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} x^n/[n]_q!, with the usual notation for q-factorials: [n]_q! := [n-1]_q!*(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 (x + y +sum_{i,j >= 1} Q_{i,j} (x,y)). Our expansion is consistent with the well-known result by Schutzenberger 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}.

### The q-deformed Campbell-Baker-Hausdorff-Dynkin Theorem

#### 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} x^n/[n]_q!, with the usual notation for q-factorials: [n]_q! := [n-1]_q!*(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 (x + y +sum_{i,j >= 1} Q_{i,j} (x,y)). Our expansion is consistent with the well-known result by Schutzenberger 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}.
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2015
Achilles, R.; Bonfiglioli, A.; Katriel, J.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/522689`
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