Constructing non-semisimple modular categories with relative monoidal centers

This paper is a contribution to the construction of non-semisimple modular categories. We establish when M\"uger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoidal centers give (non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give (non-semisimple) modular categories.


Introduction
The purpose of this article is to establish new constructions of modular tensor categories in the non-semisimple setting. We work over an algebraically closed field k.
To begin, let us recall the main structure of interest in this work, which is due to Kerler-Lyubashenko [KL01]. We refer the reader to Section 2 for a discussion of various types of tensor categories relevant here. Take vect k to be the tensor category of finite-dimensional k-vector spaces, and for a braided tensor category C, let C 1 be the Müger center of C (see (2.4)).
Definition 1.1 (Definitions 2.10, 2.11). Take C a braided finite tensor category. We call C a modular tensor category (MTC) if C is non-degenerate (i.e., C 1 » vect k ) and ribbon.
Note that the definition above does not require semisimplicity, as the commonly used definition of an MTC (see, e.g., [BK01]). MTCs provide actions of the modular group though their modular data, the S-and T -matrices, a structure that emerged from mathematical physics [MS89]. Semisimple MTCs have appeared in various fields such as low-dimensional topology [Tur92], conformal field theory [MS89,Hua05,Gan05], and subfactor theory [KLM01]; they have been under intense investigation towards classification results by rank (see, e.g., [RSW09]).
The definition of a non-semisimple MTC of [KL01] has been given further justification through equivalent characterizations in [Shi19]. Moreover, non-semisimple MTCs are gaining traction due to their growing list of applications, starting with non-semisimple topological quantum field theories [KL01], most recently in [DRGG`19], to the study of logarithmic conformal field theories [HLZ14], modular functors [FSS19], and mapping class group actions [LMSS20]. Some module categories of small quantum groups (and of related quasi-Hopf algebras) have been shown to yield examples of non-semisimple MTCs [Lus10,GLO18,LO17,Neg21]. But, in general, non-semisimple MTCs are the full image GpBq is a topologizing subcategory of ZpCq, then the relative monoidal center Z B pCq is modular.
For B a rigid braided category and H a Hopf algebra in B, Theorem 1.4 can be used to study the modularity of the category of finite-dimensional H-Yetter Drinfeld modules in B [Example 4.12]. If, further, B is a representation category of a quasi-triangular Hopf algebra K, then Theorem 1.4 can also be used to study the modularity of the representation category of the braided Drinfeld double Drin K pH, H˚q [Example 4.13]. From this, we show, as a first example, that the representation category of the small quantum group u q psl 2 q, for q a root of unity of odd order, is an MTC [Example 5.3]. Generalizations of this (non-semisimple) MTC will be given in Proposition 1.7 below.
Motivated again by work of Müger in the semisimple case, we next consider the decomposition of modular tensor categories into Deligne tensor product of modular subcategories. We obtain the result below; cf., [Müg03b,Theorem 4.2].
Theorem 1.5 (Theorem 4.17). Let D be a modular tensor category, with a topologizing nondegenerate braided tensor subcategory E. Then, there is an equivalence of ribbon categories: In particular, under the conditions of Theorem 1.4, the relative monoidal center is related to the monoidal center through the factorization ZpCq » B ⊠ Z B pCq.
Continuing an example mentioned above, for H a Hopf algebra in the braided tensor category K-mod, we have that DrinpH ⋊ Kq-mod » K-mod ⊠ Drin K pH, H˚q-mod as modular categories under the hypotheses of Theorem 1.4; see Example 4.18(2).
As in [Müg03b], we call a modular tensor category C in the non-semisimple setting prime if every topologizing non-degenerate braided tensor subcategory is equivalent to either C or vect k . We obtain the result below as an immediate consequence of the theorem above, cf. [Müg03b,Theorem 4.5].
Corollary 1.6 (Corollary 4.20). Every modular tensor category is equivalent to a finite Deligne tensor product of prime modular categories.
Although primality is difficult to detect in the semisimple case (see [Müg03b, Section 4]), we inquire when it holds in the non-semisimple case, particularly for Drin K pH, H˚q-mod in the example above; see Question 4.21.
Finally, we construct several examples of non-semisimple MTCs, via Theorem 1.4, by using Nichols algebras of diagonal type in braided categories of comodules over finite abelian groups.
Proposition 1.7 (Proposition 5.15). Take K :" kG, for G a finite abelian group, assume that char k " 0, and take B a finite-dimensional Nichols algebra of diagonal type in a certain braided category B of K-comodules. Consider the relative monoidal center, D :" Z B pB-modpBqq, or equivalently the category of finite-dimensional modules over the braided Drinfeld double Drin K˚p B˚, Bq.
Then, D is modular when (i) the canonical symmetric bilinear form b on the coquasi-triangular Hopf algebra K is non-degenerate, and (ii) certain conditions involving elements of the top degree of B and on the dual R-matrix of K are satisfied.
Note that the Drinfeld double of the bosonization of Nichols algebras has been studied in the literature, see e.g. [Hec10], where two copies of the group algebra consitute the Cartan part. In this paper, an approach is used where the Cartan part consists of a simple (dual) group algebra K˚.
We end the paper by constructing, via Proposition 1.7, examples of non-semisimple modular tensor categories attached to Nichols algebras of Cartan type [Example 5.17] and not of Cartan type [Example 5.18]. The former includes the representation category of the small quantum group u q pgq at an odd root of unity. Thus, the methods developed in this paper provide an alternative argument showing that the category of finite-dimensional u q pgq-modules is a non-semisimple MTC, which was previously obtained in [Lyu95,Section A.3]. See also [LO17,GLO18] for more general results on the modularity of representation categories of small quantum groups. On the other hand, the non-semisimple MTCs in Example 5.18 illustrate that our methods can be used to analyze the modularity of representation categories attached to a broader class of Nichols algebras beyond small quantum groups.

Preliminaries on Monoidal Categories
In this section, we review terminology pertaining to monoidal categories. We refer the reader to [BK01], [EGNO15], and [TV17]  We assume that all categories here are locally small (i.e., the collection of morphisms between any two objects is a set), and that all categories here are abelian. A full subcategory of a category is called topologizing if it is closed under finite direct sums and subquotients [Ros95, Section 3.5.3], [Shi19, Definition 4.3]. Given a functor F : C Ñ D between two categories C and D, the full image of F is the full subcategory of D on all objects isomorphic to an object of the form F pCq for C in C. A monoidal category consists of a category C equipped with a bifunctor b : Zq for each X, Y, Z P C, an object 1 P C, and natural isomorphisms l X : 1 b X " Ñ X and r X : X b 1 " Ñ X for each X P C, such that the pentagon and triangle axioms hold. By MacLane's coherence theorem, we will assume that all monoidal categories are strict in the sense that pX b Y q b Z " X b pY b Zq and 1 b X " X " X b 1, for all X, Y, Z P C; that is, α X,Y,Z , l X , r X are identity maps. For a monoidal category pC, b, 1q, define the opposite monoidal category A (strong) monoidal functor pF, F´,´, F 0 q between monoidal categories pC, b C , 1 C q to pD, b D , 1 D q is a functor F : C Ñ D equipped with a natural isomorphism F X,Y : F pXq b D F pY q " Ñ F pX b C Y q for all X, Y P C, and an isomorphism F 0 : 1 C " Ñ F p1 C q in D, that satisfy associativity and unitality constraints. An equivalence of monoidal categories is provided by a monoidal functor between the two monoidal categories that yields an equivalence of the underlying categories.
Representations of monoidal categories are provided by the next notion. A left C-module category is a category M equipped with a bifunctor b : CˆM Ñ M, natural isomorphisms for associativity m X,Y,M : pX b Y q b M Ñ X b pY b M q, for all X, Y P C, M P M satisfying the pentagon axiom, and for each M P M a natural isomorphism 1 b M Ñ M satisfying the triangle axiom.
A monoidal category pC, b, 1q is rigid if it comes equipped with left and right dual objects, i.e., for each X P C there exist, respectively, an object X˚P C with co/evaluation maps ev X : X˚b X Ñ 1 and coev X : 1 Ñ X b X˚, as well as an object˚X P C with co/evaluation maps r ev X : X b˚X Ñ 1, Ć coev X : 1 Ñ˚X b X, satisfying the usual coherence conditions of left and right duals. An object X in a rigid monoidal category C is invertible if ev X and coev X are isomorphisms.
