BRAIDED COMMUTATIVE ALGEBRAS OVER QUANTIZED ENVELOPING ALGEBRAS

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Vol.26, No. 3, 2021, pp.957 993–

BRAIDED COMMUTATIVE ALGEBRAS OVER QUANTIZED ENVELOPING ALGEBRAS

ROBERT LAUGWITZ^∗ School of Mathematical Sciences

University of Nottingham University Park

Nottingham NG7 2RD, UK

robert.laugwitz@nottingham.ac.uk

CHELSEA WALTON^∗∗

Department of Mathematics Rice University

P.O. Box 1892 Houston TX 77005-1892, USA

notlaw@rice.edu

Abstract. We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centersZ_B(C) from algebras inB-central monoidal categoriesC, whereBis an arbitrary braided monoidal category; Davydov’s (and previous works of others) take place in the special case when B is the category of vector spaces Vect_k over a field k. Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras inB- central monoidal categories. Moreover, for a large class ofB-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in Vect_k serve as Morita invariants. Many examples are provided throughout.

1. Introduction

Let k be a field and note that all algebraic structures in this manuscript are k-linear. The purpose of this work is to systematically produce and study braided commutative algebras (or, commutative algebras, for short) in a certain well- behaved class of braided monoidal categories. This is achieved by generalizing Davydov’sfull centerconstruction in [11], [12] for commutative algebras in centers of monoidal categoriesZ(C), which was built on works of Fr¨ohlich–Fuchs–Runkel–

Schweigert [19] and of Kong–Runkel [27] in their studies of algebras in modular tensor categories.

Braided commutative algebras are interesting mathematically for several reasons.

For instance, they can be used to provide natural examples of bialgebroids, which DOI: 10.1007/S

∗Partially supported by the Simons Foundation.

∗∗Partially supported by the Alfred P. Sloan foundation and the US National Science Foundation.

Received February 18, 2019. Accepted May 17, 2020.

00031-020-09599-9

Corresponding Author: Robert Laugwitz, e-mail: robert.laugwitz@nottingham.ac.uk Published onlineJuly 20, 2020.

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ROBERT LAUGWITZ, CHELSEA WALTON

are generalizations of k-bialgebras with a base algebra possibly larger than k. Namely, for a bialgebra H and an algebra A in the (braided monoidal) category of H-Yetter–Drinfeld modules, we have that A is commutative if and only if the smash product algebra AoH admits the structure of a bialgebroid with base algebra A[4, Thm. 4.1], [32, Thm. 5.1].

Commutative algebras in braided monoidal categories also have applications to physics. For instance, extended chiral algebras in rational conformal field theory (RCFT) arise as commutative algebras in modular tensor categories [20, Sect. 5.5].

These algebras were shown to be Morita invariants in modular tensor categories, and are used to prove that in two-dimensional RCFT there cannot be several incompatible sets of boundary conditions for a given bulk theory [27].

Moreover, commutative algebras in braided monoidal categories C have been used to classify certain extensions of vertex operator algebras; see [21] and [22] for more details.

We anticipate that our construction of braided commutative algebras here will have similar and new implications both in mathematics and physics. For now, note that we deliver a supply of commutative algebras in (braided monoidal) representation categories of quantized enveloping algebras, a result that extends beyond work in [11, 12] as we discuss below.

In this work, we build commutative algebras inrelative monoidal centersZ_B(C) [Definition 3.3], which is a class of braided monoidal categories studied by the first author [28], [30] (motivated by [2], [36], and related to [39, Def. 2.6]; see also [14, Sect. 4]). Here,Bis a braided monoidal category, andCis a monoidal category that is B-central[Definition 3.1]. For instance, whenBis the category of k-vector spacesVect_k, we have thatZB(C) is the usual monoidal centerZ(C) ofC[23], [33].

In general,ZB(C) is a proper subcategory ofZ(C) [Proposition 3.5, Example 3.8].

Analogous to Davydov’s full center construction for commutative algebras inZ(C) [11], we show that if there exists a functor

R_B:C → Z_B(C) (1.1) that is a right adjoint to the forgetful functor Z_B(C)→ C, then it is lax monoidal [Lemma 3.9] (so it sends algebras in C to algebras in ZB(C)). Our method for producing commutative algebras inZB(C) is called theB-centerconstruction; see Section 3.3 and Theorem 3.18.

Key examples of Davydov’s work occur when C = H-Mod, the category of modules²over a Hopf algebraH; ifH is finite-dimensional, thenZ(C) is equivalent to the category of modules over the Drinfeld double of H [15], [16]. Now we construct a larger class of commutative algebras in braided monoidal categories, including commutative algebras in representation categories of braided Drinfeld doubles [28] (or, of double bosonizations [36]), including those in our title. In particular, takega semisimple Lie algebra overkwith positive/negative nilpotent

2Throughout the paper, all modules are left modules unless stated otherwise.

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partn^+/− and positive/negative Borel partg^+/−. Then,

• ZB(C)'Uq(g)-Mod^lfw, the category of locally n⁺-finite weight modules over a quantized enveloping algebra ofgoverk(q), forqa generic variable, when

• C=U_q(g⁻)-Mod^w, the category of weight modules over the negative Borel part, and

• B=K-CoMod, whereK =U_q(h) is the quantized enveloping algebra of the Cartan subalgebra hofg, also realized as a group algebra of a lattice;

(†)

see [30, Sect. 4.3]. One can also work with small quantum groups, cf. [30, Sect. 4.4]:

• Z_B(C) ' uq(g)-Mod, the category of modules over a finite-dimensional quantized enveloping algebra ofg, forqa root of unity, when

• C=uq(g⁻)-Mod, and

• B=K-Mod, whereKis a group algebra of a certain finite abelian group.

(‡)

Let us consider a setting more general than (†,‡) as follows. Take:

(?)

• K, a quasi-triangular Hopf algebra;

• B=K-Mod, the braided monoidal category ofK-modules with a braiding Ψ,

• H, a braided Hopf algebra inB; and

• C=H-Mod(B), the monoidal category ofH-modules inB[Example 3.2(3)].

Then, by [30, Example 3.35 and Prop. 3.36] (as recorded in [Proposition 3.7]):

• ZB(C)'^H_HYD(B), the category ofH-Yetter–Drinfeld modules overB [1], [2].

