Frobenius algebras associated with the $\alpha$-induction for equivariantly braided tensor categories

Let $G$ be a group. We give a categorical definition of the $G$-equivariant $\alpha$-induction associated with a given $G$-equivariant Frobenius algebra in a $G$-braided multitensor category, which generalizes the $\alpha$-induction for $G$-twisted representations of conformal nets. For a given $G$-equivariant Frobenius algebra in a spherical $G$-braided fusion category, we construct a $G$-equivariant Frobenius algebra, which we call a $G$-equivariant $\alpha$-induction Frobenius algebra, in a suitably defined category called neutral double. This construction generalizes Rehren's construction of $\alpha$-induction Q-systems. Finally, we define the notion of the $G$-equivariant full center of a $G$-equivariant Frobenius algebra in a spherical $G$-braided fusion category and show that it indeed coincides with the corresponding $G$-equivariant $\alpha$-induction Frobenius algebra, which generalizes a theorem of Bischoff, Kawahigashi and Longo.


Introduction
In rational conformal field theory, we can construct two-dimensional conformal field theories from a chiral conformal field theory. Mathematically, it corresponds to finding nonnegative integral combinations of some functions on the complex upper half plane which is invariant under the action of the modular group SL(2, Z). Such combinations are called modular invariants. It is wellknown that the modular invariants for the SU (2) Wess-Zumino-Witten models admit an A-D-E classification i.e. they correspond to the simply-laced Dynkin diagrams [CIZ87] [Kat87].
In the framework of algebraic quantum field theory, a chiral conformal field theory is modeled by a conformal net. When a conformal net A satisfies the property called complete rationality [KLM01,Definition 8], it produces a modular tensor category as its Doplicher-Haag-Roberts (DHR) category [DHR71] [DHR74], which is denoted by Rep A in this article, see [KLM01,Corollary 37]. In this case, modular invariants can be formulated as matrices with nonnegative integer coefficients indexed by the simple objects of Rep A that commute with the S and T matrices of the modular tensor category.
It was shown by Böckenhauer, Evans and Kawahigashi [BE98][BE99a][BE99b] [BEK99][BEK00] that α-induction [LR95, Proposition 3.9] is a powerful categorical tool for producing modular invariants. Namely, when a finite index standard extension A ⊂ B of a completely rational conformal net A [LR95, Section 3] is given, the α-induction for the extension assigns to each DHR endomorphism λ of A some (not necessarily DHR) endomorphisms α ± (λ) of B and indeed ⟨α + (λ), α − (µ)⟩ := dim Hom(α + (λ), α − (µ)) for simple objects λ and µ of Rep A is a modular invariant matrix [BEK99,Theorem 5.7]. Moreover, Rehren [Reh00] showed that the matrix is physical i.e. he constructed a commutative Q-system λ,µ ⟨α + (λ), α − (µ)⟩λ ⊠ µ in Rep A ⊠ (Rep A) rev , and it was shown by Kong  When a group G acts on A, we can consider the notion of a G-twisted representation of A, which generalizes the notion of a DHR endomorphism of A [Müg05]. Recently, the notion of the αinduction for G-twisted representations, which we call G-equivariant α-induction, was introduced by Nojima [Noj20]. Then, it is natural to expect that G-equivariant α-induction is also a powerful tool for producing modular invariants in some sense. In this article, we generalize some of the classical results introduced above, which gives some evidence for this expectation. Indeed, in some cases, our construction can produce modular invariants of the fixed point net A G (Theorem 4.25).
The main results of this article are twofold. First, we generalize Rehren's construction of αinduction Q-systems [Reh00] to G-equivariant α-induction (Theorem 4.15). Indeed, this theorem can be stated in a purely algebraic setting. Namely, we construct the G-equivariant α-induction Frobenius algebra for a neutral symmetric special simple G-equivariant Frobenius algebra in a split spherical G-braided fusion category (see Subsections 2.2 and 2.3 for terminologies). For this, we have to introduce the notion of the neutral double of a split semisimple G-braided multitensor category as an ambient category (Definition 4.7). Second, we generalize [BKL15,Proposition 4.18], which is an important ingredient in the characterization of physical modular invariants in terms of α-induction [BKL15, Proposition 5.2]. Namely, we introduce the notion of the G-equivariant full center of a neutral symmetric special G-equivariant Frobenius algebra in a split spherical G-braided fusion category with nonzero dimension (Definition 5.11) and show that it indeed coincides with the corresponding G-equivariant α-induction Frobenius algebra when both of them are well-defined (Theorem 5.25).
This article is structured as follows. In Section 2, we review some preliminaries on bicategories and multitensor categories equipped with group actions. In particular, we recall the notions of a G-braided multitensor category and a G-equivariant Frobenius algebra for a group G. We also introduce an appropriate notion of equivalence between bicategories with group actions. In Section 3, we give the categorical definition of G-equivariant α-induction. For this, we define an action of G on the bicategory of G-equivariant Frobenius algebras. In Section 4, we define the notion of the neutral double of a split semisimple G-braided multitensor category and construct the G-equivariant α-induction Frobenius algebras. Finally, in Section 5, we define the notion of a Gequivariant full center and show the coincidence of G-equivariant α-induction Frobenius algebras and G-equivariant full centers.

Group actions on bicategories
In this subsection, we recall the notion of a group action on a monoidal category and that on a bicategory and give an appropriate notion of equivalence between bicategories with group actions, which turns out to be equivalent to the notion introduced in [BGM19]. Moreover, we prove a coherence theorem for G-actions on bicategories, see Theorem 2.6, cf. [BGM19, Theorem 3.1].
An action of a group G on a monoidal category C is a monoidal functor from the monoidal category G of the elements of G with only identity morphisms to the monoidal category of monoidal endofunctors on C.
We also have to consider group actions on bicategories in this article (see Subsections 3.1 and 5.2 below). For basic notions of bicategories, see e.g. [Lei04, Section 1.5]. Here we fix our notation: the composition of 1-cells of C is denoted by ⊗ C or ⊗, for we regard bicategories as generalizations of monoidal categories, and often omitted. The associativity constraint is denoted by a C . The unit is denoted by 1 C , and the left and right unit constraints are denoted respectively by l C and r C . By stating the coherence theorem for pseudofunctors [Gur13,Subsection 2.3.3] in the form of "all diagrams commute", we can see that the 2-cells obtained by vertically and horizontally composing components of a C , l C and r C of bicategories C and J F 's and φ F 's of pseudofunctors F are indeed canonical, which allows us to suppress these 2-cells. In particular, we may suppress the constraints of the tricategory of bicategories. We can also introduce some graphical representations in the tricategory of bicategories, which are used only in this subsection. A pseudonatural transformation is represented as an arrow from top to bottom, and the vertical and horizontal compositions of pseudonatural transformations are represented respectively by the vertical and horizontal concatenation of arrows. Note that we do not have to be careful of relative vertical positions when taking horizontal concatenations since the comparison constraints between pseudonatural transformations (see [BGM19, Section 1.1]) are canonical by coherence.
For a 1-cell λ of C and g ∈ G, the 1-cell γ(g)(λ) is often denoted by g λ. Moreover, g λµ denotes g (λ) ⊗ µ for g ∈ G and 1-cells λ and µ, and g f ⊗ f ′ denotes g (f ) ⊗ f ′ for g ∈ G and 2-cells f and f ′ in this article.
An appropriate notion of morphisms between (strict) 2-categories with unital (see [BGM19, Definition 2.1]) group actions is given in [BGM19, Definition 2.3]. Here, we define the notion of an equivalence in a general setting.
First, recall that for a biequivalence F : C → D, we can take a pseudofunctor F −1 : D → C with pseudonatural equivalences ev F : F −1 F ≃ id C and coev F : id D ≃ F F −1 by fixing data that consist of 0-cells F −1 (D) ∈ Obj(C), equivalence 1-cells coev F,0 D : D → F (F −1 (D)), left adjoint inverses (coev F,0 D ) ∨ of coev F,0 D (i.e. left duals of coev F,0 D in the bicategory D with invertible evaluation and coevaluation maps) for D ∈ Obj(D) and left adjoint functors F ∨ C,C ′ of F C,C ′ for C, C ′ ∈ Obj(C), see e.g. [Lei04,Proposition 1.5.13]. The set of these data is also denoted by F −1 and referred to as an adjoint inverse of F . Another choice of an adjoint inverse only yields a pseudonaturally equivalent pseudofunctor F −1 by a standard duality argument. The pseudonatural equivalences ev F and coev F are graphically represented by arcs as in the case of duality in bicategories. Their adjoint inverses are represented by opposite arcs.
We also recall that we have a natural isomorphism J F ∨ A,B,C : ) for A ∈ Obj(C) with the coherence conditions as that for pseudofunctors. Indeed, they are defined by putting (J F ∨ A,B,C ) λ,µ := F ∨ A,C (( coev  F (B). Then, as in the case of pseudofunctors, we may suppress J F ∨ 's and φ F ∨ 's. By the definition of J F ∨ 's and φ F ∨ 's, the naturality of ev F and coev F and conjugate equations for F , we can see that ev F and coev F are monoidal i.e. coev    (and similar statements for ev F ), where (J F ∨ A,B,C ) λ,µ and φ F ∨ A are suppressed. When a biequivalence F : C → D is given, we can transport H ∈ End(C) to End(D) by fixing an adjoint inverse F −1 of F and putting Ad(F )(H) := F HF −1 ∈ End(D).
Lemma 2.1. Let F : C → D be a biequivalence between bicategories. Then Ad(F ) : End(C) → End(D) can be regarded as a monoidal pseudofunctor. Another choice of an adjoint inverse of F only yields a monoidally equivalent one (i.e. there exists a triequivalence with an identical 1-cell).
Then, put κ is an adjoint inverse. Thus, Ad(F ) is a monoidal pseudofunctor.
Definition 2.3. Let (C, γ C ) and (D, γ D ) be pairs of bicategories and actions of a group G. A G-biequivalence between (C, γ C ) and (D, γ D ) is the pair F = (F, η F ) of a biequivalence F : C → D and a monoidal equivalence The existence of a G-biequivalence between bicategories does not depend on a choice of F −1 by Lemma 2.1.
By definition [GPS95, Section 3.3], for a G-biequivalence F : (C, γ C ) → (D, γ D ), the monoidal equivalence η F consists of pseudonatural equivalences η F g : Let η F g be graphically represented by a fork with three inputs F , γ C (g) and F −1 and one output γ D (g).
Next, we compare our definition with that in [BGM19, Definition 2.3]. Suppose we are given a G-biequivalence F : (C, γ C ) → (D, γ D ). Then, we obtainη F g : F γ C (g) ≃ γ D (g)F for g ∈ G by duality: namely, we putη F g : Figures 4 and 5. We can check that they satisfy the conditions in Figures 6, 7 and 8, where a crossing from F γ C (g) to γ D (g)F denotesη F g for g ∈ G, with some standard computations, but note that we use id ev F ⊗ (id F −1 * ξ F ) = id ev F ⊗ (ξ F * id F ) in the check for Figure 8. Conversely, when a biequivalence F and a triple (η F ,Π F ,M F ) with the conditions in Figures 6, 7 and 8 are given, we can construct a triple (η F , Π F , M F ) by putting η F g := (id γ D (g) * (coev F ) ∨ ) • (η F g * id F −1 ) for g ∈ G and defining Π F g,h for g, h ∈ G and M F by Figures 9 and 10. We can see that (F, η F , Π F , M F ) is a G-biequivalence, using the following lemma to check the condition in [GPS95,p. 24].
Thus, we obtain the statement since we may compose ξ F −1 * ξ F −1 with the modifications in the statement.
Thus, we can equivalently define a G-equivalence to be a tuple (F,η F ,Π F ,M F ) with the conditions in Figures 6, 7 and 8, which reduces to [BGM19, Definition 2.3] when the action is unital.
Then, we state the coherence theorem for group actions on bicategories [BGM19, Theorem 3.1] in the form of "all diagrams commute" for our convenience.
Theorem 2.6. Let C be a bicategory with an action γ C of a group G. Define a set W = n≥0 W n of words recursively by the following rules: 1 C C , (χ γ C g,h ) 0 C , (ι γ C ) 0 C ∈ W 0 for any 0-cell C and g, h ∈ G, − ∈ W 1 , ⊗ ∈ W 2 , γ C (g) ∈ W 1 for g ∈ G and w ′ ((w i ) i ) ∈ W i ni for w ′ ∈ W n and a family (w i ) n i=1 with w i ∈ W ni . Let the functor C n → C, where C 0 denotes the category with only one object and its identity morphism, corresponding naturally to a word w ∈ W n be denoted again by w. Define a set I of morphisms recursively by the following rules: the components of a C , l C , r C , J γ C (g) and φ γ C (g) are in I, (χ γ C g,h ) λ and (ι γ C ) λ for any 1-cell λ and g, h ∈ G are in I, Then, for any w, w ′ ∈ W n and 1-cells Proof. By the proof of [BGM19, Theorem 3.1], there exists a G-biequivalence F from C to a 2category D with a strict action γ D (see [BGM19, Definition 2.2]) such that every equivalence 1-cell is indeed an isomorphism. We regard F as data (F,η F ,Π F ,M F ) as above. We define 2-cells C v (λi)i for v ∈ W n and 1-cells (λ i ) n i=1 recursively by the following rules: , which uniquely determines f . It is enough to prove this when f is a generator of I. The statement for a C , l C , r C , J γ C (g) and φ γ C (g) follows since F is a pseudofunctor andη F g is a pseudonatural transformation for any g ∈ G, which is standard. The statement for (χ γ C g,h ) λ and (ι γ C ) λ follows sinceΠ F g,h andM F are modifications. The statement for ω γ C g,h,k , κ γ C g and ζ γ C g follows from Figures 6, 7 and 8.
Thanks to this theorem, we may hereafter suppress the 2-cells in I in this article. Finally, we return to the case of monoidal categories.
Definition 2.7. Let (C, γ C ) and (D, γ D ) be monoidal categories with actions of a group G. A monoidal G-equivalence is a G-biequivalence with monoidal natural isomorphisms η F g : Ad(F ) • γ C (g) ∼ = γ D (g) (see [EGNO15,Definition 2.4.8]) and identical Π F and M F .
As we have already seen, we can give an equivalent definition of a monoidal G-equivalence with η F , whereΠ F andM F are identical. Namely, we can define a G-biequivalence to be the pair (F,η F ) withη F gh = (id * η F h ) • (η F g * id) for g, h ∈ G andη F e = id, where we suppressed some natural Figure 10: Construction of M F fromM F isomorphisms by Theorem 2.6. Since these equations do not include an adjoint inverse F −1 , we can give the following definition, which recovers that in [Gal17, Section 3.1] for unital actions.
Definition 2.8. Let (C, γ C ) and (D, γ D ) be monoidal categories with actions of a group G. A monoidal G-functor is the pair (F,η F ) of a monoidal functor F and a familyη F of monoidal isomorphismsη F g : By definition, a monoidal G-functor is a monoidal G-equivalence if and only if it is an equivalence as a functor. One can also define the notion of a G-pseudofunctor, recovering [BGM19, Definition 2.3], with Figures 6, 7 and 8 by replacing F by a general pseudofunctor, which is not necessary for this article.

