1 4 Ju n 20 06 EXPONENTIAL BAKER - CAMPBELL - HAUSDORFF FORMULA AND APPLICATIONS TO FORMAL VECTOR FIELDS

The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series H = log(e X e Y) for non-commuting X, Y. Formally H lives in a completionˆL of the free Lie algebra L generated by X, Y. We prove that there are F, G ∈ [ ˆ L, ˆ L] such that H = e F Xe −F + e G Y e −G. We give a closed expression for H in the Lie algebra of formal vector fields on the line.