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A formula for computing
 
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
 +
$$
 +
\ln: y \mapsto \ln(y) = \sum_{n = 1}^{\infty} \frac{(-1)^{n - 1}}{n} (y - 1)^{n}.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200901.png" /></td> </tr></table>
+
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 $:
 
+
$$
in the algebra of formal power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200902.png" />, where the latter are assumed to be associative but non-commutative. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200903.png" /> be a free associative algebra with unit over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200904.png" />, with free generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200906.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200907.png" /> be the Lie subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200908.png" /> generated by these elements relative to the commutation operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c0200909.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009011.png" /> be the natural power series completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009013.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009014.png" /> is the ring of power series in the associative but non-commutative variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009016.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009018.png" />. Then the mapping
+
\sum_{m = 1}^{\infty}
 
+
\left[
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009019.png" /></td> </tr></table>
+
\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}}
is a continuous bijection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009020.png" /> onto the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009022.png" /> is the set of series without constant term. The inverse of this mapping is
+
\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 ]]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009023.png" /></td> </tr></table>
+
\right] \qquad (\ast)
 
+
$$
The restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009024.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009025.png" /> is a bijection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009026.png" /> onto the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009027.png" />. One can thus introduce a group operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009028.png" /> in the set of elements of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009029.png" />, and it can be shown that the subgroup of this group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009031.png" /> is free. The Campbell–Hausdorff formula provides an expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009032.png" /> as a power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009034.png" />:
+
or (in terms of the adjoint representation $ (\operatorname{ad} x)(y) = [x,y] $)
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
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],
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009036.png" /></td> </tr></table>
 
 
 
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009037.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009039.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009040.png" /><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009041.png" /><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009042.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009044.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009045.png" />, in the general term of this series) or (in terms of the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009046.png" />):
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009047.png" /></td> </tr></table>
 
 
 
 
where
 
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009048.png" /></td> </tr></table>
+
The first investigation of the expression $ w $ is due to J.E. Campbell. F. Hausdorff ([[#References|[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} $.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009049.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009050.png" /></td> </tr></table>
 
 
 
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009051.png" /> denotes summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009054.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009055.png" /> denotes summation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009058.png" />.
 
 
 
The first investigation of the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009059.png" /> is due to J.E. Campbell . F. Hausdorff [[#References|[2]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009060.png" /> can be expressed in terms of the commutators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009062.png" />, i.e. it is an element of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009063.png" />.
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009064.png" /> is a normed Lie algebra over a complete non-discretely normed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009065.png" />, the series (*), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009066.png" />, is convergent in a neighbourhood of zero. Near the zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009067.png" /> one can thus define the structure of a local Banach Lie group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009068.png" /> (in the ultrametric case — the structure of a Banach Lie group), with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009069.png" />. 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.
+
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====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  J.E. Campbell,  ''Proc. London Math. Soc.'' , '''28'''  (1897)  pp. 381–390</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  J.E. Campbell,  ''Proc. London Math. Soc.'' , '''29'''  (1898)  pp. 14–32</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Die symbolische Exponential Formel in der Gruppentheorie"  ''Leipziger Ber.'' , '''58'''  (1906)  pp. 19–48</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Ecole Norm. Sup.  (1955)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)</TD></TR></table>
 
 
  
 +
<table>
 +
<TR><TD valign="top">[1a]</TD><TD valign="top">
 +
J.E. Campbell, ''Proc. London Math. Soc.'', '''28''' (1897), pp. 381−390.</TD></TR>
 +
<TR><TD valign="top">[1b]</TD><TD valign="top">
 +
J.E. Campbell, ''Proc. London Math. Soc.'', '''29''' (1898), pp. 14−32.</TD></TR>
 +
<TR><TD valign="top">[2]</TD><TD valign="top">
 +
F. Hausdorff, “Die symbolische Exponential Formel in der Gruppentheorie”, ''Leipziger Ber.'', '''58''' (1906), pp. 19−48.</TD></TR>
 +
<TR><TD valign="top">[3]</TD><TD valign="top">
 +
N. Bourbaki, “Elements of mathematics. Lie groups and Lie algebras”, Addison-Wesley (1975). (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[4]</TD><TD valign="top">
 +
J.-P. Serre, “Lie algebras and Lie groups”, Benjamin (1965). (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[5]</TD><TD valign="top">
 +
''Theórie des algèbres de Lie. Topologie des groupes de Lie'', ''Sem. S. Lie'', '''1e année 1954−1955''', École Norm. Sup. (1955).</TD></TR>
 +
<TR><TD valign="top">[6]</TD><TD valign="top">
 +
W. Magnus, A. Karrass, B. Solitar, “Combinatorial group theory: presentations in terms of generators and relations”, Wiley (Interscience) (1966).</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009070.png" /> be the component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009071.png" /> consisting of non-commutative polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009072.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009073.png" />. Similarly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009074.png" />.
 
  
The formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009075.png" /> is also known as the Baker–Campbell–Hausdorff formula or Campbell–Baker–Hausdorff formula. The first few terms are:
+
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} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009076.png" /></td> </tr></table>
+
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).
  
The formula in terms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020090/c02009078.png" /> is known as the explicit Campbell–Hausdorff formula (in Dynkin's form).
+
====References====
  
====References====
+
<table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.F. Baker,   "Alternants and continuous groups"  ''Proc. London Math. Soc. (2)'' , '''3''' (1905) pp. 24–47</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Varadarajan,   "Lie groups, Lie algebras, and their representations" , Springer (1984) pp. Section 2.15</TD></TR></table>
+
<TR><TD valign="top">[a1]</TD><TD valign="top">
 +
H.F. Baker, “Alternants and continuous groups”, ''Proc. London Math. Soc. (2)'', '''3''' (1905), pp. 24−47.</TD></TR>
 +
<TR><TD valign="top">[a2]</TD><TD valign="top">
 +
V.S. Varadarajan, “Lie groups, Lie algebras, and their representations”, Springer (1984), Section 2.15.</TD></TR>
 +
</table>

Latest revision as of 17:41, 4 May 2017

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.
How to Cite This Entry:
Campbell-Hausdorff formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Campbell-Hausdorff_formula&oldid=18290
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article