View source for Campbell-Hausdorff formula

A formula for computing

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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

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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

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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" />:

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(<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" />):

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where

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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.

====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>



====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:

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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====
<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>