On G -crossed Frobenius ⋆ -algebras and fusion rings associated with braided G -actions
aa r X i v : . [ m a t h . QA ] S e p On G -crossed Frobenius ⋆ -algebras and fusion rings associated withbraided G -actions Prashant Arote and Tanmay Deshpande
Abstract
For a finite group G , Turaev introduced the notion of a braided G -crossed fusion category. The clas-sification of braided G -crossed extensions of braided fusion categories was studied by Etingof, Nikshychand Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a G -crossed Frobenius ⋆ -algebra and give a classification of (strict) G -crossed extensions of a commutativeFrobenius ⋆ -algebra R equipped with a given action of G , in terms of the second group cohomology H ( G, R × ). Now suppose that B is a non-degenerate braided fusion category equipped with a braidedaction of a finite group G . We will see that the associated G -graded fusion ring is in fact a (strict) G -crossed Frobenius ⋆ -algebra. We will describe this G -crossed fusion ring in terms of the classificationof braided G -actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusioncoefficients. Let G be a finite group. The goal of this paper is to define and study the notion of a G -crossed Frobenius ⋆ -algebra and classify the G -crossed extensions of a commutative Frobenius ⋆ -algebra in terms of somecohomological data. The next goal is to use these results to describe the G -crossed fusion ring associatedwith a braided G action on a non-degenerate braided fusion category. All algebras considered in this paperare over C , though for some definitions and results it would also be possible to work over slightly more generalcommutative rings equipped with an involution. However most of the results make use of the properties of C . We will now give some motivation behind the results of this paper.Let us begin with the motivation behind the notion of a Frobenius ⋆ -algebra (see Definition 2.3). Let C be fusion category over C , namely a C -linear finite semisimple abelian rigid monoidal category whose unitobject is simple. For more details about fusion and multi-fusion categories we refer to [ENO05]. Thenthe complexified Grothendieck ring K ( C ) of C has the structure of a Frobenius ⋆ -algebra, with the ⋆ -anti-involution being induced by the duality in C , extended semi-linearly to K ( C ). In particular, for any finitegroup G , the group algebra C [ G ] is naturally a Frobenius ⋆ -algebra. More generally, any (complexified)based ring, in the sense of [Lus87], is naturally a Frobenius ⋆ -algebra.Let us now study the motivation for G -graded and G -crossed Frobenius ⋆ -algebras. A G -extension of afusion category C is a G -graded fusion category D = L g ∈ G C g whose trivial component C is equivalent to C .The G -extensions of fusion categories were studied by Etingof, Nikshych and Ostrik in [ENO10]. The notionof a braided G -crossed fusion category was introduced by Turaev, see [Tur00],[Tur08]. Roughly speaking, itis a G -graded fusion category D = L g ∈ G C g , equipped with a monoidal action of G such that g ( C h ) = C ghg − and a family of natural isomorphisms c X,Y : X ⊗ Y → g ( Y ) ⊗ X for any g ∈ G, X ∈ C g , Y ∈ D called G -braiding isomorphisms. The above action and G -braiding are required to satisfy certain compati-bility conditions. In particular, its trivial component C is a braided fusion category equipped with a braidedaction of G . A G -crossed extension of braided fusion category B is a G -crossed fusion category D whose1rivial component is equivalent to B . We refer to [DGNO10, Section 4.4.3] for a detailed discussion of braided G -crossed categories. The classification of G -crossed extensions of a (non-degenerate) braided fusion categorywas carried out in [ENO10, Theorem 7.12].Let K ( D ) denote the complexified Grothendieck ring of a braided G -crossed fusion category D . Then K ( D ) is a G -graded Frobenius ⋆ -algebra with an action of G . This action of G on K ( D ) satisfies certainconditions induced by the G -braiding. Moreover, if D is a G -crossed extension of braided fusion category B then the trivial graded component of K ( D ) is equal to K ( B ). The complexified Grothendieck ring K ( D ) isan example of G -crossed Frobenius ⋆ -algebra (see § G -crossedFrobenius ⋆ -algebra is to study the fusion ring K ( D ).We will define G -crossed Frobenius ⋆ -algebra by abstracting the properties satisfied by complexifiedGrothendieck ring of a G -crossed braided fusion category. Roughly, a G -crossed Frobenius ⋆ -algebra is a G -graded Frobenius ⋆ -algebra equipped with an action of G which satisfies certain conditions (see Definition4.1). In particular its identity component is a commutative Frobenius ⋆ -algebra equipped with a G -action.A G -crossed extension of a commutative Frobenius ⋆ -algebra R is a G -crossed Frobenius ⋆ -algebra whosetrivial component is isomorphic to R . One of our main goals in this paper is to classify such G -crossedextensions.We now briefly describe the organization and main results of this paper. In § ⋆ -algebra and that they are semi-simple algebras. A basic example of a Frobenius ⋆ -algebra isa matrix algebra M n ( C ). We prove that there is a bijective correspondence between the set of isomorphismclasses of commutative Frobenius ⋆ -algebras and the set of tuples of positive real numbers upto permutation.In § G -graded Frobenius ⋆ -algebra.In § G -crossed Frobenius ⋆ -algebra and that of a strict G -crossed Frobenius ⋆ -algebra. In § G -crossed Frobenius ⋆ -algebras that arisefrom braided G -actions on non-degenerate braided fusion categories. We will prove some results about thestructure and classification of strict G -crossed Frobenius ⋆ -algebras (see Prop. 4.10). In particular, we willprove the following main result: Theorem 1.1.
Let G be a finite group and let R be a commutative Frobenius ⋆ -algebra equipped with anaction of G by Frobenius ⋆ -algebra automorphisms. Then there is a bijective correspondence between the setof isomorphism classes of strict G -crossed extensions of R , denoted by Frob(
G, R ) , and the second groupcohomology H ( G, R × ) i.e. Frob( G, R ) ←→ H ( G, R × ) . We will also state and prove an analogue of the Verlinde formula in case of a strict G -crossed Frobenius ⋆ -algebra (see Corollary 4.15). The twisted categorical Verlinde formula in case of braided G -crossed categorieswas proved in [DM19]. We will then derive a slight generalization of this result from [DM19] as describedbelow. We will also relate the above result to the classification of braided G -actions and braided G -crossedfusion categories of [ENO10].Let B be a non-degenerate braided fusion category and let c : G → EqBr( B ) ∼ = Pic( B ) be a grouphomomorphism from G to the group EqBr( B ) of braided auto-equivalences of B up to natural isomorphismswhich is isomorphic to the group Pic( B ) of equivalence classes of invertible B -module categories. We refer to[ENO10] for more details about these and related results. Note that such a group homomorphism inducesan action of G by Frobenius ⋆ -algebra automorphisms on K ( B ). Let O ×B denote the abelian group ofisomorphism classes of invertible (in particular they will also be simple) objects in B . Note that c alsoinduces an action of G on the commutative group O ×B . We have the short exact sequence of G -modules1 → O ×B → K ( B ) × → K ( B ) × / O ×B → . Now given such a c , [ENO10] construct an element T ∈ H ( G, O ×B ). The lifts of c to a braided G action on B , namely morphisms of 1-groups c : G → EqBr( B ) ∼ = Pic( B ) lifting c , are classified by a certain H ( G, O ×B )-torsor T which is non-empty if and only if T ∈ H ( G, O ×B ) vanishes. Given such a c , we willconstruct an element t ∈ H ( G, K ( B ) × / O ×B ) which is mapped to T ∈ H ( G, O ×B ) under the connecting2omomorphism (see Corollary 4.25) coming from the above short exact sequence. When the obstruction T vanishes, we construct a mapping Φ : T → H ( G, K ( B ) × ) . (1)Let c : G → EqBr( B ) ∼ = Pic( B ) define an action of G by braided auto-equivalences on B lifting c . Such abraided G -action corresponds to an element [ c ] ∈ T . Using the results of [ENO10], we can define an associatedstrict G -crossed Frobenius ⋆ -extension of K ( B ), which we call the fusion algebra associated with the braided G -action, see Proposition 4.30. We will see that the element (given by Theorem 1.1) of H ( G, K ( B ) × )corresponding to this strict G -crossed Frobenius ⋆ -extension is given by Φ([ c ]) (see also Remark 4.31). Wewill derive a Verlinde formula to compute the fusion rules in the above G -crossed Frobenius ⋆ -algebra. Wewill also prove an analogue of this result in case B is a modular fusion category (i.e. B is also equippedwith a ribbon structure, see [DGNO10]) and the G -action is modular, and express the fusion coefficients interms of the associated crossed S-matrices. We refer to Corollaries 4.32 and 4.41 for these categorical twistedVerlinde formulae. ⋆ -algebra We will begin by recalling the definition of Frobenius algebras, Frobenius ⋆ -algebras and some of theirproperties. For more details see [Aro18]. Definition 2.1 (Frobenius Algebra) . (i) A Frobenius algebra A is a finite dimensional associative unital C -algebra equipped with a linear functional λ : A → C such that the bilinear form on A defined by ( a, b ) = λ ( ab )is nondegenerate.(ii) Moreover if λ ( ab ) = λ ( ba ) , ∀ a, b ∈ A , then A is called as symmetric Frobenius algebra. (A linearfunctional λ : A → C on an algebra satisfying this condition is said to be a class functional.) Remark 2.2.
There is an equivalent definition of a Frobenius algebra: A Frobenius algebra A is a finite-dimensional, unital, associative C -algebra equipped with a left A -module isomorphism Θ : A → A ∨ .The equivalence between the two definitions is given by:( A, λ ) Θ( a ) = λ ( − · a )( A, φ ) λ ( x ) = Θ(1)( x ) . Definition 2.3 (Frobenius ⋆ -algebra) . A Frobenius ⋆ -algebra A is a finite dimensional associative unital C -algebra equipped with a class functional λ : A → C and a map ⋆ : A → A , such that the following holds: • ( a + b ) ⋆ = a ⋆ + b ⋆ ∀ a, b ∈ A, • ( a ⋆ ) ⋆ = a ∀ a ∈ A, • ( ab ) ⋆ = b ⋆ a ⋆ ∀ a, b ∈ A, • ( αa ) ⋆ = αa ⋆ ∀ α ∈ C , a ∈ A, • λ ( a ⋆ ) = λ ( a ) ∀ a ∈ A, • The Hermitian form h a, b i := λ ( ab ⋆ ) admits an orthonormal basis in A . Remark 2.4.
