Examples of (non)-braided tensor categories

TL;DR

This paper introduces six non-braidable tensor categories extending comodules over super-group algebras and two braided categories with only trivial braiding, expanding the landscape of tensor category examples.

Contribution

It provides new explicit examples of non-braidable and trivial-braiding tensor categories, enriching the classification and understanding of tensor categories.

Findings

Six non-braidable tensor categories introduced

Two braided categories with only trivial braiding

Extensions of comod(H) for super-group algebra

Abstract

Are introduced six examples of non-braidable tensor categories which are extensions of the category Comod(H), for H a super-group algebra; and two examples of braided categories where the only possible braiding is the trivial braiding.

Equations

$${\mathcal{F}}_A:\mathrm{Comod}(L)\to \mathrm{Comod}(H),{\mathcal{F}}_A=A{\square }_L-.$$

$$f\otimes {\mathrm{id}}_X:H{\square }_{H}X\to H{\square }_{H}X,(X,{\rho }_X)\in \mathrm{Comod}(H),$$

$$(\Delta \otimes \mathrm{id})R=R^{13}{R}^{23},(\mathrm{id}\otimes \Delta )R=R^{13}{R}^{12},{\Delta }^{op}(h)=R\Delta (h)R^{-1},h\in H.$$

$$r(c\otimes ab)=r(c_1\otimes b)r(c_2\otimes a),r(ab\otimes c)=r(a\otimes c_1)r(b\otimes c_2),$$

$$r(a_1\otimes b_1)a_2{b}_2=r(a_2\otimes b_2)b_1{a}_1.$$

$$\Delta (v)=v\otimes 1+u\otimes v;\Delta (g)=g\otimes g;S(v)=-uv;S(g)=g^{-1},v\in V,g\in C_2.$$

$${\alpha }_{V,W,U}{c}_{U,V\otimes W}{\alpha }_{U,V,W}=(\mathrm{id}\otimes c_{U,W}){\alpha }_{V,U,W}(c_{U,V}\otimes \mathrm{id}),$$

$${\alpha }_{W,U,V}^{-1}{c}_{U\otimes V,W}{\alpha }_{U,V,W}^{-1}=(c_{U,W}\otimes \mathrm{id}){\alpha }_{U,W,V}^{-1}(\mathrm{id}\otimes c_{V,W}),$$

$$L_e=H,(g(e,a),f^{e,a})=(e,{\mathrm{id}}_{L_a})=(g(a,e),f^{a,e});$$

$$\varphi (L_a,g(b,c))g(a,bc)=g(a,b)g(ab,c);$$

$$f^{ab,c}(f^{a,b}\otimes {\mathrm{id}}_{L_c})=f^{a,bc}({\mathrm{id}}_{L_a}\otimes f^{b,c}).$$

$$[V,a]\otimes [W,b]:$$

$$[V,1{]}^{\ast }:$$

$$[\mathbb{k},a{]}^{\ast }:$$

$$[V,e][W,g]$$

$$[V,e{]}^{\ast }$$

$$[V,e][W,e]$$

$$[V,e][W,u]$$

$$[V,e{]}^{\ast }$$

$$[V,e][W,g]$$

$$[V,e{]}^{\ast }$$

$$[V,e][W,e]$$

$$[V,e][W,u]$$

$$[V,e{]}^{\ast }$$

$$v^1=w^1={\mathrm{id}}_H,$$

$$(g(a,b),f^{a,b})=(g(b,a),f^{b,a}),$$

$$W^b(x_{-3})W^a(x_{-2})(W^{ab}{)}^{-1}(x_{-1})x_0=F^{a,b}(x_{-2})r(x_{-1}\otimes g(a,b))x_0,x\in X,$$

$$V^b(x_{-3})V^a(x_{-2})(V^{ab}{)}^{-1}(x_{-1})x_0=r(x_{-2}\otimes g(a,b))F^{a,b}(x_{-1})x_0,x\in X,$$

$$V^a(g(b,c))=({\gamma }_{a,b,c}{\gamma }_{b,c,a}{)}^{-1}{\gamma }_{b,a,c},$$

$$W^b(g(c,a))={\gamma }_{c,a,b}{\gamma }_{b,c,a}{\gamma }_{c,b,a}^{-1};$$

$${\mathbf{c}}_{[V,a],[W,b]}=c_{V,W}((V^a\otimes \mathrm{id}){\rho }_V\otimes (W^a\otimes \mathrm{id}){\rho }_W)\otimes \mathrm{id},V,W\in \mathrm{Comod}(H),a,b\in \Gamma .$$

