JHEP Journal of High Energy Physics 1029-8479 JHEP05(2015)113 10.1007/JHEP05(2015)113 An algorithm for the Baker-Campbell-Hausdorff formula Marco Matone matone@pd.infn.it Dipartimento di Fisica e Astronomia “G. Galilei”, Istituto Nazionale di Fisica Nucleare, Università di Padova, Via Marzolo, 8-35131 Padova, Italy 21052015 2015 05 113 06032015 16042015 27042015 OPEN ACCESS, © The Authors 2015 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 1502.06589

A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the Baker-Campbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if , , and, consistently with the Jacobi identity, , then where , , and are solutions of four equations. In particular, the Van-Brunt and Visser formula extends to cases when contains also elements different from and . Such a closed form of the BCH formula may have interesting applications both in mathematics and physics. As an application, we provide the closed form of the BCH formula in the case of the exponentiation of the Virasoro algebra, with following as a subcase. We also determine three-dimensional subalgebras of the Virasoro algebra satisfying the Van-Brunt and Visser condition. It turns out that the exponential form of has a nice representation in terms of its eigenvalues and of the fixed points of the corresponding Möbius transformation. This may have applications in Uniformization theory and Conformal Field Theories.

Conformal and W Symmetry Gauge Symmetry Space-Time Symmetries

Article funded by SCOAP3

 The algorithm

Implementation of the algorithm

Exponentiating the Virasoro algebra

Geometrical constructions

Acknowledgments

http://dx.doi.org/10.1088/1751-8113/48/22/225207 http://arxiv.org/abs/1501.05034 http://arxiv.org/abs/1503.08198 http://arxiv.org/abs/1504.05174