Compressed Drinfeld associators

Explicit Description of Compressed Logarithms of All 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]. We describe explicitly the images of the log...

On Drinfel'd associators

In 1986 [6], in order to study the linear representations of the braid group Bn coming from the monodromy of the Knizhnik-Zamolodchikov differential equations, Drinfel’d introduced a class of formal power series Φ on noncommutative variables over the finite alphabet X = {x0, x1}. Such a power series Φ is called an associator. For n = 3, it leads to the following fuchsian noncommutative differen...

Buchsteiner Loops: Associators and Constructions

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpoten...

On Associators and the Grothendieck - Teichmullergroup

On Associators and the Grothendieck - Teichmuller

We present a formalism within which the relationship (discovered by Drinfel’d in [Dr1, Dr2]) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a great simplification of Drinfel’d’s original work. In particular, we re-prove that rational associators exist and can be constructed iteratively.