A rigid monoidal category is pivotal if it is equipped with isomorphisms j X : X " Ñ X˚˚natural in X and satisfying j XbY " j X b j Y for all X, Y P C. Equivalently, a pivotal category is a rigid monoidal category such that the functors of left and right duality coincide as monoidal functors [TV17, Section 1.7].
The quantum dimension of an object X of a pivotal (rigid) monoidal category pC, b, 1, jq is defined to be dim j pXq " ev X˚p j X bId X˚q coev X P End C p1q. A pivotal monoidal category pC, b, 1, jq is trace-spherical if dim j pXq " dim j pX˚q for each X P C.
2.3. Finite tensor categories. Recall that k is an algebraically closed field. We now discuss certain k-linear monoidal categories following the terminologies of [EGNO15, Sections 1.8, 7.1-7.3, 7.9].
A k-linear abelian category C is locally finite if, for any two objects V, W in C, Hom C pV, W q is a finite-dimensional k-vector space and every object has a finite filtration by simple objects. Moreover, we say that C is finite if there are finitely many isomorphism classes of simple objects. Equivalently, C is locally finite if it is equivalent to the category of finite-dimensional comodules over a k-coalgebra (or, to modules over a finite-dimensional k-algebra if C is finite). A tensor category is a locally finite, rigid, monoidal category pC, b, 1q such that b is k-linear in each slot and 1 is a simple object of C. A tensor functor is a k-linear exact monoidal functor between tensor categories.
An example of a finite tensor category is vect k , the category of finite-dimensional k-vector spaces. More generally, the category H-mod of finite-dimensional k-modules over a (finite-dimensional) Hopf algebra H is a (finite) tensor category.
We will use the following tensor product of finite tensor categories. The Deligne tensor product of two finite abelian categories is the abelian category C ⊠ D equipped with a bifunctor ⊠ : CˆD Ñ C ⊠ D, pX, Y q Þ Ñ X ⊠ Y , right exact in both variables so that for any abelian category A and any bifunctor F : CˆD Ñ A right exact in both slots, there exists a unique right exact functor F : C ⊠ D Ñ A with F˝⊠ " F [Del90, Section 5]. It is monoidal when both C and D are so, via for all X, X 1 P C and Y, Y 1 P D, and with the unit object 1 C ⊠ 1 D . If C, D are finite tensor categories, then so is C ⊠ D. Given two tensor functors F : C Ñ D and F 1 : C 1 Ñ D 1 between finite tensor categories, there exists an induced tensor functor F ⊠ F 1 : For a tensor category C over k, a left module category over C is a module category M as in Section 2.1 with the requirement that M is also k-linear and abelian so that the underlying bifunctor is k-linear on morphisms and exact in the first variable (it is always exact in the second variable).
An internal Hom object for a module category M over a k-linear, finite, tensor category C is an object HompM 1 , M 2 q in C, for M 1 , M 2 P M, that represents the left exact functor C Ñ vect k , defined by X Þ Ñ Hom M pX b M 1 , M 2 q. Namely, we have a natural isomorphism: A braided tensor category pC, b, 1, cq is a tensor category equipped with a natural isomorphism c X,Y : X b Y " Ñ Y b X for each X, Y P C such that the hexagon axioms hold. By a braided tensor subcategory of a braided tensor category C we mean a subcategory of C containing the unit object of C, closed under the tensor product of C, and containing the braiding isomorphisms. A braided tensor functor between braided tensor categories C and D is a tensor functor pF, F˚,˚, F 0 q : C Ñ D so that F Y,X c D F pXq,F pY q " F pc C X,Y q F X,Y for all X, Y P C. An equivalence of braided tensor categories is a braided tensor functor between the two tensor categories that yields an equivalence of the underlying categories.
An important example of a braided tensor category is the monoidal center (or Drinfeld center) ZpCq of a tensor category pC, b, 1q: its objects are pairs pV, c V,´q where V is an object of C and c V, Proposition 2.2 (see [EGNO15,Section 7.13]). If C is a (finite) tensor category, then ZpCq is a braided (finite) tensor category.
Given two braided finite tensor categories pC, b C , 1 C , c C q and pD, b D , 1 D , c D q, the Deligne tensor product C ⊠ D is a braided finite tensor category. The braiding is obtained from We need to consider later the Müger center of a braided tensor category pC, b, 1, cq, which is the full subcategory on the objects (2.4) ObpC 1 q :" tX P C | c Y,X c X,Y " Id XbY for all Y P Cu.
2.5. Algebraic structures in tensor categories. In this section, let C :" pC, b, 1q be a tensor category over k. Assume that all structures below are k-linear as well.
2.5.1. (Co)algebras and their (co)modules. We discuss in this part algebras and coalgebras in C and their (co)modules. More information is available in [EGNO15, Section 7.8] and [TV17, Section 6.1].
An algebra in C is an object A P C equipped with two morphisms m : We denote by AlgpCq the category of algebras in C, where morphisms in AlgpCq are morphisms f : Given an algebra A in C, a left A-module in C is a pair pV, a V q for V an object in C and This way, we define the category A-modpCq of left A-modules in C. Analogously, we define mod-ApCq, the category of right A-modules in C.
A coalgebra in C is an object C P C equipped with two morphisms ∆ : C Ñ C b C (comultiplication) and ε : C Ñ 1 (counit) satisfying p∆bId C q∆ " pId C b ∆q∆ and pεbId C q∆ " Id C " pId C b εq∆. Dual to above, we can define the category CoalgpCq of coalgebras and their morphisms in C, and given C P CoalgpCq we can define categories, C-comodpCq and comod-CpCq, of left and right Ccomodules in C, respectively. For V P C-comodpCq, the left C-coaction map is denoted by 2.5.2. Bialgebras and Hopf algebras. In this part, let C :" pC, b, 1, cq be a braided tensor category over k. We define bialgebras and Hopf algebras in C here, and more details can be found [TV17, Sections 6.1 and 6.2].
A bialgebra in C is a tuple H :" pH, m, u, ∆, εq where pH, m, uq P AlgpCq and pH, ∆, εq P CoalgpCq so that ∆m " pm b mqpId b c b Idqp∆ b ∆q, ∆u " u b u, εm " ε b ε, and εu " Id 1 . We denote by BialgpCq the category of bialgebras in C, where morphisms in BialgpCq are morphisms in C that belong to AlgpCq and CoalgpCq simultaneously.
A Hopf algebra is a tuple H :" pH, m, u, ∆, ε, Sq, where pH, m, u, ∆, εq P BialgpCq and S : H Ñ H is a morphism in C (called an antipode) so that mpS b Id H q∆ " mpId H b Sq∆ " uε. We denote by HopfAlgpCq the category of Hopf algebras in C, where morphisms are morphisms in BialgpCq. We assume that all Hopf algebras in this work have an invertible antipode, that is, there exists a morphism S´1 : H Ñ H is a morphism in C so that SS´1 " Id H " S´1S.
If V, W are left H-modules in C, then so is the tensor product V b W , via the action (2.5) below: This makes the category H-modpCq a monoidal category, with unit object p1 " k, a 1 " ε H b Id 1 q. Assume that C is rigid, and take pV, a V q P H-modpCq. Then its left dual pV˚, a V˚q P H-modpCq is defined using S H , and its right dual p˚V, a˚V q P H-modpCq is defined using S´1 H . It follows that H-modpCq is a (finite) tensor category provided C is a (finite) braided tensor category.
