The \(q\)-Zassenhaus formula
The authors consider the \(q\)-analogue \(E_q(x)\) of the usual exponential function \(\exp(x)\). The function \(E_q(x)\) is obtained from \(\exp(x)\) by replacing \(i!\) in the power series expansion of \(\exp(x)\) by \((q^i- 1)/(q- 1)\). The paper now centers around the problem of generalizing the well-known formula \(\exp(x+ y)= \exp(x)\exp(y)\) to non-commuting variables and \(q\)-analogues. By results of \textit{M. P. Schützenberger} [C. R. Acad. Sci., Paris 236, 352-353 (1953; Zbl 0051.09401)] \(E_q(x+ y)= E_q(x) E_q(y)\) if \(xy= qyx\) (i.e., \(x\) and \(y\) are \(q\)-commuting variables). For non-commuting variables \(x\) and \(y\) there exists a formula due to Zassenhaus, mentioned in the work of Magnus, that expresses \(\exp(x+ y)\) as a product of exponentials whose arguments are homogeneous parts of the Campbell-Baker-Hausdorff formula. As main result the authors provide a \(q\)-analogue of this formula. The relevance of this work in the theory of quantum groups, \(q\)-oscillators and physics is stressed, but no application is provided within the letter.
scientific article; zbMATH DE number 888949