Quasi-reflection algebras and cyclotomic associators

Original · EN

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 Bₙ¹ -> (ZZ/N ZZ)ⁿ rtimes Sₙ, where Bₙ¹ is a braid group of type B. The formality isomorphism depends algebraically on a series PsiₖZ, 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 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 grtmd(N,k) of grtm(N,k); it is a graded Lie algebra with an action of (ZZ/N ZZ)*, and we give a lower bound for the character of its space of generators.