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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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In this set of notes, we describe the basic analytic structure theory of Lie groups, by relating them to the simpler concept of a Lie algebra. Roughly speaking, the Lie algebra encodes the “infinitesimal” structure of a Lie group, but is a simpler object, being a vector space rather than a nonlinear manifold. Nevertheless, thanks to the fundamental theorems of Lie, the Lie algebra can be used to reconstruct the Lie group (at a local level, at least), by means of the exponential map and the Baker-Campbell-Hausdorff formula. As such, the local theory of Lie groups is completely described (in principle, at least) by the theory of Lie algebras, which leads to a number of useful consequences, such as the following:

  • (Local Lie implies Lie) A topological group {G} is Lie (i.e. it is isomorphic to a Lie group) if and only if it is locally Lie (i.e. the group operations are smooth near the origin).
  • (Uniqueness of Lie structure) A topological group has at most one smooth structure on it that makes it Lie.
  • (Weak regularity implies strong regularity, I) Lie groups are automatically real analytic. (In fact one only needs a “local {C^{1,1}}” regularity on the group structure to obtain real analyticity.)
  • (Weak regularity implies strong regularity, II) A continuous homomorphism from one Lie group to another is automatically smooth (and real analytic).

The connection between Lie groups and Lie algebras also highlights the role of one-parameter subgroups of a topological group, which will play a central role in the solution of Hilbert’s fifth problem.

We note that there is also a very important algebraic structure theory of Lie groups and Lie algebras, in which the Lie algebra is split into solvable and semisimple components, with the latter being decomposed further into simple components, which can then be completely classified using Dynkin diagrams. This classification is of fundamental importance in many areas of mathematics (e.g. representation theory, arithmetic geometry, and group theory), and many of the deeper facts about Lie groups and Lie algebras are proven via this classification (although in such cases it can be of interest to also find alternate proofs that avoid the classification). However, it turns out that we will not need this theory in this course, and so we will not discuss it further here (though it can of course be found in any graduate text on Lie groups and Lie algebras).

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Let {G} be a Lie group with Lie algebra {{\mathfrak g}}. As is well known, the exponential map {\exp: {\mathfrak g} \rightarrow G} is a local homeomorphism near the identity. As such, the group law on {G} can be locally pulled back to an operation {*: U \times U \rightarrow {\mathfrak g}} defined on a neighbourhood {U} of the identity in {G}, defined as

\displaystyle  x * y := \log( \exp(x) \exp(y) )

where {\log} 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 {x*y} is provided by the Baker-Campbell-Hausdorff (BCH) formula

\displaystyle  x*y = x+y+ \frac{1}{2} [x,y] + \frac{1}{12}[x,[x,y]] - \frac{1}{12}[y,[x,y]] + \ldots

for all sufficiently small {x,y}, where {[,]: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}} is the Lie bracket. More explicitly, one has the Baker-Campbell-Hausdorff-Dynkin formula

\displaystyle  x * y = x + \int_0^1 F( \hbox{Ad}_x \hbox{Ad}_{ty} ) y\ dt \ \ \ \ \ (1)

for all sufficiently small {x,y}, where {\hbox{Ad}_x = \exp( \hbox{ad}_x )}, {\hbox{ad}_x: {\bf R}^d \rightarrow {\bf R}^d} is the adjoint representation {\hbox{ad}_x(y) := [x,y]}, and {F} is the function

\displaystyle  F( t ) := \frac{t \log t}{t-1}

which is real analytic near {t=1} 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 {C^2} 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 {C^{1,1}} 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 ({C^{1,1}} Baker-Campbell-Hausdorff formula) Let {{\bf R}^d} be a finite-dimensional vector space, and suppose one has a continuous operation {*: U \times U \rightarrow {\bf R}^d} defined on a neighbourhood {U} around the origin, which obeys the following three axioms:

  • (Approximate additivity) For {x,y} sufficiently close to the origin, one has

    \displaystyle  x*y = x+y+O(|x| |y|). \ \ \ \ \ (2)

    (In particular, {0*x=x*0=x} for {x} sufficiently close to the origin.)

  • (Associativity) For {x,y,z} sufficiently close to the origin, {(x*y)*z = x*(y*z)}.
  • (Radial homogeneity) For {x} sufficiently close to the origin, one has

    \displaystyle  (sx) * (tx) = (s+t)x \ \ \ \ \ (3)

    for all {s,t \in [-1,1]}. (In particular, {x * (-x) = (-x) * x = 0} for all {x} 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 {\hbox{Ad}_x} appropriately) in this setting; see below the fold.

The reason that we call this a {C^{1,1}} Baker-Campbell-Hausdorff formula is that if the group operation {*} has {C^{1,1}} regularity, and has {0} 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 {C^{1,1}} 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).

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