Note that by the given conditions, λ ( ab ⋆ ) = λ (( ab ⋆ ) ⋆ ) = λ ( ba ⋆ ). Hence h· , ·i is indeed aHermitian form on A . Since all our algebras are over C , the existence of an orthonormal basis in the lastcondition is equivalent to the Hermitian form being positive definite. Definition 2.5.
Let ( A , λ , ⋆ ) and ( A , λ , ⋆ ) be two Frobenius ⋆ -algebras. An algebra homomorphism f : A → A is called as Frobenius ⋆ -algebra homomorphism if λ ( a ) = λ ◦ f ( a ) and f ( a ⋆ ) = f ( a ) ⋆ for all a ∈ A . We will denote the set of all Frobenius ⋆ -algebra automorphisms of ( A, λ, ⋆ ) by Aut
Frob ( A ). Example 2.6. (1) A = M n ( C ) with linear functional λ = trace and the ⋆ -map is conjugate transpose i.e. X ⋆ = X t .(2) Let G be a finite group. Then A = C [ G ] with linear functional λ defined by λ ( g ) = ( g = e ;0 otherwise . The ⋆ -map is defined by g ⋆ = g − and extend to C [ G ] by conjugate linearity.3 emark 2.7. Let R be a commutative ring with unity which is equipped with an involution ( · ) : R → R .One can also define the notion of Frobenius algebra and Frobenius ⋆ -algebra for R -algebras, but in this paperwe stick to C with the involution being complex conjugation. Remark 2.8.
Let (
A, λ, ⋆ ) be a Frobenius ⋆ -algebra. Then the following holds:1. ( A, λ ) is symmetric Frobenius algebra.2. A is a semisimple algebra (see [Aro18, Theorem 2.4]).3. Let Sim( A ) denote the set of irreducible representations of A . Let { e M : M ∈ Sim( A ) } be the set ofprimitive central orthogonal idempotents of A then e ⋆M = e M , for all M ∈ Sim( A ) (see [Aro18, Lemma2.5]).Next, we will see the possible Frobenius ⋆ -algebra structures on End( V ) for a finite dimensional C -vectorspace V . Proposition 2.9.
Let V be a finite dimensional vector space over C . Let (End( V ) , λ, ⋆ ) be a Frobenius ⋆ -algebra. Then λ = α · trace, for some positive real number α and ⋆ : End( V ) → End( V ) is an adjoint mapcorresponding to some positive definite Hermitian form on V . Proof. As λ is a non-degenerate class functional on End( V ) it must be a non-zero multiple of the trace: λ ( X ) = α · trace( X ) ∀ X ∈ End( V )for some α ∈ C × . As Id ⋆ = Id in End( V ), by positive definiteness we have0 < h Id, Id i = λ ( Id · Id ⋆ ) = α · trace( Id ) = α · n. Thus λ = α · trace for some α ∈ R + .We know that End( V ) = V ∗ ⊗ V ∼ = V ⊕ n as a left End( V )-module. Fix a left End( V )-module embedding f : V ֒ → End( V ). By the definition of a Frobenius ⋆ -algebra, End( V ) is equipped with a positive definiteHermitian form. Define a positive definite Hermitian form on V using the embedding f as: h v, w i f = λ ( f ( v ) f ( w ) ⋆ ) ∀ v, w ∈ V. Then by the definition of a Frobenius ⋆ -algebra, h· , ·i f is a positive definite Hermitian form on V . Let T ∈ End( V ) be any endomorphism, then for v, w ∈ V , h T v, w i f = λ ( f ( T v ) f ( w ) ⋆ ) = λ ( T f ( v ) f ( w ) ⋆ ) = λ ( f ( v ) f ( w ) ⋆ T )= λ ( f ( v )( T ⋆ f ( w )) ⋆ ) = λ ( f ( v ) f ( T ⋆ w ) ⋆ ) = h v, T ⋆ w i f . Thus ⋆ is an adjoint map corresponding to positive definite Hermitian form h· , ·i f . This proves the proposi-tion. Theorem 2.10.
Let A be any Frobenius ⋆ -algebra then A ∼ = r M i =1 M n i ( C ) where each M n i ( C ) is a Frobenius ⋆ -algebra with a linear functional λ i = α i · trace for some α i ∈ R + andthe ⋆ -map is conjugate transpose.Proof. We know that any Frobenius ⋆ -algebra is a semisimple algebra and the ⋆ -map preserve the primitivecentral simple idempotents. Then the statement follows from the Prop. 2.9.We obtain the following from the above result: 4 orollary 2.11. There is a bijection between the set of isomorphism classes of n -dimensional commutativeFrobenius ⋆ -algebras and the set of n -tuples of positive real numbers up to permutation. Corollary 2.12.
Let A be a commutative Frobenius ⋆ -algebra and let f : A → A be any algebra automor-phism. Then f commute with ⋆ -map i.e. f ( a ⋆ ) = f ( a ) ⋆ ∀ a ∈ A. We begin by briefly recalling the notion of group graded algebras and some results about them. For moredetails we refer to [Dad86].
Definition 3.1. A G -graded ring A is a ring with an internal direct sum decomposition: A = ⊕ g ∈ G A g as(additive groups), where the additive subgroups A g of A satisfy: A g A h ⊆ A gh , ∀ g, h ∈ G .The above decomposition is called as a G -grading of A , and its summands A g are called as g -componentsof A . The component of A corresponding to 1 ∈ G is a subring of A and A is an unital bimodule over A .Also A g is an unital bimodule over A for each g ∈ G . If A is a G -graded ring and H is a subgroup of G then we naturally obtain a H -graded ring from A , A H = ⊕ h ∈ H A h . Remark 3.2.
We will use the same notation i.e. “1” to denote the identity element of a finite group G andfor the unity in an algebra A . Definition 3.3. A G -graded module over a G -graded ring A is an A -module M with an internal direct sumdecomposition: M = P g ∈ G M g , (as A -modules), where A -modules M g satisfy: A g M σ ⊆ M gσ , ∀ g, σ ∈ G .Suppose M is a G -graded module over a G -graded ring A then we get a natural H -graded A H -module M H = P h ∈ H M h . A G -graded A -submodule N of M is an A -submodule which has G -grading with g -component, N g = M g ∩ N, ∀ g ∈ G .Let M be a G -graded A -module. We say that an G -graded A -submodule U of M is H-null if U H = 0.Define the H -null Socle , S H ( M ) of M as the largest (under inclusion) H -null G -graded A -submodule of M . The tensor product A ⊗ A H N is the most natural G -graded A -module associated with a H -graded A H -module N . We define induced module A ⊗ A H N to be a G -graded A -quotient module: A ⊗ A H N = ( A ⊗ A H N ) /S H ( A ⊗ A H N ) . Definition 3.4.
A simple G -graded A -module M is a G -graded A-module with no proper G -graded sub-module. i.e. , the only G -graded A -submodules of M are 0 and itself. Remark 3.5. [Aro18, Lemma 3.6 and Lemma 3.8] Let M be a simple G -graded A -module then M H is either0 or a simple H -graded A H -module. Also if N is a simple H -graded A H -module then A ⊗ A H N is either 0 ora simple G -graded A -module with ( A ⊗ A H N ) H = 0. Remark 3.6.
Define the support of a simple G -graded A -module M as,Supp( M ) = { g ∈ G : M g = 0 } . We define a partial action of G on simple A -modules as follows:For g ∈ G and for simple A -module M , g M = ( A ⊗ A M ) g by Remark 3.5 g M is a simple A -module, if g ∈ Supp( A ⊗ A M ) and g M = 0, if g / ∈ Supp( A ⊗ A M ). Formore details see [Dad86, Section 7], [Aro18, Section 3.3].5 efinition 3.7 ( G -graded Frobenius ⋆ -algebra) . A G -graded Frobenius ⋆ -algebra is a G -graded finite di-mensional associative unital C -algebra A = L g ∈ G A g equipped with a semilinear anti-involution ⋆ : A → A and a class functional λ : A → C (which is extended linearly to λ : A → C by zero on all non-trivialcomponents), such that the following conditions hold: • ( A, λ, ⋆ ) is a Frobenius ⋆ -algebra • ( A g ) ⋆ = A g − , ∀ g ∈ G . Remark 3.8.
Let A be a G -graded Frobenius ⋆ -algebra then the trivial component A of A is itself aFrobenius ⋆ -algebra. Definition 3.9.
Let A be a G -graded Frobenius ⋆ -algebra. The centralizer of A in A is denoted by Z A ( A )and defined as Z A ( A ) = { a ∈ A : ab = ba ∀ b ∈ A } . Moreover we have, Z A ( A ) = M g ∈ G Z A ( A g )and Z A ( A ) is a G -graded Frobenius ⋆ -algebra under the restrictions of ⋆ and λ . Remark 3.10. (cf.[Aro18, Lemma 4.3]) Let A be a G -graded Frobenius ⋆ -algebra and let M be any A -module. Then the null socle of G -graded A -module A ⊗ A M is zero. Thus the A -module induced by the A -module M is equal to A ⊗ A M . Remark 3.11.
