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,
- (ISBN 052136034X)
