Campbell-Hausdorff formula
The Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to
z = ln(exey) for non-commuting x and y.
Specifically, let G be a simply-connected Lie group with Lie algebra  . Let the map exp:  be an exponential map, defining,
The general formula is given by:
 .
Here ad(A) B = [A,B]
The first few terms are well-known:
 .
There is no expression in closed form.
For a matrix Lie algebra  the Lie algebra is the tangent space of the identity I, and the commutator is simply [X,Y] = XY - YX; the exponential map is the standard exponential map of matrices,
 .
When we solve for Z in eZ = eX eY, we obtain a simpler formula:
 .
We note first, second, third and fourth order terms are:
References and external links
http://mathworld.wolfram.com/Baker-Campbell-HausdorffSeries.html
- L. Corwin & F.P Greenleaf (1990) Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples,
Referenced By
Continuous group | Hilbert's fifth problem | Hilberts fifth problem | Infinitesimal generator | LieGroup | Lie group | Lie groups | List of Lie group topics | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | Transformation groups
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