Representations and Conjugacy Classes of Semisimple Quasitriangular Hopf Algebras

In this paper we give two general formulae for the M\"uger centralizers in the category of representations of a semisimple quasitriangular Hopf algebra. The first formula is given in the terms of the Drinfeld map associated to the quasitriangular Hopf algebra. The second formula for the M\"uger centralizer is given in the terms of the conjugacy classes introduced by Cohen and Westreich in [J. Algebra 283 (2005), 42-62]. In the case of a factorizable Hopf algebra these formulae extend some particular cases obtained by the author in [Math. Z. 279 (2015), 227-240].


Introduction
The notion of centralizer in a braided fusion category was introduced by Müger in [25]. It was shown in [12,Theorem 3.13] that the centralizer of a nondegenerate fusion subcategory of a braided category is a categorical complement of the nondegenerate subcategory. This principle is the basis of many classification results of braided fusion categories, see for example papers [11,12,15] and references therein.
Despite its importance, in the current literature there is no concrete known formula for the Müger centralizer of all fusion subcategories of a given fusion category. Only few particular cases are completely known in the literature. For instance, in the same paper [25], Müger described the centralizer of all fusion subcategories of the category of finite-dimensional representations of a Drinfeld double of a finite abelian group. More generally, for the category of representations of a (twisted) Drinfeld double of an arbitrary finite group, not necessarily abelian, a similar formula was then given in [26]. For the braided center of Tambara-Yamagami categories, in [16], the centralizer can be described by computing completely the S-matrix of the modular category. In [5] a different approach gave a partial formula for the centralizer of fusion subcategories of a braided equivariantized fusion category.
Given a fusion subcategory D of a braided fusion category C, the notion of Müger centralizer of D was introduced in [12]. The centralizer D is defined as the fusion subcategory D of C generated by all simple objects X of C satisfying c X,Y c Y,X = id X⊗Y for all objects Y ∈ O(D) (see also [25]). For a fusion category C as usually, we denote by O(C) the set of isomorphism classes of simple objects of C.
If (A, R) is a quasitriangular Hopf algebra then the category Rep(A) of finite-dimensional A-modules is a braided category with the braiding given by c M,N : M ⊗ N → N ⊗ M, m ⊗ n → R 21 (n ⊗ m) = R (2) n ⊗ R (1) m, for any two objects M, N ∈ Rep(A). Given a quasitriangular Hopf algebra (A, R) one can also define the Drinfeld map where Q = R 21 R is the monodromy matrix. We prove the following theorem which gives a general description for the centralizer of any fusion subcategory of the category of representations of a quasitriangular Hopf algebra: We denote by F 0 , F 1 , . . . , F r the central primitive idempotents of the character ring C(A) where F 0 = t is the idempotent integral of A * . Following [7] one can define the conjugacy classes C j of A as C j := Λ F j A * , where Λ is an idempotent integral of A and a f = f, a 1 a 2 for all a ∈ A and f ∈ A * . It is well known that these conjugacy classes are the simple D(A)submodules of the induced D(A)-module k ↑