$${\theta }^1={\tau }^1=\mathrm{id},{\theta }_1^a={\mathrm{id}}_1={\tau }_1^a,t_{a,1}=t_{1,a}={\mathrm{id}}_1,$$

$$c_{U_{a,b},X}{\sigma }_X^{a,b}=(({\tau }_X^{ab}{)}^{-1}{\tau }_X^a{\tau }_X^b)\otimes {\mathrm{id}}_{U_{a,b}},$$

$${\sigma }_X^{a,b}{c}_{U_{a,b},X}={\mathrm{id}}_{U_{a,b}}\otimes (({\theta }_X^{ab}{)}^{-1}{\theta }_X^a{\theta }_X^b),$$

Full text

1991 Mathematics Subject Classification 18D10

Keywords: supergroup algebra, braidings, tensor category

1. Introduction

Braided categories were introduced by Joyal and Street [5]. They are related to knot invariants, topology and quantum groups, since they can express symmetries. Some examples of braided categories are:

• •

graded modules over a commutative ring,

• •

(co)modules over a (co)quasi-triangular Hopf algebra,

• •

the Braid category, [5, Section 2.2],

• •

the center of a tensor category.

In the last example, we begin with a tensor category and construct a braided one. In a general scenario, a natural question is it is possible to construct braidings starting with tensor categories. In particular, if $G$ is a finite group, can a $G$-extension of a tensor category be braided? In this work we show that this can be done in very few cases. Then, an extension of a braided category is not necessarily braided, so it is really complicated to extend that property.

However, constructing examples of non-braided categories is also important. A big family of these come from the category of (co)modules of a Hopf algebra without a (co)quasi-triangular structure, see [9, T 10.4.2]. Masuoka in [6] and [7] constructs explicit examples of non- Quasi-triangular or non-CoQuasi-triangular Hopf algebras. In particular these Hopf algebras can not be obtained from any group algebra by twist (or cocycle) deformation. Other examples were constructed in [4].

In the literature there are a few explicit examples of tensor categories, for this reason we construct in [8] eight tensor categories, following the description introduced in [3] of Crossed Products. These categories extend the module category over certain quantum groups, called supergroup algebras. In a few words, a crossed product tensor category is, as Abelian category, the direct sum of copies of a fixed tensor category, and the tensor product comes from certain data. Then founding all possible data, we explicitly construct tensor categories.

In the same work [3], the author also describes all posible braidings over a crossed product. Following this, three conditions were introduced to decide if a $G$-crossed product is braidable:

1. (1)

the base category has to be braided, 2. (2)

$G$ has to be Abelian, and the biGalois objects associated to each crossed product have to be trivial, 3. (3)

the $3$-cocycle associated to each crossed product over an specific supergroup algebra has to be trivial, if $G$ is the cyclic group of order $2$.

The goal in the present paper is to obtain all possible braidings over the categories introduced in [8]. With this, only two categories of the eight found in [8] are braided with the trivial braiding only, and the other 6 are not braidable.

In [8, Thm 6.3], using the Frobenius-Perron dimension, we proved that these eight categories are the module category of a quasi-Hopf algebra. Although we do not know how to explicitly compute these algebras, as a Corollary of this work, we know that six of these algebras are non-Quasi-triangular and two are Quasi-triangular only. In particular, we are obtaining information about certain quasi-Hopf algebras without knowing them explicitly; showing how useful it is to work in the category world. In a near future, when we can explicitly describe these quasi-Hopf algebras, we will already know how their Quasi-triangular structures are.

2. Preliminaries and notation

Throughout this paper we shall work over an algebraically closed field $\Bbbk$ of characteristic zero. For basic knowledge of Hopf algebras see [9]. Let $H$ be a finite-dimensional Hopf algebra and $A$ be a left $H$-comodule. Then $A$ is also a right $H$-comodule with right coaction $a\mapsto a_{0}\otimes S(a_{-1})$, see [1, Prop 2.2.1(iii)]. A left $H$-Galois extension of $A^{co(H)}$ is a left $H$-comodule algebra $(A,\rho)$ such that $A\otimes_{A^{co(H)}}A\to H\otimes A$, $a\otimes b\mapsto(1\otimes a)\rho(b)$ is bijective. Similarly, we define right $H$-Galois extension.

