Some ribbon elements for the quasi-Hopf algebra $D^\omega(H)$

TL;DR

This paper constructs explicit isomorphisms and ribbon elements for the quasi-Hopf algebra $D^omega(H)$, advancing the understanding of its structure and properties in quantum algebra.

Contribution

It provides explicit isomorphisms and new characterizations for ribbon structures in quasi-Hopf algebras, specifically for $D^omega(H)$.

Findings

Explicit isomorphism between $D^omega(H)$ and a quantum double quasi-Hopf algebra

New characterizations for ribbon quasi-Hopf algebras

Construction of ribbon elements for $D^omega(H)$

Abstract

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra $D^\omega(H)$ defined in \cite{bp} and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for $D^\omega(H)$.

Equations

$$S(h_1)\alpha h_2=\epsilon (h)\alpha \mathrm{and}{h}_1\beta S(h_2)=\epsilon (h)\beta ,$$

$$X^1\beta S(X^2)\alpha X^3=1_H\mathrm{and}S(x^1)\alpha x^2\beta S(x^3)=1_H.$$

$$p_R=p^1\otimes p^2:=x^1\otimes x^2\beta S(x^3)\text{mbox}and{q}_R=q^1\otimes q^2:=X^1\otimes S^{-1}(\alpha X^3)X^2.$$

$$\epsilon (R^1)R^2=\epsilon (R^2)R^1=1_H.$$

$$u=S(R^2{x}^2\beta S(x^3))\alpha R^1{x}^1.$$

$$S^2(h)=uh{u}^{-1},\forall h\in H,$$

$$S(\alpha )u=S(R^2)\alpha R^1,$$

$$\Delta (u)=f^{-1}(S\otimes S)(f_{21})(u\otimes u)(R_{21}R{)}^{-1}.$$

$$\Delta (\nu )=(\nu \otimes \nu )(R_{21}R{)}^{-1}\text{mbox}andS(\nu )=\nu .$$

$${\nu }^2=uS(u),$$

$$\alpha =\epsilon (\eta )\alpha =S({\eta }_1)\alpha {\eta }_2=S(\eta R^2{r}^1)\alpha \eta R^1{r}^2\stackrel{(\text{ref}eq:ribbonqHa)}{=}{\eta }^{2}S(R^2{r}^1)\alpha R^1{r}^2$$

$$\stackrel{(\text{ref}eq:extr)}{=}{\eta }^{2}S(\alpha r^1)u{r}^2\stackrel{(\text{ref}eq:ssinau)}{=}{\eta }^{2}S(S(r^2)\alpha r^1)u\stackrel{(\text{ref}eq:extr)}{=}{\eta }^{2}S(S(\alpha )u)u\stackrel{(\text{ref}eq:ssinau)}{=}{\eta }^{2}S(u)u\alpha .$$

$$A\alpha B=A{\eta }^{2}S(u)u\alpha B={\eta }^{2}S(u{S}^{-1}(A))u\alpha B$$

$$\stackrel{(\text{ref}eq:ssinau)}{=}{\eta }^{2}S(u)S^2(A)u\alpha B\stackrel{(\text{ref}eq:ssinau)}{=}{\eta }^{2}S(u)uA\alpha B.$$

$$G(H)=\{l\in H\mid\mbox{$l$ is invertible with $l^{-1}=S(l)$ and $\Delta(l)=(l\otimes l)(S\otimes S)(f_{21}^{-1})f$}\}.$$

$$R(H):=\{l\in H\mid l^2=u^{-1}S(u),\Delta (l)=(l\otimes l)(S\otimes S)(f_{21}^{-1})f\text{mbox}\mathrm{and}l{S}^2(h)=hl,\forall h\in H\}$$

$$u^{-1}S(u)=\mu(X^{1}R^{2}p^{2}S(X^{3})f^{1})\mbox{$S^{-1}$}(S(X^{2})f^{2})R^{1}p^{1}S(\underline{g}^{-1}),$$

$$\{l\in H\mid l^2=\underline{g},\Delta (l)=(l\otimes l)(S\otimes S)(f_{21}^{-1})f\text{mbox}\mathrm{and}l{S}^2(h)=hl,\forall h\in H\}$$

$$S^2({\underline{g}}^{-1})=S(u^{-1}S(u))=S^2(u)S(u^{-1})=uS(u{)}^{-1}=S(u{)}^{-1}u=(u^{-1}S(u){)}^{-1}=S(\underline{g}).$$

$$\Delta (h)=h\otimes h,({\mathrm{Id}}_H\otimes \Delta )(\Delta (h))=(\Delta \otimes {\mathrm{Id}}_H)(\Delta (h))=h\otimes h\otimes h,$$

$$S(h)h=hS(h)=\epsilon (h)1_H,\epsilon (h)h=h\epsilon (h)=h.$$

$$\sigma (x,y)\sigma (xy,z)=[x\cdot \sigma (y,z)]\sigma (x,yz),\forall x,y,z\in H.$$

$$\omega (x,y,zt)\omega (xy,z,t)=\omega (y,z,t)\omega (x,yz,t)\omega (x,y,z),\forall x,y,z,t\in H,$$

$$\omega (1_H,x,y)=\omega (x,1_H,y)=\omega (x,y,1_H)=\epsilon (x)\epsilon (y),\forall x,y\in H.$$

$$\theta (g;x,y)=\omega (g,x,y)\omega (x,y,g◃(xy)){\omega }^{-1}(x,g◃x,y),$$

$$\theta (g;x,y)\theta (g;xy,z)=\theta (g◃x;y,z)\theta (g;x,yz),\forall g,x,y,z\in H.$$