Equivariantly braided tensor categories
In this subsection, we recall the notions of a G-crossed multitensor category and a G-braided multitensor category.
Our terminologies on tensor category theory follow those in [EGNO15]. In particular, a multitensor category over a field k is a locally finite k-linear abelian rigid monoidal category with a bilinear monoidal product as in [EGNO15, Definition 4.1.1]. A tensor functor is an exact faithful k-linear monoidal functor as in [EGNO15, Definition 4.2.5].
Remark 2.9. Although tensor categories that arise in algebraic quantum field theory do not have zero objects as remarked in [Müg05, Remark 2.11], because we are interested only in their rigid subcategories, which are semisimple, we regard such a subcategory as an abelian category by adding a zero object. Note that a linear functor on such a category is automatically exact.
When C is moreover a multitensor category, an action of G is a monoidal functor from G to the monoidal category of the tensor autoequivalences on C. When C is moreover pivotal (see e.g. [EGNO15, Definition 4.7.8]), a pivotal action of G is a monoidal functor from G to the monoidal category of pivotal tensor autoequivalences on C. Recall that a monoidal functor F : C → D is pivotal when δ D F = F δ C , where δ C denotes the pivotal structure of C.
First, we recall the notion of a grading on a multitensor category. Let {C i } i∈I be a family of additive categories. Its direct sum C = i∈I C i is an additive category with a family of additive functors {I i : C i → C} i∈I which has the universal property that for any additive category D and a family of additive functors {F i : C i → D} i∈I there exists an additive functor F : C → D with a family of natural isomorphisms σ F = {σ F,i : F I i ∼ = F i } i∈I such that for any other such pair (F ,σ F ) and a family of natural transformations {τ i : F i →F i } i∈I we have a unique natural transformation τ : F →F with τ i • σ F,i = σF ,i • τ I i for any i ∈ I. Such a category C always exists: indeed, we can explicitly give C as the category of families of objects whose all but finite components are zero, see e.g. [EGNO15, Section 1.3]. The category C is unique up to a unique equivalence by universality. We can indeed take σ F,i 's to be identities, and in this case, we refer to the natural transformation τ obtained by universality as the extension of the original family of natural transformations {τ i } i∈I .
We can equivalently define the direct sum C = i∈I C i to be an additive category which has C i 's as its subcategories such that every object is a direct sum of objects of C i 's and Hom C (λ, µ) = {0} for λ ∈ Obj(C i ) and µ ∈ Obj(C j ) with i ̸ = j. Indeed, the explicit construction above satisfies this property, and conversely by decomposing the objects of C we obtain an additive equivalence from C to i∈I C i .
The objects in the set Homog(C) := i∈I Obj(C i ) are called homogeneous. We write ∂ C λ = i or simply ∂λ = i if λ ∈ Obj(C i ). Since Obj(C i ) ∩ Obj(C j ) consists of the zero objects of C for i ̸ = j, ∂ is single-valued on nonzero objects.
By definition, we can always decompose an object into a direct sum of homogeneous objects. The following lemma shows that such a decomposition, which we call a homogeneous decomposition, is essentially unique.
Lemma 2.10. Let λ be an object of an I-graded multitensor category and let λ = i∈F λ i = j∈F ′ λ ′ j be decompositions in two ways with finite subsets F and F ′ of I. Then, for every where ι i and ι ′ i denote the morphisms that embeds respectively λ i and λ ′ i into λ) by a unique subobject isomorphism. Moreover, λ i ∼ = 0 for every i ∈ F \ F ′ and λ ′ j ∼ = 0 for every j ∈ F ′ \ F .
Proof. Let p i and p ′ i denote the projections of the decompositions λ = i λ i and λ = j λ ′ j respectively, and let ι i and ι ′ j denote the corresponding inclusions. Since Definition 2.11. Let G be a group. A G-grading on a multitensor category C is its decomposition into a direct sum C = g∈G C g of a family {C g } g∈G of abelian full subcategories of C with Obj(C g )⊗ Obj(C h ) ⊂ Obj(C gh ) for any g, h ∈ G. A multitensor category with a G-grading is called a Ggraded multitensor category. A G-graded tensor functor between G-graded multitensor categories is a tensor functor F : C → D with F (Obj(C g )) ⊂ Obj(D g ) for every g ∈ G.
A G-crossed multitensor category is the pair C = (C, γ C ) of a pivotal G-graded multitensor category C and a pivotal action γ C of G on C such that γ C (g)(Obj(C h )) ⊂ Obj(C ghg −1 ) for g, h ∈ G. A G-crossed functor between G-crossed multitensor categories is a tensor G-functor (i.e. a monoidal G-functor which is a tensor functor as a monoidal functor, see Definition 2.8) which is G-graded and pivotal. A G-crossed functor is a G-crossed equivalence if it is an equivalence as a functor.
A G-braided multitensor category is the pair C = (C, b C ) of a G-crossed multitensor category C and a family b C of isomorphisms b C λ,µ : λµ ∼ = g µλ for λ ∈ Obj(C g ), µ ∈ Obj(C) and g ∈ G which is natural in λ and µ such that , λ 2 ∈ Obj(C g2 ), µ, µ 1 , µ 2 ∈ Obj(C) and g, h, g 1 , g 2 ∈ G, where we suppressed some isomorphisms by Theorem 2.6. We call b C the G-braiding of C. A G-braided functor between G-braided multitensor categories C and D is a G-crossed functor F : For a G-braided multitensor category C, we can define a family of isomorphisms b C− by putting b C− µ,λ := b C−1 λ,µ : g µλ ∼ = λµ for λ ∈ Obj(C g ), µ ∈ Obj(C) and g ∈ G. We call b C− the reverse of b C .
Note that we automatically have 1 C ∈ Obj(C e ) for a G-graded multitensor category C. We also have λ ∨ , ∨ λ ∈ Obj(C g −1 ) for λ ∈ Obj(C g ). The objects in C e are said to be neutral.
A multitensor category that arises in algebraic quantum field theory is indeed a * -multitensor category i.e. a multitensor category over C with a contravariant strict monoidal antilinear involution endofunctor * that is the identity on objects and has positivity, see [Müg03,Section 2.4]. Note that we assume rigidity in our definition. For an action γ of a group G on a * -multitensor category, we always assume that γ(g) commutes with the * -involution for any g ∈ G. For a G-braided * -multitensor category, we always assume that the components of the G-braiding are unitary.
Here, we give our motivating example of a G-braided ( * -)tensor category. Let A be an irreducible Möbius covariant net (on S 1 ) with Haag duality on R, see e.g. [ where U denotes the group of unitary operators, H A denotes the Hilbert space of A, I denotes the set of intervals in S 1 , and Ω A denotes the vacuum of A. We do not consider topologies. A group Remark 2.13. For any I ∈ I, the adjoint action Ad : Aut(A) → Aut(A(I)) is injective by Reeh-Schlieder property.
The following notions were defined in [Müg05, Definition 2.8].
Definition 2.14. Let A be a Möbius covariant net on S 1 with an action β of a group G. For g ∈ G, an endomorphism λ of , where I ′ L (resp. I ′ R ) denotes the left (resp. right) connected component of the open complement I ′ of I. A g-localized endomorphism λ is a g-twisted DHR endomorphism of A if for anyĨ ∈ I R , there exists a unitary u ∈ A ∞ such that Ad u • λ is localized inĨ. The * -category of rigid g-twisted DHR endomorphisms is denoted by g-Rep A, and an object of the * -tensor category G- Remark 2.15. In [Müg05, Definition 2.8], the category G-Rep A is defined to be the category generated by g-Rep A's in End A ∞ , and the faithfulness of an action β is assumed to assure that g-Rep A's are mutually disjoint. Here, we have defined G-Rep A as a direct sum category from the beginning, and therefore the faithfulness of an action β is not needed.
An action β of G on A induces the adjoint action on the * -tensor category G-Rep A: g λ := Ad β(g) • λ • Ad β(g −1 ) for λ ∈ G-Rep A and g ∈ G, which makes G-Rep A into a G-crossed * -tensor category. If moreover A satisfies Haag duality on R, the category G-Rep A turns into a G-braided * -tensor category by [Müg05, Theorem 2.21], see also [Noj20, Section 3]. The argument recovers the classical DHR theory when G is trivial. We go back to general theory. We collect here some graphical calculi for G-braided multitensor categories. Our graphical notation follows that in [BEK99] i.e. a morphism in a monoidal category is represented as an arrow from top to bottom. Now, let C be a G-braided multitensor category. We do not draw the isomorphisms in Theorem 2.6. For λ ∈ Homog(C) and µ ∈ Obj(C), the component b C λ,µ of the G-braiding is represented by the crossing in Figure 11. The component b C− µ,λ of the reverse is represented by the reverse crossing in Figure 12. The naturality of b C is represented in Figure 13, where we may assume ∂λ = ∂λ ′ since otherwise the morphisms in the equation are zero. We also have a similar representation for b C− . The first axiom of a G-braiding says that h b C λ,µ and b C h λ, h µ are represented by the same diagram. The second and third axioms for b C are represented in Figure 14. Figure 13: The naturality of a G-braiding Figure 14: Axioms for the G-braiding b C Since coevg µ and g coev µ (resp. evg µ and g ev µ , and similar for right duals) are represented by the same diagram for µ ∈ Obj(C) and g ∈ G (see e.g. [EGNO15, Exercise 2.10.6]), for example, we can perform the calculation in Figure 15 by Figures 13 and 14. Lemma 2.16. Let C be a G-braided multitensor category. Let λ ∈ Homog(C) and let µ ∈ Obj(C). Then, the equations in Figure 16 hold. Proof. By the graphical calculation in Figure 17, the leftmost diagram in Figure 16 is the right inverse of b C− µ,λ . One can similarly show that it is also the left inverse and therefore the left equality in Figure 16. The proof of the right equality is similar. Let C be a G-braided multitensor category. For any λ, µ ∈ Obj(C) and g ∈ G, λ not necessarily homogeneous, let a dashed crossing labeled by g from λµ to g µλ denote the morphism in Figure  18, where i g and p g respectively denote the inclusion and projection of λ g in a homogeneous decomposition λ = g λ g . Then, this morphism does not depend on the choice of i g and p g by Lemma 2.10 and by the naturality of b C . Note also that a dashed crossing is natural. Finally, we consider the notion of the G-equivariant version of a ribbon structure. Let C be a G-braided multitensor category.
For a G-ribbon multitensor category C and λ ∈ Homog(C), we can do the graphical calculations in Figure 19, where we identify the left dual λ ∨ and right dual ∨ λ of λ using the pivotal structure δ C . Indeed, the equalities follow from (θ C Figure 19: A Reidemeister move for a framed tangle