For one supply of braided tensor categories, take the category H-modpvect k q for H a finitedimensional k-Hopf algebra. We say that H is quasi-triangular if it comes equipped with an where ∆ op is the opposite coproduct. It follows that H-modpvect k q is a braided tensor category if and only if the finite-dimensional Hopf algebra H is quasi-triangular; here, the braiding is given by for pV, a V q, pW, a W q P H-modpvect k q. We say that H is coquasi-triangular if it comes equipped with a convolution-invertible bilinear form r : H b H Ñ k satisfying rph, kℓq " rph p1q , ℓqrph p2q , kq, rpℓh, kq " rpℓ, k p1q qrph, k p2q q, . It follows that H-comodpvect k q is a braided tensor category if and only if the finite-dimensional Hopf algebra H is coquasi-triangular; here, the braiding is given by For another supply of braided tensor categories, take a Hopf algebra H in C, and consider the category of H-Yetter-Drinfeld modules in C, denoted by H H YDpCq, which consists of objects pV, a V , δ V q, where pV, a V q P H-modpCq with left H-coaction in C denoted by δ V : V Ñ H b V , subject to compatibility condition: given by a morphism f : V Ñ W in C that belongs to H-modpCq and H-comodpCq. Given two objects pV, a V , δ V q and pW, a W , δ W q in H H YDpCq, their tensor product is given by pV b W, a V bW , δ V bW q, where a V bW as in (2.5) and The category H H YDpCq is braided with braiding given by c YD Further, when C " vect k and dim k H ă 8, we get that H H YDpvect k q is equivalent to the braided tensor category of modules over the Drinfeld double, DrinpHq, see e.g. [Maj00, Theorem 7.1.2] and cf. Example 4.13 below with K " k.
A braided tensor category pC, b, 1, cq is ribbon (or tortile) if it is equipped with a natural isomorphism θ X : X " Ñ X (a twist) satisfying θ XbY " pθ X b θ Y q˝c Y,X˝cX,Y and pθ X q˚" θ X˚f or all X, Y P C. A functor (or, equivalence) of ribbon categories is a functor (respectively, equivalence) F : C Ñ D of braided tensor categories such that F pθ C X q " θ D F pXq , for any X P C, cf. [Shu94, Section 1].
In a ribbon category pC, b, 1, c, θq, consider the Drinfeld isomorphism: defines a pivotal structure on C.
For a supply of ribbon categories, consider the category H-modpvect k q for H " pH, Rq a finitedimensional quasi-triangular k-Hopf algebra. We say that H is a ribbon Hopf algebra if there exists a central invertible element v P H satisfying This definition is equivalent to the one given in [RT90, Section 3.3], [Rad12, Definition 12.3.5]. It follows that H-modpvect k q is a ribbon category if and only if H is a ribbon Hopf algebra [Maj00, Corollary 9.3.4]. In this case, the ribbon twist is given by the action of v´1.
The following lemma will be of use.
Lemma 2.9. Take D a braided full tensor subcategory of a braided tensor category C. If C is ribbon, then so is D.
This result is obtained by restricting the ribbon structure from C to D. Moreover, the ribbon structure of the monoidal center ZpCq will be discussed later in Section 3.2.
2.7. Modular tensor categories. In the section, we discuss a notion of a modular tensor category for the non-semisimple setting. This is based on work of Kerler-Lyubashenko [KL01] and recent work of Shimizu [Shi19]. To proceed, we adopt the definition of non-degeneracy below, which extends the notion of non-degeneracy in the semisimple setting; see [EGNO15, Definition 8.19.2 and Theorem 8.20.7].
Next we discuss a characterization of non-degeneracy. Let pC, c X, Xq be a braided tensor category, and take the braided tensor category: The assignments C Ñ ZpCq, X Þ Ñ pX, c X,´q , and C Ñ ZpCq, X Þ Ñ pX, c´1 ,X q, extend to a braided tensor functor C ⊠ C Ñ ZpCq. If this functor yields an equivalence between the braided tensor categories C ⊠ C and ZpCq, then we say that pC, b, 1, cq is factorizable. A braided finite tensor category is non-degenerate if and only if it is factorizable [Shi19, Theorem 4.2]; note that this article also provides a third equivalent characterization of non-degeneracy in terms of a non-degenerate form on the coend.
Moreover, the following type of tensor categories are of primary interest in this work.
Definition 2.11 ([KL01, Definition 5.2.7], [Shi19, Section 1]). A braided finite tensor category is called modular if it is non-degenerate and ribbon. Now consider the braided finite tensor category H-modpvect k q for H a finite-dimensional, quasitriangular Hopf algebra over k. We get that H-modpvect k q is modular precisely when H is ribbon and factorizable [EGNO15, Proposition 8.11.2 and Example 8.6.4].
Remark 2.12. By Lemma 2.9, we obtain that a topologizing non-degenerate braided tensor subcategory of a modular category is also modular.
Remark 2.13. It is straight-forward to show that if C and D are modular, then so is C ⊠ D via the monoidal structure (2.1), the braiding (2.3), and with ribbon structure θ C⊠D :" θ C ⊠ θ D .
3. Non-semisimple spherical categories and ribbon structures on the center Let C be a finite tensor category over an algebraically closed field k. The purpose of this section is to review sufficient conditions for the monoidal center ZpCq to be a modular tensor category. First, we recall the notion of a distinguished invertible object and the Radford isomorphism of C in Section 3.1. This allows us to recall, in Section 3.2, Shimizu's necessary and sufficient conditions for ZpCq to be ribbon, generalizing a result of Radford-Kauffman in the case when C " H-modpvect k q for H a finite-dimensional Hopf algebra. In Section 3.3, we recall the concept of a spherical category introduced in the work of Douglas-Schommer-Pries-Snyder [DSPS18], expanding the semisimple notion in [BW99] to the non-semisimple setting; it is then applied to describe when ZpCq is modular. Consider C as a C ⊠ C bop -module category. In this case, the canonical algebra is defined as A can :" Homp1, 1q P AlgpC ⊠ C bop q. For example, if H is a Hopf algebra over k, then the canonical algebra in C :" H-modpvect k q is H˚P AlgpC ⊠ C bop q viewed as an H-bimodule over k with left and right H-actions given by translation. The category HopfBimodpCq :" mod-A can pC ⊠ C bop q of right A can -modules in C ⊠ C bop is called the category of Hopf bimodules in C. Both A can and its dual object Ac an belong to HopfBimodpCq. Moreover, HopfBimodpCq is a tensor subcategory of Xq and ρ is the natural action of vect k on C. Continuing the example above, for C :" H-modpvect k q with H P HopfAlgpvect k q, and A can " H˚, we get that HopfBimodpCq is the usual category of Hopf bimodules over H. Moreover, for M, N P HopfBimodpCq, we obtain that M d N " pM˚b H N˚q˚, for pq˚denoting the k-linear dual here.
By [ENO04,Theorem 3.3] and [DSPS18, Theorem 3.3.4], there exists a invertible object D P C so that pD ⊠ 1q d A can -Ac an as objects in HopfBimodpCq. This isomorphism is unique up to a scalar, and D is indeed an invertible object of C. We call D the distinguished invertible object of C.
We also get a canonical natural tensor isomorphism defined as follows. Let F, G : C Ñ C be two tensor functors, and consider the category ZpF, Gq with objects: where the compatibility conditions are [Shi18, (3.1),(3.2)]. Two objects pV, σ V q and pV 1 , This category is not always monoidal, but it is always a finite abelian category [Shi18,Theorem 3.4]. We also have that ZpId C , Id C q is the monoidal center ZpCq. By [Shi18, Lemma 3.3, (4.3)], we get equivalences The first equivalence is an isomorphism, and the second equivalence is induced by C " Ñ HopfBimodpCq given by Y Þ Ñ pY ⊠ 1q d A can . Now the object Ac an in A˚c an -HopfBimod(C) corresponds to pair pD, ξ D q in ZpId C , p´q 4C q. Here, ξ D is called the Radford isomorphism of C. Now consider the case C " H-modpvect k q for a Hopf algebra H over k, and consider the distinguished grouplike elements of H and H˚defined as follows (see [KR93, Section 1] or [Rad12, Section 10.5]). In this case, D is a one-dimensional module, and so the action is given through an invertible character α H P H˚, i.e. h¨d " α´1 H phqd for any d P D. By virtue of D being the distinguished invertible object in C, for a choice of non-zero left integral Λ for H. The Radford isomorphism is now given by the action of an element where λ is a non-zero right integral of H˚. Explicitly, if D " kv, then for any X P C and x P X we implies that pD, ξ D q defines an object in ZpId C , p´q 4˚q .