D(A) A
A, see [31]. Let (A, R) be a semisimple quasitriangular Hopf algebra and V 0 = k, . . . , V r be a complete set of isomorphism classes of irreducible A-modules. Let also Irr(A) = {χ 0 = , χ 1 , . . . , χ r } be the set of irreducible characters afforded by these modules and E i ∈ Z(A) be the associated central primitive idempotent of the irreducible character χ i . Since the Drinfeld map φ R : C(A) → Z(A) is an algebra map we may suppose that φ R (F j ) = i∈A j E i for some subset A j ⊆ {0, . . . , r}.
Since φ R (1) = 1 we obtain a partition for the set of indices of all irreducible representations {0, 1, . . . , r} = j∈J A j . For any 0 ≤ i ≤ r we denoted by m(i) the unique index j ∈ J such that i ∈ A j . Therefore in this way we obtain a unique function m : {0, 1, . . . , r} → J with the property that E i φ R (F m(i) ) = 0 for all i ∈ {0, 1, . . . , r}.
Our second main result is the following: is a semisimple quasitriangular Hopf algebra and L is a left normal coideal subalgebra of A. With the above notations one has Recall that the quasitriangular Hopf algebra (A, R) is called factorizable if the Drinfeld map φ R : A * → A is an isomorphism of algebras. In this case, its restriction φ R | C(A) : C(A) → Z(A) is an isomorphism of algebras. For a factorizable semisimple Hopf algebra we can record the primitive central idempotents F j of C(A) such that F j := φ −1 R (E j ) any 1 ≤ j ≤ r. With these notations, m(i) = i for any 0 ≤ i ≤ r and Theorem 1.2 implies the following: Corollary 1.3. Let (A, R) be a semisimple factorizable Hopf algebra and L be a left normal coideal subalgebra of A. Then with the above notations one has that Shortly, this paper is organized as follows. In Section 2 we recall the basic notions of Hopf algebras and fusion categories that are used throughout this paper. In this section we also prove a canonical decomposition of a left normal coideal subalgebra in terms of the decomposition of its integral, see equation (2.2). In Section 2.5 we recall the main properties of quasitriangular Hopf algebras and their associated Drinfeld maps. In Section 3 we prove Theorem 1.1 and some consequences of it. In particular we apply Theorem 1.1 to the adjoint subcategory of the category of representations of a factorizable Hopf algebra. In this way we obtain a relation, via the Drinfeld map, between the Hopf center and the first commutator of a factorizable Hopf algebra.
In Section 4 we prove Theorem 1.2. Some consequences of this result are also described. In Section 5 we give an example, by considering the semisimple quasitriangular Hopf algebra H 8 of dimension 8. Based on our results we are able to compute the function m in this case and therefore the centralizer of any fusion subcategory of Rep(H 8 ).
We work over an algebraically closed field k of characteristic zero. The comultiplication and antipode of a Hopf algebra are denoted by ∆ and S respectively. We use Sweedlers notation for comultiplication with the sigma symbol dropped. All the other Hopf algebra notations are those used in [24].

Preliminaries
Let A be a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic zero. Then A is also cosemisimple and S 2 = id [21]. The character ring C(A) := G 0 (A) ⊗ Z k is a semisimple subalgebra of A * and it has a vector space basis given by the set Irr(A) of irreducible characters of A, see [30]. Moreover, C(A) = Cocom(A * ), the space of cocommutative elements of A * . By duality, the character ring of A * is a semisimple subalgebra of A and C(A * ) = Cocom(A). If M is an A-representation with character χ then M * is also an A-representation with character χ * = χ • S. This induces an involution " * ": C(A) → C(A) on C(A).
Throughout of this paper we denote by Λ an idempotent integral of A and by t an idempotent integral of A * . Moreover one has that t(Λ) = 1 dim k (A) . Recall also [20] that Recall that a left coideal subalgebra of A is a subalgebra L ⊆ A with ∆(L) ⊂ A ⊗ L. Then L is called left normal coideal subalgebra if L is closed under the left adjoint action of A, i.e., a 1 lS(a 2 ) ∈ L for any l ∈ L and any a ∈ A. Recall also from [1] that given a left coideal subalgebra L of A there is a unique element Λ L ∈ L (called integral) such that lΛ L = (l)Λ L for all l ∈ L, see also [19]. Then the coideal subalgebra L is normal if and only if Λ L is central, i.e., Λ L ∈ Z(A).
For any left normal coideal subalgebra L of A the augmentation ideal AL + is a Hopf ideal and it has the following form AL + = A(1 − Λ L ) = Ann A (Λ L ). Thus one can define the Hopf quotient A//L := A/AL + . It is well known that any fusion subcategories of Rep(A) can be written as Rep(A//L) for some Hopf quotient of A.
Remark 2.1. Since A is free as left L-module [29] it follows that the map is an isomorphism of A-modules. Moreover, by [2,Proposition 3.11] it follows that the regular module of the quotient Hopf algebra A//L is isomorphic to the induced module A ⊗ L k.

