THE q-DEFORMED CAMPBELL-BAKER-HAUSDORFF-DYNKIN THEOREM

We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the q-exponential exp(q)(x)= Sigma(infinity)(n=0) x(n)/[n]q!, [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 + Sigma(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 vertical bar 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).