Campbell-Hausdorff formula

A formula for computing

in the algebra of formal power series in , where the latter are assumed to be associative but non-commutative. More precisely, let be a free associative algebra with unit over the field , with free generators and ; let be the Lie subalgebra of generated by these elements relative to the commutation operation ; and let and be the natural power series completions of and , i.e. is the ring of power series in the associative but non-commutative variables and is the closure of in . Then the mapping

is a continuous bijection of onto the multiplicative group , where is the set of series without constant term. The inverse of this mapping is

The restriction of to is a bijection of onto the group . One can thus introduce a group operation in the set of elements of the Lie algebra , and it can be shown that the subgroup of this group generated by and is free. The Campbell–Hausdorff formula provides an expression for as a power series in and :

(*)

( times , then times times , then times , in the general term of this series) or (in terms of the adjoint representation ):

where

Here denotes summation over , , ; and denotes summation over , , .

The first investigation of the expression is due to J.E. Campbell . F. Hausdorff [2] proved that can be expressed in terms of the commutators of and , i.e. it is an element of the Lie algebra .

If is a normed Lie algebra over a complete non-discretely normed field , the series (*), with , is convergent in a neighbourhood of zero. Near the zero of one can thus define the structure of a local Banach Lie group over (in the ultrametric case — the structure of a Banach Lie group), with Lie algebra . 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 , Ie année 1954–1955 , Ecole 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 be the component of consisting of non-commutative polynomials of degree . Then . Similarly .

The formula for is also known as the Baker–Campbell–Hausdorff formula or Campbell–Baker–Hausdorff formula. The first few terms are:

The formula in terms of the and 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) pp. Section 2.15