where . One easily verifies that these expressions are well-defined (and depend smoothly on and ) when and are sufficiently small.
We have the famous Baker-Campbell-Hausdoff-Dynkin formula:
Theorem 1 (BCH formula) Let be a finite-dimensional Lie group over the reals with Lie algebra . Let be a local inverse of the exponential map , defined in a neighbourhood of the identity. Then for sufficiently small , one has
See for instance these notes of mine for a proof of this formula (it is for , but one easily obtains a similar proof for ).
for sufficiently small , and the inverse operation by (one easily verifies that for all small ).
It is tempting to reverse the BCH formula and conclude (the local form of) Lie’s third theorem, that every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some local Lie group, by using (3) to define a smooth local group structure on a neighbourhood of the identity. (See this previous post for a definition of a local Lie group.) The main difficulty in doing so is in verifying that the definition (3) is well-defined (i.e. that is always equal to ) and locally associative. The well-definedness issue can be trivially disposed of by using just one of the expressions or as the definition of (though, as we shall see, it will be very convenient to use both of them simultaneously). However, the associativity is not obvious at all.
With the assistance of Ado’s theorem, which places inside the general linear Lie algebra for some , one can deduce both the well-definedness and associativity of (3) from the Baker-Campbell-Hausdorff formula for . However, Ado’s theorem is rather difficult to prove (see for instance this previous blog post for a proof), and it is natural to ask whether there is a way to establish these facts without Ado’s theorem.
After playing around with this for some time, I managed to extract a direct proof of well-definedness and local associativity of (3), giving a proof of Lie’s third theorem independent of Ado’s theorem. This is not a new result by any means, (indeed, the original proofs of Lie and Cartan of Lie’s third theorem did not use Ado’s theorem), but I found it an instructive exercise to work out the details, and so I am putting it up on this blog in case anyone else is interested (and also because I want to be able to find the argument again if I ever need it in the future).
The key is to observe that the right and left BCH laws commute with each other:
Note that this commutativity has to hold if (3) is to be both well-defined and associative. Assuming Proposition 2, we can set in (4) and use the easily verified identities , to conclude that for small , ensuring that (3) is well-defined; and then inserting (3) into (4) we obtain the desired (local) associativity.
It remains to prove Proposition 2. We first make a convenient observation. Thanks to the Jacobi identity, the adjoint representation is a Lie algebra homomorphism from to the Lie algebra . As this latter Lie algebra is the Lie algebra of a Lie group, namely , the Baker-Campbell-Hausdorff formula is valid for that Lie algebra. In particular, one has
for sufficiently small . But as is a Lie algebra homomorphism, one has
This would already give what we want if the adjoint representation was faithful. We unfortunately cannot assume this (and this is the main reason, by the way, why Ado’s theorem is so difficult), but we can at least use (5) to rewrite the formulae (1), (2) as
This leads to the important radial homogeneity identities
for all sufficiently small and , as can be seen by a short computation.
for fixed small , a large natural number , and with the understanding that the operations are only applied to sufficiently small elements . From radial homogeneity we have and , and so a large number (, to be more precise) of iterations of (7) (using uniform smoothness to control all errors) gives
and the claim (4) then follows by sending .
It remains to prove (6). When , then is the identity map and the claim is trivial; similarly if . By Taylor expansion, it thus suffices to establish the infinitesimal commutativity law
(One can interpret this infinitesimal commutativity as a commutativity of the vector fields corresponding to the infinitesimal generators of the left and right BCH laws, although we will not explicitly adopt that perspective here.) This is a simplification, because the infinitesimal versions of (1), (2) are simpler than the non-infinitesimal versions. Indeed, from the fundamental theorem of calculus one has
for any fixed , and similarly
Thus it suffices (by Clairaut’s theorem) to show that
It will be more convenient to work with the reciprocals , of the functions . Recall the general matrix identity
for any smoothly varying invertible matrix function of a real parameter . Using this identity, we can write the left-hand side of (8) as
If we write and , then from Taylor expansion we have
and so we can simplify the above expression as
Similarly, the right-hand side of (8) is
Now, we write
and thus expand (9) as
We write as the exponential of . Using the Duhamel matrix identity
for any smoothly varying matrix function of a real variable , together with the linearity of , we see that
For any , the adjoint map is a derivation in the sense that
thanks to the Jacobi identity. Exponentiating, we conclude that
(thus each is a Lie algebra homomorphism) and thus
Using this, we can simplify (11) as
which we can rewrite as
But by an appropriate change of variables (and the anti-symmetry of the Lie bracket), both sides of this equation can be written as
and the claim follows.
Remark 1 The above argument shows that every finite-dimensional Lie algebra can be viewed as arising from a local Lie group . It is natural to then ask if that local Lie group (or a sufficiently small piece thereof) can in turn be extended to a global Lie group . The answer to this is affirmative, as was first shown by Cartan. I have been unable however to find a proof of this result that does not either use Ado’s theorem, the proof method of Ado’s theorem (in particular, the structural decomposition of Lie algebras into semisimple and solvable factors), or some facts about group cohomology (particularly with regards to central extensions of Lie groups) which are closely related to the structural decompositions just mentioned. (As noted by Serre, though, a certain amount of this sort of difficulty in the proof may in fact be necessary, given that the global form of Lie’s third theorem is known to fail in the infinite-dimensional case.)