Compressed Drinfeld associators

Drinfeld associator is a key tool in computing the Kontsevich integral of knots. A Drinfeld associator is a series in two non-commuting variables, satisfying highly complicated algebraic equations—hexagon and pentagon. The logarithm of a Drinfeld associator lives in the Lie algebra L generated by the symbols a,b,c modulo [a,b]=[b,c]=[c,a]. The main result is a description of compressed associators that obey the compressed pentagon and hexagon in the quotient L/[[L,L],[L,L]]. The key ingredient is an explicit form of Campbell–Baker–Hausdorff formula in the case when all commutators commute.