Duality between the character ring and the center
Let A be a semisimple Hopf algebra over the ground field k. Let us denote by Irr(A) the set of irreducible characters of A. We suppose that Irr(A) = {χ 0 , χ 1 , . . . , χ r }. Without loss of generality we may suppose that χ 0 = . Let also E 0 , E 1 , . . . , E r be the corresponding central primitive central idempotents in A. The evaluation form is nondegenerate. A pair of dual bases for this form is given by

Another pair of dual basis in the commutative case
Let A be a semisimple Hopf algebra with a commutative character ring. According to [7] in the case of a commutative ring C(A) there is another pair of dual bases corresponding to this nondegenerate form. This pair of dual bases is given in terms of the conjugacy class sums as defined in [7].
The conjugacy classes C j of A are defined as C j = Λ F j A * , where Λ = Λ A is a two-sided idempotent integral of A and {F j } j is the (complete) set of central primitive idempotents of the semisimple algebra C(A). This notion of conjugacy classes generalizes the usual notion of conjugacy classes in finite groups.
Example 2.2. Let G be a finite group and A = kG be the associated group algebra. It is easy to see that the conjugacy classes as defined above coincide with the usual notion of conjugacy class in a group. Indeed, let C 0 , C 1 , . . . , C r be the usual conjugacy classes of G. Then the set of central primitive idempotents of C(kG) can be described as is the vector sub-space of kG generated by all group elements of C j .
Recall that the Fourier transform F : One can also define the corresponding conjugacy class sums h is the usual class sum of a conjugacy class C j .
Remark 2.4. By the class equation for semisimple Hopf algebras, see [22], one has that the value n j : is an integer. Moreover as in [7, equation (11)] one can deduce that F j (Λ) = 1 n j .
Example 2.5. If A = kG then n j = |G| |C j | is the order of the centralizer of any group element g j ∈ C j . This implies that a second pair of dual bases for the form of equation (2.1) can be given by [7, equation (17)]. This can be written as F i ,

Decomposition of the integral
Let L be a left normal coideal subalgebra of a semisimple Hopf algebra A with a commutative character ring C(A). We shall use the notation λ L ∈ (A//L) * for the idempotent integral of the Hopf algebra (A//L) * . Clearly λ L ∈ C((A//L) * ) ⊂ C(A * ) and we may suppose that for some subset of indices I L ⊆ {0, 1, . . . , r}. Note that by [10, Lemma 1.1] Λ L := Λ λ L is a left integral for L. It follows from above that Then one has Note also that It follows then that is a formula for the idempotent integral of L.
Define the functional p C j ∈ A * as the unique linear functional that coincides to on C j and it is equal to zero on the other conjugacy classes C l with l = j. The following lemma was proven in [3, Theorem 5.13].
Lemma 2.6. Suppose that A is a semisimple Hopf algebra with a commutative character ring C(A). Let {F j } 0≤j≤r be a complete set of central primitive idempotents of C(A). Then F j = p C j for all 0 ≤ j ≤ r.

