleason metric —
We can now give a slightly alternate derivation of Theorem 7 from the previous post, which asserted that every locally compact group with a Gleason metric was isomorphic to a Lie group. As in those notes, one begins by constructing the space of one-parameter subgroups, demonstrating that it is isomorphic to a finite-dimensional vector space , constructing the exponential map , and then showing that this map is locally a homeomorphism. Thus we can identify a neighbourhood of the identity in with a neighbourhood of the origin in , thus giving a locally defined multiplication operation in . By construction, this map is continuous and associative, and obeys the homogeneity (3) by the definition of the exponential map. Now we verify the estimate (2). From Lemma 8 in the previous post, one can verify that the exponential map is bilipschitz near the origin, and the claim is now to show that
for sufficiently close to the identity in . By definition of , it suffices to show that
for all ; but this follows from Lemma 8 of the previous post (and the observation, from the escape property, that and ).
Applying Theorem 1, we now see that is smooth, and so the group operations are smooth near the origin. Also, for any , conjugation by is an (local) outer automorphism of a neighbourhood of the identity, hence also an automorphism of . Since linear maps are automatically smooth, we conclude that conjugation by is smooth near the origin in exponential coordinates. From this, we can transport the smooth structure from a neighbourhood of the origin to the rest of (using either left or right translations), and obtain a Lie group structure as required.