Equivariant Frobenius algebras
In this subsection, we recall the notion of a G-equivariant Frobenius algebra.
A Frobenius algebra in a monoidal category is an algebra object with a compatible coalgebra structure, see e.g. [FFRS06, Section 2.3] for the precise definition and more terminologies. The product, unit, coproduct, and counit of a Frobenius algebra A are denoted by m A , η A , ∆ A , and ε A respectively.
For a pivotal multitensor category C, we say a Frobenius algebra In this article, we have to consider equivariant Frobenius algebras. To define this notion, note that if A is an algebra (resp. Frobenius algebra) in a monoidal category with an action of a group G, ) is again an algebra (resp. Frobenius algebra).
Definition 2.18. Let C be a monoidal category with an action of a group G.
Similarly, a G-equivariant Frobenius algebra in C is a pair A = (A, z A ) of a Frobenius algebra A in C and a G-equivariant structure z A on A such that z A g is a Frobenius algebra isomorphism for every g ∈ G. A G-equivariant Q-system in a * -multitensor category is a G-equivariant Frobenius algebra A such that A is a Q-system and z A g is unitary for every g ∈ G. A homomorphism between G-equivariant (Frobenius) algebras is a (Frobenius) algebra homomorphism that is G-equivariant. By an isomorphism between G-equivariant Q-systems, we always mean a unitary isomorphism.
Example 2.19. It is well-known [Lon94] (see also [BKLR15,Theorem 3.11]) that a finite index extension of a type III factor N gives a Q-system in the * -tensor category End 0 (N ) of finite index unital * -endomorphisms of N (see e.g. [BKLR15, Chapter 2]) up to isomorphisms and vice versa. Indeed, when a group G acts on N , we have the adjoint action of G on End(N ), and a finite index extension of N with an extension of the action of G corresponds to a G-equivariant Q-system in End 0 (N ), see [Noj20, Section 2.4].
Here, we give examples coming from algebraic quantum field theory.
I is of finite index for any I ∈ J . An isomorphism between standard extensions A ⊂ B 1 and A ⊂ B 2 of A is a unitary u : Let A be an irreducible local Möbius covariant net. Then, the restriction of A on R, which is again denoted by A, is a family of type III factors indexed by the directed set of intervals in R for which the vacuum Ω A is a common cyclic separating vector. Moreover, for an inclusion A ⊂ B of irreducible local Möbius covariant nets, there exists a unique family of faithful normal conditional expectations that makes A ⊂ B into a standard extension of A by Bisognano-Wichmann property and Takesaki's theorem [Tak72].
By [LR95,Theorem 4.9], a finite index standard extension of A bijectively corresponds to a Q-system in Rep A up to isomorphisms. If moreover A satisfies Haag duality on R, then a finite index local standard extension of A (i.e. a finite index standard extension B of A such that B(I) and B(J) commute if I ∩ J = ∅) bijectively corresponds to a commutative Q-system in Rep A up to isomorphisms.
Remark 2.21. We consider not only local extensions but also nonlocal extensions i.e. B is not necessarily a local Möbius covariant net in the example above, which is crucial for the theory of α-induction, see [BEK00, Section 5]. Note also that we do not assume even the relative locality of B with respect to A (i.e. A(I) and B(J) commute if I ∩ J = ∅), which is indeed automatic by [LR95, Theorem 4.9].
Definition 2.22. Let A be an irreducible local Möbius covariant net on S 1 . Suppose we have an action β A : G → Aut(A) of a group G on A. For a standard extension A ⊂ B, let Aut(B) denote the group which consists of the unitaries preserving the local algebras by adjoint action and the vacuum as that for the Möbius covariant nets. We say an action β B : G → Aut(B) of G on B extends the action β A on A if Ad β B (g)(a) = Ad β A (g)(a) for any interval I in R, a ∈ A(I) and g ∈ G, see [BJLP19, Definition 6.2]. We refer to the pair A ⊂ B = (A ⊂ B, β B ) of a standard extension and an extension of β A as a G-equivariant standard extension of (A, β A ). An isomorphism between two G-equivariant extensions A ⊂ B 1 and A ⊂ B 2 is a unitary u between the Hilbert spaces H B1 and H B2 on which B 1 and B 2 act respectively such that u is an isomorphism of standard extensions and Ad u • β B1 = β B2 .
1. When B is the restriction of a local Möbius covariant net with strong additivity, the local algebras on S 1 are generated by the local algebras on R and therefore Aut(B) defined above coincides with that of a local Möbius covariant net.
2. If Ad β B (g)(a) = Ad β A (g)(a) for any interval I and a ∈ A(I), then Ad β B (g)(A(I)) ⊂ A(I) and therefore β B (g) commutes with the Jones projection e A of A ⊂ B. Hence indeed Similarly, for an isomorphism u between G-equivariant extensions, we have Ad u • E B1 I • Ad u automatically for any interval I. Proposition 2.24. Let A be an irreducible local Möbius covariant net on S 1 with Haag duality on R. Then, a finite index G-equivariant standard extension A ⊂ B bijectively corresponds to a G-equivariant Q-system in Rep A up to isomorphisms.
Proof. Suppose a finite index G-equivariant standard extension A ⊂ B is given. Then, a corresponding Q-system θ is localized in some interval I and can be restricted to a dual canonical endomorphism of A(J) ⊂ B(J) for any interval J including I. By applying an argument in [Noj20, Lemma 2.8] for the inclusion A(I) ⊂ B(I), we can give z g ∈ Hom End(A(I)) ( g θ| A(I) , θ| A(I) ) for every g ∈ G, which is what we want since the argument for A(J) ⊂ B(J) gives the same morphism z g and therefore z g ∈ Hom End A∞ ( g θ, θ). If we change θ for an isomorphic one, then we only get isomorphic G-equivariant Q-system by [Noj20, Lemma 2.8]. If we change B for an isomorphicB with a unitary u, then U := Ad u • − • Ad u * is a strict 2-functor between the 2-categories of morphisms of A(I) ⊂ B(I) and A(I) ⊂B(I), see e.g. [Müg03, Section 1.3] for the 2-category of morphisms. Since u and therefore U intertwine group actions, U (θ) = θ, which follows since u commutes with A ∞ , is a G-equivariant Q-system corresponding toB. Thus, changing B does not affect the resulting G-equivariant Q-systems.
Conversely, suppose we are given a G-equivariant Q-system (θ, w, x, z) in Rep A that is localized in I. Then, w, x, z g ∈ A(I) by the Haag duality assumption on R of A, and we can construct a finite index standard extension ι : A ⊂ B on the GNS Hilbert space L 2 (B(I)) of B(I) associated with the vacuum state, see the proof of [LR95, Theorem 4.9]. By [Noj20, Lemma 2.10], we have an extension β B I : G → Aut(B(I)) of an action Ad β A | A(I) : G → Aut(A(I)). Then, for every g ∈ G, define a linear operator β B (g) on L 2 (B(I)) by putting for a ∈ A(I), by definition. We show that β B (g) ∈ Aut(B) and therefore β B is an extension of β. First, β B (g) is unitary since it is an automorphism that preserves the vacuum. Next, we check that Ad β B (g)(B(J)) ⊂ B(J) for an arbitrary interval J. For this, we show Ad β B (g)(ã) = Ad β A (g)(ã) forã ∈ A ∞ . Let K be an interval containingã. By construction, we have an element v ∈ B(I) with L 2 (B(I)) = v * L 2 (A(I)) as a representation of A(K), see the proof of [LR95, Theorem 4.9]. We also have g for every a ∈ A(I) and therefore gã : is used in the third equality and z e = e z e = id θ in the penultimate equality. Hence in particular g A(J) ⊂ A(J). Then, recall that we have a Q-system isomorphism u : θ ∼ =θ, wherẽ θ is localized in J, with B(J) = A(J)uv by construction. We can make u into a G-equivariant Q-system isomorphism by putting zθ g := uz g g u * . Then, zθ g ∈ A(J) by Haag duality and therefore Thus β B is an extension of β A . If we change θ for an isomorphicθ localized in an interval J with a unitary u then we construct A ⊂B on L 2 (B(J)). We have the counterpartṽ ∈B(J) of v and U : L 2 (B(J)) → L 2 (B(I)); [aṽ] → au[v] (a ∈ A(J)) defines an isomorphism of standard extensions, see the proof of [LR95, Theorem 4.9]. Moreover, U intertwines group actions and therefore is an isomorphism of G-equivariant standard extensions since for any g ∈ G and a ∈ A(J). Thus, changing θ only yields an isomorphic G-equivariant extension. Note that in particular, we can harmlessly replace an interval in which a given G-equivariant Qsystem is localized by another (say larger) one. It is easy to see that our constructions are mutually inverse.
Remark 2.25. In the proposition above, the action β A does not need to be faithful. That is, β A can have a nontrivial kernel. We encounter such a situation e.g. when A = B G for a Möbius covariant net B.
See also a result [BJLP19, Proposition 6.3] for finite group actions on completely rational conformal nets. Note that our proof does not need either the finiteness of a group or the locality of a net.
We go back to general theory. We collect here some graphical calculi for equivariant algebras. We follow the graphical notations for algebras and Frobenius algebras in [FFRS06,Equation 2.22]. Namely, the product and unit of an algebra are represented respectively by a fork and a small circle.
Let A be a G-equivariant algebra. Then, it is graphically represented in Figure 20 that z A g is an algebra homomorphism for g ∈ G. The first equality in Figure 20 is often used in the form of Figure 21. When A is moreover a G-equivariant Frobenius algebra, we have similar representations of the coassociativity and counit property of A. It is often used in the form of Figure 22 that z A is a G-equivariant structure. In particular, by putting h = g −1 , we obtain Figure 23 since Finally, note that when A is an algebra in a G-braided multitensor category, we can move the algebra structures through crossings as in Figure 24 by the definition of λ A.

