Resumen de Quasi-reflection algebras and cyclotomic associators

Benjamin Enriquez

• 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 B1 n → (Z/NZ) n Sn, where B1 n is a braid group of type B. The formality isomorphism depends algebraically on a series ΨKZ, 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–Teichm¨uller 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 (Z/NZ) ×, and we give a lower bound for the character of its space of generators.