Eur. Phys. J. C (2016) 76: 610
https://doi.org/10.1140/epjc/s10052-016-4455-4

Regular Article - Theoretical Physics

Closed form of the Baker–Campbell–Hausdorff formula for the generators of semisimple complex Lie algebras

¹ Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo, 8-35131, Padua, Italy
² Sezione di Padova dell’Istituto Nazionale di Fisica Nucleare, Istituto Nazionale di Fisica Nucleare, Via Marzolo, 8-35131, Padua, Italy

^* e-mail: matone@pd.infn.it

Received: 12 July 2016
Accepted: 24 October 2016
Published online: 7 November 2016

Abstract

Recently it has been introduced an algorithm for the 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 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 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 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, a linear combination of X, Y and Z, such that It turns out that the relevant commutator algebras are type 1c-i, type 4 and 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.