Consider $L$ another finite-dimensional Hopf algebra. An (H, L)-biGalois object [10] is an algebra $A$ that is a left $H$-Galois extension and a right $L$-Galois extension of the base field $\Bbbk$ such that the two comodule structures make it an $(H,L)$-bicomodule. Two biGalois objects are isomorphic if there exists a bijective bicomodule morphism that is also an algebra map. For $A$ an $(H,L)$-biGalois object, define the tensor functor

[TABLE]

By [10], every tensor functor between comodule categories is one of these, and $\mathcal{F}_{A}\simeq\mathcal{F}_{B}$ as tensor functors if and only if $A\simeq B$ as biGalois objects.

If $A=H$, then every natural monoidal equivalence $\beta:\mathcal{F}_{H}\to\mathcal{F}_{H}$ is given by

[TABLE]

where $f:H\to H$ is a bicomodule algebra isomorphism.

Lemma 2.1**.**

Every natural monoidal equivalence $\operatorname{id}_{\operatorname{Comod}(H)}\to\operatorname{id}_{\operatorname{Comod}(H)}$ is given by $(\varepsilon f\otimes\operatorname{id}_{X})\rho_{X}.$

Proof.

For $X\in\operatorname{Comod}(H)$, the coaction induces an isomorphism $X\simeq H\Box_{H}X$ with inverse induced by $\varepsilon$, the counit. Then $\operatorname{id}_{\operatorname{Comod}(H)}\simeq\mathcal{F}_{H}$ as tensor functors. Since all natural monoidal autoequivalences of $\mathcal{F}_{H}$ are given by $f\otimes\operatorname{id}_{X}$ then all natural monoidal autoequivalences of $\operatorname{id}_{\operatorname{Comod}(H)}$ are given by $(\varepsilon f\otimes\operatorname{id}_{X})\rho_{X}.$ ∎

Definition 2.2**.**

[9, Defn 10.1.5]** $(H,R)$ is a Quasi-triangular (or QT) Hopf algebra if $H$ is a Hopf algebra and there exist $R\in H\otimes H$, called the $R$-matrix, invertible such that

[TABLE]

Dualizing we can define, $(H,r)$ is a CoQuasi-triangular (or CQT) Hopf algebra if $H$ is a Hopf algebra and $r:H\otimes H\to\Bbbk$, called the $r$-form, is a linear functional which is invertible with respect to the convolution multiplication and satisfies for arbitrary $a,b,c\in H$

[TABLE]

[TABLE]

Remark 2.3**.**

Drinfeld defined a quantum group as a non-commutative, non-cocommutative Hopf algebra. Examples of these are the QT Hopf algebras. The importance of quantum groups lies in they allow to construct solutions for the quantum Yang-Baxter equation in statistical mechanics (the R-matrix is a solution of the Yang-Baxter equation). An example of quantum group are the supergroup algebras.

A supergroup algebra is a supercocommutative Hopf superalgebra of the form $\Bbbk[G]\ltimes\wedge V$, where $G$ is a finite group and $V$ is a finite-dimensional $\Bbbk$-module of $G$. They appear and have an interesting role in the classification of triangular algebras, see [2, Thm 4.3].

Example 2.4**.**

Consider $H=\Bbbk C_{2}\ltimes\Bbbk V$, for $V$ a 2-dimensional vector space and $C_{2}$ the 2-cyclic group generated by $u$ with $u\cdot v=-v$ for $v\in V$. As an algebra, it is generated by elements $v\in V,g\in C_{2}$ subject to relations $vw+wv=0;gv=(g\cdot v)g$ for all $v,w\in V,g\in C_{2}$. The coproduct and antipode are determined by

[TABLE]

Taking $R=\frac{1}{2}(1\otimes 1+1\otimes u+\otimes 1-u\otimes u)$, $(H,R)$ is a QT-Hopf algebra. We can construct a CoQuasi-triangular structure taking $r=R^{*}$ since $H$ is auto-dual. Then $(H,R^{*})$ is a CQT-Hopf algebra.