Equation (2.2) and the above lemma implies the following:
Lemma 2.7. Let L be a left normal coideal subalgebra of a semisimple Hopf algebra A. With the above notations one has j ∈ I L ⇐⇒ F j (Λ L ) = 0.
Let A be a semisimple Hopf algebra with a commutative character ring C(A). Then {F j } form a k-linear basis for C(A) and for any character χ ∈ C(A) one can write χ = r j=0 α χ,j F j with α χ,j ∈ k. Previous lemma implies the following result, see also [8,Theorem 1.12].
Proposition 2.8. Let A be a semisimple Hopf algebra with a commutative character ring and χ ∈ C(A). Then one has On the other hand note that Example 2.9. If A = kG then a left normal coideal subalgebra of A is of the form L = kN for some normal subgroup N ¢ G. Then

Left kernels and Burnside formula
Let M be an A-module and let LKer A (M ) be the left kernel of M . Recall [1] that LKer A (M ) is defined by Then by [1] it follows that LKer A (M ) is the largest left coideal subalgebra of A that acts trivially on M . It is also a left normal coideal subalgebra. For example, if A = kG is the group algebra of a finite group G and M is a kG-module then LKer Next theorem generalizes a well known result of Brauer in the representation theory of finite groups. This implies that for any left normal coideal subalgebra L of A one has that The previous theorem also implies that any fusion subcategory of Rep(A) is of the type Rep(A//L) for some left normal coideal subalgebra L of A. Moreover, for any V ∈ Rep(A) one has (2.4)

Quasitriangular and factorizable Hopf algebras
Recall that a Hopf algebra A is called quasitriangular if A admits an R-matrix, i.e., an element R ∈ A ⊗ A satisfying the following properties: Here is a quasitriangular Hopf algebra then the category of representations is a braided fusion category with the braiding given by for any two left A-modules M, N ∈ Rep(A) (see [18]). Recall that R 21 := R (2) ⊗ R (1) . Denote Q := R 21 R. Then the monodromy of two objects is defined as A quasitriangular Hopf algebra (A, R) is called factorizable if and only if the Drinfeld map is an isomorphism of vector spaces. In this situation, following [28, Theorem 2.3] φ R maps the character ring C(A) onto the center Z(A) of A and the restriction φ R | C(A) is an isomorphism of algebras.
Remark 2.11. By [27, Lemma 4.1] one has that φ R (C) is a left normal coideal for any subcoalgebra C of A * .
One can also define the map R φ : In the case of a factorizable Hopf algebra R φ is also bijective and moreover by [9] the two maps R φ and φ R coincide on the character ring C(A).

Proof of Theorem 1.1 on Müger centralizer
In this section we prove the first main theorem mentioned in the introduction. Given a fusion subcategory D of a braided fusion category C, recall that the Müger centralizer D is defined as the fusion subcategory of C generated by all simple objects X of C satisfying c X,Y c Y,X = id X⊗Y for all objects Y ∈ O(D) (see also [25]). Recall that O(D) denotes the set of isomorphism classes of simple objects of D.
Let A be a semisimple quasitriangular Hopf algebra over k and D = Rep(A//L) be a fusion subcategory of Rep(A) where L is a left normal coideal subalgebra of A.
On the other hand note that

On the commutators and Hopf centre
Given a fusion category C we denote by C pt the fusion subcategory generated by the invertible objects of C. In the case C = Rep(A) for a semisimple Hopf algebra we have that C pt is the full abelian subcategory generated by one-dimensional modules. It was shown in [2]  Moreover by [2] A is the smallest left normal coideal subalgebra L with the property that A//L is a commutative Hopf algebra. A is called the the commutator of A. For a fusion category C, recall that the adjoint subcategory C ad is defined as the smallest fusion subcategory generated by all objects of the type X ⊗ X * with X a simple object of C. Recall that a braided fusion subcategory C is called nondegenerate if its Müger center is trivial, i.e., C = Vec. If C is a nondegenerate braided fusion category then by [11,Corollary 3.11] one has (C ad ) = C pt . (3.4) Since D = D for any fusion subcategory D of a nondegenerate braided category C we can also write that (C pt ) = C ad .

