Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas

Abstract: We show that there are {\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ ,$ derived in arXiv:1502.06589, JHEP {\bf 1505} (2015) 113. This includes, as a particular case, $\exp(X) \exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $$\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({\alpha Y}) \exp({(1-\alpha) Y}) \exp(Z)$$, with $\alpha$ fixed in such a way that it reduces to $\exp({\tilde X})\exp({\tilde Y})$, with $\tilde X$ and $\tilde Y$ satisfying the Van-Brunt and Visser condition $$[\tilde X,\tilde Y]=\tilde u\tilde X+\tilde v\tilde Y+\tilde cI$$. It turns out that $e<sup>\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\it types} of commutator algebras, such an equation leads to rational solutions for $\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.