We also recall in Definition 5.2 and Proposition 5.3 that there is a functor Φ from ^H_HYD(B) to the category of representations of the braided Drinfeld double DrinK(H^∗, H) . Here, DrinK(H^∗, H) is the usual Drinfeld double ofHwhenK=k andH is a finite-dimensional Hopf algebra. In a special case,

U_q(g)∼= DrinU_q(h)(U_q(n⁺), U_q(n⁻)) as shown in [36, Sect. 4], see also [29, Sect. 3.6].

We verify the functor RB from (1.1) exists in setting (?) [Theorem 3.10]. We also establish under the setting (?) that the image of an algebra A in C under RB is a braided version of the centralizer algebra Cent^l_A_o_H(A)^Ψ⁻¹ ofA inAoH [Theorem 4.5]. This is analogous to the main result of [12] in the caseB=Vect_k. Our main constructions and results are summarized in Figure 1 below for setting (?) whenC=H-Mod(B),although much of the work below holds for arbitraryB- central monoidal categories.

TheB-centerwas discussed above after (1.1). Theleft center, considered initially in [43], [41], [19] for a given braided monoidal category D, was used in [11] to produce commutative algebras inDfrom algebras inD.

In comparison with [11], [12], to achieve our constructions above we must use more involved techniques of graphical calculus sensitive to the order of crossing strands, since we work in braided monoidal categoriesBmore general thanVect_k. In any case, withB-centers we are able to produce Morita invariants of algebras inB-central monoidal categories as discussed below.

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C=H-Mod(B) ^∃^R^B(lax monoidal)

[Theorem 3.10] //^H_HYD(B) ^{Φ (∼}^if^H^{fin. dim.)}

[Prop. 5.3] //DrinK(H^∗, H)-Mod

Alg(C) ^Alg(R^B⁾ //

B-centerZB(−) [Def. 3.14, Prop. 3.16]

((

Alg(^H_HYD(B)) ^Alg(Φ) //

left centerC^l(−) [Def. 2.3]

[Prop. 2.4]

Alg(DrinK(H^∗, H)-Mod)

induced by Alg(Φ) andC^l(−)

∈

A

ZB(A) [Theorem 3.18]

= C^l(RB(A)) ∈ComAlg(^H_HYD(B)) ^Alg(Φ)//

Forgetful

ComAlg(DrinK(H^∗, H)-Mod)

Cent^l_AoH(A)^Ψ⁻¹

[Theorem 4.5]

∈ Alg(B)

Figure 1. Main constructions for setting (?). Straight arrows are functors and dotted arrows are algebra assignments.

Theorem 3.20. TakeC aB-central monoidal category, and algebrasA,A⁰ inC.

Suppose that the categories of right modules overAand overA⁰ inCare equivalent as leftC-module categories. Then, theB-centersZ_B(A)andZ_B(A⁰)are isomorphic as commutative algebras in ZB(C).

In reference to setting (‡), for instance, one can employ the theorem above to produce Morita invariants foru_q(n⁻)-module algebras by using braided commutative uq(g)-module algebras.

In addition to Davydov’s work [11], [12] and the first author’s work on comodule algebras over braided Drinfeld doubles [29], our results have connections to several other articles in the literature. See several works on braided commutative algebras in Yetter–Drinfeld categories, including [6, 9, 10]. See also work of Montgomery–

Schneider [38], of Cline [8], and of Kinser and the second author [26] on extending module algebras over Taft algebras to those over their Drinfeld double and over uq(sl2). On another related note, Etingof–Gelaki realized the representation category of a small quantum group uq(g) as the monoidal center of a representation category of a certain quasi-Hopf algebraAq(g) [17].

This paper is organized as follows. We discuss categorical preliminaries in Sec- tion 2, including the left center construction; we also introducebraided centralizer algebras there. Next, we introduce and study the B-center construction and the functor R_Bin Section 3, and we also verify Theorems 3.18 and 3.20 and discuss the special case whenC =B, which is a monoidal category central over itself. Then, we restrict our attention to setting (?) in Section 4 and show that algebra images under R_B are braided centralizer algebras. Towards constructing examples for the material in Sections 2–4, we discuss braided Drinfeld doubles and Heisenberg doubles in Section 5. Finally, we provide examples of our results for braided commutative algebras in the representation categories of the small quantum group

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uq(sl2) in Section 6, and of the Sweedler Hopf algebra T2(−1) in Section 7. We discuss how to generalize the detailed work in Section 6 to Uq(g) and uq(g) in Section 8, and we end by listing several directions for further investigation there.

Acknowledgements.The authors thank Alexandru Chirvasitu, Alexei Davydov, J¨urgen Fuchs, and Christoph Schweigert for insightful discussions and references and two anonymous referees for helpful comments. R. Laugwitz was partially supported by an AMS-Simons travel grant. C. Walton was partially supported by a research fellowship from the Alfred P. Sloan foundation, and by the US National Science Foundation grants #DMS-1663775, 1903192. Part of this work was carried out during a visit of R. Laugwitz to University of Illinois, Urbana-Champaign, and hospitality of the hosting institution is gratefully acknowledged.

2. Categorical preliminaries

In Section 2.1, we first set up notation and conventions that we will use throughout this work. Next, we recall terminal objects and the comma category in Secti- on 2.2, and then discuss in Section 2.3 the left and right center construction that produces commutative algebras in braided monoidal categories from algebras in such categories. Finally, we introduce and study braided versions of centralizer algebras in Section 2.4.

2.1. Notation and conventions

All categories in this work are abelian, complete under arbitrary countable bipro- ducts, and enriched over the category of k-vector spaces Vect_k. The reader may wish to refer to [18] or [42] for further background information.

Throughout, C = (C,⊗) denotes a monoidal category; later in Section 3.1, C will beB-centralin the sense of Definition 3.1 for a braided monoidal categoryB.

The tensor unit of each monoidal category is denoted by I. We usually omit the associativity and unitarity isomorphisms for monoidal categories which is justified by MacLane’s coherence theorem.

Unless stated otherwise, we assume that all monoidal functors F:C → C⁰ are strong monoidal, i.e., there exists a natural isomorphism

FX,Y:F(X)⊗F(Y)→^∼ F(X⊗Y)

that is compatible with the associativity constraints of C and C⁰, and so that F(IC)∼=IC⁰.

We denote byAlg(C) the category ofalgebras(A, m, u) inC; here,Ais an object ofC with associative multiplication m: A⊗A→Aand unitu:I→A. As usual, given an algebra A in C, a left A-module is a pair (V, a_V) for V an object in C anda_V:A⊗V →V a morphism inC satisfyinga_V(m⊗Id_V) =a_V(Id_A⊗a_V) and a_V(u⊗Id_V) = Id_V. A morphism ofA-modules (V, a_V)→(W, a_W) is a morphism V →W in C that intertwines with aV andaW. This way, we define the category A-Mod(C) of leftA-modules inC. Analogously, we defineMod-A(C), the category ofright A-modules inC.