Let A be a G -graded Frobenius ⋆ -algebra and let Sim( A ) denote the set of simple A -modules upto isomorphism. Then by Remark 3.10, the partial action of G on Sim( A ) is given as follows:for g ∈ G and M ∈ Sim( A ), g M = ( ( A ⊗ A M ) g if g ∈ Supp( A ⊗ A M ) , g ∈ G on Sim( A ) by Sim( A ) g i.e. Sim( A ) g = { M ∈ Sim( A ) : g M = M } . Remark 3.12. [Aro18, Section 4] Let M be a simple A -module such that g M ∼ = M as A -module. Thenwe can make M into a simple A
Note that χ gM , and hence α gM are well-defined up to scaling by n th -roots of unity, where n is the order of g in G . If g = 1 then A is a Frobenius ⋆ -algebra, so one can identify dual of A with A .Using the above identification we have the well- defined element α M ∈ A corresponding to M ∈ Sim( A ). Definition 3.15 (Twisted class functional) . A linear functional f on A -bimodule E is said to be a twistedclass functional if f ( am ) = f ( ma ) ∀ m ∈ E, a ∈ A . We will denote the set of all twisted class functionalson E by cf A ( E ) . Remark 3.16. [Aro18, Lemma 4.12, Corollary 4.16] Let A be a G -graded Frobenius ⋆ -algebra and letΘ : A → A ∨ be the corresponding A -module isomorphism. Then Θ − ( cf A ( A g )) = Z A ( A g − ) and for g ∈ G , the set of g -twisted characters forms an orthogonal basis of cf A ( A g ).6 emark 3.17. Let A be a G -graded Frobenius ⋆ -algebra. Let { e M : M ∈ Sim( A ) } denote the set ofprimitive central orthogonal idempotents of A and let E M denote the simple Z ( A )-module C · e M . By theRemark 2.8, we have Z ( A ) = M M ∈ Sim( A ) E M and any simple Z ( A )-module is isomorphic to E M for some M ∈ Sim( A ). Remark 3.18.
1. By [Aro18, Lemma 4.15], for any g ∈ G , there is a bijection between the sets Sim( A ) g and Sim( Z ( A )) g (which is well-defined using Definition 3.9).2. For every M ∈ Sim( A ) and for any g ∈ G , we have g ( E M ) = 0 or g ( E M ) ∼ = E M . Moreover, Z A ( A g − ) ∼ = M M ∈ Sim( A ) g C · α gM as Z ( A )-module. For more details see [Des19, Lemma 3.3]. Lemma 3.19.
Let V be a finite dimensional C -vecor space and A be a G -graded Frobenius ⋆ -algebra with A = End( V ). Then for each g ∈ G , A g = 0 or A g ∼ = End( V ) as a A -bimodule. Proof.
Note that for any ring R , ( R, R )-bimodule is a R ⊗ R op -module. For any finite dimensional vectorspace V , End( V ) ⊗ End( V ) op ∼ = End( V ⊗ V ∨ ). Therefore, any (End( V ) , End( V ))-bimodule is a semisimplemodule and any simple (End( V ) , End( V ))-bimodule is isomorphic to End( V ). Then the result follows fromthe Remark 3.18. Lemma 3.20.
Let A be a G -graded Frobenius ⋆ -algebra. Then we have the direct sum decomposition of A g − , A g − = M M ∈ Sim( A ) g M r m as a A − module (2)for some r m ∈ N . Proof. As A is a semisimple algebra, so A g − is a semisimple A -module i.e A g − = M M ∈ Sim( A ) M r m . Then the result follows from the Remark 3.18.
Remark 3.21.
Let R be a commutative Frobenius ⋆ -algebra equipped with an action of G by Frobenius ⋆ -algebra automorphisms. Let us define, U := { x ∈ R : x ⋆ = x − } then we have a short exact sequence of G -modules 1 → U → R × → R × /U → . Note that R × ∼ = ( C × ) dim R , U ∼ = ( S ) dim R , hence the quotient R × /U is isomorphic to ( R × > ) dim R . But R × > is an uniquely divisible group. Hence H n ( G, R × /U ) = 0 for all n ≥
1. Using the long exact sequence ofcohomology, we have H k ( G, U ) ∼ = H k ( G, R × ) ∀ k ≥ . Proposition 3.22.
Let (
A, λ, ⋆ ) be a G -graded Frobenius ⋆ -algebra. Let ⋆ : A → A be an anti-involutionsuch that ( A, λ, ⋆ ) is also a G -graded Frobenius ⋆ -algebra. Suppose a ⋆ = a ⋆ for all a ∈ A . Then x ⋆ = x ⋆ for all x ∈ A . 7 roof. Let Z A ( A ) = ⊕ g ∈ G Z A ( A g ) be a A center of ( A, λ, ⋆ ). From the Remark 3.18, we can choose theelements e gM ∈ Z A ( A g ), e g − M ∈ Z A ( A g − ) corresponding to each M ∈ Sim( A ) g such that Z A ( A g ) = M M ∈ Sim( A ) g C · e gM and Z A ( A g − ) = M M ∈ Sim( A ) g C · e g − M . Also one can choose these elements such that ( e gM ) ⋆ = e g − M . By the definition of a G -graded Frobenius ⋆ -algebra, ( e gM ) ⋆ = f M ( g ) e g − M for some f M ( g ) ∈ C × . We have 0 < λ ( e gM ( e gM ) ⋆ ) = λ ( e gM f M ( g ) e g − M ) this implies that f M ( g ) >
0. Suppose G M denotes thestabilizer of M ∈ Sim( A ) in G under the partial action. Then the anti-involution implies that f M : G M → C × is a group homomorphism. This implies that f M ( g ) = 1 for all g ∈ G M and for all M ∈ Sim( A ).Therefore ( e gM ) ⋆ = e g − M = ( e gM ) ⋆ for all g ∈ G M and for all M ∈ Sim( A ).Let x ∈ A g be any element then x = P M ∈ Sim( A ) g a m e gM for some a m ∈ A . So x ⋆ = X M ∈ Sim( A ) g ( e gM ) ⋆ a ⋆ m = X M ∈ Sim( A ) g ( e gM ) ⋆ a ⋆m = x ⋆ (since a ⋆ = a ⋆ ∀ a ∈ A ) . This proves that x ⋆ = x ⋆ for all x ∈ A . G -crossed Frobenius ⋆ -algebras Now we will define G -crossed Frobenius ⋆ -algebras and prove some results about them. Definition 4.1 ( G -crossed Frobenius ⋆ -algebra) . A G -crossed Frobenius ⋆ -algebra A is a G -graded Frobenius ⋆ -algebra equipped with an action of G on A by Frobenius ⋆ -algebra automorphism such that following holds: • For each g ∈ G, g : A → A is a Frobenius ⋆ -algebra automorphism of A . • g ( A h ) ⊆ A ghg − ∀ g, h ∈ G. • ab = g ( b ) a ∀ a ∈ A g , ∀ b ∈ A (Crossed commutativity condition). Remark 4.2.
Let A be a G -crossed Frobenius ⋆ -algebra then the following holds:1. A is commutative Frobenius ⋆ -algebra.2. For each g ∈ G , g ( A ) ⊆ A .3. For any g ∈ G , Z A ( A g ) = A g . Example 4.3. . The complexified Grothendieck algebra K ( D ) of a G -crossed braided fusion category D isa G -crossed Frobenius ⋆ -algebra.2 . Let A be a strongly G -graded extension of a commutative Frobenius ⋆ -algebra R then Z R ( A ) is a G -crossed Frobenius ⋆ -algebra. (For more details see § G -crossed Frobenius ⋆ -algebras Let A be a G -crossed Frobenius ⋆ -algebra. By Schur’s lemma, every irreducible representation of A is1-dimensional. Suppose Sim( A ) denotes the set of irreducible representations of A i.e. Sim( A ) = { χ : A → C : χ is algebra homomorphism } . From the Remark 4.2, there is a natural action of G on Sim( A ) as follows: for χ ∈ Sim( A ) and g ∈ G , g χ = χ ◦ g − . (3)8 emma 4.4. Let A be a G -crossed Frobenius ⋆ -algebra. Suppose χ ∈ Sim( A ) is fixed by the partial actionof some g ∈ G on Sim( A ) coming from a G -grading (as in Remark 3.11). Then it is also fixed by the actionof g defined above (3). In other words, the set of fixed points for the partial action of g on Sim( A ) iscontained in the set of fixed points for the action of g on Sim( A ) defined above (3). Proof.
Let e χ denote the primitive central orthogonal idempotent of A corresponding to character χ ∈ Sim( A ). By definition of G -crossed Frobenius ⋆ -algebra, we have e χ · A g = g ( e χ ) · A g . If χ ◦ g − = χ thenby Remark 3.18, we must have e χ · A g = 0 and hence, χ is not fixed by partial action. This proves theresult. Definition 4.5 (Strict G -crossed Frobenius ⋆ -algebra) . Let A be a G -crossed Frobenius ⋆ -algebra. We saythat A is a strict G -crossed Frobenius ⋆ -algebra if for each g ∈ G the set of fixed points for the partial actionof g on Sim( A ) is equal to the set of fixed points for the action of g on Sim( A ) from (3), i.e. if the inclusionfrom Lemma 4.4 is an equality for each g ∈ G . In this case the notation Sim( A ) g is unambiguous. Lemma 4.6.
Let A be a strict G -crossed Frobenius ⋆ -algebra.(i) Let χ, χ ′ ∈ Sim( A ) then χ ( α χ ′ ) = 0 if χ = χ ′ and the element e χ := α χ χ ( α χ ) is the primitive centralorthogonal idempotent corresponding to χ ∈ Sim( A ).(ii) For χ, χ ′ ∈ Sim( A ), we have the orthogonality relations: h α χ , α χ ′ i = ( χ = χ ′ ; χ ( α χ ) λ ( α χ ) if χ = χ ′ . (iii) Let χ ∈ Sim( A ) g and α gχ be the corresponding g -twisted character (3.13). Then for any χ ′ ∈ Sim( A ), α χ ′ · α gχ = α gχ · α χ ′ = ( χ = χ ′ ; χ ( α χ ) α gχ if χ = χ ′ . (iv) We have the direct sum decomposition of A g − , A g − ∼ = M χ ∈ Sim( A ) g C · α gχ as a A − module . (4) Proof.