Recall that a finite tensor category C is unimodular if D " 1 [EGNO15, Section 6.5]. When C is a factorizable finite tensor category, then C is unimodular [EGNO15, Proposition 8.10.10].
3.2. Ribbon structures on the center. In this section, we recall results of [Shi18] and [KR93] on the existence of ribbon structures on the center ZpCq of a finite tensor category, using the pair pD, ξ D q defined in the previous section.
. Let C be a finite tensor category and recall pD, ξ D q from Section 3.1. We define Sqrt C pD, ξ D q to be the set of equivalence classes of invertible objects pV, σ V q in ZpId C , p´q˚˚q such that there exists an isomorphism ν : V˚˚b V " Ñ D such that the following diagram commutes: (3.4) Theorem 3.5 ([Shi18, Sections 5.2, 5.4, 5.5]). The set of ribbon structures on ZpCq is in bijection with the set Sqrt C pD, ξ D q. In particular, if we take pV, σ V q P Sqrt C pD, ξ D q, then the corresponding ribbon structure on ZpCq is given by θ " φ´1j, where φ is the Drinfeld isomorphism of ZpCq from (2.6) and j is a pivotal structure on ZpCq given by for X P C.
In the case when C " H-modpvect k q, Shimizu's theorem specializes to the following classical result of Kauffman-Radford.

2). Then there is a bijection between the sets
and the set of ribbon elements of the Drinfeld double, DrinpHq, cf. (2.8).
The bijection is given by sending a pair pζ, aq to SpR p2q qR p1q pζ´1ba´1q, for R and S the R-matrix and antipode of DrinpHq, respectively.
The precise connection between the above results of Kauffman-Radford and Shimizu is given by the following proposition.
Proposition 3.8. Let H be a finite-dimensional Hopf algebra and C " H-mod. Then there is a bijection between the set of pairs pζ, aq of Theorem 3.6 and the set Sqrt C pD, ξ D q of Definition 3.3.
Proof. Given a pair pζ, aq as in Theorem 3.6, we define pV, σ V q P ZpId, p´q˚˚q of Sqrt C pD, ξ D q as follows. First, V is the one-dimensional H-module with action h¨v " ζ´1phqv for any v P V, h P H.
This isomorphism defines an element in Sqrt C pD, ξ D q provided that pζ, aq satisfy the conditions of Theorem 3.6. In particular, (3.7) implies that σ V pXq is a morphism of H-modules.
Conversely, assume given a pair pV, σ V q P Sqrt C pD, ξ D q. Then V is an invertible H-module and thus is 1-dimensional. Fix a generator v P V . Then we obtain ζ such that h¨v " ζ´1phqv. The isomorphism ν : V˚˚b V Ñ D of H-modules implies that ζ´2 " α´1 H and hence ζ 2 " α H . We obtain an element a P GpHq so that a 2 " g H , satisfying, (3.7) as follows. Recall that there is an isomorphism of Hopf algebras H -EndpF q, where F : C Ñ vect k is the forgetful functor, for details, see e.g. [EGNO15, Sections 5.2-5.3]. Here, h P H gets sent to th X : F pXq Ñ F pXq, x Þ Ñ h¨xu XPObpCq . Further, for the 1-dimensional H-module V fixed above, there are isomorphisms of k-vector spaces f 1 So by identifying the k-vector spaces F pXq and F pX˚˚q, we obtain that the natural isomorphism σ V pXq : F pV q b F pXq " Ñ F pXq b F pV q is of the form f 2 X˝a X˝f 1 X and must be given by v b x Þ Ñ pa¨xq b v for some a P H. The assumption that σ V defines a natural isomorphism V b p´q " Ñ p´q˚˚b V of H-modules implies condition (3.7). The diagram in (3.4) implies that a 2 " g H .
3.3. Non-semisimple spherical categories. Using the distinguished invertible object D defined in Section 3.1, we obtain a notion of sphericality for non-semisimple finite tensor categories.
Definition 3.9 ([DSPS18, Definition 3.5.2]). A pivotal finite tensor category pC, b, 1, jq is spherical if there is an isomorphism ν : 1 " Ñ D so that the following diagram commutes (3.10) In fact, if C is semisimple; then C is spherical precisely when C is trace-spherical; see [DSPS18, Proposition 3.5.4].
Remark 3.11. On the one hand, a spherical category in the sense above gives a special case of a tensor category C satisfying the assumption that Sqrt C pD, ξ D q from Definition 3.3 is non-empty; namely, p1, jq P Sqrt C pD, ξ D q. On the other hand, Example 5.2 later illustrates that there are categories C satisfying Sqrt C pD, ξ D q ‰ ∅ that do not have a spherical structure.
In this case, there is a bijection between pivotal structures j on C such that pC, jq is spherical and the set SPivpHq.
Proof. First assume that C " H-modpvect k q is spherical. Then, by definition, D -1, which implies that α H " ε. By Remark 3.11, p1, jq P Sqrt C pD, ξ D q. Using Proposition 3.8, this element of Sqrt C pD, ξ D q corresponds to a pair pζ, aq satisfying (3.7), with ζ " ε, such that a 2 " g H . Thus, a P SPivpHq. From Proposition 3.8 it further follows that there is a bijection between the subset of Sqrt C pD, ξ D q of pairs pV, σ V q such that V -1 and pairs pε, aq satisfying the conditions of Theorem 3.6 (i.e a 2 " g H and (3.7), or equivalently, a P SPivpHq). Conversely, assume α H " ε and a P SPivpHq ‰ ∅. Again under the bijection of Proposition 3.8, the pair pε, aq corresponds an element p1, σ 1 q P Sqrt C pD, ξ D q. In particular, p1, σ 1 q is an element of ZpId C , p´q˚˚q which implies, using the convention 1 b X " X " X b 1, that j :" σ 1 is a pivotal structure for C, which, by construction, satisfies (3.10). Thus, C is spherical.
Examples of spherical categories obtained from Nichols algebras can be found later in the text, see Remark 5.16(3) and Example 5.18.
Next, we show that a source of non-semisimple spherical categories is given by unimodular ribbon categories, cf. [EGNO15,Proposition 8.10.12] in the semisimple case. Recall that in the semisimple case, every finite tensor category is unimodular [EGNO15, Remark 6.5.9].
Proposition 3.13. Any unimodular finite ribbon category is spherical in the sense of Definition 3.9.
Proof. Assume C is a ribbon category with braiding c and twist θ. Then C is a pivotal category via the pivotal structure j of (2.7). Consider the following computation: Here, the first equation uses that j X˚" jX . The second equation uses (2.7) and that θ X˚" θX. The third equality uses (2.6) and the naturality of θ. The fourth equation follows from θ XbY " The next equation holds by the naturality of θ and the fact that θ 1 " Id 1 . The last equality then follows a sequence of arguments using the naturality of the braiding and rigidity axioms. So, j˚X j X " pφX˚q´1φ X .
Now assume that C is unimodular so that D " 1. Using [Shi18, Theorem A.6], cf. [EGNO15, Theorem 8.10.7], we find that Thus, using η " Id : 1 Ñ D, this shows that j satisfies the conditions of Definition 3.9. Thus, C is spherical.
Finally, recent results of Shimizu provide, in the non-semisimple framework, a sufficient condition for the monoidal center ZpCq to be modular.

Modularity of Müger centralizers and relative monoidal centers
This section contains the main categorical results of this paper. We first use the double centralizer theorem of [Shi19] to prove that Müger centralizers of non-degenerate topologizing subcategories in a modular category are again modular [Section 4.1]. Then we recall the construction of the relative center Z B pCq [Section 4.2] and conclude, as a main application, that it produces modular categories under conditions on C identified in [Shi18] and assuming that B is non-degenerate [Section 4.3]. We also produce an analogue of Müger's decomposition theorem of modular categories in the nonsemisimple setting [Section 4.4].

Modularity of Müger centralizers.
In the following we apply the double centralizer theorem of Shimizu [Shi19, Theorem 4.9] (which is a non-semisimple version of [Müg03b, Theorem 3.2(i)]) to obtain a generalization of the result of Müger [Müg03b, Corollary 3.5] that centralizers of non-degenerate subcategories in modular categories are again modular in the non-semisimple setting. For this, we require the following notion of centralizer.