Moreover, we reserve D to be an arbitrary braided monoidal category with braiding Ψ_D, or Ψ ifDis understood. In this case, we can considerbialgebrasand

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Hopf algebras in D, and we assume that all such Hopf algebras in our work have an invertible antipode.

If (A, m) is an algebra in (D,Ψ), thenAisbraided commutative(or,commutative) if mΨ = m (or, equivalently, if mΨ⁻¹ = m) as morphisms in D. We denote the full subcategory ofAlg(D) of commutative algebras byComAlg(D). Thebraided opposites of (A, m) are A^Ψ := (A, mΨ) and A^Ψ⁻¹ := (A, mΨ⁻¹) with respect to the braiding and inverse braiding ofD, respectively. IfA is braided commutative, thenA=A^Ψ=A^Ψ⁻¹.

We use the graphical calculus as in [29], similar to that used in [34], for compu- tations in (braided) monoidal categoriesD. The braiding inDis denoted by

Ψ = .

Moreover, for a Hopf algebraH := (H, m, u,∆, ε, S) inD, we denote:

m= , ∆ = , u= , ε= , S= + , S⁻¹=−. Combining these symbols, we can display all axioms of a Hopf algebra in D(see, e.g., [29, Eqns. 1.2–1.9]). For example, the bialgebra condition becomes

= ⇐⇒ ∆m= (m⊗m)(Id⊗Ψ⊗Id)(∆⊗∆).

A left H-action onV ∈ Dis denoted by

a_V = : H⊗V →V.

Moreover, the H-module structure a_V_⊗W on the tensor product V ⊗W of left H-modulesV,W becomes:

aV⊗W := (aV ⊗aW)(IdH⊗ΨH,V ⊗IdW)(∆⊗IdV⊗W) = . (2.1) Now for H a Hopf algebra in a braided monoidal category (D,Ψ), we have via (2.1) that H-Mod(D) is a monoidal category. Similarly, Alg(D) is a monoidal category: The tensor product of two algebras A, B in D is given by the triple (A⊗B, mA⊗B, uA⊗B), where

m_A⊗B:= (mA⊗mB)(IdA⊗ΨA,B⊗IdB) and u_A⊗B:=uA⊗uB. 2.2. Terminal objects and the comma category

We record some useful results about terminal objects and the comma category that we will use below. Recall that an objectT ofCisterminalif, for any objectX ∈ C, there exists a unique morphism X → T in C. If a terminal object exists, then it is unique up to unique isomorphism. By considering the morphisms T ⊗T →T (multiplication) and I →T (unit) inC, one can verify the associativity and unit axioms to obtain the following fact.

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Lemma 2.1. The terminal object of a (braided) monoidal category is a (commutative) algebra.

Moreover, we get the following fact from the uniqueness of morphisms to the terminal object.

Lemma 2.2. IfAis an algebra in a monoidal categoryC, then the unique morphism A→T to the terminal object is a morphism of algebras inC.

We will also need to use the next construction later. Take two monoidal categories C andC⁰, a monoidal functorF:C → C⁰, and an objectA∈ C⁰. Then, thecomma category F↓A is the category of pairs (X, x) with X ∈ C and x: F(X) → A in C⁰, with morphisms being morphisms of the first components that are compatible with the morphisms of the second components. IfF is monoidal andA∈Alg(C⁰), thenF↓Ais also monoidal with

(X, x)⊗(Y, y) := (X⊗Y, mA(x⊗y)F_X,Y⁻¹ :F(X⊗Y)→A).

2.3. Left and right center

Next, we describe how to associate to a given algebra A in D certain braided commutative algebras inD, analogous to the center. The definition below appears in [11, Sect. 5], following (and equivalent to) [41, Def. 5.1]; see also, [43, Def. 4.3].

Definition 2.3. Let (A, m) be an algebra in (D,Ψ). Theleft center C^l(A)→A ofAis the terminal object in the category of morphisms γ:Z→A such that the following diagram commutes:

Z⊗A ^Ψ^Z,A //

γ⊗Id

A⊗Z

Id⊗γ

A⊗A

m ''

A⊗A

ww m

A

. (2.2)

Equivalently, the left center is defined as the maximal subobjectC^l(A) ofAsuch that mΨ_Cl(A),A=m as maps fromC^l(A)⊗AtoAin D.

Similarly, we define theright center C^r(A)→Ausing Ψ⁻¹ instead of Ψ.

Proposition 2.4 ([11, Prop. 5.1]). The left centerC^l(A)has an algebra structure in D, unique up to unique isomorphism of algebras, such that C^l(A) → A is a morphism in Alg(D). In addition, C^l(A) ∈ ComAlg(D). Similarly, C^r(A) ∈ ComAlg(D).

2.4. Left and right centralizers

We generalize the left and right center constructions in the previous section as follows.

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Definition 2.5. Let (A, m) be an algebra, S be an object, and φ:S → A be a morphism in (D,Ψ). Theleft centralizerCent^l_A(S)→AofAis a terminal object in the category of morphismsγ:C→Asuch that the following diagram commutes:

C⊗S ^Ψ^C,S //

γ⊗φ

S⊗C

φ⊗γ

A⊗A

m ''

A⊗A

ww m

A

. (2.3)

Similarly, we define theright centralizer Cent^r_A(S)→Ausing Ψ⁻¹ instead of Ψ.

Example 2.6. LetAbe an algebra inD.

(1) LetS=Aandφ= Id_A. In this case, Cent^l_A(A) =C^l(A), and Cent^r_A(A) = C^r(A).

(2) LetS=I, and letφ=u_A:I→Abe the unit ofA. In this case, Cent^l_A(I) = Cent^r_A(I) =A.

Definition 2.7. Let A be an algebra, S be an object, and φ: S → A be a morphism inD. We denote byC_A^l(S) the category consisting of

• objects which are pairs (C, γ), where C is an object and γ:C → A is a morphism in Dthat make Diagram 2.3 commute; with

• morphisms (C, γ) → (C⁰, γ⁰) that are morphisms f: C → C⁰ in D such that the diagram below commutes:

C ^f //

γ

C⁰

γ⁰

~~A

. (2.4)

From Example 2.6(1), we denoteC_A^l(A) byC^l(A).