Equivariant α-induction
The α-induction for twisted representations of Möbius covariant nets (see Example 2.14) was introduced by Nojima in [Noj20]. In this section, we reformulate this notion in terms of tensor categories (Definition 3.9 and Remark 3.7), which is an equivariant generalization of Ostrik's work [Ost03, Section 5.1], see also [BKLR15,Section 4.6]. For this, we begin with crossed structures on bimodule categories.

Group actions on the bicategories of equivariant Frobenius algebras
In this subsection, we induce a group action on the bicategory of equivariant Frobenius algebras from that on the ambient multitensor category (Proposition 3.1). We also see that this indeed makes bimodule categories G-crossed (Proposition 3.5).
It is known [Yam04, Sections 4 and 5] that the special Frobenius algebras in a multitensor category C form a rigid bicategory, whose 1-cells are bimodules as in ordinary ring theory. Note that this bicategory is defined only up to equivalences since we have to fix relative tensor products of bimodules to obtain a composition of 1-cells. In particular, for a special Frobenius algebra A in C, the category Bimod C (A) of A-bimodules in C is a multitensor category. A special Frobenius algebra A is called simple if Bimod C (A) is moreover a tensor category i.e. End Bimod C (A) (A) is one-dimensional, see [FFRS06,Definition 2.26].
Let C be a multitensor category with an action of G and let A and B be algebras in C. For an A-B-bimodule λ = (λ, m L λ , m R λ ) in C, g λ = ( g λ, g m L λ , g m R λ ) is an g A-g B-bimodule. Note that we can move module products through crossings as in Figure 25 for λ ∈ Homog(C) and an A-Bbimodule µ by this definition. We modify this procedure to obtain an action that restricts to the multitensor categories of bimodules.

Figure 25: Module products and crossings
Proposition 3.1. Let C be a multitensor category equipped with an action γ C of a group G. Let Frob G (C) denote the bicategory of special G-equivariant Frobenius algebras in C. Then, for a 1-cell λ = (λ, m L λ , m R λ ) : A → B in Frob G (C) and g ∈ G, we can give a 1-cell g λ : This assignment naturally defines an action of G on Frob G (C). Another choice of relative tensor products only yields a G-biequivalent action.
Proof. The assignment γ(g) : λ → g λ defined above is well-defined since the graphical calculations in Figures 26 and 27, where z := z B and small half circles denote module products as in [BKLR15, Section 3.6], shows that g λ is indeed a left B-module, and the proof for right A-modularity is similar. We regard γ(g) as a family of functors by putting γ(g)(f ) := γ C (g)(f ) for a 2-cell f . We make γ(g) into a pseudofunctor. For 1-cells λ : A → B and µ : B → C, let (µ⊗ B λ, s µ,λ , r µ,λ ) denote a retract of µ ⊗ C λ with the idempotent e µ,λ := s µ,λ • r µ,λ . Note that s µ,λ and r µ,λ are natural in µ and λ by the definition of the bifunctor ⊗ B . By the graphical calculation in Figure  28, we have eg µ, g λ = g e µ,λ and therefore a unique subobject isomorphism J γ(g) µ,λ := g (r µ,λ ) • sg µ, g λ : . We can show that the bicategory version of [EGNO15, Diagram 2.23] commutes by eg µ, g λ = g e µ,λ and the naturality of s and r. We also put φ γ(g) A := (z A g ) −1 for A ∈ Obj(Frob G (C)), which is an A-bimodule morphism since z A g is an algebra homomorphism. Then, by the definition of the left and right unit isomorphisms of Frob G (C), the commutativity of the bicategory versions of [EGNO15, Diagrams 2.25 and 2.26] follows from the definition of γ and the naturality of s. Thus, (γ(g), J γ(g) , φ γ(g) ) is a pseudofunctor.
g B Figure 28: eg µ, g λ = g e µ,λ Since z gh = z g g z h , we have a canonical invertible 2-cell (χ γ g,h ) λ : g ( h λ) ∼ = gh λ by the coherence in C, which gives a pseudonatural equivalence χ γ g,h : γ(g)γ(h) ≃ γ(gh) with χ γ,0 g,h := 1. Similarly, z e = id B gives a pseudonatural equivalence ι γ : id ≃ γ(e) with ι γ,0 = 1. We can define all the remaining modifications to be canonical invertible 2-cells, and then γ is an action of G by coherence. Another choice of relative tensor products only yields a G-biequivalent action since we get the same χ γ and ι γ .
It is known [BKLR15, Proposition 3.33] that for a type III factor N , there is a biequivalence between the bicategory of Q-systems in End 0 (N ) and the 2-category of finite index extensions of N whose 1-cells from M 1 to M 2 are the subobjects of ι 2 λι 1 for λ ∈ End 0 (N ), where ι i : N ⊂ M i for i = 1, 2. When a group G acts on N , in the same way, we have a biequivalence between the bicategory Q G (End 0 (N )) of the G-equivariant Q-systems in End 0 (N ) and the 2-category Ext G (N ) of finite index extensions of N with G-actions whose 1-cells are as above.
Proposition 3.2. This biequivalence can be made into a G-biequivalence, where G acts on Q G (End 0 (N )) by Proposition 3.1 and on Ext G (N ) by the adjoint action.
By definition, the restriction of the action defined in Proposition 3.1 to C coincides with γ C , and that to Bimod C (A) for A ∈ Obj(Frob G (C)) is an action on a multitensor category. We show that Bimod C (A) is indeed G-crossed (when A is symmetric and neutral, see Propositions 3.3 and 3.5).
Proposition 3.3. Let A be a symmetric special Frobenius algebra in a pivotal multitensor category C. Then, Bimod C (A) is again pivotal. Another choice of relative tensor products only yields an isomorphic pivotal structure. If C is moreover a tensor category and A is simple, then Tr L (δ , and if C is spherical, then Bimod C (A) is again spherical. If a group G acts pivotally on C (not necessarily tensor) and A ∈ Obj(Frob G (C)), then the induced action of G on Bimod C (A) is again pivotal.
Proof. Let λ ∈ Obj(Bimod C (A)) and let m λ denote either the left or right module product of λ. By the proof of the rigidity of Bimod C (A) in [Yam04, Section 5], we have m λ ∨∨ = (m λ ) ∨∨ in C when A ∨ is taken to be A. Therefore, δ C λ is an A-bimodule morphism by the naturality and monoidality of δ C if δ C A = id A . Let us put δ Bimod C (A) λ := δ C λ , which is natural in λ by definition. The monoidality of δ Bimod C (A) follows from the naturality and monoidality of δ C . Thus, δ Bimod C (A) is a pivotal structure on Bimod C (A). Another choice of relative tensor products only yields an isomorphic pivotal structure since an equivalence from the new bimodule category to the original one is defined to be an identity as a functor. Now, suppose C is a tensor category and A is simple. By the proof of rigidity [Yam04, Section 5], evaluation and coevaluation maps of λ ∈ Obj(Bimod C (A)) are respectively given in Figures 29  and 30 as morphisms in C. Then, we find that e λ,λ ∨ , the idempotent for the subobject λ ⊗ A λ ∨ of λ ⊗ C λ ∨ (see the proof of Proposition 3.1), in Tr L (δ by the definition of the left module product of λ ∨ , see [Yam04, Section 5]. Therefore, we have The proof for Tr R is similar. Finally, the last statement follows since the group action on the morphisms of Bimod C (A) coincides with that of C. Proposition 3.5. Let C be a G-crossed multitensor category and let A ∈ Obj(Frob G (C)) be symmetric neutral (recall that neutral means A ∈ Obj(C e ), see the sentence right after Definition 2.11). Then, we can define a G-grading on Bimod C (A) by putting Obj(Bimod where F denotes the forgetful functor Bimod C (A) → C. Combined with the action defined in Proposition 3.1, Bimod C (A) becomes a Gcrossed multitensor category. Another choice of relative tensor products only yields an isomorphic G-crossed structure.
Proof. By Propositions 3.1 and 3.3, the category Bimod C (A) is equipped with a pivotal G-action. By the definition of the G-grading, the homogeneous decomposition in C gives that in Bimod C (A). For λ ∈ Obj(Bimod C (A) g ) and µ ∈ Obj(Bimod C (A) h ), since A is neutral, λ ⊗ A µ is the cokernel of a morphism λ ⊗ C A ⊗ C µ → λ ⊗ C µ in C gh , and therefore λ ⊗ A µ ∈ Obj(C gh ). Finally, k (Obj(Bimod C (A) g )) ⊂ Obj(Bimod C (A) kgk −1 ) for k ∈ G since C is G-crossed. Thus, Bimod C (A) is G-crossed. Another choice of relative tensor products only yields an isomorphic G-crossed structure as in the proof of Proposition 3.3.
Proposition 3.6. Let A be a neutral G-equivariant algebra in a G-braided multitensor category C. Put m L Aλ := m A ⊗id λ ∈ Hom(AAλ, Aλ) and define m R± Aλ ∈ Hom(AλA, Aλ) to be the morphisms given in Figure 31 for every λ ∈ Obj(C). Then α G± A (λ) := (Aλ, m L Aλ , m R± Aλ ) are A-bimodules. Proof. We only show the statement for α G+ A (λ) because the proof for α G− A is similar. m L Aλ and m R+ Aλ are denoted respectively by m L and m R in this proof. First, (Aλ, m L ) are left A-modules since A is an algebra. Then, the right A-modularity of (Aλ, m R ) follows from the graphical calculation in Figure 32. Note that we have by definition two dashed crossings labeled by, say, g, h ∈ G, but only the components labeled by g = h survive. Note also that we used Figure 24 at the second equation. The right unit property can also be checked easily by a graphical calculation, and therefore (Aλ, m R ) is a right A-module. Finally, the bimodularity of α G+ A (λ) follows from the associativity of A.
Remark 3.7. When A is a neutral G-equivariant Q-system in a G-braided * -multitensor category, the bimodules α G± A (λ) defined above are standard. In particular, when A ∈ Rep A and is localized in an interval I for a local Möbius covariant net A with Haag duality on R and a group G acting on A, for every λ ∈ Obj(G-Rep A) we can regard α G± A (λ) as standard A-bimodules in End A(I). Therefore, they correspond to objects in End B(I), where B is the extension of A corresponding to A, by [BKLR15, Proposition 3.32(ii)] via the formula (3.6.3) in [BKLR15, Section 3.6], which gives the equations (4.7) in [Noj20, Section 4.3]. Thus in this case our definition coincides with what is considered in [Noj20].
Note that here we do not assume that the action of G on A is faithful, see Remarks 2.15 and 2.25. Such a situation was already considered in [Noj20]. Namely, a group G acts on B, and the restriction of this action on A can have a nontrivial kernel, by which the quotient of G is denoted by G ′ . For g 1 , g 2 ∈ G with p(g 1 ) = p(g 2 ) = g ′ , where p : G → G ′ is the quotient map, we have two induced homomorphisms α g1;+ (λ) and α g2;+ (λ) for λ ∈ g ′ -Rep A. In our framework, they are just the induced homomorphisms of λ ∈ g 1 -Rep A and λ ∈ g 2 -Rep A. Thus, the framework in [Noj20] is included in ours.
We regard α G± A as functors by putting α G± A := id A ⊗ − on morphisms. Proposition 3.8. Let A be a neutral special symmetric G-equivariant Frobenius algebra in a G-braided multitensor category C. Note that in this setting Bimod C (A) is a G-crossed multitensor category by Proposition 3.5. Then, α G± A : C → Bimod C (A) can be regarded as G-crossed tensor functors.

