# ct.category theory – Can a braided fusion category have an order-2 Morita equivalence which has no representative that is both connected and isomorphic to its opposite?

Let $$mathcal{B}$$ be a braided fusion category over $$mathbb{C}$$. Let me write $$mathrm{Alg}(mathcal{B})$$ for the set of isomorphism classes of unital associative algebra objects in $$mathcal{B}$$, and $$mathrm{Mor}(mathcal{B}) = mathrm{Alg}(mathcal{B})/sim_M$$ for the set of Morita equivalence classes. These sets are naturally associative (but usually noncommutative) monoids under the tensor product operation. (The formula for multiplication in a tensor product of algebra objects explicitly references the braiding in $$mathcal{B}$$.)

An algebra $$X in mathrm{Alg}(mathcal{B})$$ is called connected if the unit object $$1inmathcal{B}$$ appears with multiplicity $$1$$ in the direct sum decomposition of (the underlying object of) $$mathcal{B}$$. Every simple separable algebra is Morita equivalent to a connected one: given a separable algebra $$X in mathrm{Alg}(mathcal{B})$$, you choose a simple $$X$$-module, and consider its algebra of endomorphisms.

Every algebra object $$X in mathrm{Alg}(mathcal{B})$$ has an opposite algebra $$X^{mathrm{op}}$$ — the formula for multiplication in $$X^{mathrm{op}}$$ uses the braiding in $$mathcal{B}$$. An algebra $$X$$ is called Azumaya if its Morita equivalence class $$(X) in mathrm{Mor}(mathcal{B})$$ is invertible. If $$X$$ is Azumaya, then $$(X)^{-1}$$ is the Morita-equivalence class of $$X^mathrm{op}$$.

I am interested in Azumaya algebras which are Morita self-dual in the sense that $$(X) = (X)^{-1} in mathrm{Mor}(mathcal{B})$$, or in other words there exists a Morita equivalence $$X sim_M X^{mathrm{op}}$$. One way this can occur, of course, is if $$X$$ is isomorphic as an algebra to $$X^{mathrm{op}}$$. Let’s call such an algebra antiautomorphic. Every Morita self-dual algebra is Morita equivalent to an antiautomorphic one: what you do is to look at the category of $$X$$-modules; to take a direct sum of all simple $$X$$-modules, with each isomorphism class appearing with multiplicity one; and take the endomorphism algebra of this sum.

This universal representative, however, will typically fail to be connected. On the other hand, I don’t know very many examples, and in the examples I do know, I can always find an antiautomorphic connected representative of each self-dual Azumaya class.

What is an example of a braided fusion category $$mathcal{B}$$ and an order-2 element in $$mathrm{Mor}(mathcal{B})$$ which has no representative that is simultaneously connected and antiautomorphic?