Campbell-Hausdorff formula

A formula for computing $$w = \ln(e^{u} e^{v})$$ in the algebra of formal power series in $u$ and $v$, where the latter are assumed to be associative but non-commutative. More precisely, let $A$ be a free associative algebra with unit over the field $\mathbb{Q}$, with free generators $u$ and $v$; let $L$ be the Lie subalgebra of $A$ generated by these elements relative to the commutation operation $[x,y] = x y - y x$; and let $\widehat{A}$ and $\widehat{L}$ denote, respectively, the natural power-series completions of $A$ and $L$, i.e., $\widehat{A}$ is the ring of power series in the associative but non-commutative variables $u$ and $v$, and $\widehat{L}$ is the closure of $L$ in $\widehat{A}$. Then the mapping $$\exp: x \mapsto e^{x} = \sum_{n = 0}^{\infty} \frac{1}{n!} x^{n}$$ is a continuous bijection of $\widehat{A}$ onto the multiplicative group $1 + \widehat{A}_{1}$, where $\widehat{A}_{1}$ is the set of series without a constant term. The inverse of this mapping is $$\ln: y \mapsto \ln(y) = \sum_{n = 1}^{\infty} \frac{(-1)^{n - 1}}{n} (y - 1)^{n}.$$

The restriction of $\exp$ to $\widehat{L}$ is a bijection of $\widehat{L}$ onto the group $1 + \widehat{L}$. One can thus introduce a group operation, $x \circ y \stackrel{\text{df}}{=} \ln(e^{x} e^{y})$, on the set of elements of the Lie algebra $\widehat{L}$, and it can be shown that the subgroup of this group that is generated by $u$ and $v$ is free. The Campbell-Hausdorff formula provides an expression for $u \circ v$ as a power series in $u$ and $v$: $$\sum_{m = 1}^{\infty} \left[ \frac{(-1)^{m - 1}}{m} \sum_{\substack{p_{1},\ldots,p_{m} \in \mathbf{N}_{0} \\ q_{1},\ldots,q_{m} \in \mathbf{N}_{0} \\ p_{1} + q_{1} > 0 \\ \vdots \\ p_{m} + q_{m} > 0}} \frac{1}{\sum_{i = 1}^{m} (p_{i} + q_{i}) \cdot \prod_{i = 1}^{m} p_{i}! q_{i}!} [\underbrace{u,[u,\ldots [u}_{p_{1}},[\underbrace{v,[v, \ldots [v}_{q_{1}}, \ldots [\underbrace{u,[u, \ldots [u}_{p_{m}},[\underbrace{v,[v, \ldots [v,v}_{q_{m}}]] \ldots ]] \right] \qquad (\ast)$$ or (in terms of the adjoint representation $(\operatorname{ad} x)(y) = [x,y]$) $$w = \sum_{n = 1}^{\infty} \left[ \frac{1}{n} \sum_{\substack{r + s = n \\ r,s \geq 0}} (w'_{r,s} + w''_{r,s}) \right],$$ where $$w'_{r,s} \stackrel{\text{df}}{=} \sum_{m = 1}^{\infty} \left[ \frac{(-1)^{m - 1}}{m} \sum^{\ast} \left( \! \left( \prod_{i = 1}^{m - 1} \frac{(\operatorname{ad} u)^{r_{i}}}{r_{i}!} \frac{(\operatorname{ad} v)^{s_{i}}}{s_{i}!} \right) \frac{(\operatorname{ad} u)^{r_{m}}}{r_{m}!} \right) \! (v) \right]$$ and $$w''_{r,s} \stackrel{\text{df}}{=} \sum_{m = 1}^{\infty} \left[ \frac{(-1)^{m - 1}}{m} \sum^{\ast \ast} \left( \prod_{i = 1}^{m - 1} \frac{(\operatorname{ad} u)^{r_{1}}}{r_{i}!} \frac{(\operatorname{ad} v)^{s_{i}}}{s_{i}!} \right) \! (u) \right].$$ Here, $\displaystyle \sum^{\ast}$ denotes summation over $r_{1} + \cdots + r_{m} = r$, $s_{1} + \cdots + s_{m - 1} = s - 1$, $r_{1} + s_{1} \geq 1,\ldots,r_{m - 1} + s_{m - 1} \geq 1$; and $\displaystyle \sum^{\ast \ast}$ denotes summation over $r_{1} + \cdots + r_{m - 1} = r - 1$, $s_{1} + \cdots + s_{m - 1} = s$, $r_{1} + s_{1} \geq 1,\ldots,r_{m - 1} + s_{m - 1} \geq 1$.

The first investigation of the expression $w$ is due to J.E. Campbell. F. Hausdorff ([2]) proved that $w$ can be expressed in terms of the commutators of $u$ and $v$, i.e., it is an element of the Lie algebra $\widehat{L}$.

If $\mathfrak{g}$ is a normed Lie algebra over a complete non-discretely normed field $\mathbb{K}$, then the series $(\ast)$, with $u,v \in \mathfrak{g}$, is convergent in a neighbourhood of $0_{\mathfrak{g}}$. Near $0_{\mathfrak{g}}$, one can thus define the structure of a local Banach Lie group over $\mathbb{K}$ (in the ultrametric case — the structure of a Banach Lie group), with Lie algebra $\mathfrak{g}$. This yields one of the existence proofs for a local Lie group with a given Lie algebra (Lie’s third theorem). Conversely, in any local Lie group, multiplication can be expressed in canonical coordinates by the Campbell-Hausdorff formula.

References

 [1a] J.E. Campbell, Proc. London Math. Soc., 28 (1897), pp. 381−390. [1b] J.E. Campbell, Proc. London Math. Soc., 29 (1898), pp. 14−32. [2] F. Hausdorff, “Die symbolische Exponential Formel in der Gruppentheorie”, Leipziger Ber., 58 (1906), pp. 19−48. [3] N. Bourbaki, “Elements of mathematics. Lie groups and Lie algebras”, Addison-Wesley (1975). (Translated from French) [4] J.-P. Serre, “Lie algebras and Lie groups”, Benjamin (1965). (Translated from French) [5] Theórie des algèbres de Lie. Topologie des groupes de Lie, Sem. S. Lie, 1e année 1954−1955, École Norm. Sup. (1955). [6] W. Magnus, A. Karrass, B. Solitar, “Combinatorial group theory: presentations in terms of generators and relations”, Wiley (Interscience) (1966).

Let $A^{n}$ denote the component of $A$ consisting of non-commutative polynomials of degree $n$. Then $\displaystyle \widehat{A} = \prod_{i = 1}^{\infty} A^{n}$. Similarly, $\displaystyle \widehat{L} = \prod_{i = 1}^{\infty} L^{n}$.
The formula for $w = u \circ v$ is also known as the Baker-Campbell-Hausdorff formula or the Campbell-Baker-Hausdorff formula. The first few terms are: $$w = u + v + \frac{1}{2} [u,v] + \frac{1}{12} [u,[u,v]] + \frac{1}{12} [v,[v,u]] + \cdots.$$ The formula in terms of the $w'_{r,s}$’s and $w''_{r,s}$’s is known as the explicit Campbell-Hausdorff formula (in Dynkin’s form).