On a quantum analog of the Grothendieck-Teichmueller group

Abstract

We introduce a noncommutative and noncocommutative Hopf algebra which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role as the Grothendieck- Teichmueller group for quasitensor categories. We also give a result which highly restricts the possibility for similar structures for even higher weak n-categories than bicategories by showing that these structures would not allow for any nontrivial deformations. Finally, we give an explicit description of the elements of this Hopf algebra.