Generating \(q\)-commutator identities and the \(q\)-BCH formula
type: Item
instance of: scholarly article
title: Generating \(q\)-commutator identities and the \(q\)-BCH formula
zbMATH Open document ID: 1353.05024
DOI: 10.1155/2016/9598409
author: Andrea Bonfiglioli
author: Jacob Katriel
published in: Advances in Mathematical Physics
publication date: 2017-01-17T00:00:00Z
review text: Summary: Motivated by the physical applications of \(q\)-calculus and of \(q\)-deformations, the aim of this paper is twofold. Firstly, we prove the \(q\)-deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with the \(q\)-exponential function \(\mathrm{exp}_q(x) = \sum_{n = 0}^{\infty}(x^n / [n]_q!)\), where \([n]_q = 1 + q + \cdots + q^{n - 1}\) denotes, as usual, the \(n\)th \(q\)-integer. We prove that if \(x\) and \(y\) are any noncommuting indeterminates, then \(\mathrm{exp}_q(x) \mathrm{exp}_q(y) = \mathrm{exp}_q(x + y + \sum_{n = 2}^{\infty} Q_n(x, y))\), where \(Q_n(x, y)\) is a sum of iterated \(q\)-commutators of \(x\) and \(y\) (on the right and on the left, possibly), where the \(q\)-commutator \([y, x]_q := y x - q x y\) has always the innermost position. When \([y, x]_q = 0\), this expansion is consistent with the known result by \textit{M. P. Schützenberger} [C. R. Acad. Sci., Paris 236, 352--353 (1953; Zbl 0051.09401)] and \textit{J. Cigler} [Monatsh. Math. 88, 87--105 (1979; Zbl 0424.05007)]: \(\mathrm{exp}_q(x) \mathrm{exp}_q(y) = \mathrm{exp}_q(x + y)\). Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iterated \(q\)-commutators (of any length) of \(x\) and \(y\). These results can be used to obtain simplified presentation for the summands of the \(q\)-deformed Baker-Campbell-Hausdorff formula.
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 6675500
zbMATH Keywords: Baker-Campbell-Hausdorff formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2016/9598409
OpenAlex ID: W2536322773
Wikidata QID: Q59122599
cites work: Topics in noncommutative algebra. The theorem of Campbell, Baker, Hausdorff and Dynkin
cites work: Exponential Operators and Parameter Differentiation in Quantum Physics
cites work: The \(q\)-deformed Campbell-Baker-Hausdorff-Dynkin theorem
cites work: Quantum calculus
cites work: A Comprehensive Treatment of q-Calculus
cites work: Quantum groups
cites work: Hilbert spaces of analytic functions and generalized coherent states
cites work: Ordering relations for q-boson operators, continued fraction techniques and the q-CBH enigma
cites work: A q-analog of the Campbell-Baker-Hausdorff formula
cites work: Exponential mapping for non-semisimple quantum groups
cites work: On some unsolved problems in quantum group theory
cites work: Coherent states of the \(q\)-canonical commutation relations
cites work: Q4016344
cites work: \(q\)-deformed Baker-Campbell-Hausdorff formula
cites work: The \(q\)-Zassenhaus formula
cites work: Disentangling \(q\)-exponentials: a general approach
cites work: On the \(q\)-analogues of the Zassenhaus formula for disentangling exponential operators.
cites work: A Magnus- and Fer-type formula in dendriform algebras.
cites work: Dendriform equations
cites work: Twisted dendriform algebras and the pre-Lie Magnus expansion
cites work: Quantum Chevalley groups
cites work: Quantum groups via Hall algebras of complexes.
cites work: Hall algebras, hereditary algebras and quantum groups
cites work: Hall algebras and quantum groups
cites work: The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q), S n, SUq(N), and SU(N)
cites work: Q4201879
cites work: \(q\)-exponentials on quantum spaces
cites work: Bose-Einstein condensation of a quon gas.
cites work: Q5819726
cites work: Operatormethoden für q-identitäten
cites work: A q-analogue of the Campbell-Baker-Hausdorff expansion
cites work: Q5824395
cites work: A no-go theorem for a Lie-consistent q-Campbell–Baker–Hausdorff expansion
cites work: Q3660083
description: scientific article; zbMATH DE number 6675500