$$(\phi \otimes h)(\phi {}^{\prime }\otimes h{}^{\prime })=\phi (h\rightharpoonup \phi {}^{\prime }\leftharpoonup S(h))\sigma (h,h{}^{\prime })\otimes hh{}^{\prime }.$$

$$w(h\bowtie \phi )={\overline{\omega }}_2(h){\overline{\omega }}_3(S(h)){\overline{\omega }}_1(h\rightharpoonup \phi \leftharpoonup S(h))\#h,\forall h\in H,\phi \in H^{\ast },$$

$$W(\phi \#h)=p_1^1(h)p^2(S(h))({\phi }_1\rightharpoonup h\leftharpoonup S({\phi }_3))\bowtie p_2^1{\phi }_2,\forall \phi \in H^{\ast },h\in H,$$

$$\gamma :H_{\omega }^{\ast }\to H^{\ast }{\#}_{\sigma }H,\gamma (\phi )=\phi \#1_H,$$

$$v:H=(H_{\omega }^{\ast }{)}^{\ast }\to H^{\ast }{\#}_{\sigma }H,v(h)=\epsilon \#h.$$

$$\gamma ({\phi }_1)v(h\leftharpoonup {\phi }_2)=v({\phi }_1\rightharpoonup h)\gamma ({\phi }_2),v(1_H)=\epsilon \#1_H,\forall \phi \in H^{\ast },h\in H.$$

$$\epsilon \#h{h}^{\prime }=({\overline{\overline{\omega }}}_1\#1_H)(\epsilon \#{\omega }_1{\overline{\omega }}_1\rightharpoonup h\leftharpoonup {\overline{\overline{\omega }}}_2)({\omega }_2\#1_H)(\epsilon \#{\overline{\omega }}_2\rightharpoonup h^{\prime }\leftharpoonup {\overline{\overline{\omega }}}_3{\omega }_3)({\overline{\omega }}_3\#1_H),$$

$$({\overline{\overline{\omega }}}_1\#1_H)(\epsilon \#{\omega }_1{\overline{\omega }}_1\rightharpoonup h\leftharpoonup {\overline{\overline{\omega }}}_2)({\omega }_2\#1_H)(\epsilon \#{\overline{\omega }}_2\rightharpoonup h^{\prime }\leftharpoonup {\overline{\overline{\omega }}}_3{\omega }_3)({\overline{\omega }}_3\#1_H)$$

Full text

1. Introduction

Dijkgraaf, Pasquier and Roche constructed in [11], from a finite group $G$ and a normalized $3$-cocycle $\omega$ on it, the famous quasi-Hopf algebra $D^{\omega}(G)$ (called the twisted quantum double of $G$). The importance of this construction stems from the fact that the irreducible representations of $D^{\omega}(G)$ allow to recover the fusion rules, the $S$ matrix and the conformal weights of a certain Rational Conformal Field Theory described in [12]. On the other hand, ribbon (quasi-) quantum groups give rise to topological invariant of knots and links: following the constructions of Reshetikhin and Turaev [23, 24], for such an algebra one can define regular isotopy invariants of coloured ribbon graphs, the colours being finite-dimensional representations. This result was applied to the ribbon quasi-Hopf algebra $D^{\omega}(G)$ in [1] by considering surgery on the ribbon graphs coloured by a representation of $D^{\omega}(G)$, leading thus to a $3$-manifold invariant.

In [6] we generalized the construction of $D^{\omega}(G)$ to an arbitrary finite-dimensional cocommutative Hopf algebra $H$ and $\omega:H\otimes H\otimes H\rightarrow k$ a normalized 3-cocycle in the cohomology of commutative algebras over cocommutative Hopf algebras introduced by Sweedler in [26]. We denoted this new quasi-Hopf algebra by $D^{\omega}(H)$ and showed that $D^{\omega}(H)$ is always a quasitriangular (QT for short) quasi-Hopf algebra. We expected $D^{\omega}(H)$ to always be ribbon, as $D^{\omega}(G)$ is, but we were able to provide a ribbon element for it only if an extra condition is satisfied; see [6, Proposition 3.3]. The goal of this paper is to overcome this problem and to explain the meaning of the extra condition required in [6]. Towards this end, we identify $D^{\omega}(H)$ with the quantum double (in the sense of Hausser and Nill [14, 15]) of $H^{*}_{\omega}$, the finite-dimensional quasi-Hopf algebra introduced in [21]; see Theorem 4.2. This gives us for free the QT quasi-Hopf algebra structure of $D^{\omega}(H)$ as well as the modular elements of $D^{\omega}(H)$ in terms of the modular elements of $H$. The latter occur in the computation of the ribbon elements for $D^{\omega}(H)$, owing to Theorem 3.1 and its Corollary 3.3 below, and the fact that $D^{\omega}(H)$ is unimodular. It then comes out (see Theorem 5.3) that to any square root $\zeta$ in $G(H^{*})$ of the modular element $\mu_{H}$ we can associate a ribbon element for $D^{\omega}(H)$, where $G(H^{*})$ stands for the group of grouplike elements of $H^{*}$, that is algebra maps from $H$ to $k$. Situations when such an element $\zeta$ exists are uncovered at the end of Section 5, for instance when $H$ has odd dimension or $G(H^{*})$ is of odd order, or when the characteristic of $k$ does not divide the dimension of $H$ (and consequently when $k$ has characteristic zero). Example 5.6 says that the extra condition in [6, Proposition 3.3] that guarantees a ribbon element for $D^{\omega}(H)$ is satisfied if and only if $H$ is unimodular. As the Hopf group algebra $k[G]$ is always unimodular, this explains why for the quasi-Hopf algebra $D^{\omega}(G)$ one can always construct a ribbon element. We should also mention here that our approach provides new ribbon elements even for some of the quasi-Hopf algebras $D^{\omega}(G)$, and thus new $3$-manifold invariants as in [1].

