Quasi-reflection algebras and cyclotomic associators

By B. Enriquez. 2004

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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 Bn1 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 $$\mathfrak{grtm}$$ (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.

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Braided Picard groups and graded extensions of braided tensor categories (2020, background): “Section 4 deals with braided module categories over a braided tensor category B introduced and studied by Enriquez [21], Brochier [5], and Ben-Zvi, Brochier, and Jordan [2].”