Parts ( i ) , ( ii ) follows from the fact that A is a commutative semisimple algebra and ( iv ) followsfrom the Remark 3.18.For part ( iii ), we have a A -module isomorphism Θ : A → A ∨ . By the definition of twisted character α gχ = Θ − ( χ g ). Suppose χ = χ ′ ∈ Sim( A ) then e χ ′ · χ g = 0. Consider 0 = Θ − ( e χ ′ · χ g ) = e χ ′ · Θ − ( χ g ) = e χ ′ · α gχ . Therefore α χ ′ · α gχ = χ ( α χ ′ ) e χ ′ · α gχ = 0 . Also α χ · α gχ = χ ( α χ ) e χ · α gχ = χ ( α χ ) α gχ . This proves theresult.
Corollary 4.7.
Let A be a strict G -crossed Frobenius ⋆ -algebra then(i) Let χ ∈ Sim( A ) g , χ ′ ∈ Sim( A ) h be two twisted characters. Then for χ = χ ′ , α gχ · α hχ ′ = 0 . (ii) dim C ( A g − ) = dim C ( A g ) = | Sim( A ) g | .(iii) { α gχ : χ ∈ Sim( A ) g } forms an orthogonal basis of A g − .(iv) Let χ ∈ Sim( A ) g and α gχ be the corresponding g -twisted character. Define e gχ := α gχ χ ( α χ ) then ( e gχ ) n = e χ , where n is the order of g in G . 9 roof. ( i ) , ( ii ) and ( iii ) directly follows from the Lemma 4.6.For ( iv ), let χ ∈ Sim( A ) g then by [Aro18, (1)] we have h α gχ , α gχ i = h α χ , α χ i . By using the definition of theHermitian form and twisted characters, we have h α gχ , α gχ i = λ ( α gχ α gχ⋆ ) = λ ( α gχ⋆ α gχ ) = e χ ( α gχ⋆ ) and h α χ , α χ i = χ ( α ⋆χ ) = χ ( α χ )where e χ as in Remark 3.12. Hence e χ ( α gχ⋆ ) = χ ( α χ ). Also we have α gχ α gχ⋆ = χ ( α gχ α gχ⋆ ) e χ implies that h α gχ , α gχ i = χ ( α gχ α gχ⋆ ) λ ( e χ ) = e χ ( α gχ ) e χ ( α gχ⋆ ) λ ( α χ ) χ ( α χ )and by Lemma 4.6, h α χ , α χ i = χ ( α χ ) λ ( α χ ). Hence, e χ ( α gχ ) = χ ( α χ ). Consider, χ (( e gχ ) n ) = χ ( ( α gχ ) n χ ( α χ ) n ) = χ ( α gχ ) n ) χ ( α χ ) n = e χ ( α gχ ) n χ ( α χ ) n = 1 . By ( iii ) of Lemma 4.6, ( e gχ ) n = e χ . This proves the result.For χ ∈ Sim( A ), let G χ be the stabilizer of χ under the action of G i.e. G χ = { g ∈ G : g χ = χ } . Usingthe previous results we get a G χ -graded algebra α χ A , α χ A = M g ∈ G α χ · A g ∼ = M g ∈ G χ C · e gχ . From this G χ -graded algebra α χ A , we obtain a central extension G ′ χ of G χ by C × i.e → C × → G ′ χ → G χ → . Note that for g, h ∈ G χ the product e gχ · e hχ ∈ A gh . Our choice of elements e gχ determines a 2-cocycle φ χ : G χ × G χ → C × such that e gχ · e hχ = φ χ ( g, h ) e ghχ . Moreover one can choose these elements such that e gχ e g − χ = e χ . Hence we obtain: Corollary 4.8.
Let χ ∈ Sim( A ) and g, h ∈ G χ be any element then e gχ · e hχ = φ χ ( g, h ) e ghχ , where φ χ is the2-cocycle corresponding to an extension 1 → C × → G ′ χ → G χ → . Remark 4.9.
Let χ ∈ Sim( A ). For any g , g , · · · , g n ∈ G χ , let φ χ ( g , · · · , g n ) be such that χ g ( a ) χ g ( a ) · · · χ g n ( a n ) = φ χ ( g , · · · , g n ) χ g g ··· g n ( a a · · · a n ). These scalars will appear in our twistedVerlinde formula. Proposition 4.10.
Let A = ⊕ g ∈ G A g be a strict G -crossed Frobenius ⋆ -algebra then following holds:1. For each g ∈ G χ , we have a g -twisted character α gχ and element e gχ (determined up to scaling by rootof unity) such that A g − ∼ = M χ ∈ Sim( A ) g C · e gχ and e χ e gχ ′ = δ χ,χ ′ e gχ .
2. Let χ ∈ Sim( A ) be any element then e χ A ∼ = M g ∈ G χ C · e gχ is a G χ -crossed Frobenius ⋆ -algebra. There exists a normalized 2-cocycle φ χ ∈ H ( G χ , C × ) such thatthe multiplication and the action G χ on e χ A is determined as: for all g, h ∈ G χ , e gχ e hχ = φ χ ( g, h ) e ghχ and g ( e hχ ) = φ χ ( g, h ) φ χ ( ghg − , g ) e ghg − χ .
10. There exists a 2-cocycle φ ∈ H ( G, A × ) such that for χ ∈ Sim( A ), χ ◦ φ ( g, h ) = φ χ ( g, h ) and e gχ e hχ = φ ( g, h ) e ghχ ∀ g, h ∈ G χ . Moreover, the given action of G on A is determined as, for each g, h ∈ G , χ ∈ Sim( A ) h g ( e hχ ) = φ ( g, h ) φ ( ghg − , g ) e ghg − g χ .
4. Let ψ : G × G → A × defined as ψ ( g, h ) = φ ( g,h ) ⋆ φ ( h − ,g − ) , where φ as in (3) above. Then there exist a1-cochain θ : G → A such that ∂ ( θ ) = ψ and the ⋆ -map is given as, for each e gχ ∈ A g , ( e gχ ) ⋆ = θ ( g ) e g − χ ∀ g ∈ G, ∀ χ ∈ Sim( A ) g . Proof. (1) follows from the Lemma 4.6 and Corollary 4.7.For (2), let χ ∈ Sim( A ) be any element then from Corollary 4.8, there exists a 2-cocycle φ χ in H ( G χ , C × )such that e gχ · e hχ = φ χ ( g, h ) e ghχ ∀ g, h ∈ G χ . By the definition of a G -crossed Frobenius ⋆ -algebra, we have e gχ e hχ = g ( e hχ ) e gχ and g ( e hχ ) ∈ A ghg − , so g ( e hχ ) = β · e ghg − χ for some β ∈ C × .Now consider φ χ ( g, h ) e ghχ = e gχ · e hχ = g ( e hχ ) e gχ = β · e ghg − χ e gχ = βφ χ ( ghg − , g ) e ghχ . This implies g ( e hχ ) = φ χ ( g, h ) φ χ ( ghg − , g ) e ghg − χ ∀ g, h ∈ G χ . For (3), one can easily check that A × = ⊕ CoInd GG χ ( C × ), where the sum runs over the orbit representativesfor the induced action of G on Sim( A ). Then by Shapiro’s Lemma, we have H ( G, CoInd GG χ ( C × )) ∼ = H ( G χ , C × ) . Also H ( G, A × ) ∼ = ⊕ H ( G, CoInd GG χ ( C × )), where the sum runs over the orbit representatives for the inducedaction of G on Sim( A ). Hence we get a required φ ∈ H ( G, A × ).(4) follows from the part (2) and the fact that for any x, y ∈ A, ( xy ) ⋆ = y ⋆ x ⋆ and ( e gχ ) ⋆ is a scalarmultiple of e g − χ . This proves the result. Now we will define and classify the strict G -crossed extension of a commutative Frobenius ⋆ -algebra. Definition 4.11 ( G -crossed extension) . Let G be a finite group and R be a commutative Frobenius ⋆ -algebrasuch that G acts on R by Frobenius ⋆ -algebra automorphism. Then G -crossed extension of R is a pair ( A, j ),where A = ⊕ g ∈ G A g is a G -crossed Frobenius ⋆ -algebra, j : R ֒ → A is a Frobenius ⋆ -algebra embedding with j ( R ) = A and restriction of the action of G on j ( R ) agrees with the given action of G on R .If A is strict G -crossed Frobenius ⋆ -algebra then we will called ( A, j ) as a strict G -crossed extension of R . We will denote the set of all isomorphism classes of strict G -crossed extensions of R by Frob( G, R ).Now we will complete the proof of the main result:
Proof of Theorem 1.1.
Let A be a strict G -crossed extension of R . By the Prop. 4.10, A = ⊕ g ∈ G A g as R -module and for each g ∈ G there exists { e gχ : χ ∈ Sim( A ) g } such that A g − ∼ = ⊕ χ ∈ Sim( R ) g C · e gχ . Alsothere exists a normalized 2-cocycle φ ∈ C ( G, R × ) such that multiplication and representation of G on A isas follows: e gχ e hχ = φ ( g, h ) e ghχ , g ( e hχ ) = φ ( g, h ) φ ( ghg − , g ) e ghg − g χ .
