Is there a "killing" lemma for G-crossed braided fusion categories?

Question

Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different.

Premodular categories

In braided spherical fusion (=premodular) categories $\mathcal{C}$ with braiding $c_{-,-}$, we can define the double braiding: $$\gamma_{A,X}\colon A \otimes X \to A \otimes X$$ $$\gamma_{A,X} := c_{X,A} \circ c_{A,X}$$

Now we can define encirclings, where we braid an object around another and close the loop on the right hand side:

$$\Delta_{A,X}\colon A \to A$$ $$\Delta_{A,X} = \operatorname{tr}_X(\gamma_{A,X})$$

(The trace is defined via the pivotal structure.)

The "killing" lemma is quite well known. Assume that $A$ is simple. Then:

$$\Omega := \bigoplus_{X \text{ simple}} d(X) X$$ $$\Delta_{A,\Omega} = \left\{ \begin{matrix} d(\Omega) \cdot 1_A && \qquad A \text{ is in the symmetric centre} \\ 0 && \qquad \text{otherwise} \end{matrix}\right.$$ $A$ is in the symmetric centre if braids trivially with every object. One says that $A$ is killed if it's not in the symmetric centre, hence the name of the lemma.

$G$-crossed braided fusion categories

Now consider $G$-crossed braided fusion categories. If you think of fusion categories as some sort of categorified group (algebra), then a $G$-crossed braided fusion category is a categorified crossed module. It is, amongst other things, a $G$-graded fusion category:

$$\mathcal{C} = \bigoplus_{g \in G} \mathcal{C}_g$$

It also has a monoidal $G$-action:

$${}^g(-)\colon \mathcal{C}_h \to \mathcal{C}_{ghg^{-1}} \quad \forall g, h \in G$$

And it has a crossed braiding:

$$ c_{A,X}\colon A \otimes X \to {}^gX \otimes A\quad \forall A \in \mathcal{C}_g$$

This means that the double braiding is now not an endomorphism anymore:

$$ \gamma_{A,X}\colon A \otimes X \to {}^{ghg^{-1}}A \otimes {}^g X \quad\forall A \in \mathcal{C}_g, X \in \mathcal{C}_h$$

You can only take the trace on the right hand side, and thus define encircling, if $X \cong {}^gX$. The only canonical way I can think of is to demand that the grade of $A$ is trivial, i.e. $g = e$, since there is a coherence isomorphism $\epsilon_X\colon {}^eX \to X$. Then we can define:

$$\Omega_h := \bigoplus_{X \text{ simple } \in \mathcal{C}_h} d(X) X$$ $$\Delta_{A, \Omega_h} = \operatorname{tr}_{\Omega_h}\left((1_A \otimes \epsilon_{\Omega_h}) \circ \gamma_{A,\Omega_h} \right) \qquad \forall A \in \mathcal{C}_e$$

But now source and target aren't equal: $$\Delta_{A,\Omega_h}\colon A \to {}^hA$$

In some cases we have $A \cong {}^hA$, and then one can ask whether the endomorphism is 0 or not, which leads me to my question:

Is there a similar lemma to the standard killing lemma in the $G$-crossed case?

Answer 1

Edit: I used to believe that there is a possible generalisation stemming from work of Altschüler and Bruguières. (See Appendix C in Drinfeld, Gelaki, Nikshych, Ostrik - On braided fusion categories I). But that was based on a flawed assumption. (I didn't realise that $A$ must be in the trivial degree for it to work.) With the correct assumptions, the following can be said:

Sliding lemma

Like in braided spherical fusion categories, the sliding lemma is closely related to the killing lemma.

Lemma

Let $Y \in \mathcal{C}_g, A \in \mathcal{C}_e$. Up to coherences in the $G$-crossed fusion category, we have:

$$\Delta_{A,\Omega_h} \otimes 1_Y = c_{Y,A} \circ (\epsilon_Y \otimes \Delta_{A,\Omega_{hg^{-1}}}) \circ c_{A,Y}$$

In words, we can slide the $Y$-strand over the $\Omega_h$-encircling so that it links with the $A$-strand. If you draw a picture, it will become clear.

Killing lemma

Recall that the trivial degree $\mathcal{C}_e$ is a braided spherical fusion category, so the classical killing lemma applies to $\Delta_{A,\Omega_e}$.

Lemma

Let $A \in \mathcal{C}_e$ be a simple object, and $\mathcal{C}_h \not\simeq 0$. Then:

$$\Delta_{A,\Omega_h}\colon A \to {}^hA$$ is nonzero, and thus an isomorphism, iff $A$ is in the symmetric centre of $\mathcal{C}_e$.

Proof

The statement is equivalent to the statement with $- \otimes 1_{\Omega_{h^{-1}}}$ applied to it (recall that $\mathcal{C}_h \not\simeq 0 \implies \Omega_h \neq 0 \implies \Omega_{h^{-1}} \cong \Omega_h^* \neq 0)$. By the sliding lemma, the left hand side is then equal to $c_{\Omega_{h^{-1}}, A} \circ (\epsilon_{\Omega_{h^{-1}}} \otimes \Delta_{A,\Omega_e}) \circ c_{A, \Omega_{h^{-1}}}$. Apply the killing lemma for $\Delta_{A,\Omega_e}$, yielding:

$$\Delta_{A,\Omega_h} \otimes 1_{\Omega_{h^{-1}}} = \left\{ \begin{matrix} d(\Omega_e) \cdot \gamma_{A,\Omega_{h^{-1}}} && \qquad A \text{ is in the symmetric centre of } \mathcal{C}_e \\ 0 && \qquad \text{otherwise} \end{matrix}\right.$$

Observation

This seems to prove that if all degrees of $\mathcal{C}$ are nonzero (that is, the grading is "faithful"), all objects in the symmetric centre of $\mathcal{C}_e$ are automatically equivariant, i.e. possess a family of isomorphisms $A \cong {}^gA, \forall g \in G$. Surely, that result must have been known before?

Edit This has been commented on briefly by Etingof, Nikshych, Ostrik and Meir in Fusion categories and homotopy theory, (39).