Definition 2.5**.**

A finite tensor category is a locally finite, $\mathbf{k}$-linear, rigid, monoidal Abelian category $\mathcal{D}$ with $\mathrm{End}_{\mathcal{D}}\left(\mathbf{1}\right)\cong\mathbf{k}$. Given a finite group $\Gamma$, a (faithful) $\Gamma$-grading on a finite tensor category $\mathcal{D}$ is a decomposition $\mathcal{D}=\oplus_{g\in\Gamma}\mathcal{D}_{g}$, where $\mathcal{D}_{g}$ are full Abelian subcategories of $\mathcal{D}$ such that

• $\bullet$

$\mathcal{D}_{g}\neq 0$;

• $\bullet$

$\otimes:\mathcal{D}_{g}\times\mathcal{D}_{h}\to\mathcal{D}_{gh}$* for all $g,h\in\Gamma.$*

We have that $\mathcal{C}:=\mathcal{D}_{e}$ is a tensor subcategory of $\mathcal{D}$. The tensor category $\mathcal{D}$ is call a $\Gamma$-extension of $\mathcal{C}$. Denote by $[V,g]$ the homogeneous elements in $\mathcal{D}$, for $V\in\mathcal{D}_{g}$, $g\in\Gamma$.

A braided tensor category is a tensor category $\mathcal{C}$ with natural isomorphisms $c_{X,Y}:X\otimes Y\to Y\otimes X$ such that

[TABLE]

[TABLE]

If $(H,r)$ is a CQT-Hopf algebra then $\operatorname{Comod}(H)$ is a braided tensor category where the braiding is given by $c_{V\otimes W}(x\otimes y)=r(y_{-1}\otimes x_{-1})y_{0}\otimes x_{0}$, for all $V,W\in\operatorname{Comod}(H)$.

The following Lemma gives us the first condition to know when an extension can be braided.

Theorem 2.6**.**

Let $\mathcal{D}=\oplus_{g\in G}\mathcal{D}_{g}$ be a $\Gamma$-extension of $\mathcal{C}$. If $\mathcal{D}$ is a braided tensor category then $\mathcal{C}$ is a braided tensor category.

Proof.

Let $c$ be the braiding of $\mathcal{D}$, then $c_{[V,e],[W,e]}:[V\otimes W,e]\to[W\otimes V,e]$ and $c_{[V,e],[W,e]}=[\overline{c}_{V,W},e]$ for some natural isomorphism $\overline{c}_{V,W}:V\otimes W\to W\otimes V$, for all $V,W$ objects in $\mathcal{C}$. Since the associativity isomorphism satisfies $a_{[V,e],[W,e],[U,e]}=[\overline{a}_{V,W,U},e]$, where $\overline{a}$ is the associativity morphism for $\mathcal{C}$; then $\overline{c}$ is a braiding for $\mathcal{C}$.

∎

In [3], the author describes and classifies a family of such extensions and calls it crossed product tensor category. Fix $H$ a finite-dimensional Hopf algebra. In the case when $\mathcal{C}=\operatorname{Comod}(H)$, in [8], we described crossed products in terms of Hopf-algebraic datum. A continuation they are introduced.

If $g\in G(H)$ and $L$ is a $(H,H)$-biGalois object then the cotensor product $L\Box_{H}\Bbbk_{g}$ is one-dimensional. Let $\phi(L,g)\in\Gamma$ be the group-like element such that $L\Box_{H}\Bbbk_{g}\simeq\Bbbk_{\phi(L,g)}$ as left $H$-comodules. Assume that $A$ is an $H$-biGalois object with left $H$-comodule structure $\lambda:A\to H\otimes_{\Bbbk}A$. If $g\in G(H)$ is a group-like element we can define a new $H$-biGalois object $A^{g}$ on the same underlying algebra $A$ with unchanged right comodule structure and a new left $H$-comodule structure given by $\lambda^{g}:A^{g}\to H\otimes_{\Bbbk}A^{g}$, $\lambda^{g}(a)=g^{-1}a_{-1}g\otimes a_{0}$ for all $a\in A$.

Theorem 2.7**.**

[8, Lem 5.7, Thm 5.4]** Let $\Upsilon=(L_{a},(g(a,b),f^{a,b}),\gamma)_{a,b\in\Gamma}$ be a collection where

• •

$L_{a}$* is a $(H,H)$-biGalois object;*

• •

$g(a,b)\in G(H)$;

• •

$f^{a,b}:(L_{a}\Box_{H}L_{b})^{g(a,b)}\to L_{ab}$* are bicomodule algebra isomorphisms;*

• •

$\gamma\in Z^{3}(G(H),\Bbbk^{\times})$* normalized,*

such that for all $a,b,c\in\Gamma$:

[TABLE]

[TABLE]

[TABLE]

Then $\operatorname{Comod}(H)(\Upsilon):=\oplus_{g\in\Gamma}\operatorname{Comod}(H)$ as a structure of tensor category.