2. Preliminaries

We present, briefly, the definition an the basic properties of a (quasitriangular) quasi-Hopf algebra. For more information we refer to [13] or [18], [19]. We work over a field $k$. All algebras, linear spaces, etc. will be over $k$; unadorned $\otimes$ means $\otimes_{k}$.

A quasi-bialgebra is a $4$-tuple $(H,\Delta,\varepsilon,\Phi)$, where $H$ is an associative algebra with unit $1_{H}$, $\Phi$ is an invertible element in $H\otimes H\otimes H$, and $\Delta:\ H\rightarrow H\otimes H$ and $\varepsilon:\ H\rightarrow k$ are algebra homomorphisms such that $\Delta$ is coassociative up to conjugation by $\Phi$ and $\varepsilon$ is counit for $\Delta$; furthermore, $\Phi$ is a normalized $3$-cocycle.

In what follows we denote $\Delta(h)=h_{1}\otimes h_{2}$, for all $h\in H$, the tensor components of $\Phi$ by capital letters and the ones of $\Phi^{-1}$ by lower case letters.

$H$ is called a quasi-Hopf algebra if, moreover, there exists an anti-morphism $S$ of the algebra $H$ and elements $\alpha,\beta\in H$ such that, for all $h\in H$, we have:

[TABLE]

A quasi-Hopf algebra with $\Phi=1_{H}\otimes 1_{H}\otimes 1_{H}$ and $\alpha=\beta=1_{H}$ is an ordinary Hopf algebra.

For a quasi-Hopf algebra $H$ we introduce the following elements in $H\otimes H$:

[TABLE]

Note that our definition of a quasi-Hopf algebra is different from the one given by Drinfeld [13] in the sense that we do not require the antipode to be bijective. Anyway, the bijectivity of the antipode $S$ will be implicitly understood in the case when $S^{-1}$, the inverse of $S$, appears is formulas or computations. According to [3], $S$ is always bijective, provided that $H$ is finite dimensional.

The antipode of a Hopf algebra is an anti-morphism of coalgebras. For a quasi-Hopf algebra $H$ there is an invertible element $f=f^{1}\otimes f^{2}\in H\otimes H$, called the Drinfeld twist, such that $\varepsilon(f^{1})f^{2}=\varepsilon(f^{2})f^{1}=1_{H}$ and $f\Delta(S(h))f^{-1}=(S\otimes S)(\Delta^{\rm cop}(h))$, for all $h\in H$, where $\Delta^{\rm cop}(h)=h_{2}\otimes h_{1}$.

For $H$ a quasi-bialgebra, the category of left $H$-modules ${}_{H}{\mathcal{M}}$ is monoidal (see [18, Chapter XV] for the definition of a monoidal category). The tensor product is defined by $\otimes$ endowed with the $H$-module structure given by $\Delta$ and unit object $k$, considered as an $H$-module via $\varepsilon$; the associativity constraint is determined by $\Phi$ and the left and right unit constraints are given by the canonoical isomorphisms in the category of $k$-vector spaces.

A quasi-bialgebra $H$ is called quasitriangular (QT for short) if, moreover, the category ${}_{H}{\mathcal{M}}$ is braided in the sense of [18, Definition XIII.1.1]. This is equivalent to the existence of an invertible element $R=R^{1}\otimes R^{2}\in H\otimes H$ (formal notation, summation implicitly understood), called $R$-matrix, satisfying certain conditions. Record that any $R$-matrix obeys

[TABLE]

When we refer to a QT quasi-bialgebra or quasi-Hopf algebra we always indicate the $R$-matrix $R$ that produces the QT structure by pointing out the couple $(H,R)$.

For $(H,R)$ a QT quasi-Hopf algebra, $u$ is the element of $H$ defined by

[TABLE]

By [5], $u$ is an invertible element of $H$ and the following equalities hold ($R_{21}:=R^{2}\otimes R^{1}$):

[TABLE]

3. Ribbon quasi-Hopf algebras

The definition of a ribbon quasi-Hopf algebra is designed in such a way that the category ${}_{H}{\mathcal{M}}^{\rm fd}$ of finite-dimensional left $H$-modules is a ribbon category as in [18, Definition XIV.3.2]. More exactly, by [7] a QT quasi-Hopf algebra $(H,R)$ is ribbon if there exists an invertible central element $\nu\in H$ such that

[TABLE]

We can provide the following characterization for ribbon quasi-Hopf algebras. Note that (3.10) below was proved for the first time in [7, Proposition 5.5] under the condition $\alpha$ invertible; that it works in general was proved in [25, Section 2.3]. In what follows we present an easy proof of this, based mostly on the arguments used in [7].

Theorem 3.1**.**

A QT quasi-Hopf algebra $(H,R)$ is ribbon if and only if there exists a central element $\nu\in H$ (called ribbon element) such that the conditions in (3.9) hold and

[TABLE]

where, as before, $u$ is the element defined in (2.5).

Proof.

Suppose that $(H,R)$ is ribbon and let $\nu\in H$ be an invertible central element such that the conditions in (3.9) are satisfied. To prove that (3.10) is satisfied as well, observe first that by applying $\varepsilon\otimes\varepsilon$ to both sides of the first equality in (3.9) we get $\varepsilon(\nu)=1$; see (2.4). Denote $\eta:=\nu^{-1}$ and restate the first equality in (3.9) as $\Delta(\eta)=(\eta\otimes\eta)R_{21}R$. Thus, by this equality and the fact that $\eta$ is a central element in $H$ we deduce that

[TABLE]

This fact allows to compute, for all $A,B\in H$:

[TABLE]

In particular, by taking $A\otimes B=S(x^{1})\otimes x^{2}\beta S(x^{3})$, by (2.2) we conclude that $1_{H}=\eta^{2}S(u)u$, and therefore $\nu^{2}=\eta^{-2}=S(u)u$, as needed. Note that $S(u)u=uS(u)$ is a central element of $H$.