Equivariant α-induction Frobenius algebras
In this section, we construct equivariant α-induction Frobenius algebras (Theorem 4.15), which generalizes Rehren's work [Reh00] (see also [BKL15,Definition 4.17]) and is one of our main theorems. To give their ambient categories, we introduce the neutral double construction (Definition 4.7) for G-braided multitensor categories.

The neutral double of a G-braided tensor category
In this subsection, we introduce the neutral double construction (Definition 4.7). For this, we begin with crossed products.
Proposition 4.1. Let G be a group. Let C be a G-graded multitensor category and let D be a multitensor category with an action γ D of G. Then, we can define a bilinear bifunctor ⊗ : (C ⊠ D) × (C ⊠ D) → C ⊠ D by putting (c 1 ⊠ d 1 ) ⊗ (c 2 ⊠ d 2 ) := c 1 c 2 ⊠ c2 d 1 d 2 for c 1 ∈ Obj(C), c 2 ∈ Homog(C) and d 1 , d 2 ∈ Obj(D) and make C ⊠ D into a multitensor category, which is denoted by C ⋉ γ D D or simply C ⋉ D. Another choice of the Deligne tensor product C ⊠ D only yields a strictly isomorphic multitensor category. If C and D are tensor (resp. multifusion, resp. fusion) categories, then so is C ⋉ D.
Proof. Since C × D × C × D has a G-grading g∈G C × D × C g × D and ⊗ is linear exact in each variable (see [EGNO15, Proposition 4.2.1]), ⊗ can be extended to a linear exact functor on C × D × C × D and therefore on (C ⊠ D) × (C ⊠ D). Since for c 1 ∈ Obj(C), c 2 , c 3 ∈ Homog(C) and d 1 , d 2 , d 3 ∈ Obj(D), we can define a natural isomorphism a : (− ⊗ −) ⊗ − ∼ = − ⊗ (− ⊗ −) to be the extension of the canonical isomorphism given by coherence (Theorem 2.6), which satisfies the pentagon axiom by coherence. We put 1 := 1 C ⊠1 D and similarly define left and right unit isomorphisms by coherence. Thus, C ⋉ D turns into a monoidal category. Another choice of C ⊠ D only yields a strictly isomorphic multitensor category by universality since we define the tensor product by extension.
Next, we show that C ⋉ D is rigid and therefore a multitensor category. As in the proof of [Del90, Proposition 5.17], it is enough to check the rigidity of the objects of C g × D for g ∈ G. We show that a left dual of c ⊠ d for c ∈ Homog(C) and d ∈ Obj(D) is given by where we suppressed some isomorphisms by Theorem 2.6, we can define evaluation and coevaluation maps by putting ev c⊠d := ev c ⊠ ev d and coev := coev c ⊠ c coev d . The conjugate equations follow from those for c and d and therefore (c ⊠ d) ∨ = c ∨ ⊠ c d ∨ . We can also show that ∨ (c ⊠ d) = ∨ c ⊠ c∨ d and therefore C ⋉ D is rigid.
We call C ⋉ D the crossed product of C and D. Let us see that C ⋉ D inherits structures on C and D.
Proposition 4.2. Let G be a group, let C be a G-graded pivotal multitensor category, and let D be a pivotal multitensor category with a pivotal action of G. Then, the pivotal structures on C and D induce a pivotal structure δ C⋉D on C ⋉ D. Another choice of C ⊠ D only yields an isomorphic pivotal structure.
Proof. Since (c ⊠ d) ∨∨ = c ∨∨ ⊠ d ∨∨ for c ∈ Obj(C) and d ∈ Obj(D) by the proof of Proposition 4.1, we can define a natural isomorphism δ C⋉D : id ∼ = (−) ∨∨ by putting δ C⋉D for c 1 ∈ Obj(C), c 2 ∈ Homog(C) and d 1 , d 2 ∈ Obj(D) by the monoidality of δ C and δ D . Thus, δ C⋉D is monoidal and therefore a pivotal structure on C ⋉ D. Another choice of C ⊠ D only yields an isomorphic pivotal structure since we define the pivotal structure by extension. Proposition 4.4. Let C be a G-crossed multitensor category and let D be a multitensor category with an action γ D of G. Then, an action of G is naturally induced on C ⋉ D. Another choice of C ⊠ D only yields a G-tensor isomorphic one. When D is moreover pivotal and γ C and γ D are pivotal actions, the induced action on C ⋉ D is again pivotal.
Proof. Define a linear exact functor γ(g) for every g ∈ G by putting γ(g)(c ⊠ d) := g c ⊠ g d for c ∈ Obj(C) and d ∈ Obj(D). Since for c 1 ∈ Obj(C), c 2 ∈ Homog(C) and d 1 , d 2 ∈ Obj(D) by ∂ g c 2 = g∂c 2 g −1 , we can define a canonical natural isomorphism J γ(g) by coherence. We similarly have canonical natural isomorphisms χ γ g,h for g, h ∈ G and ι γ by coherence. Thus, γ is an action of G. Another choice of C ⊠ D only yields a G-tensor isomorphic action since we define the action by extension. The last statement follows from the definitions of γ and the pivotal structure δ C⋉D , see the proof of Proposition 4.2.
Proposition 4.5. Let C and D be split semisimple G-crossed multitensor categories. Then, a G-crossed structure is naturally induced on C ⋉ D. Another choice of C ⊠ D only yields a G-crossed isomorphic one.
Proof. We already know that C ⋉ D is a pivotal category with a pivotal action of G by Proposition 4.4. Since C ⊠ D = g h C h ⊗ D h −1 g follows from [LF13, Theorem 27] by the assumption of split semisimplicity, we put (C ⋉ D) g := h C h ⊠ D h −1 g for g ∈ G. Then, we have (C ⋉ D) g ⊗ (C ⋉ D) h ⊂ (C ⋉ D) gh for g, h ∈ G since ∂((c 1 ⊠ d 1 )(c 2 ⊠ d 2 )) = ∂(c 1 c 2 ⊠ c2 d 1 d 2 ) = ∂c 1 ∂d 1 ∂c 2 ∂d 2 for c 1 , c 2 , d 1 , d 2 ∈ Homog(C) by definition. Thus, C ⋉ D is a G-graded multitensor category. Moreover, C ⋉ D is G-crossed since ∂ C⋉D ( g c ⊠ g d) = g∂cg −1 g∂dg −1 = g∂c∂dg −1 for c ∈ Homog(C), d ∈ Homog(D) and g ∈ G. Another choice of C ⊠ D only yields an isomorphic G-crossed structure since changing C ⊠ D preserves C h ⊠ D h −1 g and therefore yields an isomorphic G-grading.
Next, we show that if C and D are moreover G-braided, then we can obtain a G-braiding on a subcategory of C ⋉ D.
Theorem 4.6. For a G-braided multitensor category C, define C rev to be C as an abelian category with an action of G. Put C rev g := C g −1 for g ∈ G. Then, by putting λ ⊗ C rev µ := ∂ C µ λ ⊗ C µ for λ ∈ Obj(C rev ) and µ ∈ Homog(C rev ) = Homog(C), and putting 1 C rev := 1 C , we obtain a multitensor structure on C rev . Moreover, by putting b C rev λ,µ := b C− λ,µ (see Definition 2.11), we obtain a G-braiding on C rev . We call C rev the reverse of C.
Next, we show that C rev is rigid and therefore a multitensor category. For λ ∈ Homog(C rev ), put λ ∨ := λ λ ∨ C , where λ ∨ C denotes the left dual of λ in C. Then, ev λ := ev C λ : λ ∨ ⊗ C rev λ = λ ∨ C λ → 1 and coev λ : give desired duality, where ev C λ and coev C λ denote evaluation and coevaluation maps of λ in C. The proof of right duality is similar. Moreover, since λ ∨∨ = (λ ∨ C ) ∨ C , we can put δ C rev := δ C , which indeed defines a pivotal structure on C rev since the action of G on C is pivotal. The shows that C rev is a G-graded multitensor category and therefore a G-crossed category since ∂ C rev g µ = (g∂ C µg −1 ) −1 = g∂ C rev µg −1 for g ∈ G. Finally, it follows from the axioms for b C that b C rev is a G-braiding.
Definition 4.7. Let C and D be split semisimple G-crossed multitensor categories. Let D(C, D) denote the full multitensor subcategory of C ⊠ D rev with G-grading restricted through the diagonal embedding G ⊂ G × G. We regard D(C, D) as a G-braided multitensor category by restricting the G × G-braiding of C ⊠ D rev . We refer to the G-braided multitensor category D(C) := D(C, C) as the neutral double of C.
D is for Double. Note that D(C, D) = (C ⋉ D) e as a pivotal multitensor category with an action of G. Note also that when G is trivial, D(C) is equal to C ⊠ C rev as an (ordinary) braided multitensor category, where C rev is the reverse of the (ordinary) braided multitensor category C.
Finally, we give an application of these constructions to algebraic quantum field theory. Let For a local Möbius covariant net A on R 1,1 , we can define its automorphism group Aut A and the notion of a group action on A as in the case of local Möbius covariant nets on S 1 .
denotes the left (resp. right) connected component of the causal complement O ′ of O. A g-localized endomorphism λ is a g-twisted DHR endomorphism of A if for anyÕ ∈ DC, there exists a unitary u ∈ A ∞ such that Ad u • λ is localized inÕ. The * -category of rigid g-twisted DHR endomorphisms is denoted by g-Rep A, and an object of the * -tensor category G-Rep A := g∈G g-Rep A is called a G-twisted DHR endomorphism of A.
The following proposition gives a physical interpretation of C ⊠ D rev and D(C, D).
Proposition 4.9. Let A 1 (resp. A 2 ) be a completely rational irreducible local Möbius covariant net on S 1 with an action of a group G 1 (resp. G 2 ). Then, (G 1 ×G 2 )-Rep(A 1 ⊗A 2 ) ≃ (G 1 -Rep A 1 )⊠ (G 2 -Rep A 2 ) rev as G 1 ×G 2 -crossed * -tensor categories, where G 1 ×G 2 acts on A 1 ⊗A 2 by the tensor product representation. In particular, when G 1 = G 2 = G, we have an equivalence G- h (µ(b)) for a ∈ A 1 (I) and b ∈ A 2 (J) and I ∈ I + , J ∈ I − , where I + (resp. I − ) denotes the set of intervals in the positive (resp. negative) half line. This assignment defines a fully faithful * -functor F : by universality. This functor F is a strict tensor functor since 1, 2). Moreover, F is a strict G-equivariant functor and therefore a strict G-crossed functor since such that ρ ∼ = λ⊠µ and therefore F is essentially surjective. By the complete rationality assumption of A 1 and A 2 , their vacuum Hilbert spaces H A1 and H A2 are separable by [KLM01,Proposition 15] and Bisognano-Wichmann property. By the proof of [KLM01, Lemma 27], the factoriality of ρ| A1⊗C and ρ| C⊗A2 follows from the simplicity of ρ. Suppose ρ is (g, h)-localized in I × J ∈ DC. Then, by Reeh-Schlieder property, ρ| A1(I ′ )⊗C is a faithful normal representation. Since A 1 (I ′ ) = A 1 (I) ′ by Haag duality on R, which is included in the complete rationality assumption, A 1 (I ′ ) is a type III factor and therefore ρ| A1(I ′ )⊗C is unitarily equivalent to a DHR endomorphism of A 1 by [Tak02, Corollary V.3.2], which is of type I by [KLM01,Corollary 14]. Then, by the proof of [KLM01, Lemma 27], there are faithful normal representations π 1 , π 2 of (A 1 ) ∞ and (A 2 ) ∞ on H A1 and H A2 respectively such that ρ is unitarily equivalent to π 1 ⊗ π 2 . By replacing π 1 and π 2 with endomorphisms, ρ is unitarily equivalent to λ⊠µ for some λ ⊗ µ ∈ Obj((g-Rep A 1 ) ⊠ (h −1 -Rep A 2 )). This unitary is indeed in A 1 (I) ⊗ A 2 (J) by Haag duality on R.
Thanks to this proposition, we can regard (G 1 × G 2 )-Rep(A 1 × A 2 ) as a G 1 × G 2 -braided * -tensor category. Note that we cannot apply [Müg05, Proposition 2.17] to this category due to the lack of Haag duality on R 1,1 .