Proof.

We give an sketch of the proof. Let $\Upsilon$ be a collection as in the Theorem. For $V,W\in\operatorname{Comod}(H)$, $a,b\in\Gamma$, define

[TABLE]

Using [8, Eq (5.8)], we obtain the pentagon diagram and therefore $\operatorname{Comod}(H)(\Upsilon)$ is a monoidal category. Since $\operatorname{Comod}(H)$ is finite tensor category, then $\operatorname{Comod}(H)(\Upsilon)$ is also finite tensor category. ∎

The following Lemma gives us a second condition to decided if our extensions can be braided.

Theorem 2.8**.**

If $\operatorname{Comod}(H)(\Upsilon)$ is braided with braiding $c$ then the following conditions have to hold

1. (1)

$L_{a}\simeq H$* for all $a\in\Gamma$,* 2. (2)

$\Gamma$* is Abelian,* 3. (3)

$\Upsilon$* comes from a data $(g,f^{a,b},\gamma)_{a,b\in\Gamma}$ with*

• •

$g\in Z^{2}(\Gamma,G(H))$* normalized,*

• •

$f^{a,b}:H^{g(a,b)}\to H$* a bicomodule algebra isomorphism with $f^{ab,c}f^{a,b}=f^{a,bc}f^{b,c}$,*

• •

$\gamma\in Z^{3}(G(H),\Bbbk^{\times})$* normalized.*

Proof.

(1) Take, for any $V\in\operatorname{Comod}(H)$, $c_{[V,e][\mathbf{1},a]}:[V,a]\to[L_{a}\Box_{H}V,a]$, this defines a natural isomorphism $\overline{c}_{a}:\operatorname{id}_{\mathcal{C}}\to L_{a}\Box_{H}-$ wich is monoidal since $c$ is a braiding. Then $L_{a}\simeq H$ as bicomodule algebras for all $a\in\Gamma$.

(2) Consider $c_{[\mathbf{1},a][\mathbf{1},b]}:[\Bbbk_{g(a,b)},ab]\to[\Bbbk_{g(b,a)},ba]$ then $ab=ba$ for all $a,b\in\Gamma$ and $\Gamma$ is Abelian.

(3) Since $L_{a}$ is trivial, then condition Theorem 2.7(4) is equivalent to $g\in Z^{2}(\Gamma,G(H))$ and it is normalized by condition Theorem 2.7(3). Moreover $f^{a,b}:H^{g(a,b)}\to H$ is a bicomodule algebra isomorphism that satisfies $f^{ab,c}f^{a,b}=f^{a,bc}f^{b,c}$ which is equivalent to condition Theorem 2.7(5).

∎

Remark 2.9**.**

By definition of bicomodule morphism, $f^{a,b}:H\to H$ has to be an algebra isomorphism such that $f^{a,b}(h)_{1}\otimes f^{a,b}(h)_{2}=g^{-1}h_{1}g\otimes f^{a,b}(h_{2})$ and $f^{a,b}(h)_{1}\otimes f^{a,b}(h)_{2}=f^{a,b}(h_{1})\otimes h_{2}$, then $g^{-1}h_{1}g\otimes f^{a,b}(h_{2})=f^{a,b}(h_{1})\otimes h_{2}$.

In the case when $H=\wedge V\#\Bbbk C_{2}$, as Example 2.4, using the previous Theorem we obtained eight tensor categories non-equivalent pairwise, [8, Sect 6.3], named $\mathcal{C}_{0}(1,\operatorname{id},\pm 1)$, $\mathcal{C}_{0}(u,\iota,\pm 1)$, $\mathcal{D}(1,\operatorname{id},\pm 1)$, $\mathcal{D}(u,\iota,\pm 1)$.

In all cases, the underlying Abelian category is $\operatorname{Comod}(H)\oplus\operatorname{Comod}(H)$ and for $V,W,Z\in\operatorname{Comod}(H)$ they are defined in the following way:

$\bullet$ The tensor product, dual objects and associativity in $\mathcal{C}_{0}(1,\operatorname{id},\pm 1)$ are given by

[TABLE]

$\alpha_{[V,u],[W,u],[Z,u]}$ is not trivial, and $\textbf{U}_{0}$ is certain BiGalois object, see [8, Section 4].