Conversely, let $\nu$ be a central element of $H$ such that the relations in (3.9) and (3.10) are fulfilled. Then $\nu$ is invertible because so is $u$, and therefore $\nu$ is an invertible central element of $H$. It follows now that $(H,R,\nu)$ is a ribbon quasi-Hopf algebra. ∎

It is a well established result that ribbon categories are pivotal (or, equivalently, sovereign) braided categories satisfying a certain condition related to some canonical twists; see [17, Proposition A.4]. In the case when the category is ${}_{H}{\mathcal{M}}^{\rm fd}$, with $(H,R,\nu)$ a ribbon quasi-Hopf algebra as in Theorem 3.1, this result is encoded in [6, Theorem 3.5]. It asserts that $l\mapsto ul$ defines a one to one correspondence between $\{l\in G(H)\mid l^{2}=u^{-1}S(u)~{}\mbox{and}~{}S^{2}(h)=l^{-1}hl,~{}\forall~{}h\in H\}$ and central invertible elements $\nu\in H$ satisfying (3.9) and (3.10), i.e. ribbon elements of $H$. Here

[TABLE]

Due to the proof of [10, Proposition 3.12], the conditions $l$ invertible and $l^{-1}=S(l)$ in the definition of an element $l\in G(H)$ are redundant, provided that $lS^{2}(h)=hl$, for all $h\in H$. Otherwise stated, we have the following.

Corollary 3.2**.**

Let $(H,R)$ be a QT quasi-Hopf algebra and $u$ the element defined in (2.5). Then $l\mapsto ul$ defines a one to one correspondence between

[TABLE]

and ribbon structures $\nu$ on $H$ as in Theorem 3.1.

We end this section by pointing out that in the finite-dimensional case the element $u^{-1}S(u)$ can be computed in terms of the modular elements $\underline{g}\in H$ and $\mu\in H^{*}$, and the $R$-matrix $R=R^{1}\otimes R^{2}$ of $H$ (we refer to [16] for the definitions of $\underline{g}$ and $\mu$). Namely, by the formula (6.21) in [9] we have

[TABLE]

where $\underline{g}^{-1}$ is the inverse of $\underline{g}$ in $H$ and $p_{R}=p^{1}\otimes p^{2}$ is the element defined in (2.3).

Corollary 3.3**.**

If $(H,R)$ is a finite-dimensional unimodular QT quasi-Hopf algebra and $u$ is as in (2.5) then $l\mapsto ul$ defines a one to one correspondence between

[TABLE]

and ribbon elements of $H$, where $\underline{g}$ is the modular element of $H$.

Proof.

When $H$ is unimodular, i.e. $\mu=\varepsilon$, the formula in (3.11) gives the equality $u^{-1}S(u)=S(\underline{g}^{-1})$. As $uS(u)=S(u)u$ and $S^{2}(u)=u$ this implies

[TABLE]

By using the bijectivity of $S$, we obtain that $S(\underline{g}^{-1})=\underline{g}$, and so $u^{-1}S(u)=\underline{g}$. Now everything follows from Corollary 3.2. ∎

4. The quasi-Hopf algebras $D^{\omega}(H)$ and $D^{\omega}(G)$

Let $H$ be a cocommutative Hopf algebra with antipode S over a base field $k$. As $H$ is cocommutative, we can introduce a simplified version of Sweedler’s sigma notation: for $h\in H$, we denote

[TABLE]

and so on. With this notation, the antipode and counit axioms read:

[TABLE]

We recall some facts concerning Hopf crossed products and cohomology (see [26]). Let $H$ be a cocommutative Hopf algebra and $A$ a commutative left $H$-module algebra, with $H$-action denoted by $H\otimes A\rightarrow A$, $h\otimes a\mapsto h\cdot a$. Assume that we are given a linear map $\sigma:H\otimes H\rightarrow A$, which is normalized (that is, $\sigma(1_{H},h)=\sigma(h,1_{H})=\varepsilon(h)1_{A}$ for all $h\in H$) and convolution invertible. Suppose that, moreover, $\sigma$ satisfies the 2-cocycle condition:

[TABLE]

Then, if we define a multiplication on $A\otimes H$ by $(a\#h)(b\#g)=a(h\cdot b)\sigma(h,g)\#hg$ (for $a\in A$ and $h\in H$ we write $a\#h$ in place of $a\otimes h$ in order to distinguish this structure), this multiplication is associative and $1_{A}\otimes 1_{H}$ is a unit. We denote $A\otimes H$ with this algebra structure by $A\#_{\sigma}H$ and called it the Hopf crossed product of $A$ and $H$.

From now on, for the rest of this section, we assume that $H$ is a finite dimensional cocommutative Hopf algebra. Thus, $H^{*}$ is a commutative Hopf algebra with unit $\varepsilon$, counit $\varepsilon(\varphi)=\varphi(1_{H})$, multiplication $(\varphi\psi)(h)=\varphi(h)\psi(h)$, comultiplication $\Delta(\varphi)=\varphi_{1}\otimes\varphi_{2}$ if and only if $\varphi(hg)=\varphi_{1}(h)\varphi_{2}(g)$, and antipode $\overline{S}(\varphi)=\varphi\circ S$, where $\varphi,\psi\in H^{*}$ are arbitrary as well as $h,g\in H$.