Let S be a subset of objects of a braided category C, the Müger centralizer C C pSq [Müg03b, Definition 2.6] of S in C is defined as the full subcategory of C with objects (4.1) ObpC C pSqq :" tX P C | c Y,X c X,Y " Id XbY for all Y P Su.
Note that C C pSq is a topologizing monoidal subcategory of C and, thus, braided. For a single object X in C, we denote C C ptXuq " C C pXq. If C is rigid, then C C pSq is a rigid monoidal subcategory of C [Müg03b, Lemma 2.8]. The following result is straight-forward.

Lemma 4.2. If C is a (finite) tensor category and S a subset of its objects, then C C pSq is a (finite) tensor category.
The result below is the main method of this paper used to construct modular tensor categories.
Theorem 4.3. Let D be a modular category in the sense of Definition 2.11, let E be a topologizing braided tensor subcategory of D, and consider the Müger centralizer C D pEq. Then,

As a consequence, C D pEq is modular if and only if E is modular.
Proof. First, let V be an object in E 1 . Then, for any object W in E, the equation Id V bW " c W,V c V,W holds in D using that E is a full braided subcategory of D. By definition, this shows that V is an object in C D pEq. Now let X be an object in C D pEq. Again, by definition and since V P E, we get that V centralizes X. Hence, V is contained in the Müger center C D pEq 1 . Conversely, let X be an object in C D pEq 1 . Then, using C D pEq Ď D, we have that X is in the Müger centralizer C D pC D pEqq. Using the double centralizer theorem [Shi19, Theorem 4.9], it follows that C D pC D pEqq equals E. Note that this result uses that E is a topologizing subcategory of D. Hence, X is an object of E. But as X was assumed to be an object of C D pEq it centralizes all objects of E and is thus isomorphic to an object in E 1 . Therefore, C D pEq 1 » E 1 , as desired.
Hence, C D pEq is non-degenerate as in Definition 2.10 if and only E is. Since both C D pEq and E are ribbon subcategories of D [Lemma 2.9], the consequence holds.
If E is not a topologizing subcategory that we can replace E by its subquotient completion, which is also a braided tensor subcategory of D, in the statement of the theorem above. Note that if D is semisimple, then E is a topologizing subcategory provided that it is a full subcategory closed under direct summands. (This follows as the simple objects of E are also simple in D and cannot have any non-trivial subquotients.) 4.2. B-central monoidal categories and relative monoidal centers. Let B :" pB, b B , 1 B , ψq be an abelian braided monoidal category throughout this section. Also, recall the braided monoidal category B :" pB, ψ´1 Y,X : Definition 4.4. A monoidal category C is B-central if there exists a faithful braided monoidal functor G : B Ñ ZpCq. In this case, we refer to the functor G as B-central as well.
Likewise, if B is a braided (finite) tensor category, then we say that a (finite) tensor category C is B-central if there exists a faithful braided tensor functor G : B Ñ ZpCq. (1) Denote by F : ZpCq Ñ C the forgetful functor. We have that for a B-central functor G, the functor T :" F˝G : B Ñ C is central in the sense of [DNO13, Definition 2.3] and, in addition, faithful. Later in [DNO13] only central functors such that T is fully faithful are considered. While faithfulness of G is equivalent to faithfulness of T we do not require that T is full.
(2) Recall from [Lau20, Section 3.3] that a monoidal category C is B-augmented if it comes equipped with monoidal functors F 1 : C Ñ B and T 1 : B Ñ C and natural isomorphisms τ : T 1 q such that σ descends to ψ under F 1 , τ and σ are coherent with the structure of C and B. So a B-augmented monoidal category C is B-central. In fact, we may define a functor of braided monoidal categories G : B Ñ ZpCq, by B Þ Ñ pT 1 pBq, σ´1q and by T 1 on morphism spaces. Since T 1 has a right inverse, it is faithful, and thus G is faithful.
(3) Note that, in the semisimple case, if B is non-degenerate, then G is fully faithful; see [DMNO13, Corollary 3.26].
Example 4.6. Given H a Hopf algebra in a braided monoidal category (respectively, braided (finite) tensor category) B, we have that C " H-modpBq is a B-central monoidal category (respectively, Bcentral (finite) tensor category) [Lau20, Example 3.17]. Define a braided monoidal functor G : B Ñ ZpCq by sending V P B to ppV, a triv V q, ψ´1 ,V q, where a triv V :" ε b Id V : H b V Ñ V is the trivial H-action on V and ψ is the braiding of B. The two conditions of the action being trivial, and the half-braiding equaling ψ´1, are stable under taking subquotients in ZpCq. Hence, the image of B in ZpCq is a topologizing subcategory.
Definition 4.7. Given a B-central monoidal category C, we define the relative monoidal center Z B pCq to be the braided monoidal full subcategory consisting of objects pV, cq of ZpCq, where V is an object of C, and the half-braiding c :" c V,´: V b Id C " Ñ Id C b V is a natural isomorphism satisfying the two conditions below: That is, Z B pCq is the full subcategory of ZpCq of all objects that centralize GpBq for any object B of B.  Proof. First, recall that ZpCq is a braided finite tensor category by Proposition 2.2. Next, Remark 4.8 implies that the braiding in Z B pCq is the restriction of the braiding in ZpCq. Finally, with Lemma 4.2, the full braided subcategory Z B pCq " C ZpCq pGpBqq of ZpCq is a tensor category, and it is finite provided that C is finite.
Then there is an equivalence of tensor categories Here, Drin K pH, H˚q is a quasi-triangular Hopf algebra called the braided Drinfeld double of H.
It is due to [Maj99] where it is referred to as the double bosonization. For details, including a presentation of Drin K pH, H˚q, see [Lau19, Section 3.2]. See Lemma 5.9 for a presentation in the case that H is a Nichols algebra of diagonal type. The case when K " k is discussed at the end of Section 2.5.3.

4.3.
Main application: Modularity of Z B pCq. In this part, we provide the main application of Theorem 4.3 to establish when a relative monoidal center is modular. This result provides sufficient conditions that Z B pCq is a modular category.
Theorem 4.14. Let B be a non-degenerate braided finite tensor category, and C a B-central finite tensor category so that the set Sqrt C pD, ξ D q from Definition 3.3 is non-empty. Assume that the full image GpBq in ZpCq is a topologizing subcategory. Then the relative monoidal center Z B pCq is a modular tensor category.
Here, the braiding of Z B pCq is restricted from that of ZpCq, see Proposition 4.9, and the ribbon structure is restricted to Z B pCq from that of ZpCq by Lemma 2.9.
Proof of Theorem 4.14. We have that ZpCq is modular by Theorem 3.14. By assumption, the full image GpBq is a topologizing subcategory of ZpCq, and since G : B Ñ ZpCq is faithful, GpBq is nondegenerate. Now, apply Theorem 4.3 with D " ZpCq and E " GpBq, together with Remark 4.8, to conclude that Z B pCq is modular, as desired.
Remark 4.15. The statement of Theorem 4.14 can be varied to requiring that the subquotient completion of the full image of G (with braiding obtained from being a tensor subcategory of ZpCq) is non-degenerate instead of requiring that B is non-degenerate and GpBq topologizing.
In the setting of Example 4.12, Shimizu achieved Theorem 4.14 for Z B pCq » H H YDpBq in [Shi19b, Theorem 4.2] using [Shi19, Theorem 6.2]. Next, by Theorem 3.14 we obtain the following result as a special case.
Corollary 4.16. Let B be a non-degenerate braided finite tensor category, and C a B-central finite tensor category which is spherical in the sense of Definition 3.9 and such that the full image GpBq in ZpCq is a topologizing subcategory. Then the relative monoidal center Z B pCq is modular.
Proof. This follows directly from Theorem 4.14 using Remark 3.11.
Observe that this result is a relative generalization of Corollary 3.15 above due to Shimizu. Theorem 4.17. Let D be a modular tensor category, with a topologizing non-degenerate braided tensor subcategory E. Then there is an equivalence of ribbon categories: Here, E is modular [Remark 2.12]; the results in [Müg03b] have E being modular as a hypothesis.