$\bullet$ The tensor product, dual objects and associativity in $\mathcal{C}_{0}(u,\iota,\pm 1)$ are given by

[TABLE]

$\alpha_{[V,u],[W,u],[Z,u]}$ is not trivial.

$\bullet$ The tensor product, dual objects and associativity in $\mathcal{D}(1,\operatorname{id},\pm 1)$ are given by

[TABLE]

$\alpha_{[V,u],[W,u],[Z,u]}=[\pm\operatorname{id}_{V\otimes W\otimes Z},u]$ and the others are trivial.

$\bullet$ The tensor product, dual objects and associativity in $\mathcal{D}(u,\iota,\pm 1)$ are given by

[TABLE]

$\alpha_{[V,u],[W,u],[Z,u]}=[\pm\operatorname{id}_{V\otimes W}\otimes\tau(\varepsilon\iota\rho_{Z}\otimes\operatorname{id}_{Z\otimes\Bbbk_{u}}),u],$ where $\iota:H^{u}\to H$ is the unique bicomodule algebra isomorphism which satisfies $\iota(u)=-u$ and $\iota(x)=-x$ for $x\in V$; and $\tau:X\otimes Y\to Y\otimes X$, $\tau(z\otimes k)=k\otimes z$ for all $X,Y\in\operatorname{Comod}(H)$, see [8, Rem 2.2].

Remark 2.10**.**

By Lemma 2.8(1), we obtain that only the categories $\mathcal{D}(1,\operatorname{id},\pm 1)$ and $\mathcal{D}(u,\iota,\pm 1)$ could be braided, since the BiGalois objects have to be trivials.

By direct calculation on Equation (1), $\mathcal{D}(u,\iota,-1)$ is not braided with trivial braiding. So, in this case, we want to know if there exist another possible braidings.

3. Braided crossed product

Let $\Gamma$ be an Abelian group. In [8], following the ideas developed in [3], we described all $\Gamma$-crossed product tensor categories which are extensions of $\operatorname{Comod}(H)$ for $H$ a Hopf algebra in terms of certain Hopf-algebraic datum. Fix $(H,r)$ a CQT-Hopf algebra. In the first Lemma of this Section, we do the same for the braiding of crossed products that are $\Gamma$-extensions of $\operatorname{Comod}(H)$.

Remark 3.1**.**

If $v:H\to H$ is a left $H$-comodule morphism, since the coaction is the coproduct, $v$ satisfies $v(x)_{1}\otimes v(x)_{2}=x_{1}\otimes v(x_{2})$, for all $x\in H$. In particular, $v$ is not a coalgebra morphism and if $g\in G(H)$, $v(g)=g\varepsilon(v(g))$.

Lemma 3.2**.**

Fix a datum $(g,f^{a,b},\gamma)_{a,b\in\Gamma}$, as in Lemma 2.8, and let $\mathcal{C}$ be the associated tensor category. Consider a pair $(v^{a},w^{a})_{a\in\Gamma}$ where $v^{a},w^{a}:H\to H$ are left $H$-comodule algebra isomorphisms. Let $W^{a}=\varepsilon w^{a}$, $V^{a}=\varepsilon v^{a}$ and $F^{a,b}=\varepsilon f^{a,b}$. If for all $a,b,c\in\Gamma$ and $X\in\operatorname{Comod}(H)$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

then we obtain a braiding over $\mathcal{C}$ given by

[TABLE]

All braidings over $\mathcal{C}$ come from a pair $(v^{a},w^{a})_{a\in\Gamma}$ which satisfies (7) to (11).

Proof.