11e will denote A with the above operations by A φ . Let A φ ′ be an another strict G -crossed extension of R isomorphic to A φ then there exists a grade preserving isomorphism f : A φ ′ → A φ which is identity on 1-grade.Using this isomorphism we get a function θ : G → R × such that f ( e ′ gχ ) = θ ( g ) e gχ , for all χ ∈ Sim( R ) g . Also f is a homomorphism implies that φ ′ ( g, h ) = ∂ ( θ )( g, h ) φ ( g, h ) for all g, h ∈ G i.e. [ φ ′ ] = [ φ ] ∈ H ( G, R × ).Thus, we get a well-defined map F : Frob( G, R ) → H ( G, R × ) defined as F ( A φ ) = [ φ ] . Let φ ∈ H ( G, R × ) be a normalized 2-cocycle. Let A g − be a C -vector space with basis { e gχ : χ ∈ Sim( R ) g } . We define a R -bimodule structure on A g − as, for χ ∈ Sim( R ) , χ ′ ∈ Sim( R ) g e χ e gχ ′ = e gχ ′ e χ = δ χ,χ ′ e gχ ′ extend C -linearly to A g − .Suppose A φ = ⊕ g ∈ G A g then A φ is a G -graded R -bimodule. Now we define a multiplication, action of G on basis elements using φ as, e gχ e hχ = φ ( g, h ) e ghχ , and g ( e hχ ) = φ ( g, h ) φ ( ghg − , g ) e ghg − g χ and extend it R -linearly to A φ . Define ( e gχ ) ⋆ = e g − χ extend it R -conjugate linearly to A φ then in view ofRemark 3.21, ⋆ -map is an anti-involution. Then A φ becomes a G -crossed Frobenius ⋆ -algebra. We will showthat [ A φ ] in Frob( G, R ) is depends only on [ φ ] in H ( G, R × ). Let φ ′ be another 2-cocycle such that [ φ ] = [ φ ′ ]in H ( G, R × ) then there exists θ : G → R × such that φ = ∂ ( θ ) φ ′ . Define f : A φ → A φ ′ as f ( e gχ ) = θ ( g ) e gχ for χ ∈ Sim( R ) , g ∈ G and extend R -linearly to A then one can easily check that f is an isomorphism of G -crossed extensions. This proves that [ A φ ] in Frob( G, R ) is depends only on [ φ ] ∈ H ( G, R × ). Thus, weget a well-defined map F : H ( G, R × ) → Frob(
G, R ) defined as F ([ φ ]) = A φ . One can check that F and F are inverses of each other. This proves the theorem. We will now state and prove a twisted Verlinde formula for the computation of the linear functional λ on astrict G -crossed Frobenius ⋆ -algebra. The twisted Verlinde formula in case of braided G -crossed categorieswas proved in [DM19]. Lemma 4.12.
Let A be a strict G -crossed Frobenius ⋆ -algebra. Let g , · · · , g n be any n -element of G and let a i ∈ A g i , so that a a · · · a n ∈ A g g ··· g n . Let χ ∈ Sim( A ) g g ··· g n such that ( g g · · · g n -twisted character) χ g g ··· g n ( a a · · · a n ) = 0. Then χ ∈ Sim( A ) h g ,g , ··· ,g n i . Proof.
Consider any a ∈ A . Then by the definition of G -crossed Frobenius ⋆ -algebra, for each i we have a · a a · · · a i = a a · · · a i · a = g g · · · g i ( a ) · a a · · · a i . Hence for each i , a ( a a · · · a n ) = ( g g · · · g i )( a )( a a · · · a n ) as an element of A g g ··· g n . This implies that, χ g g ··· g n ( aa a · · · a n ) = χ g g ··· g n (( g g · · · g i )( a ) a a · · · a n ) . Hence, χ ( a ) χ g g ··· g n ( a a · · · a n ) = χ ◦ ( g g · · · g i )( a ) χ g g ··· g n ( a a · · · a n ). Since we have assumed that χ g g ··· g n ( a a · · · a n ) = 0. Therefore, χ ( a ) = χ ◦ ( g g · · · g i )( a ) for any a ∈ A and 1 ≤ i ≤ n. This implies that χ is fixed by each g i . This proves the result.12 heorem 4.13 (twisted Verlinde formula in genus 0) . Let A be a strict G -crossed Frobenius ⋆ -algebra. Let g , g , · · · , g n ∈ G such that g g · · · g n = 1 . Let a i ∈ A g i then λ ( a a · · · a n ) = X χ ∈ Sim( A ) h g ,g , ··· ,gn i χ g ( a ) χ g ( a ) · · · χ g n ( a n ) χ ( α χ ) · χ ◦ φ ( g , g , · · · , g n ) where χ ◦ φ ( g , g , · · · , g n ) = φ χ ( g , g , · · · , g n ) are the scalar defined in Remark 4.9.Proof. As A is a semisimple algebra. Therefore,1 = X χ ∈ Sim( A ) e χ = X χ ∈ Sim( A ) α χ χ ( α χ ) . Then by definition of Frobenius ⋆ -algebra, we have λ = X χ ∈ Sim( A ) χχ ( α χ ) . Thus λ ( a a · · · a n ) = X χ ∈ Sim( A ) χ ( a a · · · a n ) χ ( α χ )= X χ ∈ Sim( A ) h g ,g , ··· ,gn i χ ( a a · · · a n ) χ ( α χ ) (by Lemma 4 . X χ ∈ Sim( A ) h g ,g , ··· ,gn i χ g ( a ) χ g ( a ) · · · χ g n ( a n ) χ ( α χ ) · χ ◦ φ ( g , g , · · · , g n ) . The last equality follows from the Remark 4.9.Let A be a strict G -crossed Frobenius ⋆ -algebra. Let g, h ∈ G be any elements. Suppose P g denotes anorthonormal basis of A g . We define the elementΩ g,h := X x ∈ P g x · h ( x ⋆ ) ∈ A ghg − h − . Let P ′ g be an another orthonormal basis of A g and let U be the change of basis matrix. Then U is a P ′ g × P g -unitary matrix. Consider, X y ∈ P ′ g y · h ( y ⋆ ) = X x,x ′ ∈ P g X y ∈ P ′ g U y,x U y,x ′ x · h ( x ′ ⋆ ) = X x ∈ P g x · h ( x ⋆ ) . The last equality follows from the unitarity of U . This implies that the element Ω g,h is independent of choiceof orthonormal basis. Lemma 4.14.
Let g, h ∈ G and let χ ∈ Sim( A ) [ g,h ] be a character fixed by the commutator [ g, h ] = ghg − h − . Then χ [ g,h ] (Ω g,h ) = ( χ ( α χ ) χ ◦ φ ( g,h,g − ,h − ) if χ ∈ Sim( A )
Corollary 4.15 (twisted Verlinde formula for any genus) . Let r, s be non-negative integers and let g , g , · · · ,g r , h , h , · · · , h r , m , · · · , m s ∈ G be any elements such that [ g , h ] · · · [ g r , h r ] · m · · · m s = 1. Let G ◦ ≤ G be a subgroup of G generated by g i , h i , m j . Let a j ∈ A m j then λ (Ω g ,h · · · Ω g r ,h r · a · · · a s ) = X χ ∈ Sim( A ) G ◦ χ ( α χ ) r − χ m ( a ) · · · χ m s ( a s ) χ ◦ φ ( g , h , g − , h − , · · · , m , · · · , m s ) . In this subsection we will see that an important class examples of G -crossed Frobenius ⋆ -algebra arises fromthe theory of fusion categories. We will begin by recalling some definition and results from [DGNO10],[ENO05],[ENO10], [Tur00].A tensor category (or monoidal C -linear category ) is a C -linear abelian category C with a structure ofmonoidal category such that the bifunctor ⊗ : C × C → C is bilinear on morphisms. By a fusion category we14ean a C -linear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects,finite-dimensional spaces of morphisms, and such that the unit object of C is simple. A fusion category is braided if it is equipped with a natural isomorphism c X,Y : X ⊗ Y ∼ −→ Y ⊗ X satisfying the hexagon axioms. Remark 4.16.
Let C be a fusion category. The tensor product on C induces the ring structure onGrothendieck ring. We will denote complexified Grothendieck ring of C by K ( C ). We have the non-degeneratesymmetric linear functional λ : K ( C ) → C which represents the coefficient of the class of the unit [ ] in anyelement of K ( C ). The duality in C induces a ⋆ -map on K ( C ), which is defined to be semilinear. Thus K ( C ) becomes a Frobenius ⋆ -algebra. Moreover, if B is braided fusion category then K ( B ) is a commutativeFrobenius ⋆ -algebra. Definition 4.17.
Let G be a finite group. A G - grading on a tensor category C is a decomposition C = M g ∈ G C g into a direct sum of full abelian subcategories such that tensor product ⊗ maps C g ×C h to C gh for all g, h ∈ G .In this case, trivial component C is a full monoidal subcategory and each C g is C -bimodule category. Remark 4.18.
Let C be a fusion category, and M a right C -module category. Let M op be the categoryopposite to M . Then M op is a left C -module category with the C -action ⊙ given by X ⊙ M := M ⊗ ⋆ X .Similarly, if N is a left C -module category, then N op is a right C -module category, with the C -action · givenby N · X := X ⋆ ⊗ N . Note that ( M op ) op is canonically equivalent to M as a C -module category. If M is a C -(left)module category then K ( M ) is a K ( C )-(left)module. Definition 4.19.
A ( C , D )-bimodule category M is said to be invertible if there exist bimodule equivalences: M ⊠ C M op ∼ = D and M op ⊠ D M ∼ = C . Let B be a braided fusion category, and M be a left B -module category. Then M is automatically a B -bimodule category(using the braiding on B ). We will say a B -module category is invertible if it is invertibleas a B -bimodule category. Let Pic( B ) denotes the group of invertible B -module categories and Pic( B ) denotesthe categorical 1-group whose objects are invertible B -module categories and 1-morphisms are equivalencesof B -module categories, see [ENO10]. Now we will state some results without proof. Lemma 4.20. [ENO10, § G be a finite group and B be a non-degenerate braided fusion category.Let c : G → Pic( B ) be a group homomorphism i.e. for each g ∈ G there is an invertible B -module category C g and for each pair g, h ∈ G there is an equivalence of B -module categories M g,h : C g ⊠ B C h ∼ = C gh . Thenthe following diagram: C f ⊠ B C g ⊠ B C h C fg ⊠ B C h C f ⊠ B C gh C fghM f,g ⊠ Id C h Id C f ⊠ M g,h M fg,h M f,gh commutes upto an element T ∈ H ( G, O ×B ), where O ×B denotes the abelian group of isomorphism classes ofinvertible objects in B . Remark 4.21. [ENO10, §
5] Let B be a non-degenerate braided fusion category then Pic( B ) ∼ = EqBr( B ),where EqBr( B ) denotes a categorical 1-group whose objects are braided autoequivalences of B and morphismsare natural isomorphisms between braided autoequivalences. Remark 4.22.