Proof of Theorem 4.17. As in the discussion before [Shi19, Lemma 4.8], let D 1 and D 2 be topologizing subcategories of D, and let T : D 1 ⊠ D 2 Ñ D be the functor induced by b D . Set D 1 _ D 2 to be the closure under subquotients of the image of T . We then get that D 1 _ D 2 and D 1 X D 2 are topologizing full subcategories of D. Applying this to D 1 " E and D 2 " C D pEq we see that as ribbon tensor categories via the functor T above. Indeed we see by construction that T is essentially surjective. Applying [Shi19, Lemma 4.8] with E X C D pEq » vect k we get that So by [EGNO15, Proposition 6.3.4], T is an equivalence of categories. Moreover, E and C D pEq centralize each other by definition. So, as in the proof of [Müg03, Proposition 7.7], T is a functor of braided tensor categories such that T pθ V ⊠ θ W q " θ V bW , for all V P E, W P C D pEq. Thus, T is an equivalence of ribbon tensor categories as the braiding and twist are preserved.
Finally, we get the desired result by the computation below, which follows from the double centralizer theorem [Shi19, Theorem 4.9]: Example 4.18. Let B be a non-degenerate braided finite tensor category, and let C be a B-central finite tensor category, with B-central functor G : B Ñ ZpCq. Assume that GpBq is a topologizing subcategory of ZpCq and that Sqrt C pD, ξ D q ‰ ∅.
(1) Then, by the Theorems 4.14 and 4.17, we have a decomposition of modular tensor categories: (2) If, further, B " K-mod and C " H-modpK-modq, for a quasi-triangular Hopf algebra K, and for a finite-dimensional Hopf algebra H in B, then by Example 4.13, we have a decomposition of modular tensor categories: In comparison with [Müg03b, Definition 4.4], consider the following terminology.
Definition 4.19. A modular tensor category C is prime if every topologizing non-degenerate braided tensor subcategory is equivalent to either C or vect k .
As a consequence of Theorem 4.17, we immediately obtain the result below; cf., [Müg03b, Theorem 4.5].
Corollary 4.20. Every modular tensor category is equivalent to a finite Deligne tensor product of prime modular categories.
Question 4.21. Continuing Example 4.18(2), when is the (not necessarily semisimple) modular category Drin K pH, H˚q-mod prime?
In Example 5.3 below, we recall that the (non-semisimple, factorizable, ribbon) small quantum group u q psl 2 q arises as a braided Drinfeld double Drin K pH, H˚q. It is an interesting question to determine the primality of its module category, and as well as of the module categories of other examples of non-semisimple braided Drinfeld doubles in the next section. In the semisimple case, Müger offers examples of prime and non-prime module categories over Drinfeld doubles of groups [Müg03b, Theorem 4.7, Table 1].

Examples of modular categories
In this part, we provide several examples of modular tensor categories using the relative center construction [Theorem 4.14]. In some cases, we also illustrate our decomposition result Theorem 4.17 above. We start in Section 5.1 by discussing relative centers over vect k , monoidal centers of modules over Taft algebras, and their relation to u q psl 2 q-mod. Next, we provide preliminary information about braided doubles of Nichols algebras of diagonal type in Section 5.2. We study the modularity of braided doubles of such Nichols algebras in Section 5.3; the main result is Proposition 5.15 there. Finally, in Section 5.4 we apply this result to module categories of small quantum groups (of Cartan type) and also to module categories of a braided Drinfeld double of Nichols algebras not of Cartan type. Throughout this section, we additionally assume that k has characteristic zero.

First examples.
Here, we include some first examples of non-semisimple modular categories obtained from the general result of Theorem 4.14.
Example 5.1. Take B " vect k , the category of finite-dimensional k-vector spaces with its usual symmetric structure. Let C be a finite tensor category over k (i.e., C is vect k -central) so that the set Sqrt C pD, ξ D q is non-empty. For instance, if C is unimodular finite ribbon category, then C is spherical [Proposition 3.13] and hence Sqrt C pD, ξ D q ‰ ∅ [Remark 3.11]. Then Theorem 4.14 specializes to the result that Z vect k pCq " ZpCq is modular, recovering Theorem 3.14.
Example 5.2 ((Drinfeld double of) the Taft algebra T n pq´2q). Let n ě 3 be an integer, and let q be a primitive root of unity so that q 2 has order n. Take K " kZ n , for Z n " xg | g n " 1y, and set B q " K-mod. Here, B q is braided using the R-matrix R " 1 n ř n´1 i,j"0 q´2 ij g i b g j . Next, take the monoidal category C :" H-modpB q q, for H :" krxs{px n q P HopfAlgpB q q, with ∆pxq " x b 1`1 b x and εpxq " 0. As in [LW20, Section 6], consider the Taft algebra T n pq´2q, which is the k-Hopf algebra: T :" T n pq´2q " kxg, xy{pg n´1 , x n , gx´q´2xgq, with ∆pgq " g b g, ∆pxq " g´1 b x`x b 1, εpgq " 1, εpxq " 0, Spgq " g´1, Spxq "´gx. Then, we get an equivalence of monoidal categories: Computations as in [KR93,Proposition 7] show that if n is even, then Sqrt C pD, ξ D q " ∅; see Proposition 3.8. If n " 2m´1 is odd, then the distinguished grouplike elements (see (3.1), (3.2)) are given by g T " g and α T : T Ñ k with α T pgq " q´2, α T pxq " 0. Thus, a " g m T and ζ " α m T are the unique elements satisfying the equations of Theorem 3.6. Hence, using Proposition 3.8, the unique element pV, σ V q of Sqrt C pD, ξ D q is given by the 1-dimensional T -module V " kv, where g¨v " q 2m , x¨v " 0, and σ V pW q : Thus, Sqrt C pD, ξ D q ‰ ∅ if and only if n is odd. In this case, by Theorem 3.14, we get that ZpCq is modular with a unique ribbon structure. We further have an equivalence of modular categories ZpCq » DrinpT n pq´2qq-mod, where modularity of the right-hand side is inherited from that on the left-hand side. We note that C is not a spherical category in the sense of Definition 3.9 since D ≇ 1, cf. [DSPS18, Section 3.5.2] for the case n " 3. In particular, α T fails to equal ε as in Proposition 3.12.
Example 5.3 (The small quantum group u q psl 2 q). For q as in Example 5.2, with n odd, consider the small quantum group u q psl 2 q which is generated by k, e, f , subject to the relations k n " 1, e n " f n " 0, ke " q 2 ek, kf " q´2f k, ef´f e " k´k´1 q´q´1 , with coproduct and counit determined on generators by For the categories B q and C as in Example 5.2, the relative center Z Bq pCq is equivalent as a monoidal category to u q psl 2 q-mod, see [LW20, Proposition 6.7(i)]. Thus, u q psl 2 q-mod is a braided category. We have seen in Example 5.2 that since n is odd, Sqrt C pD, ξ D q contains a unique element. Moreover, in this case one checks that the underlying braided category B q is non-degenerate using [EGNO15, Exercise 8.6.4]. Thus, by Theorem 4.14, Z Bq pCq » u q psl 2 q-mod is modular. Continuing Example 4.18, we see that there is a decomposition of modular categories DrinpT n pq´2qq-mod » B q ⊠ u q psl 2 q-mod.
A vast generation of this example, using braided Drinfeld doubles of Nichols algebras of diagonal type, will be given in the following sections. 5.2. Braided Drinfeld doubles of Nichols algebras of diagonal type. In this part, we discuss Nichols algebras of diagonal type, a large class of Hopf algebras in braided categories kG-comod, for G a finite abelian group. To start, consider the following notation that will be used throughout the rest of the section.
‚ Let G " xg 1 , . . . , g n y be a finite abelian group, where g i has order m i .
‚ Take Λ to denote the lattice Z m 1ˆ¨¨¨ˆZ mn , and let e i be the i-th elementary vector.
‚ For i " pi 1 , . . . , i n q P Λ, write g i :" g i 1 1 . . . g in n and use additive notation on indices i, so e.g. g´e i " g´1 i . ‚ Take K to be the group algebra kG.