Assume that we are given a $k$-linear map $\omega:H\otimes H\otimes H\rightarrow k$ that is convolution invertible and satisfies the conditions:

[TABLE]

Note that such a map $\omega$ is noting but a normalized 3-cocycle in the Sweedler cohomology of $H$ with coefficients in $k$, as defined in [26].

Since $H$ is finite dimensional, we can identify $(H\otimes H\otimes H)^{*}$ with $H^{*}\otimes H^{*}\otimes H^{*}$, so we can regard $\omega\in H^{*}\otimes H^{*}\otimes H^{*}$; we denote $\omega=\omega_{1}\otimes\omega_{2}\otimes\omega_{3}$ and its convolution inverse $\omega^{-1}=\overline{\omega}_{1}\otimes\overline{\omega}_{2}\otimes\overline{\omega}_{3}$.

We define the element $\Phi\in H^{*}\otimes H^{*}\otimes H^{*}$ by $\Phi:=\omega^{-1}=\overline{\omega}_{1}\otimes\overline{\omega}_{2}\otimes\overline{\omega}_{3}$. Since $H^{*}$ is a commutative algebra, $(H^{*},\Delta,\varepsilon,\Phi)$ is a quasi-bialgebra, where $\Delta$ and $\varepsilon$ are the ones that give the usual coalgebra structure of $H^{*}$ (dual to the algebra structure of $H$). Moreover, if we define $\beta\in H^{*}$ by the formula $\beta(h)=\omega(h,S(h),h)$, then $(H^{*},\Delta,\varepsilon,\Phi,\overline{S},\alpha=\varepsilon,\beta)$ is a quasi-Hopf algebra, which will be denoted by $H^{*}_{\omega}$, see [21].

We can consider the diagonal crossed product $(H^{*}_{\omega})^{*}\bowtie H^{*}_{\omega}$ as in [8, 14]. On the other hand, we will construct a certain Hopf crossed product $H^{*}\#_{\sigma}H$ as follows, see [6].

We introduce first the following notation: $g\triangleleft x=S(x)gx$, for all $g,x\in H$. Next, we define the linear map $\theta:H\otimes H\otimes H\rightarrow k$, by

[TABLE]

for all $g,x,y\in H$, where $\omega^{-1}$ is the convolution inverse of $\omega$.

It is easy to see that $\theta$ is also normalized and convolution invertible. By [6], we have

[TABLE]

Since $H$ is cocommutative, $H^{*}$ becomes a commutative left $H-$module algebra, with action $H\otimes H^{*}\rightarrow H^{*}$, $h\otimes\varphi\mapsto h\bullet\varphi$, where $h\bullet\varphi=h\rightharpoonup\varphi\leftharpoonup S(h)$, where we denoted by $\rightharpoonup$ and $\leftharpoonup$ the left and right regular actions of $H$ on $H^{*}$ given by $(h\rightharpoonup\varphi)(a)=\varphi(ah)$ and $(\varphi\leftharpoonup h)(a)=\varphi(ha)$ for all $h,a\in H$ and $\varphi\in H^{*}$. Hence, $(h\bullet\varphi)(a)=\varphi(a\triangleleft h)$ for all $h,a\in H$ and $\varphi\in H^{*}$.

Define now the linear map $\sigma:H\otimes H\rightarrow H^{*}$ by $\sigma(x,y)(g)=\theta(g;x,y)$. Since $\theta$ is normalized and convolution invertible, $\sigma$ is also normalized and convolution invertible, and the relation (4.5) is equivalent to the fact that $\sigma$ is a 2-cocycle, that is (4.1) holds if we replace the action $\cdot$ by $\bullet$. Hence, we can consider the Hopf crossed product $H^{*}\#_{\sigma}H$, denoted by $D^{\omega}(H)$, which is an associative algebra with unit $\varepsilon\#1_{H}$. Its multiplication is given, for all $\varphi,\varphi{{}^{\prime}}\in H^{*}$, $h,h{{}^{\prime}}\in H$, by

[TABLE]

Theorem 4.1**.**

The linear map $w:(H^{*}_{\omega})^{*}\bowtie H^{*}_{\omega}\rightarrow H^{*}\#_{\sigma}H$ defined by

[TABLE]

is an algebra isomorphism, with inverse $W:H^{*}\#_{\sigma}H\rightarrow(H^{*}_{\omega})^{*}\bowtie H^{*}_{\omega}$ given by

[TABLE]

where we denoted by $p^{1}\otimes p^{2}=x^{1}\otimes x^{2}\beta S(x^{3})$ the element for $H^{*}_{\omega}$ given by (2.3) and by $\rightharpoonup$ and $\leftharpoonup$ the regular actions of $H$ on $H^{*}$ and of $H^{*}$ on $H$.

Proof.

We will construct the map $w$ by using the Universal Property of the diagonal crossed product ([2, Proposition 8.2]). We define the linear maps

[TABLE]

One can easily see that $\gamma$ is an algebra map and the relations (8.2) and (8.4) in [2, Proposition 8.2] are satisfied, that is we have

[TABLE]

So the only thing left to prove is the relation (8.3) in [2, Proposition 8.2], namely

[TABLE]

where we denoted by $\omega^{-1}=\overline{\overline{\omega}}_{1}\otimes\overline{\overline{\omega}}_{2}\otimes\overline{\overline{\omega}}_{3}$ another copy of $\omega^{-1}$. We compute:

[TABLE]

When we evaluate this in $g\otimes\varphi\in H\otimes H^{*}$ we obtain:

[TABLE]

Thus, Proposition 8.2 from [2] yields an algebra map $w:(H^{*}_{\omega})^{*}\bowtie H^{*}_{\omega}\rightarrow H^{*}\#_{\sigma}H$, defined by

[TABLE]

where $q^{1}\otimes q^{2}=X^{1}\otimes S^{-1}(\alpha X^{3})X^{2}$ is the element for $H^{*}_{\omega}$ given by (2.3). An easy computation shows that this map is identical to the one given by (4.7).

