An algorithm for the Baker-Campbell-Hausdorff formula

Abstract

A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the Baker-Campbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if $[X,Y]=uX+vY+cI$, $[Y,Z]=wY+zZ+dI$, and, consistently with the Jacobi identity, $[X,Z]=mX+nY+pZ+eI$, then $$ \exp(X)\exp(Y)\exp(Z)=\exp({aX+bY+cZ+dI}) $$ where $a$, $b$, $c$ and $d$ are solutions of four equations. In particular, the Van-Brunt and Visser formula $$\exp(X)\exp(Z)=\exp({aX+bZ+c[X,Z]+dI}) $$ extends to cases when $[X,Z]$ contains also elements different from $X$ and $Z$. Such a closed form of the BCH formula may have interesting applications both in mathematics and physics. As an application, we provide the closed form of the BCH formula in the case of the exponentiation of the Virasoro algebra, with ${\rm SL}_2({\rm C})$ following as a subcase. We also determine three-dimensional subalgebras of the Virasoro algebra satisfying the Van-Brunt and Visser condition. It turns out that the exponential form of ${\rm SL}_2({\rm C})$ has a nice representation in terms of its eigenvalues and of the fixed points of the corresponding M\"obius transformation. This may have applications in Uniformization theory and Conformal Field Theories.