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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.

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  • $\begingroup$ I quote Owen Gwilliam: "I kinda hate the word [non-pointed]. It's like studying non-elephant mammals". $\endgroup$ Dec 27, 2022 at 15:05
  • $\begingroup$ As to your actual question, which is really specifically about quantum groups and not about general braided fusion categories: what examples have you tried? I think you'll find that when the level is small, there will be lots of exotic behaviour. $\endgroup$ Dec 27, 2022 at 15:08
  • $\begingroup$ Even at large level, you'll have to contend with level-rank duality. For example, there is a braided-reversing isomorphism between Rep(Sp(n), k) and Rep(Sp(k), n). So if you take your group G to be Sp(n) x Sp(k), with the level (k,n), then the diagonal is Lagrangian. I don't see any Lagrangian subalgebra of Sp(n) x Sp(k). $\endgroup$ Dec 27, 2022 at 15:12
  • $\begingroup$ On the other hand, as the level becomes very large compared to the rank, then Rep(G,k) approximates Rep(G) in a way that knows something about the symplectic geometry of g. So some version of your statement will most likely hold in the level >> rank regime. $\endgroup$ Dec 27, 2022 at 15:14

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