To prove that $w$ is bijective with inverse $W$, since the underlying vector spaces have the same (finite) dimension, it is enough to prove that $w\circ W={\rm Id}$. We compute:

[TABLE]

When we evaluate this in $g\otimes\psi\in H\otimes H^{*}$ we obtain:

[TABLE]

To finish the proof it will be enough to prove that

[TABLE]

The 3-cocycle condition for $\omega$ applied to the elements $x=h$, $y=S(h)$, $z=h$, $t=S(h)$ yields

[TABLE]

So it is enough to prove that

[TABLE]

But this relation follows immediately by applying the 3-cocycle condition for $\omega$ to the elements $x=g$, $y=h$, $z=S(h)$, $t=h$. ∎

The quantum double $D(H^{*}_{\omega})$ of the quasi-Hopf algebra $H^{*}_{\omega}$ has as underlying algebra structure the diagonal crossed product $(H^{*}_{\omega})^{*}\bowtie H^{*}_{\omega}$, so Theorem 4.1 implies:

Theorem 4.2**.**

$D^{\omega}(H)=H^{*}\#_{\sigma}H$* is a QT quasi-Hopf algebra.*

It turns out that the QT quasi-Hopf algebra structure otained on $D^{\omega}(H)$ by transferring the structure from $D(H^{*}_{\omega})$ via the isomorphism (4.7) coincides with the one introduced in [6]. Namely:

$\bullet$ the reassociator: $\Phi=(\overline{\omega}_{1}\#1_{H})\otimes(\overline{\omega}_{2}\#1_{H})\otimes(\overline{\omega}_{3}\#1_{H})\in D^{\omega}(H)\otimes D^{\omega}(H)\otimes D^{\omega}(H)$.

$\bullet$ the comultiplication: define the linear map $\gamma:H\otimes H\otimes H\rightarrow k$ by

[TABLE]

for all $g,h,x\in H$. Then define the linear map $\nu:H\rightarrow(H\otimes H)^{*}$, $\nu(h)(x\otimes y)=\gamma(x,y;h)$. Identifying $(H\otimes H)^{*}$ with $H^{*}\otimes H^{*}$, we will write, for any $h\in H$, $\nu(h)=\nu_{1}(h)\otimes\nu_{2}(h)\in H^{*}\otimes H^{*}$. Then the comultiplication of $D^{\omega}(H)$ is defined, for all $\varphi\in H^{*}$, $h\in H$, by

[TABLE]

$\bullet$ the counit: $\varepsilon:D^{\omega}(H)\rightarrow k$, $\varepsilon(\varphi\#h)=\varphi(1_{H})\varepsilon(h)$, for all $\varphi\in H^{*}$, $h\in H$.

$\bullet$ the antipode: $\alpha_{D^{\omega}(H)}=\varepsilon\#1_{H}$, $\beta_{D^{\omega}(H)}=\beta\#1_{H}$ and $s:D^{\omega}(H)\rightarrow D^{\omega}(H)$ given by

[TABLE]

for all $\varphi\in H^{*}$, $h\in H$, where we denoted by $\nu^{-1}$ the convolution inverse of $\nu$, with notation $\nu^{-1}(h)=\nu_{1}^{-1}(h)\otimes\nu_{2}^{-1}(h)\in H^{*}\otimes H^{*}$.

$\bullet$ the $R$-matrix:

[TABLE]

where $\{e_{i},e^{i}\}_{i}$ are dual bases in $H$ and $H^{*}$.

Moreover, by [6], $\beta$ is convolution invertible with inverse $\beta^{-1}=\beta\circ S$ and we have the relation:

[TABLE]

Let now $G$ be a finite group, with multiplication denoted by juxtaposition and unit denoted by $e$. Let $\omega$ be a normalized 3-cocycle on $G$, i.e. $\omega:G\times G\times G\rightarrow k^{*}$ is a map such that $\omega(x,y,z)\omega(tx,y,z)^{-1}\omega(t,xy,z)\omega(t,x,yz)^{-1}\omega(t,x,y)=1$ for all $t,x,y,z\in G$, and $\omega(x,y,z)=1$ whenever $x$, $y$ or $z$ is equal to $e$. We can take $H=k[G]$, the group algebra of $G$, which is a finite dimensional cocommutative Hopf algebra, and extend $\omega$ by linearity to a map $\omega:H\otimes H\otimes H\rightarrow k$, which turns out to be a Sweedler 3-cocycle on $H$. So, we can consider the QT quasi-Hopf algebra $D^{\omega}(H)$, which will be denoted by $D^{\omega}(G)$. This QT quasi-Hopf algebra structure of $D^{\omega}(G)$ was introduced in [11].

5. Some ribbon elements for $D^{\omega}(H)$ and $D^{\omega}(G)$

Let $H$ be a finite dimensional cocommutative Hopf algebra, $\omega$ a normalized $3$-cocycle on $H$ and $D^{\omega}(H)$ the quasi-Hopf algebra constructed in Section 4. So $D^{\omega}(H)$ is a QT quasi-Hopf algebra isomorphic to the quantum double $D(H^{*}_{\omega})$.

