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Given a finite-dimensional Lie algebra, and a representation by derivationson the completed symmetric algebra of its dual, a number of interesting twistedconstructions appear: certain twisted Weyl algebras, deformed Leibniz rules,quantized ``star'' product. We first illuminate a number of interrelationsbetween these constructions and then proceed to study a special case in certainprecise sense corresponding to the symmetric or Weyl ordering. This case hasbeen known earlier to be related to computations with Hausdorff series, forexample the expression for the star product is in such terms. For the deformedLeibniz rule, hence a coproduct, we present here a new nonsymmetric expression,which is then expanded into a sum of expressions labelled by a class of planartrees, and for a given tree evaluated by Feynman-like rules. These expressionsare filtered by a bidegree and we show recursion formulas for the sums ofexpressions of a given bidegree, and compare the recursions to recursions forHausdorff series, including the comparison of initial conditions. This way weshow a direct corespondence between the Hausdorff series and the expression fortwisted coproduct.

Leibniz rules for enveloping algebras in symmetric ordering