We recall some results from group cohomology.(1) Let M be a G -module. The mapsΨ i : Hom Z [ G ] ( Z [ G i +1 ] , M ) → C i ( G, M )15efined by Ψ i ( φ )( g , g , · · · , g i ) = φ (1 , g , g g , · · · , g g · · · g i )are isomorphisms for all i ≥
0. This provides isomorphisms between complexes · · · d i − −−−→ Hom Z [ G ] ( Z [ G i ] , M ) d i −→ Hom Z [ G ] ( Z [ G i +1 ] , M ) d i +1 −−−→ · · · and · · · d i − −−−→ C i ( G, M ) d i −→ C i +1 ( G, M ) d i +1 −−−→ in the sense that Ψ i +1 ◦ d i +1 = d i ◦ Ψ i for all i ≥
0. Moreover, these isomorphisms are natural in the G -module M .(2) Let H be a subgroup of G and let M be a H -module. The mapsΦ i : Hom Z [ G ] ( Z [ G i ] , CoInd GH ( M )) → Hom Z [ H ] ( Z [ G i ] , M )defined by Φ i ( φ )( x ) = φ ( x )(1)are isomorphisms for all i ≥ H be a subgroup of G then Z [ G ] is a free Z [ H ]-module with basis parametrise by right cosetrepresentatives of H in G . Consider · · · Z [ G i +1 ] Z [ G i ] · · · Z [ G ] Z · · · Z [ H i +1 ] Z [ H i ] · · · Z [ H ] Z d i α i +1 α i ǫα Idd i ǫ where d i ( g , g , · · · , g i ) = i X j =0 ( − j ( g , · · · , g j − , g j +1 , · · · , g i )for all i ≥ ǫ is the augmentation map. Then there exists α i a Z [ H ]-module map such that ǫ ◦ α = ǫ and α i ◦ d i = d i ◦ α i +1 for all i ≥ → L → M → N → G -modules. Then there existsconnecting homomorphisms δ ∗ : H i ( G, L ) → H i +1 ( G, N ) and a long exact sequence of abelian groups0 → H ( G, L ) → H ( G, M ) → H ( G, N ) δ −→ H ( G, L ) → · · · Moreover, this construction is natural in the short exact sequence in the sense that any morphism of shortexact sequences gives a morphism of long exact sequences.
Remark 4.23.
Let B be a non-degenerate braided fusion category and let c : G → Pic( B ) ∼ = EqBr( B ) be agroup homomorphism. Then we have an action of G on K ( B ) by Frobenius ⋆ -algebra automorphism. Thisinduces an action of G on Sim( K ( B )). By [Des19, § K ( C g − ) has basis indexed by Sim( K ( B )) g = { χ ∈ Sim( K ( B )) : g χ = χ } . Moreover, the decomposition of K ( C g − ) into simple K ( B )-module is given by K ( C g − ) ∼ = M χ ∈ Sim( K ( B )) g C · e gχ as a K ( B )-module. For χ ∈ Sim( K ( B )), G χ denotes the stabilizer of χ in G .16ow using the definition of a tensor product of B -module categories, we have a bilinear map on K ( D ) = L g ∈ G K ( C g ) such that K ( C g ) × K ( C h ) maps to K ( C gh ). By Lemma 4.20, this map satisfies associativityproperty up to an element T ∈ H ( G, O ×B ).As O ×B is a G -stable subgroup of K ( B ) × , so we have a short exact sequence of G -modules,0 → O ×B i −→ K ( B ) × π −→ K ( B ) × / O ×B → . (6)Using this short exact sequence we get a long exact sequence, · · · → H ( G, K ( B ) × ) π ∗ −→ H ( G, K ( B ) × / O ×B ) δ −→ H ( G, O ×B ) i ∗ −→ H ( G, O ×B ) → · · · . We will prove that i ∗ ( T ) is the trivial element in H ( G, K ( B ) × ). Proposition 4.24.
Let c : G → Pic( B ) be a group homomorphism and K ( D ) = ⊕ g ∈ G K ( C g ) be thecorresponding G -graded K ( B )-bimodule. Then there exists a 2-cochain φ ∈ C ( G, K ( B ) × ) such that,for g, h ∈ G χ , e gχ · e hχ = χ ◦ φ ( g, h ) e ghχ . Moreover, φ satisfies the 2-cocycle condition modulo O ×B i.e t := π ∗ ( φ ) ∈ H ( G, K ( B ) × / O ×B ), where π as in(6). In particular, φ satisfies the following relation: φ ( f g, h ) · φ ( f, g ) = T f,g,h · φ ( f, gh ) · φ ( g, h ) for all f, g, h ∈ G. Proof.
For g, h ∈ G χ , we have e gχ · e hχ ∈ K ( C gh ). But the K ( B )-module structure on K ( C gh ) implies thatthere exists a φ χ ∈ C ( G χ , C × ) such that e gχ · e hχ = φ χ ( g, h ) e ghχ . By Lemma 4.20, φ χ satisfies the following relation:for f, g, h ∈ G χ , φ χ ( f, g ) φ χ ( f g, h ) = χ ( T f,g,h ) φ χ ( f, gh ) φ χ ( g, h ) . (7)By Remark 4.22, we get an element ψ χ ∈ C ( G, CoInd GG χ ( C × )). We have K ( B ) × = ⊕ CoInd GG χ ( C × ), wherethe sum runs over the orbit representatives for the induced action of G on Sim( K ( B )). Using this we canget an element φ ∈ C ( G, K ( B ) × ) such that for χ ∈ Sim( K ( B )) χ ( φ ( g, h )) = φ χ ( g, h ) , for all g, h ∈ G χ . This proves the first part of proposition.Let us define an element t ∈ C ( G, K ( B ) × / O ×B ) as follows: t := π ∗ ( φ ) = π ◦ φ. Using the (7), we get, φ ( f g, h ) · φ ( f, g ) = T f,g,h · φ ( f, gh ) · φ ( g, h ) for all f, g, h ∈ G. This implies that φ satisfies 2-cocycle condition modulo O ×B and t is a 2-cocycle. This proves the proposition. Corollary 4.25.
Let δ : H ( G, K ( B ) × / O ×B ) → H ( G, O ×B ) be the connecting homomorphism associatedwith (6). Then δ ( t ) = T , where T as in Lemma 4.20. Hence, i ∗ ( T ) is the trivial element in H ( G, K ( B ) × ). Proof.
By Prop. 4.24 and by the definition of connecting homomorphism δ , we have δ ( t ) = T ∈ H ( G, O ×B ) . Hence, i ∗ ( T ) = i ∗ ( δ ( t )) = i ∗ ◦ δ ( t ) = 0 ∈ H ( G, K ( B ) × ) . This proves the result. 17 emark 4.26.
Let c : G → Pic( B ) be a group homomorphism. Then [ENO10, Theorem 8.5] implies thatthe set of all extensions of c to a morphism of 1-groups c : G → Pic( B ) forms a torsor over H ( G, O ×B ).Moreover this torsor is non empty iff the obstruction T ∈ H ( G, O ×B ) as in Lemma 4.20 vanishes. Let usdenote this torsor by T . Corollary 4.27.
Let us assume that the torsor T is non-empty. Then there is a mapping Φ : T → H ( G, K ( B ) × ) such that π ∗ (Φ( T )) = t ∈ H ( G, K ( B ) × /O ×B ). In particular, we can identify the torsor T / ker( i ∗ ) with ( π ∗ ) − ( t ) ⊆ H ( G, K ( B ) × ) as a torsor over H ( G, O ×B ) / ker( i ∗ ). Proof.
The elements of the torsor T are nothing but the isomorphism classes of B -bimodule tensor productson D = ⊕ g ∈ G C g . Choose B -bimodule equivalences M g,h : C g ⊠ B C h → C gh for all g, h ∈ G . By Remark 4.26,the obstruction T ∈ H ( G, O ×B ) is vanishes. Then by Lemma 4.20, the B -bimodule equivalences { M g,h } g,h ∈ G defines an element M of T . Using the Prop. 4.24, we get a 2-cocycle φ ∈ H ( G, K ( B ) × ) associated with M . Note that if we change the element M ∈ T by ψ ∈ H ( G, O × ) i.e. replace M g,h by ψ g,h M g,h . Then thecorresponding element φ will be replaced by ψ · φ in H ( G, K ( B ) × ). Then using T ∋ M φ ∈ H ( G, K ( B ) × )we get a well defined morphism Φ : T → H ( G, K ( B ) × ) of H ( G, O ×B )-torsors. By the definition of t (as inProp 4.24), π ∗ (Φ( T )) = t . This proves the result. Remark 4.28.
Let c : G → Pic( B ) be a morphism of 1-groups which lifts c . Let D denote the corresponding G -graded B -bimodule category, D = ⊕ g ∈ G C g . Then the obstruction T ∈ H ( G, O ×B ) as in Lemma 4.20 isvanishes and we get a B -bimodule tensor product on D (cf.[ENO10, Theorem 8.4]). By Prop. 4.24, K ( D )becomes a G -graded associative unital C -algebra with trivial component equal to K ( B ). Note that K ( B ) isa Frobenius ⋆ -algebra, so we have a linear functional λ : K ( B ) → C , extend it linearly to K ( D ) by zerooutside of K ( B ) to get a linear functional on K ( D ). Using Remark 4.29 and Prop. 3.22, one can define a ⋆ -map on K ( D ). Thus K ( D ) becomes a G -graded Frobenius ⋆ -algebra. Remark 4.29. (see [Des17, § § B be a non-degenerate braided fusion category and M be aninvertible B -module category. Then there exists a Z /N Z -graded fusion category D = ⊕ a ∈ Z /N Z C a such thatthe trivial component is equal to B and C ¯1 ∼ = M as a B -bimodule category. The complexified Grothendieckring K ( D ) is a Z /N Z -graded Frobenius ⋆ -algebra. The ⋆ -maps K ( M ) to K ( C N − ) ∼ = K ( M op ). Proposition 4.30.