In what follows, in order to avoid any confusion, we denote by $\mu_{H}\in H^{*}$ and $\underline{\mathfrak{g}}_{H}\in H$ the modular elements of $H$ as a Hopf algebra. Similar notation, $\mu_{H^{*}_{\omega}}\in H$ and $\underline{\mathfrak{g}}_{H^{*}_{\omega}}\in H^{*}$, is used for the modular elements of the quasi-Hopf algebra $H^{*}_{\omega}$. As $H$ is cocommutative it follows that $\underline{\mathfrak{g}}_{H}=\mu_{H^{*}_{\omega}}=1_{H}$, i.e. $H^{*}$ and $H^{*}_{\omega}$ are unimodular as Hopf and respectively quasi-Hopf algebras. Also, it is clear that a left (and at the same time right) integral in the quasi-Hopf algebra $H^{*}_{\omega}$ is nothing but a left (and at the same time right) integral on the Hopf algebra $H$, i.e. an element $\lambda\in H^{*}$ obeying $\lambda(h)h=\lambda(h)1_{H}$, for all $h\in H$. Finally, $K$ will always denote a finite dimensional quasi-Hopf algebra and $H$ a finite dimensional cocommutative Hopf algebra.

In this section we show that the element $\nu=u(\zeta\#1_{H})\beta_{D^{\omega}(H)}=u(\zeta\beta\#1_{H})$ is a ribbon element for $D^{\omega}(H)$, provided that $\zeta:H\rightarrow k$ is an algebra map such that $\zeta^{2}=\mu_{H}$; here, as before, $u$ is the element in (2.5) corresponding to $D^{\omega}(H)$ and the other notation is as in Section 4. Note that when ${\rm dim}_{k}H\not=0$ in $k$ or $H$ is unimodular we can take $\zeta=\varepsilon$, and therefore $D^{\omega}(H)$ is ribbon with ribbon element $\nu=u(\beta\#1_{H})$. This applies for instance to a finite dimensional Hopf group algebra $H=k[G]$, and so $D^{\omega}(G)$ is always a ribbon quasi-Hopf algebra.

To this end, we start by describing the space of left cointegrals on $H^{*}_{\omega}$. By the comments made after the proof of [4, Theorem 3.7] we have that a non-zero left cointegral on a quasi-Hopf algebra $K$ for which $\alpha,\beta$ are invertible elements is a non-zero morphism $\lambda\in K^{*}$ satisfying $\lambda(t_{2})t_{1}=\lambda(t)\beta S^{-1}(\alpha)$, for any left integral $t$ in $K$ (in [4] the assumption $\beta$ invertible is omitted and in place of $S^{-1}(\alpha)$ appears $\alpha$; we correct these facts now).

Lemma 5.1**.**

Let $t\in H$ be a non-zero left integral in $H$, i.e. $ht=\varepsilon(h)t$, for all $h\in H$. Then ${\mathfrak{t}}:=\beta(S(t))t\in H$ is a non-zero left cointegral for $H^{*}_{\omega}$.

Proof.

For the quasi-Hopf algebra $H^{*}_{\omega}$ the elements $\alpha,\beta$ are invertible, so a non-zero left cointegral on $H^{*}_{\omega}$ is a non-zero element ${\mathfrak{t}}\in H$ satisfying $\lambda(h{\mathfrak{t}})=\lambda({\mathfrak{t}})\beta(h)$, for all $h\in H$.

It is well known that for any non-zero left integral $t$ in $H$ the map $H^{*}\ni h^{*}\mapsto h^{*}(t)t\in H$ is bijective (result valid for any finite-dimensional Hopf algebra, not necessarily cocommutative). Thus we find a unique element $h^{*}\in H^{*}$ such that $h^{*}(t)t={\mathfrak{t}}$; in particular, $h^{*}(t)\not=0$. We have, for all $h\in H$, that $h^{*}(t)\lambda(ht)=h^{*}(t)\lambda(t)\beta(h)$, which is equivalent to $h^{*}(S(h)ht)\lambda(ht)=\lambda(t)h^{*}(1_{H})\beta(h)$, which in turn is equivalent to $\lambda(t)h^{*}(S(h))=\lambda(t)h^{*}(1_{H})\beta(h)$.

As $\lambda(t)\not=0$, it follows that $h^{*}=h^{*}(1_{H})\beta\circ S$, which implies $\beta(S(t))\not=0$. Therefore, by rescaling, we can assume without loss of generality that ${\mathfrak{t}}=\beta(S(t))t$, as stated.

Conversely, ${\mathfrak{t}}=\beta(S(t))t$ is a left cointegral on $H^{*}_{\omega}$ since

[TABLE]

and this is equal to $\lambda({\mathfrak{t}})\beta(h)$ because

[TABLE]

So our proof ends. ∎

For a finite-dimensional quasi-Hopf algebra $K$ the modular element $\underline{g}\in K$ is defined by $\underline{g}=\lambda(S^{-1}(q^{2}t_{2}p^{2}))q^{1}t_{1}p^{1}$, where $\lambda\in K^{*}$ is a left cointegral, $t\in K$ is a left integral such that $\lambda(S^{-1}(t))=1$ and $p_{R}=p^{1}\otimes p^{2}$ and $q_{R}=q^{1}\otimes q^{2}$ are the elements defined in (2.3).

Corollary 5.2**.**

We have that $\underline{\mathfrak{g}}_{H^{*}_{\omega}}=\beta^{2}\mu_{H}$. Consequently, the modular element $\underline{\mathfrak{g}}_{D(H^{*}_{\omega})}$ of the quantum double $D(H^{*}_{\omega})$ equals $1_{H}\Join\beta^{2}\mu_{H}$.

Proof.

