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Let {{\mathfrak g}} be a finite-dimensional Lie algebra (over the reals). Given two sufficiently small elements {x, y} of {{\mathfrak g}}, define the right Baker-Campbell-Hausdorff-Dynkin law

\displaystyle  R_y(x) := x + \int_0^1 F_R( \hbox{Ad}_x \hbox{Ad}_{ty} ) y \ dt \ \ \ \ \ (1)

where {\hbox{Ad}_x := \exp(\hbox{ad}_x)}, {\hbox{ad}_x: {\mathfrak g} \rightarrow {\mathfrak g}} is the adjoint map {\hbox{ad}_x(y) := [x,y]}, and {F_R} is the function {F_R(z) := \frac{z \log z}{z-1}}, which is analytic for {z} near {1}. Similarly, define the left Baker-Campbell-Hausdorff-Dynkin law

\displaystyle  L_x(y) := y + \int_0^1 F_L( \hbox{Ad}_{tx} \hbox{Ad}_y ) x\ dt \ \ \ \ \ (2)

where {F_L(z) := \frac{\log z}{z-1}}. One easily verifies that these expressions are well-defined (and depend smoothly on {x} and {y}) when {x} and {y} are sufficiently small.

We have the famous Baker-Campbell-Hausdoff-Dynkin formula:

Theorem 1 (BCH formula) Let {G} be a finite-dimensional Lie group over the reals with Lie algebra {{\mathfrak g}}. Let {\log} be a local inverse of the exponential map {\exp: {\mathfrak g} \rightarrow G}, defined in a neighbourhood of the identity. Then for sufficiently small {x, y \in {\mathfrak g}}, one has

\displaystyle  \log( \exp(x) \exp(y) ) = R_y(x) = L_x(y).

See for instance these notes of mine for a proof of this formula (it is for {R_y}, but one easily obtains a similar proof for {L_x}).

In particular, one can give a neighbourhood of the identity in {{\mathfrak g}} the structure of a local Lie group by defining the group operation {\ast} as

\displaystyle  x \ast y := R_y(x) = L_x(y) \ \ \ \ \ (3)

for sufficiently small {x, y}, and the inverse operation by {x^{-1} := -x} (one easily verifies that {R_x(-x) = L_x(-x) = 0} for all small {x}).

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 {R_y(x)} is always equal to {L_x(y)}) and locally associative. The well-definedness issue can be trivially disposed of by using just one of the expressions {R_y(x)} or {L_x(y)} as the definition of {\ast} (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 {{\mathfrak g}} inside the general linear Lie algebra {\mathfrak{gl}_n({\bf R})} for some {n}, one can deduce both the well-definedness and associativity of (3) from the Baker-Campbell-Hausdorff formula for {\mathfrak{gl}_n({\bf R})}. 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).

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