I would like to ask for references on automorphisms of a modular tensor category, that do not change the objects. Some special cases, such as automorphisms of a quantum double, are also helpful.



Automorphisms of a modular tensor category
For most quantum group categories *all* braided autoequivalences are classified by Cain Edie-Michell in this [paper](https://arxiv.org/abs/2002.03220). The kind you're interested in, which is called "gauge auto-equivalences" there, almost never exist. One example where it does happen is Cor. 3.2 for the adjoint subcategory of sl\_3 at level 3, but I'm not sure if that one is braided.


Many of the arguments there build on an idea from [my paper with Grossman](https://arxiv.org/abs/1102.2631) Thm 5.5 which is that if you have a planar algebraic description of your tensor category then gauge autoequivalences in particular give you an automorphism of the planar algebra, and you can often see that no such automorphism exists.