Equivariant α-induction Frobenius algebras
In this subsection, we construct equivariant α-induction Frobenius algebras (Theorem 4.15) and show that it is symmetric and has the equivariant version of commutativity.
First, we introduce the notion of the conjugate of a morphism of a G-braided multitensor category.
For any g ∈ G and λ ∈ I, fix an isomorphism u λ g : g λ ∼ = λ(g), where λ(g) denotes the object in I which is isomorphic to g λ. For every g ∈ G, define z g ∈ Hom( g θ, θ) by putting where L(g) is the multi-index (λ 1 (g), λ 2 (g), m) andη g λ := z A−1 g ⊗ C idg λ for λ ∈ I. When A is a Q-system in a * -fusion category C, let w ∈ Hom(1 D(C) , θ) denote the isometric inclusion. Take {ϕ L l } l 's and {e ν,λµ i } i 's to be orthonormal bases and define x ∈ Hom(θ, θ 2 ) to be i ⊠ e ν2,λ2µ2 * j ).
Take u λ g 's to be unitaries and define z ∈ Hom( g θ, θ) by putting Theorem 4.15. Let A be a neutral symmetric special simple G-equivariant Frobenius algebra in a split G-braided fusion category C. Then, the tuple Θ G α (A) := (θ,m, η, ∆, ε, z) is a G-equivariant Frobenius algebra in D(C).
If C is moreover spherical, thenm • ∆ = dim Θ G α (A). Another choice of {e ν,λµ i } i 's and u λ g 's yields the same G-equivariant Frobenius algebra. Another choice of I, a direct sum and λ ∨ 2 's in the definition of θ and {ϕ L l } l 's yields an isomorphic G-equivariant Frobenius algebra. We call Θ G α (A) the G-equivariant α-induction Frobenius algebra associated with A.
When A is a Q-system in a * -fusion category C, Θ G α (A) := (θ, w, x, z) is a G-equivariant Qsystem. Another choice of {e ν,λµ i } i 's and u λ g 's yields the same G-equivariant Q-system. Another choice of I, a direct sum and λ ∨ 2 's in the definition of θ and {ϕ L l } l 's yields an isomorphic Gequivariant Q-system. We call Θ G α (A) the G-equivariant α-induction Q-system associated with A.
Next, we note that Θ G α (A) does not depend on a choice of {e ν,λµ i } i 's since the effect of changing {e ν,λµ i } i 's in the definition ofm and ∆ is canceled by the corresponding change of {ẽ ν,λµ i } i 's. For the same reason, it does not depend on a choice of u λ g 's. Note also that another choice of {ϕ L l } l 's only yields another direct sum in the definition of θ. Once we have shown that Θ G α (A) is a G-equivariant Frobenius algebra, we can see that another direct sum yields a canonically isomorphic one since we define structures to be direct sums. Also, another choice of left duals yields a canonically isomorphic one by the definition of conjugation. Another choice of I yields an isomorphic one by fixing isomorphisms between simple objects.
Then, we show that Θ G α (A) is a Frobenius algebra. Because the proof is similar to that of [Reh00, Theorem 1.4], we only show coassociativity here. We have j ⊗ id µ ′ 2 ) by ∂µ ′ 1 = ∂µ ′ 2 and Lemmata 4.12 and 4.13. By the cyclicity of the trace, we have .
Note that {d ν2 /d ν1 ϕ N n } n is the dual basis of {φ N n } n by the cyclicity of the trace since dim α G+ A (ν 1 ) = d ν1 and dim α G− A (ν 2 ) = d ν2 by Proposition 3.3. Then, by the Fourier expansion in the basis {φ N n } summing over the index n in N , we obtain By a similar calculation, we obtain and their dual bases are respectively given by for k = 1, 2, we can interchange these bases in the representations of (∆ ⊗ id θ )∆ and (id θ ⊗ ∆)∆ and therefore obtain (∆ ⊗ id θ )∆ = (id θ ⊗ ∆)∆. We can also showm • ∆ = d θ when C is spherical as in the proof of [Reh00, Theorem 1.4] by Proposition 3.3.
Finally, we show that Θ G α (A) is a G-equivariant Frobenius algebra, which is essentially new. First, we show that z g is invertible for every g ∈ G. Indeed, we show that the inverse is given by By Lemma 4.12, we have By the cyclicity of the trace, we have so that we obtain by the Fourier expansion in the basis {ϕ L(g) l ′ } l ′ since the group action on Bimod C (A) is pivotal by Proposition 3.3 and therefore preserves traces. We can similarly check z g z −1 g = id θ . Thus, z g is invertible.
Next, we show that (θ, z) is a G-equivariant object. For any g, h ∈ G, we have (see the proof of Proposition 3.8), we have by the cyclicity of the trace and the Fourier expansion in the basis {ϕ The right hand side is equal to z gh since u λ k (h) g g u λ k h is an isomorphism gh λ k ∼ = λ k (gh) for k = 1, 2 and z does not depend on a choice of isomorphisms. Therefore, (θ, z) is a G-equivariant object.
Finally, we show that z g is a Frobenius algebra homomorphism for every g ∈ G. Since we may take a basis of Hom(α G+ A (1 C ), α G− A (1 C )) to be the identity and take u 1 C g 's to be canonical isomorphisms, we have z g g η = η and εz g = g ε for every g ∈ G. We show ∆z g = (z g ⊗ z g ) g ∆. We have by Lemma 4.12, the cyclicity of the trace and the Fourier expansion in the basis {ϕ On the other hand, we have by ∂µ 1 = ∂µ 2 and Lemmata 4.12 and 4.13. Sinceη g is monoidal by the proof of Proposition 3.8 and by definition, we obtain by the cyclicity of the trace and the Fourier expansion in these bases {ϕ Then, comparing the bases of Hom( g ν k , λ k µ k ) for k = 1, 2, we obtain the conclusion. We can similarly show z g m = m(z g ⊗z g ). Thus, Θ G α (A) is a G-equivariant Frobenius algebra. We can similarly prove the statements for Qsystems just by taking bases to be orthonormal and replacing dual bases by * -conjugated bases.
Next, we show that Θ G α (A) satisfies the G-equivariant version of commutativity.
Definition 4.16. Let C be a G-braided multitensor category C. Then, a G-equivariant algebra A in C is G-commutative if the equation in Figure 42 holds. Similarly, when A is a G-equivariant Frobenius algebra, it is G-cocommutative if the equation in Figure 43 holds. Lemma 4.17. Let A be a neutral special G-equivariant Frobenius algebra in a G-braided multitensor category C. Let λ, λ ′ ∈ Obj(C g ) and µ, µ ′ ∈ Obj(C). Then, for f ∈ Hom(α G+ Proof. The statement for b C− follows from the graphical calculations in Figure 44, where z := z A . We used the right A-modularity of f and the left A-modularity ofη g−1 µ ′ g f ′ηg µ respectively at the second and third equalities in the upper equation. We used the left A-modularity of f and the right A-modularity ofη g−1 µ ′ g f ′ηg µ respectively at the second and third equalities in the lower equation. The proof for b C is similar. Lemma 4.18. Let C be a G-braided multitensor category. Then, b C− µ,λ = b C− λ ∨ ,µ ∨ for any λ, µ ∈ Homog(C).
Proof. The statement follows from the graphical calculation in Figure 45.
Proposition 4.19. Let A be a neutral symmetric special simple G-equivariant Frobenius algebra in a split G-braided fusion category C. Then, the G-equivariant α-induction Frobenius algebra Θ G α (A) associated with A is G-commutative and G-cocommutative.
Proof. Let us follow the notation in Theorem 4.15. We only show cocommutativity because the proof of commutativity is similar. First, note that θ = g∈G L,∂λ1=g λ 1 ⊠ λ ∨ 2 is a homogeneous decomposition. Therefore, the left-hand side of Figure 43 for θ is given by Figure 45: The conjugation of a reverse crossing Then, since by ∂µ 1 = ∂µ 2 and Lemma 4.17, the left hand side of Figure 43 is equal to ⊗ id µ k )} j is its dual basis for k = 1, 2.
Then, we show that Θ G α (A) is symmetric.
Proposition 4.20. Let A be a neutral symmetric special simple G-equivariant Frobenius algebra in a split spherical G-braided fusion category C. Then, Θ G α (A) is symmetric.
Proof. Let us follow the notation in Theorem 4.15. For every λ ∈ I, let λ denote the element in I with λ = λ ∨ . Since C is split by assumption, Hom(1 C , µλ) for λ, µ ∈ I is zero if µ ̸ = λ and one-dimensional if µ = λ. Since Θ G α (A) does not depend on a choice of a basis of Hom(1 C , λλ), we may take it to be the right coevaluation coev ′ λ . Then, the corresponding dual basis is given by ev λ . Now, let us calculate δ D(C) θ . By the proof of Proposition 4.1, we can take θ ∨ to be Since θ is a Frobenius algebra and therefore itself is a left dual, there exists a canonical isomorphism θ ∨∨ ∼ = θ, which is given in Figure 46.
Finally, we give an application of equivariant α-induction Frobenius algebras to algebraic quantum field theory. Indeed, some of them give modular invariants of fixed point nets (Theorem 4.25). Ind G H (λ) g ′ is defined as follows. For g ∈ G, there is a uniqueg ∈ G with g ′ g ∈gH. Let h ∈ H be a unique element with g ′ g =gh. Then, the component of z Note that the G-equivariant Longo-Rehren Frobenius algebra Θ G LR always satisfies the condition FPdim Θ G LR = FPdim C.
Theorem 4.25. Let A be a completely rational irreducible local Möbius covariant net on S 1 with an action of a finite group G. Let A be a simple G-equivariant Q-system in Rep A. Then, Proof. The first statement follows from [Müg05, Theorem 3.12] and Lemma 4.23. The last statement follows from [BKL15, Proposition 6.6].

