The $\boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem

Rüdiger Achilles; Andrea Bonfiglioli; Jacob Katriel

doi:10.3934/era.2015.22.32

Article Contents

2015, Volume 22: 32-45. Doi: 10.3934/era.2015.22.32 open access

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The $\boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem

1.
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna
2.
Department of Chemistry, Technion -- Isreal Institute of Technology, Haifa 32000

Received: July 31, 2014

Published: December 2014

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Abstract

We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the $q$-exponential $\exp_q(x)=\sum_{n=0}^{\infty} \frac{x^n}{[n]_q!}$, with the usual notation for $q$-factorials: $[n]_q!:=[n-1]_q!\cdot(q^n-1)/(q-1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are non-commuting indeterminates and $[y,x]_q$ is the $q$-commutator $yx-q\,xy$, then there exist linear combinations $Q_{i,j}(x,y)$ of iterated $q$-commutators with exactly $i$ $x$'s, $j$ $y$'s and $[y,x]_q$ in their central position, such that $\exp_q(x)\exp_q(y)=\exp_q\!\big(x+y+\sum_{i,j\geq 1}Q_{i,j}(x,y)\big)$. Our expansion is consistent with the well-known result by Schützenberger ensuring that one has $\exp_q(x)\exp_q(y)=\exp_q(x+y)$ if and only if $[y,x]_q=0$, and it improves former partial results on $q$-deformed exponentiation. Furthermore, we give an algorithm which produces conjecturally a minimal generating set for the relations between $[y,x]_q$-centered $q$-commutators of any bidegree $(i,j)$, and it allows us to compute all possible $Q_{i,j}$.

Keywords:

Exponential theorem,
Campbell-Baker-Hausdoff-Dynkin (CBHD) series,
$q$-calculus,
q-deformed CBHD series,
$q$-commutator identities.

Mathematics Subject Classification: 05A30, 81R50, 16T20 (Pri); 17B37, 05A15 (Sec).

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