Quasi-reflection algebras and cyclotomic associators

Abstract.

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism $B_n^1 \rightarrow ({\mathbb{Z}}/N{\mathbb{Z}})^n \rtimes {\mathfrak{S}}_n$, where B ¹ _n is a braid group of type B. The formality isomorphism depends algebraically on a series Ψ_KZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded $\mathfrak{grtm}$(N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue $\mathfrak{grtmd}$(N, k) of $\mathfrak{grtm}$(N, k); it is a graded Lie algebra with an action of $({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$, and we give a lower bound for the character of its space of generators.