‚ Denote the basis of K˚dual to tg i u by tδ i u, so that the pairing of K˚and K is given by xδ i , g j y " δ i,j .
‚ Let q " pq ij q P Mat n pkq with q ij ‰ 0, and let B q be the braided category K-comod r with dual R-matrix r of K given by rpg i b g j q " q ji .
‚ Consider the symmetric bilinear form b on K given by bpg i , g j q :" rpg i , g j qrpg j , g i q.
In particular, b is determined by bpg i , g j q " q ij q ji .
‚ For i " pi 1 , . . . , i n q P Λ, take the grouplike elements of K˚: γ i :" ř j rpg j b g i qδ j and γ i :" ř j rpg i b g j qδ j . We write γ i :" γ e i and γ i :" γ e i . Then γ i " γ i 1 i . . . γ in i and γ i " γ i 1 i . . . γ in i . We record a fact about non-degeneracy that will be used several times later. (1) The Nichols algebra BpV q is the quotient of TpV q be the unique largest homogeneously generated Hopf ideal IpV q Ď À ną1 V bn .
(2) We say that BpV q is of diagonal type if there exists a basis x 1 , . . . , x n of V so that There exists a complete classification of finite-dimensional Nichols algebras of diagonal type over a field k or characteristic zero [Hec09]. The Nichols algebras BpV q have a PBW basis [Kha99] and generalized root systems [Hec06]. For finite-dimensional Nichols algebras of diagonal type, relations for the ideal IpV q were found in [Ang13,Ang15] and are detailed in many examples in [AA17].
Lemma 5.7 (B q , Bq). Consider the Yetter-Drinfeld module V over K with action g i¨xj " q ij x j and coaction δpx i q " g i b x i . Then: (1) The Nichols algebra, BpV q P HopfAlgp K K YDq is of diagonal type, which we denote by B q .
(2) If B q is finite-dimensional, then B q and Bq are dually paired Hopf algebras in B q . In particular, the pairing ev : Bq b B q Ñ k is uniquely induced from the pairing of V˚and V .
Proof. Part (1) follows from [AS02, Section 2]. For part (2), consider the braided monoidal functor Φ r : , for all k P K and v P V . Then the Yetter-Drinfeld module V is the image of the K-comodule pV, δq under Φ r and B q is a Hopf algebra in the image of Φ r . Thus, B q , forgetting the K-action, is a Hopf algebra in B q .
It is well-known that the duality pairing ev : V˚b V Ñ k extends to a unique non-degenerate pairing ev : BpV˚q b BpV q Ñ k of Hopf algebras in K K YD, see e.g. [HS20, Corollary 7.2.8]. By the above observations, this is a non-degenerate pairing of Hopf algebras in B q .
Having viewed B q and Bq as braided Hopf algebras in B q " K-comod r , we are able to compute their braided Drinfeld double over the Hopf algebra K˚, cf. Example 4.13; see [Lau19, Section 3.2] for the presentation of general braided Drinfeld doubles used here. We fix the following notation.
Notation 5.8 (x i , y i ). For V in Definition 5.6 and in Lemma 5.7 above, we fix dual bases x 1 , . . . , x n of V , and y 1 , . . . , y n of V˚, and denote the resulting generators of BpV q, respectively, of the dual Nichols algebra BpV˚q by the same symbols.
Proposition 5.9 (Drin K˚p Bq, B q q). Retain the notation above and assume that the braided Hopf algebra B q in K-comod r from Lemma 5.7 is finite-dimensional. Then the braided Drinfeld double Drin K˚p Bq, B q q is a Hopf algebra generated as an algebra by elements tδ i u iPΛ , tx i u i"1,...,n , and ty i u i"1,...,n , subject to the relations IpV q and IpV˚q, along with Here, it is understood that δ i " δ j if i " j P Λ. The coproduct and counit are determined by

The antipode is determined by
The Hopf algebra Drin K˚p Bq, B q q is quasi-triangular with R-matrix given by where α indexes a basis tx α u of B q with dual basis ty α u of Bq with respect to the paring ev.
Proof. First, note that for pK, rq coquasi-triangular, ppK˚q cop , r˚q is a quasi-triangular Hopf algebra, where pK˚q cop " pK, m, u, ∆ op , ε, S´1q is the co-opposite Hopf algebra with R-matrix given by the dual of r, i.e.
The functor Φ : K-comod r ÝÑ pK˚q cop -mod R , pV, δq Þ ÝÑ pV, aV q, aV :" pev K b Id V qpId K˚b δq, defines an equivalence of braided tensor categories. In the case K " kG, pK˚q cop " K˚and, thus, Bq and B q can be regarded as dually paired Hopf algebras in K˚-mod R .
The result now follows from specifying the presentation from [Lau19, Section 3.2] to the case Drin K˚p Bq, B q q. For this, observe that R " r˚for K˚is given by R˚" ř i δ i b γ i P K˚b K˚and that the action and coaction of K˚on V are given by Remark 5.10 (k i , k i , G 1 ). Note that Drin K˚p Bq, B q q is a Z-graded Hopf algebra where deg δ i " 0, for i P Λ, deg x i " 1 and deg y i "´1, for i " 1, . . . , n. It has a triangular decomposition on Bq b K˚b B q . Modules over this Hopf algebra can be described as a relative monoidal center, cf. Example 4.13. For i P Λ, we denote k i :" γ i γ i and k i :" γ i γ i . When the braided category B q is non-degenerate [Lemma 5.5], K˚is isomorphic to the group algebra kG 1 , where G 1 " xk 1 , . . . , k n y is isomorphic to G. Thus, in this case, Drin K˚p Bq, B q q has a triangular decomposition B q bkG 1 bBq. We note neither r nor r op are necessarily non-degenerate pairings, so xγ 1 , . . . , γ n y and xγ 1 , . . . , γ n y are, in general, proper subgroups of G 1 (see, e.g., Example 5.18 below).
The next result, relating braided Drinfeld doubles and relative centers, will be of use later. Notation 5.12 (ℓ, x ℓ , i ℓ ). Continuing Notation 5.4 assume that the Nichols algebra B q is finitedimensional and let ℓ be the the top Z-degree of B q . Note that pB q q ℓ is one-dimensional [AS02, Lemma 1.12]. We choose a non-zero element x ℓ in pB q q ℓ and denote its G-degree by i ℓ .
Lemma 5.13 ([AA17, Section 2.12]). We have that i ℓ " ř βP∆q pm β´1 qβ P Λ, where ∆q is the set of positive roots of the Nichols algebra B q and m β is the order of the root of unity rpg β , g β q.
Lemma 5.14. Recall (3.1), (3.2), Notation 5.12, and the notation of Section 5.2. The distinguished grouplike element for H :" B q ⋊ K˚and for H˚, respectively, are Proof. We use techniques from [Bur08, Section 4] by first understanding integrals of H. These elements can be built from integrals of B q and of K˚as follows. Take a left integral of B q , i.e., an element x P B q such that hx " εphqx for any h P B q . Then for any left integral k of K˚, we get that Λ :" p1 b kqpx b 1q " pk p1q¨x q b k p2q is a left integral of H " B q ⋊ K˚[Bur08, Section 4.6].
Since pB q q 0 " k1, it follows that 1 is the only grouplike element of B q . By self-duality of B q , we conclude that the distinguished grouplike elements of B q and its dual are g " 1 P B q and α " ε P Bq, respectively. Thus, Λ " δ 0 x ℓ is a left integral for H and we compute that verifying the claimed formula for α H on generators.
To find g H we observe that the elements δi P H˚and xj P H˚satisfy the relation xj δi " rpg i , g j qδi xj , @j " 1, . . . , n, i P Λ.
This computation verifies the claimed formula for g H .
See [AA17, Proposition 2.42] for similar computations for B q ⋊kG. Next, we derive the following conditions for Drin K˚p Bq, B q q-mod to be modular.
Proposition 5.15. Recall Notation 5.4 and 5.12, and Remark 5.10. The braided tensor categories Drin K˚p Bq, B q q-mod » Z Bq pB q -modpB q qq are modular when (i) the symmetric bilinear form b on K is non-degenerate, and (ii) there exist j, a P Λ such that 2j " i ℓ , 2a " i ℓ , bpg i , g a q 2 " rpg i , g i ℓ q, and rpg j , g i qbpg i , g a q " q´1 ii for all i " 1, . . . , n.
