I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\mathcal C$ is the pointed braided category $\mathcal C(G, q)$ corresponding to a metric group $(G,q)$ then Lagrangian subcategories of $\mathcal C(G, q)$ bijectively correspond to Lagrangian subgroups of $G$.”
I would like to know if the statement remains true of the category is not pointed.
More precisely, you can also construct braided fusion categories from the representation theory of quantum groups $U_q(\frak g)$ at root of unity, for $\frak g$ a finite dimensional semisimple algebra. These are in general not pointed. Can one map Lagrangian subalgebras of $\frak g$ to Lagrangian subcategories of the category associated to $U_q(\frak g)$ (and if so, is the map bijective) ?
Caveat: the statements above might not be accurately formulated and please correct if that is case, I am new to the subject.