By [3, Defn 5.3], a datum $(g,f^{a,b},\gamma)_{a,b\in\Gamma}$ has associated a braiding if there exist a triple $(\theta^{a},\tau^{a},t_{a,b})_{a,b\in G}$ where

• •

$\theta^{a},\tau^{a}:\operatorname{id}_{\mathcal{C}}\to\operatorname{id}_{\mathcal{C}}$ are monoidal natural isomorphisms,

• •

for all $a,b\in G$, $t_{a,b}:(U_{a,b},\sigma^{a,b})\to(U_{b,a},\sigma^{b,a})$ are isomorphisms in $\mathcal{Z}(\mathcal{C})$, where $\sigma^{a,b}_{X}=\tau(\varepsilon f^{a,b}\otimes\operatorname{id}_{X})\rho_{X}$, for $(X,\rho_{X})\in\operatorname{Comod}(H)$, and $U_{a,b}=\Bbbk_{g(a,b)}$,

such that for all $a,b,c\in\Gamma$ and $X\in\mathcal{C}$, the following conditions hold

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By Lemma 2.1, each monoidal natural isomorphism of the identity functor comes from a left $H$-comodule algebra isomorphism, then $\theta^{a}_{X}:=(\varepsilon v^{a}\otimes\operatorname{id})\rho_{X}$ and $\tau^{a}_{X}:=(\varepsilon w^{a}\otimes\operatorname{id}_{X})\rho_{X}$ for all $X\in\operatorname{Comod}(H)$. Since $U_{g(a,b)}=\Bbbk_{g(a,b)}$, we can take $t_{a,b}\in\Bbbk^{*}$.

Each $t_{a,b}$ is a left $H$-comodule isomorphism if and only if $g(a,b)\otimes t_{a,b}\operatorname{id}_{\Bbbk}=g(b,a)\otimes t_{a,b}\operatorname{id}_{\Bbbk}$ wich gives $g(a,b)=g(b,a)$ for all $a,b\in\Gamma$. Moreover, each $t_{a,b}$ is a braided morphism if and only if $\sigma^{a,b}_{X}t_{a,b}=\sigma^{b,a}_{X}t_{a,b}$ for $a,b\in\Gamma$ and $X\in\operatorname{Comod}(H)$ if and only if $\sigma^{a,b}=\sigma^{b,a}$. Then $t_{a,b}$ is an isomorphism in $\mathcal{Z}(\operatorname{Comod}(H))$ if and only if Condition (7) holds.

Condition (12) is equivalent to $v^{1}=w^{1}=\operatorname{id}_{H}$ and $t_{a,1}=t_{1,a}=1$, since $\theta_{\Bbbk}^{a}=\operatorname{id}_{\Bbbk}=\tau_{\Bbbk}^{a}$ is always true. Condition (13) is equivalent to

[TABLE]

for $x\otimes k\in X\otimes\Bbbk_{g(a,b)}$, which is equivalent to Condition (8). In the same way, Condition (14) is equivalent to Condition (9). Condition (15) is equivalent to $\gamma_{a,b,c}V^{a}(g(b,c))t_{bc,a}\gamma_{b,c,a}=t_{b,c}\gamma_{b,a,c}t_{c,a}$ but if we take $c=1$ then

[TABLE]

so, this Condition is equivalent to Condition (10), and Condition (16) is equivalent to Condition (11).

By [3, Thm 5.4], this pair produces a braiding over $\mathcal{C}$ given by

[TABLE]

and all braidings come from such a pair. ∎

Now, we focus our attention into the case $\Gamma=C_{2}$. By Lemma 2.8, a datum $\Upsilon^{\prime}=(g,f,\gamma)$ with $g\in G(H)$ a group-like element, $f:H^{g}\to H$ a bicomodule algebra isomorphism and $\gamma\in\Bbbk^{\times}$, $\gamma^{2}=1;$ generates a tensor category $\mathcal{C}=\operatorname{Comod}(H)(\Upsilon^{\prime})$.

The following Theorem gives us the third and last condition to decide if our categories are braidable.

Theorem 3.3**.**

The category $\operatorname{Comod}(H)(\Upsilon^{\prime})$ is a braided $C_{2}$-extension if and only if, there exists a pair of isomorphisms of left $H$-comodule algebras $v,w:H\to H$ such that for all $X\in\operatorname{Comod}{H}$ and $x\in X$

1. a.

$\varepsilon(w(x_{-2})w^{-1}(x_{-1}))x_{0}=x$, 2. b.

$\varepsilon(w(x_{-2})w(x_{-1}))x_{0}=\varepsilon f(x_{-2})r(x_{-1}\otimes g)x_{0}$, 3. c.

$\varepsilon(v(x_{-2})v^{-1}(x_{-1}))x_{0}=x$, 4. d.

$\varepsilon(v(x_{-2})v(x_{-1}))x_{0}=r(x_{-2}\otimes g)\varepsilon f(x_{-1})x_{0}$, 5. e.

$\varepsilon(v(g))=\gamma^{-1}$, 6. f.