Now the left centralizer Cent^l_A(S) is the terminal object ofC_A^l(S), and the left center C^l(A) is the terminal object of C^l(A). Similar to Proposition 2.4, we have the result below.

Proposition 2.8. The left centralizerCent^l_A(S)has the structure of an algebra in D, which is unique up to (unique)algebra isomorphism, such that Cent^l_A(S)→A is morphism inAlg(D).

Proof. By the discussion above and the material in Section 2.2, it suffices to show that C_A^l(S) is a monoidal category. Towards this, take (C1, γ1) and (C2, γ2) in C_A^l(S) and define

(C₁, γ₁)⊗(C₂, γ₂) := (C₁⊗C₂, m_A(γ₁⊗γ₂)).

Similar to [12, Rem. 4.2], this definition satisfies (2.3) via the commutative diagram

below.

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C1⊗C2⊗S

ΨC1⊗C2,S

,,

Id⊗ΨC2,S //

γ1⊗Id⊗Id

$$

γ1⊗γ₂⊗φ

C1⊗S⊗C2

ΨC1,S⊗Id

//

γ1⊗Id⊗Id

zz

Id⊗Id⊗γ2

$$

S⊗C1⊗C2 Id⊗Id⊗γ2

zz

φ⊗γ₁⊗γ₂

A⊗C2⊗S^Id⊗Ψ//

Id⊗γ2⊗φ

yy

A⊗S⊗C2 Id⊗φ⊗γ2

%%

C1⊗S⊗A^Ψ⊗Id//

γ1⊗φ⊗Id

yy

S⊗C1⊗A

φ⊗γ1⊗Id

%%

A⊗A⊗A

m⊗Id

Id⊗m %%

A⊗A⊗A

m⊗Id **

tt Id⊗m

A⊗A⊗A

Id⊗m

yy m⊗Id

A⊗A

m

%%

A⊗A

m

yy

A⊗A

m

,,

A⊗A

m

rrA

Unlike Proposition 2.4 (for S=A), neither Cent^l_A(S) or Cent^r_A(S) necessarily belongs toComAlg(D): To see this use Example 2.6(2). In any case, consider the example below.

Example 2.9. Suppose thatDis the category of modules over a quasi-triangular Hopf algebra (so it comes equipped with a fiber functor to Vect_k and objects have elements). Then, for any subobject S ⊂A, the left centralizer Cent^l_A(S) is isomorphic to the following subalgebra of A:

Cent^l_A(S) = a∈A

m(a⊗s) =mΨ(a⊗s), ∀s∈S .

3. The functor R_B and the B-center

We present the main results of our work in this section. First, we discuss background material onB-central monoidal categories and relative monoidal centers ZB(C) adapted from [30] in Section 3.1. In Section 3.2, we study the right adjoint RB to the forgetful functor ZB(C) → C, and show that it exists when ZB(C) is a Yetter–Drinfeld category ^H_HYD(B) in B [Theorem 3.10]; the lax monoidal property of R_Bis also discussed. Next, in Section 3.3, we generalize Davydov’s full center construction [11, Sect. 4] to the B-central setting, thus producing braided commutative algebras Z_B(A) in Z_B(C) from algebras A in C [Proposition 3.16].

In Section 3.4, we show that for A ∈Alg(C) we have an isomorphism Z_B(A) ∼= C^l(R_B(A)) of algebras in ComAlg(Z_B(C)) if R_B exists and its adjunction counit is an epimorphism [Theorem 3.18]; this generalizes [11, Thm. 5.4]. In Section 3.5, we establish that B-centers serve as Morita invariants for algebras in B-central monoidal categories [Theorem 3.20]. Finally, in Section 3.6, we restrict our attention to the case whenCis braided and central over itself and present the results of the previous subsections in this setting.

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3.1. B-central monoidal categories and relative monoidal centers RecallB:= (B,Ψ) denotes a braided monoidal category. From now onC will be a monoidal category of the kind below.

Definition 3.1. We say that a monoidal category C is B-central monoidal if it comes equipped with a faithful monoidal functor, called the central functor,

T :B → C and a natural isomorphism

σ: ⊗C (Id_CT) =^∼⇒ ⊗^op_C (Id_CT)

such that the assignment B 7→(T(B), σ⁻¹_B,−) defines a braided monoidal functor B → Z(C), where B = (B,⊗,Ψ⁻¹), i.e., the monoidal category B with inverse braiding.

The definition above generalizes the concept of aB-augmented monoidal category of [30, Sect. 3.3] relaxing the condition that the functor T has a left inverse. See also [13, Def. 2.4] for the concept of a central functor.

Example 3.2. [30, Sect. 3.3] Below are examples ofB-central monoidal categories.

(1) By the assumptions in Section 2.1, all monoidal categories in this work are Vect_k-central.

(2) We have thatBisB-central with T = Id_Band σ= Ψ.

(3) TakeH a Hopf algebra inB, and consider the categoryC:=H-Mod(B) of leftH-modules inB. This category is monoidal: Take (V, a_V:H⊗V →V) and (W, aW:H⊗W →W) inC and we get that (V⊗W, aV⊗W)∈ C with aV⊗W defined as in Equation 2.1. Moreover,C isB-central monoidal with T giving an object ofBthe structure of a trivial H-module in B(via the counit ofH), andσ:= T(Ψ_F(V_),B) for allV ∈ C andB ∈ B.

Example 3.2(3) will play a crucial role in our work later. Next, we define the relative monoidal center of aB-central monoidal category C.

Definition 3.3 ([30, Def. 3.32, Props. 3.33 and 3.34]). Therelative monoidal center ZB(C) of aB-central monoidal categoryCis a braided monoidal category consisting of pairs (V, c), where V is an object ofC, and c:=c_V,−:V ⊗Id_C ⇒^∼ Id_C⊗V is a natural isomorphism of half-braidings satisfying the two conditions below:

(i) [tensor product compatibility] for any X, Y ∈ C the following diagram commutes:

V ⊗X⊗Y ^c^V,X⊗Y //

cV,X⊗IdY ''

X⊗Y ⊗V

X⊗V ⊗Y

IdX⊗cV,Y

77

,

(ii) [compatibility with the central functor] for anyB ∈ Bwe have c_V,T(B)=σV,B.

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A morphism from (V, c_V,−) to (W, c_W,−) is defined to be a morphism f:V →W in Cso that for each X∈ C we have

(Id_X⊗f)c_V,X =c_W,X(f⊗Id_X).

