
In the Lie Theory notes on my website based on the 2008 class by Mark Haiman, section 3.3, we give the following discussion. I should mention that I think of BCH and breaking into two parts. The first is purely algebraic:
Let $U$ be a bialgebra over field $\mathbb k$, and $s$ a formal (commuting) variable. Then we can form the $s$adic completion $U[\![ s ]\!]$ of $U \otimes_{\mathbb k} \mathbb k[s]$. It is an algebra, of course. It is not a bialgebra in the algebraic sense. On the other hand, we can extend the comultiplication $\Delta : U \to U\otimes U$ by linearity (and continuity – it is $s$adiccontinuous) to $U[\![ s ]\!] \to (U\otimes U)[\![ s ]\!]$, and the map $U[\![ s ]\!] \otimes_{\mathbb k[\![ s ]\!]} U[\![ s ]\!] \to (U\otimes U)[\![ s ]\!]$ realizes the latter as the $s$adiccompletion of the former. So $U[\![ s ]\!]$ is a bialgebra for this $s$adiccompleted tensor product, which I will denote by $\hat\otimes$. Denote this completed comultiplication by $\hat\Delta$. Note that $\hat\Delta$ is continuous in the $s$adic topology.
Choose $\psi \in U[\![ s ]\!]$ such that $\psi(s=0) = 0$. Then $\psi$ is primitive (i.e. $\hat\Delta(\psi) = \psi \otimes 1 + 1\otimes \psi$ iff $e^\psi$ is grouplike (i.e. $\hat\Delta(e^\psi) = e^\psi \hat\otimes e^\psi$). Note that $e^\psi = \sum_n \frac1{n!}\psi^n$ converges in the $s$adic topology since $\psi(s=0) = 0$. Proof: By continuity, $\hat\Delta(e^\psi) = e^{\hat\Delta(\psi)}$. As a remark, note that the power series $\psi \in U[\![ s ]\!]$ is primitive iff it is primitive termbyterm.
Everything above continues to work if we use two commuting formal variables $s$ and $t$. Let $\mathfrak f$ denote the free Lie algebra on two generators $x$ and $y$. Then its universal enveloping algebra is the free associative algebra $T$ on the same two generators. (Proof: "free" functors are left adjoint to "forget to vector spaces" and "universal enveloping" is left adjoint to "forget from associative to Lie", and the composition of left adjoints is left adjoint to the corresponding composition of forgetful functors.) We will work in the (completed) bialgebra $T[\![s,t]\!]$. Define the formal series $b(sx,ty)$ by the formula
$$ e^{b(sx,ty)} = e^{sx}e^{ty}. $$
Since $x$ and $y$ are primitive and the product of grouplikes is grouplike, it follows from 2. that $b(sx,ty)$ is primitive. Indeed, the coefficient on $s^mt^n$ in $b(sx,ty)$ is primitive.
The primitives in a universal enveloping algebra (in this case, $T$) are precisely the original Lie algebra (in this case, $\mathfrak f$). Thus the coefficient on $s^mt^n$ in $b(sx,ty)$ is a Lie polynomial (composition of brackets) in the noncommuting variables $x,y$. By degree counting, it is homogeneous of degree $m$ in $x$ and homogeneous of degree $n$ in $y$. More generally, $b(sx,ty)$ is a Lie series. If you work to define the correct topology so that you can talk about power series in noncommuting variables, then it makes sense to talk about the series $b(x,y)$.
The second part is manifoldtheoretic:
Let $G$ denote a Lie group (whose underlying manifold is given analytic structure, and such that multiplication is an analytic map), and $\mathfrak g$ its Lie algebra. Identify the universal enveloping algebra $U\mathfrak g$ with the leftinvariant differential operators on $G$. Consider the stalk $\mathcal C(G)_e$ of analytic functions defined in a neighborhood of the identity. Then $u\in U\mathfrak g = 0$ iff $u$ annihilates $\mathcal C(G)_e$.
Recall that we have a map $\exp: \mathfrak g \to G$ (defined, for example, by flowing along the leftinvariant vector on $G$ field defined by $x\in \mathfrak g$). It is an analytic isomorphism near the identity, and so near the identity we have an inverse map $\log : G \to \mathfrak g$. By passing to smaller neighborhoods of the identity as needed, we can define $\beta(x,y) = \log(\exp x\exp y)$. It is a $\mathfrak g$valued analytic function on a neighborhood of $(0,0)\in \mathfrak g \times \mathfrak g$.
Choose $f\in \mathcal C(G)_e$ and $x,y\in \mathfrak g$. Then $e^{sx} \in U\mathfrak g[\![s]\!]$ makes is $\mathbb k[\![s]\!]$valued differential operator on $\mathcal C(G)_e$, and $e^{sx} f$ is the Taylor expansion in $s$ of $f(\exp sx)$. Similarly, $e^{sx}e^{ty}f$ computes the twovariable Taylor expansion of $f(\exp sx \exp ty)$. Let $\tilde \beta$ denote the Taylor expansion of $\beta(sx,ty)$. Then
$$ e^{\tilde \beta(sx,ty)}f = f(\exp(\tilde\beta(sx,ty))) = f(\exp sx \exp ty) = e^{sx}e^{ty} f = e^{b(sx,ty)} f. $$
By 1., we must have $\tilde\beta(sx,ty) = b(sx,ty)$ as formal power series.
But $\tilde\beta$ is the Taylor expansion of the analytic function $\beta$. Thus in a small enough neighborhood of the origin, it converges. By definition, $\tilde \beta = b $ is the Baker–Campbell–Hausdorff series.


answered Jun 10 '12 at 14:24