$\varepsilon(w(g))=\gamma$,

Proof.

Condition (7) is always true. Condition (8) is equivalent to $r(x_{-1}\otimes 1)x_{0}=x$, and items a,b. Condition (9) is equivalent to $r(x_{-1}\otimes 1)x_{0}=x$, and items c,d. Condition (10) is equivalent to item e. Condition (11) is equivalent to item f.

Regarding condition $r(x_{-1}\otimes 1)x_{0}=x$, it is always true over a CoQuasi-triangular Hopf algebra.

∎

If $H=\wedge V\#\Bbbk C_{2}$, as Example 2.4, by [8, Prop 4.10], the isomorphisms $v$ and $w$ are identities. Then if the extension is braided the only possible braiding is the trivial, see Equation (17), since the category $\operatorname{Comod}{H}$ has a braiding giving by the r-form. With this information, Conditions a-f are equivalent to

1. a’.

$\varepsilon(x_{-2}x_{-1})x_{0}=x$, 2. b’.

$\varepsilon(x_{-2}x_{-1})x_{0}=\varepsilon f(x_{-2})r(x_{-1}\otimes g)x_{0}$, 3. c’.

$\varepsilon(x_{-2}x_{-1})x_{0}=r(x_{-2}\otimes g)\varepsilon f(x_{-1})x_{0}$, 4. d’.

$\varepsilon(g)=\gamma^{-1}$, 5. e’.

$\varepsilon(g)=\gamma$.

Since $g$ is a group-like element, d’ and e’ imply that $\gamma=1$. Thus, the only categories that could be braided are $\mathcal{D}(1,\operatorname{id},1)$ and $\mathcal{D}(u,\iota,1)$.

Corollary 3.4**.**

A $C_{2}$-extension over $\operatorname{Comod}{(\wedge V\#\Bbbk C_{2})}$ is braided if and only if, for all comodule $X$, $r(f(x_{-1})\otimes g)x_{0}=x,\text{ for all }x\in X.$

Proof.

Condition a’ is always true over comodules. Since $x_{1}y_{1}r(x_{2}\otimes y_{2})=r(x_{1}\otimes y_{1})y_{2}x_{2}$ for $x,y\in H$ we have

[TABLE]

Applying $\varepsilon f\otimes\operatorname{id}_{X}$, we obtain $r(x_{-2}\otimes g)\varepsilon f(x_{-1})x_{0}=\varepsilon f(x_{-2})r(x_{-1}\otimes g)x_{0}$. This implies that Conditions b’ and c’ are equivalent. Since

[TABLE]

we have $r(f(x_{-1})\otimes g)x_{0}=\varepsilon f(x_{-2})r(x_{-1}\otimes g)x_{0}$, then Condition b’ is equivalent to

[TABLE]

∎

We are ready for our main result.

Theorem 3.5**.**

The categories $\mathcal{D}(1,\operatorname{id},1)$ and $\mathcal{D}(u,\iota,1)$ are braided tensor categories. The remaining 6 categories found in [8] are non-braidable.

Proof.

By [3, Thm 5.4], the only possible option for $v$ and $w$ is for there to be the identity. Then the categories $\mathcal{D}(1,\operatorname{id},1)$ and $\mathcal{D}(u,\iota,1)$ have associated at most a single pair $(\operatorname{id},\operatorname{id})$, which would give it a braided structure For the remaining six categories, we already know that they are non-braidable.

Since $\mathcal{D}(1,\operatorname{id},1)$ has trivial associativity and $\operatorname{Comod}(H)$ is braided then the braiding for $\mathcal{D}(1,\operatorname{id},1)$ is

[TABLE]

Over $\mathcal{D}(u,\iota,1)$ it is enough to check Equations (1) and (2) where the associativity is not trivial. Since $(f\otimes\operatorname{id})(\operatorname{id}\otimes g)=(\operatorname{id}\otimes g)(f\otimes\operatorname{id})$ for any $f,g$ morphisms in the category, also the braiding given in (18) also satisfies the desired Equations. ∎

Corollary 3.6**.**

For $X\in\operatorname{Comod}{(\wedge V\#\Bbbk C_{2})}$, $r(\iota(x_{-1})\otimes g)x_{0}=x$, for all $x\in X$.

Remark 3.7**.**

Since $\operatorname{Comod}(H)$ is not symmetric, then these two categories are not symmetric either.