Here, the monoidal structure is given by

(V, c_V,−)⊗(W, c_W,−) := (V ⊗W, {cV⊗W,X := (cV,X⊗IdW)(IdV ⊗cW,X)}_X∈C), and the braiding is given by

Ψ_(V,c_V,−_),(W,c_W,−₎=cV,W.

Remark 3.4. The relative center ZB(C) is the M¨uger centralizer [39, Def. 2.6] of the set of objects (T(B), σ⁻¹_B,V) in Z(C).

In [30], the relative monoidal center Z_B(C) is defined (equivalently) as the monoidal category ofB-balanced endofunctors G of the regularC-bimodule category C; here, using composition as the tensor product, i.e., G⊗G⁰ = G⁰G, this is also a braided monoidal category.

Relative monoidal categories have the following properties, some of which hold by definition. Here, we use [30, Prop. 3.34] to employ other results from [30].

Proposition 3.5. LetCbe aB-central monoidal category over a braided monoidal categoryB.

(1) If C=B, thenZ_B(B)is isomorphic to Bas braided monoidal categories.

(2) [30, Example 3.30]IfB=Vect_k, thenZ_B(C)is isomorphic to the(ordinary) monoidal center Z(C) (e.g., as in[18, Def. 7.13.1]).

(3) ForB arbitrary,Z_B(C)is a full braided monoidal subcategory of Z(C).

(4) [30, Thm. 3.29]If C is rigid (or pivotal), then so isZ_B(C).

Note that the forgetful functor Z_B(C) → C does not necessarily have a right adjoint, but we show later in Lemma 3.9 that if a right adjoint exists, then it is lax monoidal. In particular, such a right adjoint exists for the B-central monoidal category H-Mod(B) from Example 3.2(3); see Theorem 3.10. Toward this result, consider the following explicit description of the relative monoidal center of H-Mod(B) in terms ofcrossed orYetter–Drinfeld modules.

Definition 3.6 ([1], [2]). Take a Hopf algebraH inB, and takeC:=H-Mod(B) from Example 3.2(3) with leftH-action inBdenoted bya:=aV :=a^H_V. Then the category ^H_HYD(B) ofH-Yetter–Drinfeld modules in Bconsists of objects (V, a, δ) where (V, a)∈ C with leftH-coaction inB denoted byδ:=δV :=δ_V^H, subject to the compatibility condition:

(mH⊗aV)(IdH⊗ΨH,H ⊗IdV)(∆H⊗δV)

= (m_H⊗Id_V)(Id_H⊗Ψ_V,H)(δ_V ⊗Id_H)

·(aV ⊗IdH)(IdH⊗ΨH,V)(∆H⊗IdV).

(3.1)

A morphism f: (V, aV, δV) → (W, aW, δW) in ^H_HYD(B) is given by a morphism f:V →W in Bthat is a morphism ofH-modules and ofH-comodules.

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Given two objects (V, aV, δV) and (W, aW, δW) in^H_HYD(B), their tensor product is given by (V ⊗W, aV⊗W, δV⊗W), whereaV⊗W is as in Equation 2.1 and

δ_V_⊗W = (m_H⊗Id_V_⊗W)(Id_H⊗Ψ_H,V ⊗Id_W)(δ_V ⊗δ_W) = . Here, the braiding of ^H_HYD(B) is given by

Ψ^YD_V,W = (a_W ⊗Id_V)(Id_H⊗Ψ^B_V,W)(δ_V ⊗Id_W) = , (3.2) with inverse given by

(Ψ^YD_V,W)⁻¹= (IdW⊗(aVΨ⁻¹_V,H))(Ψ⁻¹_V,W ⊗S⁻¹)(IdV ⊗(Ψ⁻¹_H,VδW)) = −, (3.3) cf. [1, Thm. 3.4.3].

Proposition 3.7 ([30, Prop. 3.36]). ForH a Hopf algebra in B, consider theB- central monoidal category C := H-Mod(B) from Example 3.2(3). Then, there is an equivalence of braided monoidal categories

Z_B(C)→^∼ ^H_HYD(B),

where (V, c)7→V with left H-module structure a_V :=a_V,c:H ⊗V → V from C, andH-coaction given byδ_V :=δ_V,c:=c_V,H(Id_V ⊗u_H) :V →H⊗V.

Next, we illustrate Proposition 3.5(3) forC:=H-Mod(B) on the level of objects below.

Example 3.8. LetKbe a quasi-triangular Hopf algebra inVect_kwithR-matrix R⁽¹⁾⊗R⁽²⁾, and letH be a Hopf algebra inB=K-Mod. Then the smash product algebraHoKis a Hopf algebra inVect_k(calledbosonizationorRadford biproduct) such that

C:=H-Mod(B)'HoK-Mod, see [37, Thm. 9.4.12]. In this case,

Z(C)'^H_H^o_o^K_KYD(Vect_k) =:^H_H^o_o^K_KYD.

On the other hand, by Proposition 3.7,

Z_B(C)'^H_HYD(B).

There is a functor from ^H_HYD(B) to^H_H^o^K

oKYD, where the object (V, a^H_V, a^K_V, δ_V^H)∈

H

HYD(B) with coaction δ_V^H(v) = v₍₋₁₎⊗v₍₀₎ is mapped to the object given by (V, a^H_Vô^K, δ_V^Hô^K)∈^H_Hô_o^K_KYDwith

a^H_V^o^K:=a^H_V(IdH⊗a^K_V) and δ_V^H^o^K(v) =v₍₋₁₎⊗R⁽²⁾⊗a^K_V(R⁽¹⁾⊗v₍₀₎).

Hence, the essential image of this functor consists precisely of objects of^H_H^o_o^K_KYD isomorphic to those where the coactionδrestricted toKhas the formδ(v) =R⁽²⁾⊗ aHoK(R⁽¹⁾⊗v), i.e., is induced from the action aHoK by using the universalR- matrix. This illustrates how a relative monoidal centerZ_B(C) is a proper subcategory of the monoidal centerZ(C) whenBis inequivalent toVect_k.

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3.2. A right adjoint to the forgetful functor

LetC be aB-central monoidal category. Consider the forgetful functor F_B:ZB(C)→ C.

In this section, we consider a general situation in which the forgetful functor FB

has a right adjoint RB. In this case, we label the corresponding adjunction natural isomorphisms as follows:

αW:W →RBFB(W) and βV: FBRB(V)→V, W ∈ ZB(C), V ∈ C.

Lemma 3.9. Assume that F_B: Z_B(C) → C has a right adjoint R_B. Then the adjunction natural transformationsαandβ are monoidal, andRB is lax monoidal.

