Abstract
A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the BakerCampbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if [X, Y] = uX + vY + cI, [Y, Z] = wY + zZ + dI, and, consistently with the Jacobi identity, [X, Z] = mX + nY + pZ + eI, then
where a, b, c and d are solutions of four equations. In particular, the Van-Brunt and Visser formula
extends to cases when [X, Z] contains also elements different from X and Z. 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 SL2(ℂ) 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 SL2(ℂ) 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.
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References
A. Van-Brunt and M. Visser, Special-case closed form of the Baker-Campbell-Hausdorff formula, J. Phys. A 48 (2015) 225207 [arXiv:1501.02506] [INSPIRE].
A. Van-Brunt and M. Visser, Simplifying the Reinsch algorithm for the Baker-Campbell-Hausdorff series, arXiv:1501.05034 [INSPIRE].
M. Matone, Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas, arXiv:1503.08198 [INSPIRE].
M. Matone, Closed Form of the Baker-Campbell-Hausdorff Formula for Semisimple Complex Lie Algebras, arXiv:1504.05174 [INSPIRE].
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ArXiv ePrint: 1502.06589
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Matone, M. An algorithm for the Baker-Campbell-Hausdorff formula. J. High Energ. Phys. 2015, 113 (2015). https://doi.org/10.1007/JHEP05(2015)113
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DOI: https://doi.org/10.1007/JHEP05(2015)113