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Campbell-hausdorff FormulaIn mathematics, the Campbell-Hausdorff formula (also called the Campbell-Baker-Hausdorff formula) is the solution to - z = ln(exey)
for non-commuting x and y. It is named for John Edward Campbell (1862-1924), H. F. Baker and Felix Hausdorff. Specifically, let G be a simply-connected Lie group with Lie algebra . Let - exp:
be the exponential map, defining -
The general formula is given by: -
\sum_{n>0}\frac {(-1)^{n+1}}{n} \sum_{ \begin{matrix} & {r_i + s_i > 0} \\ & {1\le i \le n} \end{matrix}} \frac{(\sum_{i=1}^n (r_i+s_i))^{-1}}{r_1!s_1!\cdots r_n!s_n!} \times(\mbox{ad} X)^{r_1}(\mbox{ad} Y)^{s_1}\cdots (\mbox{ad} X)^{r_n}(\mbox{ad} Y)^{s_n - 1}Y. Here - ad(A)B = A,B
is the adjoint endomorphism. In terms in the sum where , the last three factors should be interpreted as . The first few terms are well-known: -
{1}{48}Y,[X[X,Y]] - \frac{1}{48} X,[Y,[X,Y]] + \mbox{(commutators of five and greater terms)}. 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, -
{X^n}{n!}}. When we solve for Z in - eZ = eX eY,
we obtain a simpler formula: -
\sum_{n>0} \frac{(-1)^{n+1}}{n} \sum_{\begin{matrix} &{r_i+s_i>0} \\ & {1\le i\le n}\end{matrix}} \frac{X^{r_1}Y^{s_1}\cdots X^{r_n}Y^{s_n}}{r_1!s_1!\cdots r_n!s_n!}. We note that the first, second, third and fourth order terms are: {1}{2} (XY - YX) {1}{12} (X^2Y + XY^2 - 2XYX + Y^2X + YX^2 - 2YXY) {1}{24} (X^2Y^2 - 2XYXY - Y^2X^2 + 2YXYX). References - L. Corwin & F.P Greenleaf (1990) Representation of nilpotent Lie groups and their applications, Part 1: Basic theory and examples,
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