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 exp(X) exp(Y ) exp(Z) = exp(aX + bY + cZ + dI) where a, b, c and d are solutions of four equations. In particular, the Van-Brunt and Visser formula exp(X) exp(Z) = exp(aX + bZ + c[X, Z] + dI) 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 M¨obius transformation. This may have applications in Uniformization theory and Conformal Field Theories.

### An algorithm for the Baker-Campbell-Hausdorff formula

#### 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 exp(X) exp(Y ) exp(Z) = exp(aX + bY + cZ + dI) where a, b, c and d are solutions of four equations. In particular, the Van-Brunt and Visser formula exp(X) exp(Z) = exp(aX + bZ + c[X, Z] + dI) 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 M¨obius transformation. This may have applications in Uniformization theory and Conformal Field Theories.
##### Scheda breve Scheda completa Scheda completa (DC)
2015
File in questo prodotto:
File
Algorithm.pdf

accesso aperto

Descrizione: Articolo principale
Tipologia: Postprint (accepted version)
Licenza: Accesso libero
Dimensione 263.67 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11577/3186799`