Conjugacy classes and Müger centralizer
In this section we will prove Theorem 1.2. Suppose that (A, R) is a semisimple quasitriangular Hopf algebra. Let as above V 0 , V 1 , . . . , V r be a complete set of isomorphism classes of irreducible A-modules. Let also Irr(A) = {χ 0 , χ 1 , . . . , χ r } be the set of irreducible characters afforded by these modules and {E 0 , . . . , E r } be their associated central primitive idempotents of A. Without loss of generality we may suppose that V 0 = k is the trivial A-module and therefore χ 0 = and E 0 = Λ. Recall by [14], that Rep(A) is a ribbon category with the canonical ribbon element v = u −1 , where u := S R 1 R 2 is the Drinfeld element of (A, R). With respect to the canonical ribbon structure given by this ribbon element, the S-matrix of (A, R) has entries

It follows from [12] that one has |s
is an algebra map and we may suppose as in the introduction that Without loss of generality we may also suppose that F 0 = t, the idempotent integral of A * . For any index 0 ≤ i ≤ r we denoted by m(i) the unique index j ∈ J such that i ∈ A j . Therefore in this way we obtain a unique function m : {0, 1, . . . , r} → J with the property that E i φ R (F m(i) ) = 0 for all i ∈ {0, 1, . . . , r}.
Recall from Section 2.4 the definition of the left kernel of an A-module.
Lemma 4.1. Let (A, R) be a quasitriangular Hopf algebra and V i , V i be two irreducible Arepresentations. Then, with the above notations the following assertions are equivalent: Proof . For any character χ ∈ C(A) write as above χ = r j=0 α χ,j F j . Then one has that With these formulae note that where m(i) as above, is the unique index j ∈ J with i ∈ A j . Therefore we see that V i centralize V i if and only if The equivalence of assertions (2) and (4) follows from [9,Theorem 3.6]. The rest of the equivalences follow from the symmetry property of the centralizer.

Remark 4.2. The above lemma also shows that if
Next theorem is a generalization of Theorem 1.2.

Proof of Theorem 1.2
Proof . Using the previous lemma we have the following equalities:

Description of Φ(A)
As above denote by Φ(A) := φ R (A * ) the image of the Drinfeld map. On the other hand, Theorem 1.2 gives the following equality It is easy to see that

Proof of Corollary 1.3
This is now a particular case of Theorem 1.2.
Proof . Note that if A is factorizable then φ R is bijective and every set A j is a singleton. Moreover φ R (F 0 ) = E 0 , the integral of A in this case. Then, without loss of generality, after a permutation of the indices, we may suppose φ R (F i ) = E i and therefore m(i) = i for all 0 ≤ i ≤ r. Then the statement of Theorem 1.2 becomes Theorem 1.3.
For the rest of this subsection we suppose that A is a semisimple factorizable Hopf algebra. As explained above without loss of generality we may also assume φ R (F i ) = E i and therefore that the function m : {0, 1, . . . , r} → {0, 1, . . . , r} is the identity map.
Proposition 4.6. Suppose that (A, R) is a semisimple factorizable Hopf algebra. Then for any irreducible A-module V i one has that Proof . Since J = {0, 1, . . . , r} in this case, by equation (4.1) and [9, Theorem 3.6] one has that It follows that in this case one has 1. From the previous proposition, in the case of a semisimple factorizable Hopf algebra one can deduce that for any two irreducible characters χ i and χ i one has

Recall that in [4, Theorem 1.4] it is shown that
for any normal Hopf subalgebra K of a factorizable Hopf algebra A. Note that Proposition 4.3 generalizes the above result from normal Hopf subalgebras K to left normal coideal subalgebras L of A. It also drops the factorizability assumption on A.
Define C V i := C χ i ⊂ A * as the subcoalgebra of A * generated by χ i . By [9, Lemma 4.2(i)], in the factorizable case one has that φ R (C V i ) = C i . for all 0 ≤ i ≤ r.