Equivariant full centers
Let A be a neutral symmetric special simple G-equivariant Frobenius algebra in a split spherical G-braided fusion category C. For λ, µ ∈ Obj(C), we have by definition, where A Aλ denotes Aλ as a left A-module. Then, it is natural to ask if Θ G α (A) can be realized as a subalgebra of a G-equivariant Frobenius algebra with coefficients ⟨λ, Aµ⟩. In the case where G is trivial, it is known [BKL15,Proposition 4.18] that the answer is yes and the α-induction Frobenius algebra Θ α (A) is realized as the full center Z(A) [FFRS08, Definition 4.9] of A. In this section, we give the equivariant generalization of this theorem.

The equivariant full center of a G-equivariant Frobenius algebra
In this subsection, we define the equivariant generalization of a full center (Definition 5.11). Because it is defined to be a subalgebra of a product algebra as in the case where G is trivial, we begin with products and subalgebras.
Proposition 5.1. Let A, B be G-equivariant Frobenius algebras in a G-braided multitensor category C. Then, (AB, m AB , η A ⊗ η B , ∆ AB , ε A ⊗ ε B , z A ⊗ z B ) is a G-equivariant Frobenius algebra, where m AB and ∆ AB are given in Figure 48. If A and B are special (resp. symmetric), then so is AB. If A and B are Q-systems in a G-braided * -multitensor category, then AB is again a G-equivariant Q-system. Proof. By definition, (AB, z AB ) = (A, z A )(B, z B ) in C G and the right-hand side induces the stated Frobenius algebra structure on the left-hand side by [FRS02,Proposition 3.22]. Therefore, AB is a Frobenius algebra. We can also give a direct proof with some graphical calculations in C. It follows from the naturality of dashed crossings that z AB g is a Frobenius algebra isomorphism. The remaining statements follow easily.
Definition 5.2. Let A be a G-equivariant Frobenius algebra in a G-crossed multitensor category, and let e be a Frobenius idempotent for A i.e. an idempotent in End C (A) that satisfies [FFRS06, Equations 2.54 and 2.55]. We say e is a G-equivariant Frobenius idempotent for A if ez A g = z A g g e for every g ∈ G. If A is a G-equivariant Q-system in a G-crossed * -multitensor category and e is a projection, we say e is a G-equivariant Frobenius projection for A.
Proposition 5.3. Let A be a G-equivariant Frobenius algebra in a G-crossed multitensor category C and let e be a G-equivariant Frobenius idempotent for A with a retract (B, s, r). Then, for any nonzero scalar ζ, the tuple B ζ := (B, rm A (s ⊗ s), rη A , ζ(r ⊗ r)∆ A s, ζ −1 ε A s, {rz A g g s} g ) is a G-equivariant Frobenius algebra. If moreover C is a G-crossed * -multitensor category, A is a G-equivariant Q-system and e is a G-equivariant Frobenius projection, B ζ with r = s * is a Gequivariant Q-system.
Proof. It is already known that B ζ is a Frobenius algebra, see the proof of [FFRS06, Proposition 2.37]. Since e is a G-equivariant Frobenius idempotent, we obtain for every g ∈ G. We can similarly check ∆ B z B g = (z B g ⊗ z B g ) g ∆ B and ε B z B g = g ε B . Therefore z B g is a Frobenius algebra endomorphism of B. Moreover, and therefore z B g is an isomorphism. Finally, for any g, h ∈ G, and therefore {z B g } g gives a G-equivariant structure on B. The last statement follows from [BKLR15, Lemma 4.1] and z B−1 g = g rz A−1 g s = z B * g .
Note that the Frobenius algebra structure of B ζ depends on ζ in general. Indeed, if A is symmetric and special, then ε B ζ η B ζ = ζ −1 dim A. However, we do not write ζ when it is not important.
Proposition 5.4. Let A be a symmetric special G-equivariant Frobenius algebra in a G-braided multitensor category C. Then, P G A (λ) ∈ End(Aλ) defined in Figure 49 for every (λ, z λ ) ∈ C G is an idempotent, where C G denotes the G-equivariantization of C (see e.g. [EGNO15, Sections 2.7 and 4.15]). Moreover, P G A := P G A (1 C ) is a G-equivariant Frobenius idempotent for A. When C is a G-braided * -multitensor category and A is a Q-system, P G A (λ) is a projection. Figure 50. Then, the remaining part of the statement follows from [FRS02, Lemma 5.2], [FFRS06, Lemma 3.10] and [BKL15, Lemma 4.6] since P G A (λ) = P (A,z A ) ((λ, z λ )) in C G . Note that ev A = ε A m A when A ∨ is taken to be A itself. We can also give a direct proof with some graphical calculations, see Figures 67 and 73 below.
Definition 5.5. Let A be a symmetric special G-equivariant Frobenius algebra in a G-braided multitensor category C. We call P G A the G-equivariant left central idempotent of A. If P G A is split, then we call the corresponding reduced Frobenius algebra, which is denoted by C G (A), the G-equivariant left center of A.
Proposition 5.6. Let A be a symmetric special G-equivariant Frobenius algebra in a G-braided multitensor category C. Then, C G (A) is symmetric, G-commutative and G-cocommutative. If A is simple, then so is C G (A).
Proof. (C G (B), z C G (B) ) is equal to the left center C((B, z B )) since r and s can be regarded as morphisms in C G for a retract (C G (B), r, s) by definition (see Proposition 5.3), and sr = P G B is P (B,z B ) in C G . Then, the statement follows from [FFRS06, Proposition 2.37 (i) and (ii)].
1. We can also give the G-equivariant version of [FFRS06, Proposition 2.25(iii)]: a symmetric G-equivariant Frobenius algebra in a G-braided multitensor category is Gcommutative if and only if it is G-cocommutative.

The specialness of C G (A) is nontrivial in general, see [FFRS06, Proposition 2.37(iii)].
We need the following lemmata for the next subsection.
Lemma 5.8. For symmetric special G-equivariant Frobenius algebras A and B in a G-braided multitensor category C, we have P G AB = P G A (C G (B)). Proof. P G AB turns into the left central idempotent P (A,z A )(B,z B ) in C G . Then, the statement follows from [FFRS06, Proposition 3.14(i)].
Lemma 5.9. For a G-commutative (resp. G-cocommutative) symmetric special G-equivariant Frobenius algebra A in a G-braided multitensor category, we have P G (A) = id A .
Proof. As in the proof of Lemma 5.8, it follows from the classical result [FFRS06, Lemma 2.30] since (A, z A ) is commutative (resp. cocommutative) in C G .
Finally, note that for a neutral G-equivariant Frobenius algebra A in a G-crossed multitensor category C, the tuple (A⊠1 C , m A ⊠id, η A ⊠id, ∆ A ⊠id, ε A ⊠id, z A ⊠id) is a G-equivariant Frobenius algebra in D(C). By definition, if A is symmetric (resp. special), then so is A ⊠ 1 C . Then, we can give the following definition, which is what we want and is the G-equivariant version of [FFRS08, Definition 4.9].
Definition 5.11. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a split spherical G-braided fusion category C with dim C ̸ = 0. Then, we refer to the G-equivariant in D(C) as the G-equivariant full center of A. Note that we need the sphericality of C and dim C ̸ = 0 for Θ G LR to be symmetric special and therefore for Z G (A) to be well-defined, see Theorem 4.15 and Proposition 4.20. In particular, when C is a G-braided * -fusion category, the assumption is always satisfied.