Let c : G → Pic( B ) be a morphism of 1-groups and D denotes the corresponding G -graded B -bimodule category, D = ⊕ g ∈ G C g . Then K ( D ) is a strict G -crossed Frobenius ⋆ -algebra with trivialcomponent K ( B ). Proof.
By Remark 4.28, K ( D ) is a G -graded Frobenius ⋆ -algebra with trivial component K ( B ). To provethe theorem we have to check only for the G -action.Indeed, for all g, h ∈ G the category Fun B ( C g , C gh ) of B -module functors from C g to C gh is identified, onone hand, with functors of right tensor multiplication by objects of C h and, on the other hand, with functorsof left tensor multiplication by objects of C ghg − . So there is an equivalence g : C h → C ghg − defined by theisomorphism of B -module functors − ⊗ Y ∼ = g ( Y ) ⊗ − : C g → C gh , Y ∈ C h . (8)Extending it to D = ⊕ g ∈ G C g by linearity, we obtain an action of G by autoequivalences of D . Furthermore,evaluating (8) on X ∈ C g we obtain a natural family of isomorphisms X ⊗ Y ∼ −→ g ( Y ) ⊗ X, g ∈ G, X ∈ C g , Y ∈ D (9)which gives an action of G on K ( D ) by algebra automorphism. As each g ∈ G acts on D by autoequivalenceand on trivial component it acts by tensor autoequivalence. So dim(Hom( , X )) = dim(Hom( , g ( X )) forall X ∈ B . Note that the dual object is unique up to a unique isomorphism. This implies that the G -action commute with the ⋆ -map. Also the fix points for the partial action is equal to the fix points for theabove G -action is follows from [Des19, Lemma 3.5]. This prove that K ( D ) is a strict G -crossed extension of K ( B ). 18 emark 4.31. Let [ φ ] ∈ H ( G, K ( B ) × ) be the cohomology class corresponding to the G -crossed extension K ( D ) of K ( B ). Then we have π ∗ ( φ ) = t . Moreover the equivalence class of the lift c : G → Pic( B ) can beconsidered an element [ c ] in the H ( G, O ×B )-torsor T and we have [ φ ] = Φ([ c ]). Here Φ : T → H ( G, K ( B ) × )is the mapping from Corollary 4.27. Corollary 4.32. (Twisted Verlinde formula.) Let r, s be non-negative integers and let g , h , · · · , g r , h r , m , · · · , m s ∈ G be any elements such that [ g , h ] · · · [ g r , h r ] · m · · · m s = 1. Let G ◦ ≤ G be a subgroup of G generated by g i , h i , m j . For g ∈ G , let P g denote the set of isomorphism classes of simple objects in C g and define the object Ω g i ,h i := M X ∈ P gi X ⊗ h i ( X ⋆ ) ∈ C [ g i ,h i ] . Let M j ∈ C m j . Thendim(Hom( , Ω g ,h ⊗ · · · ⊗ Ω g r ,h r ⊗ M ⊗ · · · ⊗ M s )) = X χ ∈ Sim( K ( B )) G ◦ χ ( α χ ) r − χ m ([ M ]) · · · χ m s ([ M s ]) χ ◦ φ ( g , h , g − , h − , · · · , m , · · · , m s ) . Proof.
This is a special case of the Corollary 4.15.
Definition 4.33 (Modular Category) . A modular category B is a braided fusion category with a ribbontwist such that the corresponding S -matrix is invertible. Where a ribbon twist on a braided fusion category B is a natural isomorphism θ : id B → id B such that θ C ⋆ = θ ⋆C for all C ∈ B and θ C ⊗ C = ( θ C ⊗ θ C ) ◦ β C ,C ◦ β C ,C for any C , C ∈ B and the unnormalized S -matrix, S is an O B × O B matrix defined as S C ,C := tr( β C ,C ◦ β C ,C ) for C , C ∈ O B . Remark 4.34.
Using the unnormalized S -matrix S associated with the modular category B , we can identify P with Sim( K ( B )) as follows: for P ∋ C (cid:18) χ C : [ D ] S ( B ) D,C dim( C ) (cid:19) i.e. the S -matrix is essentially the character table of K ( B ). Remark 4.35.
Let B be a modular category. Let EqMod( B ) denote the categorical 1-group whose objectsare modular autoequivalences of B and morphisms are natural isomorphisms between modular autoequiva-lences. One can see that EqMod( B ) is a full 1-subgroup of the categorical 1-group EqBr( B ) of all braidedautoequivalences of B . Let Pictr( B ) ⊆ Pic( B ) be the full 2-subgroup formed by those invertible B -modulecategories which can be equipped with a B -module trace (see [Sch13], [Des17, § B ) ⊆ Pic( B ) . From Remark 4.21 and using [Des17, § B ) ∼ = Pictr( B ) . A B -module trace tr M on M assigns a trace tr M ( f ) ∈ C for each endomorphism f : M → M in M satisfyingsome properties and in particular we can talk of dimensions dim M ( M ) of objects of M . We will often assumethat the trace is normalized in such a way that dim(Hom( M, N )) = dim M ( M ) · dim M ( N ) for M, N ∈ M ,where Hom(
M, N ) ∈ B is the internal Hom. With this additional condition, tr M is uniquely defined uptoscaling by ± X M ∈O M dim M ( M ) = dim( B ) = X C ∈O B dim( C ) , where O M denotes the set of simple objects of M . Moreover with such a normalization, dim M ( M ) must bea totally real cyclotomic integer for each M ∈ M . For more details we refer to [Des17], [DGNO10].19et c : G → Pictr( B ) ∼ = EqMod( B ) be a morphism of 1-groups, i.e. a modular action of G on B . Then D = ⊕ g ∈ G C g is a G -graded category equipped with a modular action of G on C (= B ) and each C g is equippedwith a C -module trace. Suppose P g denotes the set of isomorphism classes of simple objects in C g . In thissetting we can define a P g × P g -matrix S g known as the unnormalized g -crossed S -matrix as follows: Definition 4.36.
For each simple object C ∈ P g , let us choose an isomorphism ψ C : g ( C ) → C such thatthe induced composition C = g m ( C ) → g m − ( C ) → · · · → g ( C ) → C is the identity, where m is the order of g in G. Note that the above conditions implies that ψ C is well-definedup to scaling by m th -root of unity. For each simple object M ∈ P g and C ∈ P g , S gM,C := tr C g ( C ⊗ M β C,M −−−→ M ⊗ C β M,C −−−→ g ( C ) ⊗ M ψ C ⊗ id −−−−→ C ⊗ M ) . Remark 4.37.
We will always choose ψ : → to be the identity. With this convention we see that S ( B , g ) M, = tr C g ( id M ) = dim C g ( M ), the categorical dimension of the object M ∈ C g .Let us choose a normalized B -module trace on each invertible B -bimodule category C g for all g ∈ G . Foreach g, h ∈ G , this induces a normalized B -module trace tr C g ⊠ B C h on the invertible B -bimodule category C g ⊠ B C h . We have B -module equivalence C g ⊠ B C h ∼ = C gh . The normalized trace tr C g ⊠ B C h on C g ⊠ B C h eitheragrees with the normalized B -module trace tr C gh on C gh or is the opposite of it. This gives us a 2-cocycle, sgn tr : G × G → {± } such thattr C g ⊠ B C h = sgn tr ( g, h ) tr C gh for all g, h ∈ G. Using this 2-cocycle we obtain a central extension1 → Z / Z → e G → G → G by Z / Z . Now consider the map of 1-groups e G → G → Pic tr ( B )induced by (10). Let D ′ = ⊕ ˜ g ∈ e G C ˜ g denote the corresponding e G -graded B -bimodule category and K ( D ′ )denote its complexified Grothendieck ring. Note that if ˜ g, ˜ g ∈ e G such that ˜ g, ˜ g g in G then there is acanonical B -module equivalences C ˜ g ∼ = C ˜ g ∼ = C g for all g ∈ G and the normalized B -module trace on C ˜ g and C ˜ g might be differ by sign. Suppose ˜ g ∈ e G such that ˜ g g in G and trace on C ˜ g , C g are equal. Then the unnormalized crossed S -matrices S g and S ˜ g are same. Remark 4.38.
We will allow slightly more general g -twisted character in this section i.e. we will allow the g -twisted character is well defined up to (2 · o ( g )) th -root of unity.In view of Remark 4.38 and using the results in [Des17],[Des19] we have: Theorem 4.39.
1. For M ∈ P g , C ∈ P g the numbers S gM,C dim( M ) = S gM,C S g ,M and S gM,C dim( C ) = S gM,C S ,C are cyclotomicintegers. For C ∈ P g the linear functional χ gC : K ( C g ) → C , [ M ] S gM,C S ,C is the g -twisted characterassociated with χ C ∈ Sim( K ( B )) g .2. The categorical dimension dim( B ) is a totally positive cyclotomic integer. We have S g · S gT = S gT · S g = dim( B ) · I. For C ∈ P g , we have χ C ( α χ C ) is equal to the tatally positive cyclotomic integer dim( B )dim ( C ) . emark 4.40. Under the above identification of the twisted characters, the 2-cocycle φ representing a G -crossed extension K ( D ) of K ( B ) changes to φ = sgn tr · φ . Corollary 4.41. (Twisted Verlinde formula) We continue with the same notations as the Corollary 4.32.Moverover assume that B is a modular category and M j ∈ P m j thendim Hom( , Ω g ,h ⊗ · · · ⊗ Ω g r ,h r ⊗ M ⊗ · · · ⊗ M s )= (dim( B )) r − X D ∈ P G ◦ ( S ,D ) s +2 r − S m M ,D · · · S m s M s ,D χ D ◦ φ ( g , h , g − , h − , · · · , m , · · · , m s ) . Remark 4.42.