Example H 8
In this section we compute the centralizer of any fusion subcategory of the quasi-triangular Hopf algebra H 8 , the unique semisimple non-trivial Hopf algebra of dimension 8. We note that the category of representations Rep(H 8 ) is a braided Tambara-Yamagami category and therefore Rep(H 8 ) ⊂ Rep (D(H 8 )). The S-matrix of the center of a Tambara-Yamagami was computed in [16]. Using this one can describe completely the centralizer of any fusion subcategory of Rep(H 8 ). However, we decided to include this example here to illustrate how Theorem 1.2 can be applied in a concrete example.
The eight-dimensional semisimple Hopf algebra (see [17,23]) is generated by {x, y, z} subject to the relations The comultiplication is given by Based on equation (5.1) one can compute that It can also be checked that Since z is a central element in H 8 , there are two central orthogonal idempotents: (1 + x + y + xy + z + zx + zy + yx), λ = p 1 .

H * 8 representations
Since H 8 is a self dual Hopf algebra [13] it has also four-1-dimensional representations given by the group like elements of H 8 and a 2-dimensional representation. One has that G(H 8 ) = {1, g 1 , g 2 , z}, with g 1 = xy(e 0 + ie 1 ), g 2 = xy(e 0 − ie 1 ). It can be easily checked that g 1 g 2 = z, zg i = g i and g 2 i = 1. Moreover, the set of central grouplike elements of H 8 is given byḠ(H 8 ) = {1, z}.

On the 2-dimensional comodule
From equation (5.1) one can compute that This shows that W = kw 1 ⊕ kw 2 is a left H 8 -comodule with the comodule structure given by Thus is

Fourier transform
Based on the comultiplication formulae one can compute the Fourier transform After some computations it follows that under F one has

Irreducible H 8 -modules and their characters
The action of the generators on these modules is given as follows.

Central primitive idempotent of the character ring
Based on the above multiplication, the central primitive idempotents of C(H 8 ) can be computed as follows Note that

Conjugacy class sums of H 8
Using the above formulae for the central primitive idempotents since C i = 8 Λ F i it follows that and

Description of the adjoint action
Using the antipode formulae one can see that the adjoint action of H 8 on itself can be given by x.a = xax, y.a = yay, z.a = a, for all a ∈ H 8 .

Conjugacy classes of H 8
Recall that the conjugacy classes are simple D(H 8 )-modules [31]. We rewrite the decomposition of H 8 into left H 8 -comodules from equation (5.2) as follows Moreover, by the above description of the left adjoint action it can be checked each of the above five subspaces is closed under the adjoint action of H 8 . Clearly, the first two subspaces are irreducible D(H 8 )-modules being one-dimensional. Since C 0 ∈ k1 and C 4 = z ∈ kz we deduce that It can be checked directly that the third D(H 8 )-module is an irreducible D(H 8 )-module since x.g 1 = g 2 , x.g 2 = g 1 , y.g 1 = g 2 , y.g 2 = g 1 .
Since C 2 = g 1 + g 2 = xy + yx it follows that Similarly one can check that the simple H 8 -comodule M 4 = kxe 0 ⊕kye 1 is an irreducible D(H 8 )module and since C 3 = 2xe 0 = x + xz ∈ M 4 we can say that By the same argument

Presentation of the central primitive idempotents of H 8
The associated central primitive idempotents of V 0 , V 1 , V 2 , V 3 , V can be computed as E 0 = e 0 4 (1 + x + y + xy) = Λ is the central primitive idempotent of χ 0 , i.e., the idempotent integral Λ H 8 of H 8 , is the central primitive idempotent of χ 1 , is the central primitive idempotent of χ 2 , is the central primitive idempotent of χ 3 , is the central primitive idempotent of χ 4 .
As above, using Theorem 1. All the four one-dimensional centralize each other. V centralizes only χ 2 .

On the first commutator and adjoint subcategory
One has that the first commutator of H 8 is given by H 8 = k 1, z and moreover, for this Hopf algebra C ad = C pt . Thus in this case