  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 -> (ZZ/N ZZ)^n rtimes S_n, where B_n^1 is a braid group of
type B. The formality isomorphism depends algebraically on a series Psi_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 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 (ZZ/N ZZ)^*, and we give a lower bound for the character of its
space of generators.