Equivariant α-induction Frobenius algebras as equivariant full centers
In this subsection, we prove our second main theorem (Theorem 5.25). Indeed, the proof is given as the equivariant generalization of that of [BKL15,Proposition 4.18], and for this we need to define crossings that arise from α-induction as in [BEK99, Proposition 3.1].
Lemma 5.12. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a Gbraided multitensor category C. Then, for λ ∈ Obj(C g ) and a left A-module µ in C h , the morphisms in Figures 51 and 52 are left A-module isomorphisms and natural in λ and µ. Similarly, for a right A-module ρ, the morphisms in Figures 53 and 54 are right A-module isomorphisms and natural in λ and ρ. If C is a G-braided * -multitensor category, A is a G-equivariant Q-system and µ and ρ are standard, then B ± λ,µ and B ± ρ,λ are unitary when we take s α G± A (λ),µ and s ρ,α G± A (λ) to be isometric. Figure 53: Figure 54: Proof. The naturality of the morphisms follows by definition. We only show the remaining statement for B + λ,µ and B + ρ,λ because the proof for B − λ,µ and B − ρ,λ can be obtained just by replacing a crossing by its reverse. The left A-modularity of B + λ,µ follows from that of s α G+ A (λ),µ and the left A-modularity of g µ. The morphism in Figure 55 is the left inverse of B + λ,µ since ,µ with f given in Figure 57, where z := z A , and is equal to (r α G+ A (λ),µ • s α G+ A (λ),µ ) 2 = id α G+ A (λ)⊗ A µ by the graphical calculation there. It is also the right inverse by Figure 58. Next, the right A-modularity of B + ρ,λ follows from Figure 59. Then, similarly to the argument for B + λ,µ , we can show by some graphical calculations that the morphism in Figure 56 is the inverse of B + ρ,λ . The final statement follows from [BKLR15, Lemma 3.23] and the definitions of (B ± λ,µ ) −1 and (B ± ρ,λ ) −1 . We give the following lemmata for later use.  Lemma 5.13. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a Gbraided multitensor category C. Then, B + g λ, g µ = g B + λ,µ (η g λ ⊗ A idg µ ) and B − g λ, g µ = g B − λ,µ ( g∂µg −1η g λ ⊗ A idg µ ), whereη g λ := z A−1 g ⊗ C idg λ , for λ ∈ Homog(C), a homogeneous left A-module µ in C and g ∈ G. Similarly, B + g ρ, g λ = g B + ρ,λ (idg ρ ⊗ Aη g λ ) and B − g ρ, g λ = g B − ρ,λ (idg ρ ⊗ Aη g λ ) for a homogeneous right A-module ρ in C.
Lemma 5.14. Let A be a neutral special symmetric G-equivariant Frobenius algebra in a Gbraided multitensor category C. Then, the equations in Figures 60 hold for λ ∈ Obj(C g ), a homogeneous left A-module µ and a homogeneous right A-module ρ.
ρ µ λ Proof. We only show the first equality because the proof of the other is similar. By explicitly writing down canonical morphisms in Frob G (C) as morphisms in C, we find that r α G− A ( µ λ),µ and s ρ,α G− A ( µ λ) in the definition of (B + λ,µ ) −1 and B − ρ,λ cancel. Then, the left-hand side of the first equation in Figure 60 is equal to (idρµ λ ⊗ r ρ,µ )b C ρµ,λ (e ρ,µ s ⊗ id λ ) = b C ρ⊗ A µ,λ .
Thanks to this lemma, we can move an arc along a crossing as in Figure 15 when it contains a thick line segment inside. On the other hand, when thick line segments are outside an arc, we need more arguments to move the arc.
Lemma 5.15. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a Gbraided multitensor category C. Then, the equations in Figure 61 hold for λ ∈ Obj(C g ), a homogeneous left A-module µ in C and a homogeneous right A-module ρ in C. We also have similar equations for α G− A .
α G+ A (λ)  Figure 61 are equal to r g µ g µ ∨ ,α G+ A (λ) f s α G+ A (λ),A with f given respectively by the leftmost and rightmost diagrams in Figure 62, which proves the statement. Note that we used at the second equality that coev µ is given by Figure 30 as in the case of bimodules. We can also move thick arcs by the following lemma since α G± A are tensor functors and therefore preserve arcs.
Lemma 5.16. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a Gbraided multitensor category C. Then, the equations in Figure 63 hold for λ 1 ∈ Obj(C g1 ), λ 2 ∈ Obj(C g2 ), a homogeneous left A-module µ in C and a homogeneous right A-module ρ. Similar statements hold for α G− A .
A λ 1 A λ 2 µ g1g2 µ λ 1 λ 2 = A λ 1 A λ 2 µ g1g2 µ λ 1 λ 2 A λ 1 A λ 2 µ g1g2 µ λ 1 λ 2 z g1 Figure 64: The proof of Figure 63 Corollary 5.17. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a G-braided multitensor category C. Then, the equations in Figure 65 hold for λ ∈ Obj(C g ), a homogeneous left A-module µ in C and a homogeneous right A-module ρ. Similar statements hold for α G− A . Proof. The same argument as that in the proof of Lemma 2.16 works by Lemmata 5.14, 5.15 and 5.16.
Definition 5.18. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a Gbraided multitensor category C. For λ ∈ Obj(C), we define a morphismP G A (λ) ∈ End(Aλ) to be the morphism in Figure 66, where z := z A . Remark 5.19. Note that the definition of P G A (λ) in Proposition 5.4 requires λ ∈ Obj(C G ) but does not require that A is neutral, while the definition ofP G A (λ) requires that A is neutral but does not require λ ∈ Obj(C G ). If λ ∈ Obj(C G ) and A is neutral, then P G A (λ) =P G A (λ) by definition.
Lemma 5.20. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a Gbraided multitensor category C. Then,P G A (λ) is an idempotent for any λ ∈ Obj(C).
Proof. The proof is similar to that of [BKLR15,Lemma 4.26]. Indeed, the statement follows from the graphical calculation in Figure 67, where we only gave calculations on equivariant structures. Then, Hom G loc (λ, Aµ) ∼ = Hom(α G+ A (λ), α G− A (µ)) by the map I A λ,µ : Hom G loc (λ, Aµ) → Hom(α G+ A (λ), α G− A (µ)) defined in Figure 68 for f ∈ Hom G loc (λ, Aµ), where A A (resp. A g A, resp. A A ) denotes A as a left A-module (resp. g A as a left A-module, resp. A as a right A-module). Note that A A ∨ = ∨ A A = A A by [Yam04, Section 5]. We similarly have the isomorphism I A λ,µ : Hom G loc (Aµ, λ) → Hom(α G− A (µ), α G+ A (λ)) defined in Figure 70 forf ∈ Hom G loc (Aµ, λ). Figure 69: Proof. We only show the statement for Hom G loc (λ, Aµ) because the proof for Hom G loc (Aλ, µ) is easier. We show that I A−1 λ,µ is given by the morphism in Figure 69 for f ′ ∈ Hom(α G+ A (λ), α G− A (µ)) up to the canonical isomorphism A A ⊗ A A A ∼ = A. By the graphical calculation in Figure 72, we can do a similar graphical calculation to that in the proof of [BKL15, Lemma 4.16] using Lemma 5.14 and Corollary 5.17 and see that I A−1 λ,µ • I A λ,µ (f ) is equal to the morphism in Figure 73 by the graphical calculation in Figure 72. Then, from the graphical calculation in Figure 73, where the third equality follows from a similar argument to the proof of [BKLR15,Lemma 4.26], we obtain I A−1 λ,µ • I A λ,µ (f ) =P G A (λ)f = f . We can also check that I A−1 λ,µ is indeed the right inverse as in the proof of [BKL15, Lemma 4.16] using Lemma 5.15 and the graphical calculation in Figure 74.
Remark 5.22. When C is a G-braided * -multitensor category and A is a G-equivariant Q-system, Figure 73 with λ = Aµ and f = id Aµ shows thatP G+ A (µ) is a projection as in [BKLR15, Lemma 4.26].
Lemma 5.24. Let A be a neutral symmetric special G-equivariant Frobenius algebra in a G-ribbon multitensor category C and let λ, µ ∈ Homog(C). Then, the nondegenerate pairing Hom(Aµ, λ) × Hom(λ, Aµ) restricts to Hom G loc (Aµ, λ) × Hom G loc (λ, Aµ). Moreover, the maps I A λ,µ andĨ A λ,µ defined in Lemma 5.21 are isometric with respect to this pairing. If C is a G-braided * -multitensor category and A is a G-equivariant Q-system, then they are unitary.
Proof. The first statement follows from Lemma 5.20. The second statement follows from a similar graphical calculation to that in the proof of [BKL15, Lemma 4.16] by Lemma 5.14. Note that α G+ A is pivotal and therefore preserves traces. The final statement follows from the unitarity of thick crossings.
Finally, we can prove our second main theorem in this article, which is the equivariant generalization of [BKL15, Proposition 4.18]. Figure 75: The proof of Lemma 5.23 Theorem 5.25. Let A be a neutral symmetric special simple G-equivariant Frobenius algebra in a split spherical G-braided fusion category C with dim C ̸ = 0. Then, the G-equivariant full center Z G (A) dim A of A (Definition 5.11), where the subscript dim A means that we take ζ in Proposition 5.3 to be dim A, is isomorphic to the G-equivariant α-induction Frobenius algebra Θ G α (A) associated with A (Theorem 4.15). If C is a G-braided * -multitensor category and A is a G-equivariant Q-system, then we have an isomorphism between Q-systems.
Proof. By Lemmata 5.8 and 5.9, we have P G ⟨λ 1 , Aλ 2 ⟩ G loc λ 1 ⊠ λ ∨ 2 as objects for a complete system ∆ of representatives of the simple objects of C. Hence, by Lemma 5.21, Z G (A) ∼ = Θ G α (A) as objects. By Lemma 5.24, we can see that for a basis {φ L l } l of Hom G loc (λ 1 , Aλ 2 ) and its dual basis {φ L l } l of Hom G loc (Aλ 2 , λ 1 ), the morphisms s Z G (A) := L φ L l ⊠ id λ2 and r Z G (A) := Lφ L l ⊠ id λ2 , where L := (λ 1 , λ 2 , l), split the idempotent P G+ A⊠1 (Θ G LR ) as in [BKL15,Lemma 4.15]. Then, by a similar argument to that in the proof of [BKL15,Proposition 4.18], we can see that Z G (A) dim A ∼ = Θ G α (A) as Frobenius algebras. Namely, their units and counits coincide up to an isomorphism thanks to the normalization ζ = dim A. In order to show the coincidence of coproducts, it suffices to calculate the quantity in Figure 76, which is the coefficient of e ν1,λ1µ1 i ⊠ẽ ν2,λ2µ2 j up to d A d λ2 d µ2 /d ν2 d ν1 , as in the proof of [BKL15,Proposition 4.18]. Then, by the graphical calculation there, the coproducts coincide. We put ϕ L l := I A λ1,λ2 (φ L l ) and used Lemma 5.24 at the first equality. Note that the second equality follows since the quantity is nonzero only if ∂ν 1 = ∂λ 1 ∂µ 1 . The third equality follows as in the proof of [BKL15, Proposition 4.18]. The proof for products is similar.