Proof. This follows from a general fact in category theory. The functor F_Bis strong monoidal, so it is oplax monoidal, and hence its right adjoint R_B is lax monoidal [25] (see also [40]). A direct proof of these results is also given in [11, Sect. 5].

Now assume that H is Hopf algebra in B, that C = H-Mod(B), and recall from Proposition 3.7 that Z_B(C) ' ^H_HYD(B). We construct a right adjoint R_B to F_B in the next result; this generalizes the construction from [5, Cor. 2.8] when B=Vect_k (which is used crucially in [12]).

Theorem 3.10. The forgetful functor F_B: ^H_HYD(B) → H-Mod(B) has a lax monoidal right adjoint R_B: H-Mod(B) → ^H_HYD(B). Moreover, the functor R_B sends an object (V, aV) to the object (H ⊗V, a^R, δ^R) with H-action a^R and H- coactionδ^Rgiven by

a^R= (m⊗IdV)(IdH⊗ΨV,H)(IdH⊗aV ⊗S)(Id_H⊗H⊗ΨH,V)

◦(m⊗∆⊗Id_V)(Id_H⊗Ψ_H,H⊗Id_V)(∆⊗Id_H⊗V), δ^R= ∆⊗IdV,

both pictured in (3.4)below. Forf:V →W inH-Mod(B), we have thatR_B(f) = IdH⊗f.

The lax monoidal structure is given by the morphism u:I −→^α^I R_BF_B(I) →^∼ R_B(I) and the natural transformation τV,W: R_B(V)⊗R_B(W) → R_B(V ⊗W) defined by

τV,W = (mH⊗IdV⊗W)(IdH⊗ΨV,H⊗IdW).

Proof. Checking that R_B(V) is an object in the category of H-Yetter–Drinfeld modules in B is carried out using graphical calculus, especially since Sweedler notation cannot be employed easily for objects ofB; this argument is quite similar to [12, Prop. 5.1] for the case whenBis symmetric monoidal. Here, theH-action andH-coaction on R_B(V) is displayed as follows:

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a^R= + H H V

V H

, δ^R=

H V H H V

. (3.4)

We leave verification of the action conditiona^R(IdH⊗a^R) =a^R(m⊗Id_H⊗V) to the reader. The Yetter–Drinfeld condition, Equation 3.1, is verified by the following graphical calculation:

+ +

+ + +

= = = =

H H V H H V H H V H H V H H V

H H V H H V H H V H H V H H V H

H R_B(V)

R_B(V)

= =

H R_B(V) H R_B(V)

Here, the first and last equalities follow from (3.4). The second and third equality use coassociativity, the bialgebra axiom, and naturality of the braiding. The fourth equality follows from a computation using the antipode axioms and bialgebra condition while the fifth equality again uses naturality of the braiding and coassociativity.

Functoriality of R_B is clear by definition. To see that R_B is right adjoint to F_B, we present the unitαand counitβ of the adjunction. For objectsV ofH-Mod(B) andW of^H_HYD(B), define

αW :=δW:W →R_BF_B(W) and βV :=ε⊗IdV: F_BR_B(V)→V. (3.5) A direct check verifies the adjunction axioms for α, β, and further one can check directly thatα_V is a morphism in^H_HYD(B) andβ_V a morphism inH-Mod(B).

The lax monoidal structure is computed as in [11, Section 5] as τV,W = R_B(βV ⊗βW) R_B((F_B)V,W) (α_R_B_(V_)⊗R_B_(W₎).

Using (3.5) and omitting associativity, we have that τV,W: RB(V)⊗RB(W) → RB(V ⊗W) is

τV,W =(IdH⊗ε⊗IdV ⊗ε⊗IdW)δ_H⊗V_⊗H⊗W

=(IdH⊗ε⊗IdV ⊗ε⊗IdW)(m⊗IdH⊗V⊗H⊗W)

◦(IdH⊗Ψ_H⊗V,H⊗Id_H⊗W)(∆⊗IdV ⊗∆⊗IdW)

=(m⊗Id_V_⊗W)(Id_H⊗ΨV,H⊗Id_W).

We have to verify associativity and unitarity squares for this lax monoidal structure, and these follow directly from the corresponding properties ofH. It is also directly verified thatuandτV,W are indeed morphisms in ^H_HYD(B).

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For any lax monoidal functorG:C → DwithAan algebra object inC, we get thatG(A) is an algebra inDwith product m_G(A)=G(mA)GA,A.Using this, we observe the following:

Corollary 3.11. The lax monoidal functorRB induces the following functor R_B:Alg(H-Mod(B))→Alg(^H_HYD(B))

on categories of algebra objects. Given an algebraAwith productmAinH-Mod(B), the algebraR_B(A)has product given by

m= (mH⊗mA)(IdH⊗ΨA,H ⊗IdA) = (IdH⊗mA)τA,A, that is, given by the tensor product algebra structure on H⊗AinB.

Example 3.12. Consider the ⊗-unit I of B. Then I is an H-module in B with trivial action, aI = ε⊗IdI, and moreover, I ∈ Alg(H-Mod(B)). The Yetter–

Drinfeld module R_B(I) =H⊗I∼=H is given by the adjointH-action a_ad=m(m⊗S)(Id_H⊗Ψ_H,H)(∆⊗Id_H),

together with the regular coaction ∆. It follows thatH, with this Yetter–Drinfeld structure, is an algebra object in^H_HYD(B).

Example 3.13. Suppose that we have a Hopf algebra^∗H inBright dually paired to a Hopf algebraH inB, that is, there is a pairing ev :H⊗^∗H→IinBsatisfying the conditions of [29, Def. 3.1]. Extending the graphical calculus by denoting ev =

, we define theleft coregular action a^cor ofH on^∗H by

a^cor= (ev⊗Id^∗_H)(Id_H⊗∆^∗_H) = H^∗H

∗H .

The following computation shows that^∗H^Ψ⁻¹ is an algebra object in H-Mod(B).

= = =

∗H ^∗H ^∗H ^∗H

∗H ^∗H

H H ^∗H ^∗H H^∗H^∗H H ^∗H ^∗H

=

∗H H ^∗H

=

∗H ^∗H H

.

∗H

Hence, RB(^∗H^Ψ⁻¹) =H⊗^∗H^Ψ⁻¹ is an algebra object in^H_HYD(B).

3.3. TheB-center

This section contains the categorical definition of the B-center, which is a direct generalization of thefull center of Davydov’s works [11], [12] relative to a braided monoidal categoryB. Davydov’s case corresponds to specializingB=Vect_k.

