On a quantum analog of the Grothendieck-Teichmüller group

We introduce a self-dual, noncommutative, and noncocommutative Hopf algebra HGT which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role as the GrothendieckTeichmüller group for quasitensor categories. We also give a result which highly restricts the possibility for similar structures for higher weak n-categories (n ≥ 3) by showing that these structures would not allow for any nontrivial deformations. Finally, give an explicit description of the elements of HGT . 1 The Hopf algebra HGT In [Dri] Drinfeld introduced the Grothendieck-Teichmüller group by considering the (formal) reparametrizations of the data (commutativity and associativity isomorphisms) of a quasitensor category. Consider now braided (weak) monoidal bicategories arising from the representations of a Hopf category (as defined in [CF]) on 2-vector spaces (see [KV]), i.e. on certain module categories. Let us assume, in addition, that the Hopf category itself is given as the category of finite dimensional representations of a quasi-trialgebra, satisfying a quasitriangularity and coquasitriangularity condition. This is analogous to understanding the Grothendieck-Teichmüller group GT as a universal symmetry of quasitriangular quasi-Hopf algebras (see e.g. [CP])