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 $[\tilde{X},\tilde{Y}]=\tilde{u}\tilde{X}+\tilde{v}\tilde{Y}+\tilde{c}I$. It turns out that $e^{\alpha }$ 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.

15 pages. Typos corrected. JGP version

References in corpus (2)

• Special-case closed form of the Baker-Campbell-Hausdorff formula
• An algorithm for the Baker-Campbell-Hausdorff formula

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