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Definition 3.14. Let A be an algebra in a B-central monoidal category C. The B-center of A is a pair (ZB(A), ζA), where ZB(A) is an object in ZB(C) with half-braiding cZ_B(A),A and ζA := ζ_A^B: ZB(A) → A is a morphism in C, which is terminal among pairs ((Z, cZ,A), ζ:Z →A) in the comma category FB↓A so that the following diagram commutes:

Z⊗A ^c^Z,A //

ζ⊗Id

A⊗Z

Id⊗ζ

A⊗A

m ''

A⊗A

ww m

A

. (3.6)

WhenB=Vect_k, the pair (Z(A) :=Z_Vect

k(A), ζ_A^Vect^k) is thefull centerofA(as in [11], [12]).

The B-center of A is realized as a terminal object of the following braided monoidal category.

Definition 3.15. Let A be an algebra in C. We denote by ZB(A) the category consisting of

• pairs (Z, ζ) with Z = (Z, c) an object inZB(C), andζ:Z →A a morphism inC, that make Diagram 3.6 commute; and

• morphisms (Z, ζ) → (Z⁰, ζ⁰) in ZB(C) that are morphisms f:Z → Z⁰ such that the diagram below commutes:

Z ^f //

ζ

Z⁰

ζ⁰

~~A

.

Given objects (Z, ζ), (Z⁰, ζ⁰)∈ Z_B(A), their tensor product is (Z⊗Z⁰, m(ζ⊗ζ⁰)), using the tensor productZ⊗Z⁰ inZ_B(C), cf. [11, Rem. 4.2]. This makesZ_B(A) a monoidal category.

The categoryZ_B(A) is braided via

Ψ_(Z,ζ),(Z⁰_,ζ⁰₎:=cZ,Z⁰:Z⊗Z⁰→Z⁰⊗Z,

which is a morphism in Z_B(A) by commutativity of the outer diagram in

Z⊗Z⁰ ^c^Z,Z⁰ //

Id⊗ζ⁰

%%

ζ⊗ζ⁰

Z⁰⊗Z

ζ⁰⊗Id

yy

ζ⁰⊗ζ

Z⊗A

yy ζ⊗Id

cZ,A //A⊗Z

Id⊗ζ %%

A⊗A _m //Aoo _m A⊗A.

.

The upper middle diagram commutes by naturality of c. We have the following result.

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Proposition 3.16. TheB-centerZ_B(A)ofA∈Alg(C)is a braided commutative algebra inZB(C)andζA:ZB(A)→Ais a morphism of algebras in C.

Proof. It follows from Lemma 2.1 that (Z_B(A), ζA), being the terminal object of the braided monoidal categoryZ_B(A), is a commutative algebra inZ_B(A). Since the forgetful functor Z_B(A)→ Z_B(C) is a braided monoidal functor, we get that Z_B(A) is a commutative algebra in Z_B(C). Moreover, the product m_Z_B_(A) is a morphism inZ_B(A). So,

ζAm_Z_B_(A)=mA(ζA⊗ζA),

and this condition means thatζ_A is a morphism of algebras inC.

Corollary 3.17. For any algebraA inC, there is a unique morphism of algebras ξA: Z_B(A) → Z(A) in Z(C), which commutes with the respective morphisms of algebras to A.

Proof. Recall from Proposition 3.5(3) thatZ_B(C) is a braided monoidal subcategory of Z(C). This implies that Z_B(A) is a braided monoidal subcategory of Z(A).

Hence,Z_B(A) is an algebra inZ(A) =ZVect_k(A). By Lemma 2.2, we see that the unique morphismξA:Z_B(A)→Z(A) is one of algebras inZ(A). In particular,ξA

is a morphism of algebras inZ(C), which commutes with the respective morphisms toAin the sense that the following diagram commutes:

ZB(A) ^ξ^A //

ζ_A^B ""

Z(A)

ζ_A^Vect^k

}}A

3.4. The B-center as a left center

We will now show that theB-center of an algebra in aB-central monoidal category can be computed as the left center of its image under the functor R_B, thus generalizing [11, Thm. 5.4].

Theorem 3.18. For aB-central monoidal category C, assume that there exists a right adjoint RB to the forgetful functor ZB(C)→ C, and that the counit is given by epimorphisms. Let A ∈ Alg(C). Then, there is a canonical isomorphism of (commutative)algebras C^l(R_B(A))∼=Z_B(A)inZB(C).

Proof. Given theB-central set-up provided in previous sections, the proof of the theorem for the relative monoidal center Z_B(C) is now analogous to Davydov’s formal proof for Z(C) in [11, Thm. 5.4]. The proof crucially uses the hypothesis thatβA is an epimorphism.

3.5. Morita invariants

Next, we turn our attention to module categories over the monoidal categories discussed above. A left module categoryover a monoidal category C is a category M equipped with a bifunctor ∗: C × M → M and natural isomorphisms for associativity

{mX,Y,M: (X⊗Y)∗M →^∼ X∗(Y ∗M)|X, Y ∈ C, M ∈ M}

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and for unitality, which are compatible with the structure ofM; see [18, Sect. 7.1]

for details. A morphism between two C-module categories Mand N is a functor F: M → N equipped with natural isomorphisms

{sX,M =s^F_X,M:F(X∗M)→^∼ X∗F(M)|X ∈ C, M ∈ M}

that are compatible with the associativity and unitality structure of M, cf. [18, Sect. 7.2]. The collection ofC-module endofunctors of a C-module category Mis a monoidal category and is denoted byEndC(M).

For an algebra A ∈ C, recall thatMod-A(C) is the category of right modules overAin C. It is a leftC-module category viaX∗(M, ρ) := (X⊗M, Id_X⊗ρ) for allX∈ C andM ∈Mod-A(C) with structure morphismρ:M⊗A→M inC.

Definition 3.19. We say that two algebrasAandA⁰in a monoidal categoryCare Morita equivalent if Mod-A(C) and Mod-A⁰(C) are equivalent as left C-module categories.

The above generalizes the notion of Morita equivalence for rings or for algebras inVect_k. We will establish the following result later in this section.

Theorem 3.20. TakeCaB-central monoidal category, and letAandA⁰be algebras in C. Suppose that Mod-A(C) and Mod-A⁰(C) are equivalent as left C-module categories. Then, theB-centersZB(A)andZB(A⁰)are isomorphic as(commutative) algebras in ZB(C). In particular, the B-center of an algebra in C is a Morita invariant.

