Campbell-Hausdorff formula

The Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to z = ln(e^xe^y) 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 e^Z = e^X e^Y, 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)