The category $\mathcal{C}\_l$ of tilting modules of the quantum group $U\_q(sl\_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when the quantum parameter $q$ is a root of unity. The module categories of $\mathcal{C}\_l$ (or equivalently, quantum subgroups of $U\_q(sl\_2)$) are classified by ADE diagrams. See for example, section 6 of [Ostrik](http://arxiv.org/abs/math/0111139).


Let $K\_0(\mathcal{C\_l})$ be the Grothendieck ring of $\mathcal{C}\_l$. The action of $\mathcal{C}\_l$ on a module category by tensor product on the objects defines a based module $M$ over $K\_0(l)$. The generating element of $K\_0(\mathcal C\_l)$ acts as a matrix $A$ on $M$, where $A$ is the adjacency matrix of the corresponding ADE Dynkin diagram.


The ADE Dynkin diagrams have graph symmetries: the A-type and E\_6 graphs can have the order of the vertices reversed, the D-type graphs can switch the two short legs, and the D\_4 graph exhibits a full $S(3)$ symmetry. All of these symmetries induce based module symmetries on the corresponding based module $M$.


In what ways can the symmetries on the based modules be extended to symmetries of the corresponding module categories? Can you construct module functors from the module category to itself that induce this symmetry on the based modules? If not, is there some lesser structure than a module category but greater than a based module that can be preserved under these symmetries?



Symmetries of module categories over the category of representations of quantum $sl(2)$
Let $\mathcal C$ be the $\mathrm{SU(2)}\_k$ modular tensor category
and let $\mathcal M$ be a module category. Ostrik showed in the same article the OP mentioned
that there is an algebra $A$ in $\mathcal C$, such that $\mathcal M\cong \mathcal C\_A$ as (left) module categories.
Let $\mathcal D=\mathrm{Fun}\_\mathcal{C}(\mathcal M,\mathcal M)$ be the dual category, which is equivalent with the bimodule category ${}\_A\mathcal C\_A$.


Now by definition every invertible object in $\mathcal D$, i.e. every object in the Picard group $\mathrm{Pic}(\mathcal D)$, gives a symmetry of the module category $\mathcal M$ and also of its based ring. Then it is easy to check in the literature (see below) that all symmetries of $G=A,D,E$ Dynkin diagrams correspond to invertible elements of $\mathcal D$. Namely, one simply has to count that there are exactly $|\mathrm{Aut}(G)|$ invertible objects, *\*edit\*: (see Noah Snyder's comment)* acting "faithful" on the Grothendieck ring. **Thus all symmetries of the Dynkin diagram are symmetries of corresponding module category**


Much of the data of the dual categories for $\mathrm{SU}(2)\_k$ has been calculated in the subfactor literature. See for example Section 6.2 of
*Jens Böckenhauer, and David E. Evans*, [**Modular invariants, graphs and $\alpha$-induction for nets of subfactors. III.**](http://link.springer.com/article/10.1007%2Fs002200050673), *Comm. Math. Phys.* **205** (1999), no. 1, 183–228.
and 
Section 6 of 
*Jens Böckenhauer, David E. Evans, and Yasuyuki Kawahigashi*, [**Chiral structure of modular invariants for subfactors**](http://dx.doi.org/10.1007/s002200050798), *Comm. Math. Phys.* **210** (2000), no. 3, 733--784..


Remark. The OP ask for quantum $sl\_2$, but by Proposition 8.2.6 in
Fröhlich, Jürg(CH-ETHZ-P); Kerler, Thomas(1-HRV)
Quantum groups, quantum categories and quantum field theory. 
Lecture Notes in Mathematics, 1542. Springer-Verlag, Berlin, 1993. viii+431 pp. ISBN: 3-540-56623-6 
it follows that all modular tensor categories of $\mathrm{SU(2)}\_k$ type are braided equivalent.