This result is a generalization of [11, Thm. 6.2 and Cor. 6.3] in the case when B = Vect_k, and see the discussion in Remark 4.10 in the next section for an example of how it can be used in practice. For the proof of the theorem above, we need the next construction.

Definition 3.21. Take C a B-central monoidal category, and letM be a left C- module category. Consider the monoidal functor

E:Z_B(C)→ End_C(M), (V, c_V,−)7→(LV, s^L^V),

where L_V:M → M is the functor given byM 7→V ∗M, ands is the collection of natural isomorphisms, for eachX ∈ C andM ∈ M, given by

s^L_X,M^V :=mX,V,M (cV,X∗IdM)m⁻¹_V,X,M:LV(X∗M)→^∼ X∗LV(M).

Then theB-center ofMis defined to be the terminal object in the comma category E↓I_End_C_(M).

Namely,Z_B(M) is the terminal object amongst pairs ((Z, c_Z,−), z) for (Z, c_Z,−)∈ Z_B(C) and {z=zM:Z∗M →M}_M∈M a natural transformation, such that for allX∈ C,M ∈ M, the following diagram commutes:

(Z⊗X)∗M ^c^Z,X^∗Id^M //

m_Z,X,M

(X⊗Z)∗M

m_X,Z,M

Z∗(X∗M)

zX∗M ))

X∗(Z∗M)

IdX∗zM

uuX∗M

.

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Proof of Theorem 3.20. Analogous to the proof of Proposition 3.16, we first have that ZB(M) is a commutative algebra in ZB(C). Moreover, if M and M⁰ are equivalent C-module categories, then ZB(M) ∼= ZB(M⁰) in ComAlg(ZB(C)).

So, it suffices to establish that ZB(Mod-A(C)) ∼= ZB(A) in ComAlg(ZB(C)).

In turn, it suffices to show that the comma category E↓I_End_C_(Mod-A(C)) used in Definition 3.21 is monoidally equivalent to the categoryZB(A) from Definition 3.15.

At this point, one can proceed exactly as in the proof of [11, Thm. 6.2] using only the half-braidings of the full braided monoidal subcategoryZ_B(C) ofZ(C) in order to finish the proof.

The converse of Theorem 3.20 holds when B = Vect_k, with C a (braided monoidal) modular tensor category, and the algebras inCin question being simple and non-degenerate, by [27, Sect. 4.4]. So we ask:

Question 3.22. In general, what conditions do we need on C, on B, and on algebras inC for a converse of Theorem3.20to hold?

We discuss the special setting of whenC is braided next.

3.6. The case when Cis braided

As mentioned in Example 3.2(2) and Proposition 3.5(1), we have that B is B- central, and that ZB(B) is isomorphic to B as braided monoidal categories. For instance, take C =k-Mod(B) — this can be identified canonically with B and is isomorphic toZB(B) as braided monoidal categories. Moreover in this case, there is a natural isomorphism RB ∼

=⇒ IdB, and theB-center ofA∈Alg(B) is given by Z_B(A)∼=C^l(R_B(A))∼=C^l(A) as commutative algebras inZB(B) by Theorem 3.18.

So, when C is braided, one does not need to work outside of C to get Morita invariants for algebras in C via Theorem 3.20, as C is isomorphic to its relative monoidal center. This is computationally more feasible than working with the full center construction of [11]; see, for example, Remark 6.2. In particular, constructing Morita invariants of algebras in modular tensor categories was one of the motivations behind Davydov’s work [11] and other previous works [19], [27] — now our construction of B-centers makes this goal more tractable computationally.

4. Connection to centralizer algebras

In this section, we restrict our attention to the situation where H is a Hopf algebra in a braided monoidal categoryB, andC=H-Mod(B). TakeA∈Alg(C).

We saw in Theorem 3.18 that the B-center Z_B(A) can be computed as the left center of R_B(A), and we will now realizeZ_B(A) as a braided version of a centralizer algebra — see Theorem 4.5 below. We begin by discussing braided smash product algebras in Section 4.1. Then Theorem 4.5 is established in Section 4.2, and consequences of this result are provided in Section 4.3.

4.1. Braided smash product algebras Consider the following terminology.

Definition 4.1 ([35, Prop. 2.3]). TakeH a Hopf algebra inB, andAan algebra in H-Mod(B). The (braided)smash product algebra or (braided) crossed product

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algebra AoH is the algebra inB that isA⊗H as an object inB equipped with multiplication

m_A_o_H= (m_A⊗mH)(Id_A⊗aA⊗IdH⊗H)(Id_A⊗H⊗ΨH,A⊗IdH)(Id_A⊗∆H⊗IdA⊗H) and with unituAoH=uA⊗uH.

Moreover, the result below describes the category of modules over braided smash product algebras.

Proposition 4.2 ([35, Prop. 2.7]). There is an equivalence of monoidal categories

A-Mod(H-Mod(B))'AoH-Mod(B),

where an objectV ofA-Mod(H-Mod(B))with leftH-actiona^H_V and leftA-action a^A_V gets sent to the objectV with AoH-action aAoH=a^A_V(IdA⊗a^H_V).

Next, we provide a preliminary result on braided smash product algebras.

Definition 4.3. We define a mapϕ:H⊗A→AoH by

ϕ= Ψ⁻¹_H,A(S⁻¹⊗IdA) = (IdA⊗S⁻¹)Ψ⁻¹_H,A = − H A

A H ,

and note thatϕis an isomorphism inB, with

ϕ⁻¹= ΨA,H(IdA⊗S) = (S⊗IdA)ΨA,H.

Lemma 4.4. The isomorphism ϕdefines onAoH the structure of anH-Yetter–

Drinfeld module inBfrom such a structure onR_B(A) =H⊗Avia Theorem3.10.

The H-action and H-coaction onAoH are given by

a^o= (Id_A⊗mH)Ψ⁻¹_H,A⊗H(S⁻¹⊗a_A⊗H)(∆_H⊗Id_A⊗H), δ^o= (S⊗Id_A⊗H)Ψ_A⊗H,H(Id_A⊗∆H),

using a_A⊗H from Equation2.1.

Proof. The action a^o and the coaction δ^o are defined from the action a^R and coactionδ^Rin Theorem 3.10 by requiring that ϕbecomes a morphism of Yetter–

Drinfeld modules. That is,

a^o=ϕa^R(Id_H⊗ϕ⁻¹), δ^o= (Id_H⊗ϕ)δ^Rϕ⁻¹.

Sincea^R, δ^Rare Yetter–Drinfeld compatible,a^o, δ^oare also Yetter–Drinfeld compatible.