Proof. To show that Z Bq pB q -modpB q qq is modular, it suffices to check that (a) the braided finite tensor category B q is non-degenerate and that (b) the set Sqrt Bq-modpBqq pD, ξ D q is non-empty, by Theorem 4.14. Then, the equivalent category, Drin K˚p Bq, B q q-mod [Proposition 5.11], is also modular. Now (a) follows from (i) using Lemma 5.5. Now we show that (ii) implies (b). Take H :" B q ⋊ K˚, a finite-dimensional Hopf algebra with H-mod » B q -modpB q q. By (ii), we can define a " 1 b k a , ζpx b δ i q " εpxqδ i,´j , and compute using Proposition 5.9 that for all i " 1, . . . , n and i P Λ. Here, α H and g H are the distinguished grouplike elements of H˚and H, respectively, of Lemma 5.14. For these elements, again using Proposition 5.9, we see that for all i. Here, γ i is from Notation 5.4 and ζ´1pδ i q " δ i,j . Using that rpg j , g i qbpg i , g a q " q´1 ii from (ii), we conclude that condition (3.7) holds for h " x i . This equation is evident for h P G since G is an abelian group and hence holds for all h P H using that S 2 is an algebra morphism. Thus, applying Proposition 3.8 with the above elements a and ζ yields (b).
(1) As a consequence of Radford's S 4 -formula [Rad76] for the Hopf algebra H " B q ⋊ K˚, we obtain that the values q " pq ij q satisfy (2) Similar conditions as in Proposition 5.15(ii) were already derived in [AA17, Proposition 2.42] to determine when DrinpB q ⋊ kGq is a ribbon Hopf algebra.
(3) Using Proposition 3.12, we derive necessary and sufficient conditions for C " B q -modpB q q to be spherical. The monoidal category is spherical if and only if i ℓ " 0 and there exist b, c P Λ such that for a :" γ b γ c , a 2 " 1, rpg i , g b qrpg c , g i q " q´1 ii for all i " 1, . . . , n.
In this case, SPivpHq is given by these elements a. Example 5.17 (The small quantum group u q pgq). Take q a root of unity of odd order l ě 3 and let g be the semisimple Lie algebra of rank t, associated to the irreducible symmetrizable Cartan matrix pa ij q t i,j"1 . We choose coprime integers d i " 1, 2, 3 so that pd i a ij q is a symmetric matrix. Associated to this data, one defines a finite-dimensional Hopf algebra u q pgq, the small quantum group (or Frobenius-Lusztig kernel) as, e.g., in [Ros93, Section 3.2]. 1 The Hopf algebras u q pgq generated by group-like elements k˘1 i , pk i , 1q-skew primitive elements e i , and p1, k´1 i q-skew primitive elements f i for i " 1, . . . t, subject to relations: for the abelian group K :" xk 1 , . . . , k t y -Zˆt l . The above datum also defines a Nichols algebra of diagonal type by setting q ij :" q d i a ij . The braiding given by q is of Cartan type, i.e. satisfies q ij q ji " q a ij ii , for all i, j " 1, . . . , t. Moreover, the associated Nichols algebra B q is isomorphic to u q pn`q, the positive part of the small quantum group associated to g, and the braided Drinfeld double Drin K˚p B q , Bqq is isomorphic to u q pgq, both via See, e.g., [Lus10], [AS02,Theorem 4.3], and references therein, for the isomorphism of B q and u q pn`q, along with [Som96, Section 5.10], [Maj99,Proposition 4.3], [Lau20, Theorem 4.9] for the isomorphisms of the braided Drinfeld doubles.
We obtain that the category Drin K˚p B q , Bqq-mod » u q pgq-mod is modular by applying Proposition 5.15 as follows. First, the pairing r : GˆG Ñ k obtained from q as in Notation 5.4 is non-degenerate using a computation as in [Ros93, proof of Proposition 3.5], assuming that the determinant of pd i a ij q is coprime to l. This implies that G is isomorphic to the group xγ 1 , . . . , γ r y Ď kG˚. Further, r is symmetric and the associated bilinear form b is given as its square. Thus, as all q d i a ij are primitive l-th roots of unity, with l odd, the same holds for q 2d i a ij and b is also non-degenerate. Therefore, Proposition 5.15(i) holds. Moreover, the proof of [Bur08, Theorem 5.4] contains a computation that shows that Proposition 5.15(ii) holds for this class of examples. 2 Therefore, Proposition 5.15 implies that Drin K˚p B q , Bqq-mod is a modular tensor category.
Continuing Example 4.18, we have a decomposition of modular categories: Drinpu q pb`qq-mod » B q ⊠ u q pgq-mod, where u q pb`q is the positive Borel part of u q pgq generated by the e i and k i . Finally, we produce an example of a relative monoidal center that gives a modular category that is not of the form u q pgq-mod. It consists of modules over a more general type of quantum group, namely, modules over the braided Drinfeld double of a Nichols algebra that is not of Cartan type.
Example 5.18. Let q P k be a primitive 2n-th root of unity, for n ě 1 an odd integer. We denote by G the abelian group xg 1 , g 2 y " Z 2nˆZ2n . Consider the Nichols algebra B q of diagonal type determined by q " pq ij q, with q 11 " q 22 "´1, q 12 " 1, q 21 " q, 2 In fact, Burciu denotes δ " ζ, h " a, χipgq " rpgi, gq and verifies the required equation δ´1pgiqχphq " χipgiq´1 using the Lie theoretic computation that, writing the j-th positive root as βj " ř t s"1 cjsαs, we have ř j ř s aiscjs " 2.
as in Section 5.2. This Nichols algebra appears in the classification of [Hec09, Table 1, Row 2]. It has by [Kha99] a PBW basis given by the set x a 1 1 x a 2 2 x a 12 12ˇ0 ď a 1 , a 2 ă 2, 0 ď a 12 ă 2n ( , where x 12 :" x 1 x 2´x2 x 1 , and is thus 8n-dimensional. Note that B q is generated by x 1 , x 2 subject to the relations x 2 1 " 0, x 2 2 " 0, x n 12 " 0, see [AA17, Section 5.1.11], and is one of the Nichols algebras of super type Ap1|1q. In this example, the symmetric bilinear form b from Notation 5.4 is given by bpg i , g j q " q ji q ij , so bpg 1 , g c 1 1 g c 2 2 q " bpg 1 , g 1 q c 1 bpg 1 , g 2 q c 2 " q c 2 , bpg 2 , g c 1 1 g c 2 2 q " bpg 2 , g 1 q c 1 bpg 2 , g 2 q c 2 " q c 1 , for all c 1 , c 2 . Hence, g c 1 1 g c 2 2 is in the radical of b if and only if c 1 , c 2 " 0 mod 2n since q is a primitive 2n-th root of unity. Thus, b is non-degenerate and Proposition 5.15(i) holds by Lemma 5.5.
Moreover, to study the sphericality of the category C :" B q -modpB q q, we apply Remark 5.16(3), and show that only Case (1) yields a spherical structure. For this, we set b " p1, 0q, c " p0, 1q, and get that a " γ b γ c " γ 1 γ 2 " k n 1 k n 2 . Then a 2 " k 2n 1 k 2n 2 " 1 and the equations rpg i , g b qrpg c , g i q " q´1 ii , for i " 1, 2 follow similarly to above, thus satisfying the conditions in Remark 5.16(3). Note that a here is the same as the one obtained in Case (1); thus, in this case, C is spherical in the sense of Definition 3.9. Finally, Cases (2)-(4) do not yield spherical structures on C. Indeed, by Proposition 3.12, having a P SPivpHq is equivalent to pζ, aq belonging to the set of Theorem 3.6 with ζ " ε. The pairs pj, aq above correspond to pairs pζ, aq in Theorem 3.6, with a " k a and ζ obtained from j via ζpx b δ i q " εpxqδ i,j . To get that ζ " ε, we need that j " 0 which only occurs in Case (1).