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).

Comments

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).

References

[a1]	H.F. Baker, “Alternants and continuous groups”, Proc. London Math. Soc. (2), 3 (1905), pp. 24−47.
[a2]	V.S. Varadarajan, “Lie groups, Lie algebras, and their representations”, Springer (1984), Section 2.15.