Closed Form of the Baker-Campbell-Hausdorff Formula for Semisimple Complex Lie Algebras

Marco Matone

Published 2015-04-20Version 1

In arXiv:1502.06589 it has been introduced an algorithm extending the Van-Brunt and Visser result, leading to new closed forms of the Baker-Campbell-Hausdorff formula. In particular, there are closed forms even when the commutator $[X,Y]$ also contains elements of the algebra different from $X$ and $Y$. In arXiv:1503.08198 it has been shown that there are {\it 13 types} of such commutator algebras. We show, by providing the explicit solutions, that these include the semisimple complex Lie algebras. In particular, if $X$, $Y$ and $Z$ are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, $W$, linear combination of $X$, $Y$ and $Z$, such that $$ e^X e^Y e^Z=e^W $$ It turns out that the relevant commutator algebras are {\it type 1c-i}, {\it type 4} and {\it type 5}.