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

TL;DR

This paper derives explicit closed-form expressions for the Baker-Campbell-Hausdorff formula within semisimple complex Lie algebras, expanding the understanding of their algebraic structure and providing new computational tools.

Contribution

It provides explicit solutions for the BCH formula for generators of semisimple complex Lie algebras, extending recent algorithms and considering the root system structure.

Findings

Explicit closed forms for two-generator BCH in semisimple Lie algebras.

Identification of specific commutator algebra types with closed BCH forms.

Iterative method for extending BCH closed forms to more generators.

Abstract

Recently it has been introduced an algorithm Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, 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 generators of the semisimple complex Lie algebras. More precisely, for any pair, $X$, $Y$ of the Cartan-Weyl basis, we find $W$, linear combination of $X$, $Y$, such that $$ \exp(X) \exp(Y)=\exp(W) $$ The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of of the root system. Furthermore, 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 $$ \exp(X) \exp(Y) \exp(Z)=\exp(W) $$ It turns out that the relevant commutator algebras are {\it type 1c-i}, {\it type 4} and {\it type 5}. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.