Algebraic Structures in Group-theoretical Fusion Categories

Abstract

It was shown by Ostrik (Int. Math. Res. Not. 2003(27), 1507–1520 2003) and Natale (SIGMA Symmetry Integrability Geom. Methods Appl. 13, 042 2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor Φ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.

Appendix: Remainder of the Proof of Theorem 4.1

In this appendix, we fill in some details for the proof of Theorem 4.1.

Proposition A.1

We have that

$$ \begin{array}{c} (P, \lambda_{P}^{\Gamma(S)}, \rho_{P}^{\Gamma(S^{\prime})}) \in {}_{\Gamma(S)}\mathcal{T}_{\Gamma(S^{\prime})}, \quad \quad (Q, \lambda_{Q}^{\Gamma(S^{\prime})}, \rho_{Q}^{\Gamma(S)}) \in {}_{\Gamma(S^{\prime})}\mathcal{T}_{\Gamma(S)}, \quad \text{ where}\\\\ {\small \begin{array}{ll} \lambda_{P}^{\Gamma(S)}= {\Gamma}(\lambda_{\overline{P}}^{S}) {\Gamma}_{S,\overline{P}}: {\Gamma}(S) \otimes_{\mathcal{T}} P \to P, & \rho_{P}^{\Gamma(S^{\prime})}= {\Gamma}(\rho_{\overline{P}}^{S^{\prime}}) {\Gamma}_{\overline{P},S^{\prime}}: P \otimes_{\mathcal{T}} {\Gamma}(S^{\prime}) \to P,\\ \lambda_{Q}^{\Gamma(S^{\prime})}= {\Gamma}(\lambda_{\overline{Q}}^{S^{\prime}}) {\Gamma}_{S^{\prime},\overline{Q}}: {\Gamma}(S^{\prime}) \otimes_{\mathcal{T}} Q \to Q, & \rho_{Q}^{\Gamma(S)}= {\Gamma}(\rho_{\overline{Q}}^{S}) {\Gamma}_{\overline{Q},S}: Q \otimes_{\mathcal{T}} {\Gamma}(S) \to Q. \end{array} } \end{array} $$

Proof

It is straight-forward to check that P is a right ${\Gamma }(S^{\prime })$-module in $\mathcal {T}$ with action given by $\rho _{P}^{\Gamma (S^{\prime })}$. In a similar way, it can be seen that P is a left Γ(S)-module in $\mathcal {T}$ with action $\lambda _{P}^{\Gamma (S)}$. Let us now check the left and right action compatibility for P. Consider the diagram, where $\otimes := \otimes _{\mathcal {S}}$ and we suppress the ⊗_∗ symbols in morphisms below.

Here, (1) commutes as Γ is a monoidal functor, and (2) commutes since $\overline {P}~\in ~ {}_{S}\mathcal {C}_{S^{\prime }}$. The diagrams (3) and (4) commute due to the naturality of Γ_∗,∗, and the triangles correspond to the definition of the left and right actions of P in ${}_{\Gamma (S)}\mathcal {T}_{\Gamma (S^{\prime })}$. Therefore, $(P, \lambda _{P}^{\Gamma (S)}, \rho _{P}^{\Gamma (S^{\prime })}) \in {}_{\Gamma (S)}\mathcal {T}_{\Gamma (S^{\prime })}$. Analogously, $(Q, \lambda _{Q}^{\Gamma (S^{\prime })}, \rho _{Q}^{\Gamma (S)}) \in {}_{\Gamma (S^{\prime })}\mathcal {T}_{\Gamma (S)}$. □

Proposition A.2

The epimorphisms

$$ \begin{array}{ll} \tau: P \otimes_{\Gamma(S^{\prime})} Q \twoheadrightarrow {\Gamma}(S) &\in {}_{\Gamma(S)}\mathcal{T}_{\Gamma(S)},\\ \mu: Q \otimes_{\Gamma(S)} P \twoheadrightarrow {\Gamma}(S^{\prime}) &\in {}_{\Gamma(S^{\prime})}\mathcal{T}_{\Gamma(S^{\prime})}, \end{array} $$

satisfy diagrams (∗) and (∗∗) in Proposition 2.20(b).

Proof

Diagram (∗) corresponds to the following; ⊗ is understood from context:

Diagram (1) is the definition of $\overline {\alpha }$ (see Definition 2.18). Diagram (2) commutes as Γ is a monoidal functor, and (3) results from applying Γ to the definition of $\overline {\alpha }$. Diagram (4) is the result of applying the functor Γ to the diagram (*). Diagrams (5) and (7) follow from Eq. ??. Diagram (6) is Eq. ??. Diagrams (8) and (9) commute from naturality of Γ_∗,∗. Diagram (10) commutes by applying Γ to Eq. ??. The proof of diagram (11) is given below. Finally, the commutativity of ($5^{\prime }$)–($11^{\prime }$) follow analogously to the proof of (5)–(11), respectively. Therefore, diagram (∗) commutes. In an analogous manner, diagram (∗∗) commutes.

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