We continue using the same notation as Proposition 4.30. Fix a g ∈ G then we have anaction of g on D . Let us form a g -equivariantization D g from this action. The objects of D g can be thought aspairs ( X, ψ X ), where X ∈ D and ψ X : g ( X ) ∼ = −→ X and D g = ⊕ h ∈ C G ( g ) C gh . The Grothendieck ring K ( B g ) is acommutative ring with identity element [( , id )] and K ( D g ) is a C G ( g )-graded ring with trivial componentequal to K ( B g ). Let us form a quotient algebras K ( B , g ) := K ( B )([( , ω )] − ω [ , id ]) and K ( D , g ) := K ( D )([( , ω )] − ω [ , id ])where ω is a primitive o ( g ) th root of unity. The ⋆ -map on K ( D ) and complex conjugation on C induces a ⋆ -map on K ( B , g ) and K ( D , g ). Furthermore, if g ′ ∈ C G ( g ) then we have an induced action of g ′ on K ( D , g ).Thus K ( D , g ) is a C G ( g )-crossed Frobenius ⋆ -algebra with trivial component equal to K ( B , g ). For moredetails about these algebras see [Des17]. Let G be a finite group and let A = ⊕ g ∈ G R g be a G -graded ring. Recall that A is called strongly graded if the multiplication map R g ⊗ Z R h → R gh is surjective. In this situation we say that A is a strongly G -graded extension of R . By [Dad80] for a strongly G -graded ring A the induced maps R g ⊗ R R h → R gh are isomorphisms. In particular R g is an invertible R -bimodule for any g ∈ G . Thus a strongly G -gradedextension defines a group homomorphism g R g from G to Pic( R ), the group of isomorphism classes ofinvertible R -bimodule. A group homomorphism ρ : G → Pic( R ) is called as realizable if ρ comes from somestrongly G -graded extension R as defined above.Let ρ : G → Pic( R ) be a group homomorphism. For each isomorphism class ρ ( g ), choose an invertible R -bimodule R g . Since ρ is a group homomorphism, the modules R g ⊗ R R h and R gh are isomorphic as R -bimodule for each pair g, h ∈ G . Then we can choose R -bimodule isomorphisms φ g,h : R g ⊗ R R h → R gh with φ ,g ( r ⊗ x ) = rx and φ g, ( x ⊗ r ) = xr for g ∈ G, r ∈ R, x ∈ R g . Then by [CG01] the following diagram: R f ⊗ R R g ⊗ R R h R fg ⊗ R R h R f ⊗ R R gh R fghφ f,g ⊠ Id Rh Id Rf ⊠ φ g,h φ fg,h φ f,gh commutes upto an element T ( ρ ) ∈ H ( G, Z ( R ) × ).The following result is proved in [CG01]: Theorem 4.43.
The set of isomorphism classes of strongly G -graded extensions of ring R corresponding toa group homomorphism ρ : G → Pic( R ) is non empty iff T ( ρ ) is trivial in H ( G, Z ( R ) × ) . Moreover, the setof isomorphism classes of strongly G -graded extensions of ring R corresponding to a group homomorphism ρ : G → Pic( R ) is torsor under the abelian group H ( G, Z ( R ) × ) . emark 4.44. Let R be a ring with unity and let α be an automorphism of R then one can associate to itan invertible R -module R α , which is the same left R -module as R with right action given by x · y = xα ( y )for all x, y ∈ R . Lemma 4.45.
Let ρ : G → Aut( R ) be a group homomorphism. Then it induces a homomorphism ρ ′ : G → Pic( R ) and the corresponding element T ( ρ ′ ) ∈ H ( G, Z ( R ) × ) is trivial. Proof.
For any automorphism ρ g := ρ ( g ) of R , we can associate an invertible R -module R g as in Remark4.44. Then φ ′ g,h : R g ⊗ R R h → R gh defined by φ ′ g,h ( x ⊗ y ) = xρ g ( y ) and extend linearly to R g ⊗ R R h is anisomorphism. Hence, g R g is a group homomorphism and A = ⊕ g ∈ G R g is a strongly graded extension of R . By Theorem 4.43, T ( ρ ′ ) ∈ H ( G, Z ( R ) × ) is trivial. Remark 4.46.
Let ρ : G → Aut( R ) be a group homomorphism and let A = ⊕ g ∈ G R g be the correspondingstrongly G -graded extension of R (constructed in the proof of Lemma 4.45). Using this strongly G -gradedextension A of R , we get a bijection between the set of isomorphism classes of strongly G -graded extensionsof R corresponding to homomorphism ρ ′ and H ( G, Z ( R ) × ). Remark 4.47.
Let R be a commutative Frobenius ⋆ -algebra. Then by [Bas68, Prop.5.4], Aut( R ) ∋ α [ R α ] ∈ Pic( R ) is an isomorphism. Proposition 4.48.
Let R be a commutative Frobenius ⋆ -algebra. Suppose ρ : G → Aut
Frob ( R ) ⊂ Aut( R ) ∼ =Pic( R ) is a group homomorphism. Let A = ⊕ g ∈ G A g be a strongly G -graded extension of R corresponding tothe homomorphism ρ . Then A has a unique structure of a G -graded Frobenius ⋆ -extension of R and Z R ( A )is a strict G -crossed extension of R . Proof.
Note that A is a strongly G -graded extension of R and A g is isomorphic to R as left R -module forall g ∈ G . Also the right action of R on A g is given by ρ g . As A is a strongly G -graded extension of R , sowe have a short exact sequence: 1 → R × → A × → G → . By Remark 3.21, every extension of G by R × comes from some extension of G by U as in the followingcommutative diagram: 1 U U ( A ) G R × A × G ⋆ -map on U ( A ) ⊆ A as x ⋆ = x − . Then one can extend this ⋆ -map to A by R -semilinearly. Thus, A with the above defined ⋆ -map becomes a G -graded Frobenius ⋆ -algebra. This proves the first part ofproposition.Now, Z R ( A ) is a G -graded Frobenius ⋆ -algebra. Note that A × acts on Z R ( A ) by conjugation and thenormal subgroup R × acts trivially on Z R ( A ). Hence we get a well defined action of G ( ∼ = A × /R × ) on Z R ( A ).With this action of G , Z R ( A ) becomes a G -crossed Frobenius ⋆ -algebra extension of R . As A is a strongly G -graded algebra, the partial action of G on Sim( R ) is the proper action of G on Sim( R ) induced by thegiven action of G on R . In particular, the fixed points for the partial action and the proper action of G onSim( R ) are equal. Moreover by Remark 3.18(1) the fixed points of the two partial actions on Sim( R ) comingfrom the G -graded algebras A and Z R ( A ) are equal. Hence by the definition of strict G -crossed extensions, Z R ( A ) is a strict G -crossed Frobenius ⋆ -extension of R . This proves the result. Remark 4.49.
Let R be a commutative Frobenius ⋆ -algebra with a group homomorphism ρ : G → Aut
Frob ( R ). Then by using Remark 4.46, Theorem 1.1 and by Prop. 4.48, A Z R ( A )22s a bijection between the set of isomorphism classes of strongly graded extensions of the commutativeFrobenius ⋆ -algebra R corresponding to the homomorphism ρ and set of all isomorphism classes strict G -crossed extensions of R (corresponding to homomorphism ρ ). Note that by [CG01], the strongly gradedextensions of R as above are also classified by H ( G, R × ). References [Aro18] Prashant Arote. Twisted orthogonality relations for certain Z /m Z -graded algebras, 2018.[Bas68] Hyman Bass. Algebraic K-theory . W. A. Benjamin, 1968.[CG01] Antonio M. Cegarra and Antonio R. Garz´on. Obstructions to Clifford system extensions ofalgebras.
Proc. Indian Acad. Sci. Math. Sci. , 111(2):151–161, 2001.[Dad80] Everett C. Dade. Group-graded rings and modules.
Math. Z. , 174(3):241–262, 1980.[Dad86] Everett C. Dade. Clifford theory for group-graded rings.
Journal f¨ur die reine und angewandteMathematik , 369:40–86, 1986.[Des17] Tanmay Deshpande. Modular categories, crossed S-matrices, and Shintani descent.
Int. Math.Res. Not. IMRN , (4):967–999, 2017.[Des19] Tanmay Deshpande. On centers of bimodule categories and induction-restriction functors.
Int.Math. Res. Not. IMRN , (2):578–605, 2019.[DGNO10] Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. On braided fusioncategories. I.
Selecta Math. (N.S.) , 16(1):1–119, 2010.[DM19] Tanmay Deshpande and Swarnava Mukhopadhyay. Crossed modular categories and the Verlindeformula for twisted conformal blocks. arXiv e-prints , page arXiv:1909.10799, September 2019.[ENO05] Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. On fusion categories.
Ann. of Math. (2) ,162(2):581–642, 2005.[ENO10] Pavel Etingof, Dmitri Nikshych, and Victor Ostrik. Fusion categories and homotopy theory.
Quantum Topol. , 1(3):209–273, 2010. With an appendix by Ehud Meir.[Lus87] G. Lusztig. Leading coefficients of character values of Hecke algebras. In
The Arcata Conferenceon Representations of Finite Groups (Arcata, Calif., 1986) , volume 47 of
Proc. Sympos. PureMath. , pages 235–262. Amer. Math. Soc., Providence, RI, 1987.[Sch13] Gregor Schaumann. Traces on module categories over fusion categories.
J. Algebra , 379:382–425,2013.[Tur00] Vladimir Turaev. Homotopy field theory in dimension 3 and crossed group-categories, 2000.[Tur08] Vladimir Turaev. Crossed group-categories.