Recall that $\mu_{H}$ is defined by $th=\mu_{H}(h)t$, for all $h\in H$ and $t\in H$ a non-zero left integral. Also, since $H^{*}$ is unimodular we have $\lambda\circ S=\lambda$. Thus, by specializing the above definition of $\underline{g}$ to $H^{*}_{\omega}$ we compute, for all $h\in H$, that

[TABLE]

But the pair $(\lambda,{\mathfrak{t}})$ obeys $\lambda({\mathfrak{t}})=1$ or, equivalently, $\beta(S(t))\lambda(t)=1$. The latter is equivalent to $\lambda(t)=1$, and so $\underline{\mathfrak{g}}_{H^{*}_{\omega}}=\beta^{2}\mu_{H}$, as desired.

Finally, we have $\mu_{H^{*}_{\omega}}=1_{H}$, hence the formula in [4, Proposition 5.9] yields

[TABLE]

since $\underline{\mathfrak{g}}^{-1}_{H^{*}_{\omega}}=\beta^{-2}\mu_{H}^{-1}$ with $\beta^{-1}=\beta\circ S$ and $\mu^{-1}_{H}=\mu_{H}\circ S$, and together with $S^{2}={\rm Id}_{H}$ this implies $\underline{\mathfrak{g}}^{-1}_{H^{*}_{\omega}}\circ S=\underline{\mathfrak{g}}_{H^{*}_{\omega}}$. ∎

The computation performed before [6, Proposition 3.3] ensures that the distinguished element $\beta_{D^{\omega}(H)}=\beta\#1_{H}\in D^{\omega}(H)$ satisfies

[TABLE]

where ${\mathfrak{f}}\in D^{\omega}(H)\otimes D^{\omega}(H)$ is the Drinfeld twist. Consequently, the same relation is satisfied by any element of the form $\zeta\beta\#1_{H}\in D^{\omega}(H)$, provided that $\zeta:H\rightarrow k$ is an algebra map.

We have now all the necessary ingredients in order to prove the following:

Theorem 5.3**.**

Let $H$ be a finite dimensional cocommutative Hopf algebra $H$, $\omega$ a normalized $3$-cocycle on $H$ and $\zeta:H\rightarrow k$ an algebra map. Then the element $\nu=u(\zeta\beta\#1_{H})$ is ribbon if and only if $\zeta^{2}=\mu_{H}$.

Proof.

The quasi-Hopf algebra $D^{\omega}(H)$ is unimodular; this follows from Theorem 4.1 and [9, Theorem 6.5]. Furthermore, by Corollary 5.2 and the definition of the isomorphism $w$ in (4.7) we can see that the modular element $\underline{\mathfrak{g}}_{D^{\omega}(H)}$ of $D^{\omega}(H)$ is

[TABLE]

So, according to Corollary 3.3, it suffices to see when ${\mathfrak{l}}:=\zeta\beta\#1_{H}\in D^{\omega}(H)$ satisfies the relations

[TABLE]

It can be easily checked that (5.2) is equivalent to $\zeta^{2}\beta^{2}=\beta^{2}\mu_{H}$, and the latter is equivalent to $\zeta^{2}=\mu_{H}$, because $H^{*}$ is commutative and $\beta$ is convolution invertible. Also, the comments made after (5.1) guarantees that (5.3) is always satisfied.

We look at (5.4). By (4.11) we have that (5.4) is equivalent to $(\zeta\#1_{H})(\varphi\#h)=(\varphi\#h)(\zeta\#1_{H})$, for all $\varphi\#h\in D^{\omega}(H)$. The last equation becomes $\zeta\varphi\#h=\varphi(h\rightharpoonup\zeta\leftharpoonup S(h))\#h$, and it holds for any $\varphi\#h\in D^{\omega}(H)$ since $\zeta$ is an algebra map and $H^{*}$ is commutative. ∎

We end this section with some concrete examples. In what follows by $G(H^{*}):=\{\zeta:H\rightarrow k\mid\zeta~{}~{}\mbox{is an algebra map}\}$ we denote the set of grouplike elements of $H^{*}$, assuming, as before, that $H$ is a finite-dimensional cocommutative Hopf algebra. $G(H^{*})$ is a group under convolution, and so the group Hopf algebra $k[G(H^{*})]$ is a Hopf subalgebra of $H^{*}$. By the freeness theorem proved in [20] it follows that $\mid G(H^{*})\mid$ divides ${\rm dim}_{k}(H)$.

Example 5.4**.**

If $\mu_{H}$ has odd order in $G(H^{*})$ then $D^{\omega}(H)$ is a ribbon quasi-Hopf algebra.

Proof.

If $\mu_{H}$ has order $2m+1$ in $G(H^{*})$ then $\zeta:=\mu_{H}^{m+1}:H\rightarrow k$ is an algebra map such that $\zeta^{2}=\mu_{H}$. By Theorem 5.3 we obtain that $u(\mu_{H}^{m+1}\beta\#1_{H})$ is a ribbon element for $D^{\omega}(H)$. ∎

Example 5.5**.**

Suppose that either ${\rm dim}_{k}(H)$ or $\mid G(H^{*})\mid$ is an odd number. Then $D^{\omega}(H)$ is a ribbon quasi-Hopf algebra.

Proof.

Since $\mid G(H^{*})\mid$ divides ${\rm dim}_{k}(H)$, in either case we get that $\mu_{H}$ has odd order, and so Example 5.4 applies. ∎

The next example shows that the condition in [6, Proposition 3.3] is equivalent to the unimodularity of $H$.

Example 5.6**.**

The element $\nu=u(\beta\#1_{H})$ is a ribbon element for $D^{\omega}(H)$ if and only if $H$ is unimodular.

Proof.

Take $\zeta=\varepsilon$ in Theorem 5.3; we obtain that $\nu=u(\beta\#1_{H})$ is a ribbon element if and only if $\mu_{H}=\varepsilon$, i.e. $H$ is unimodular. ∎

The next example refers precisely to the ribbon structure on $D^{\omega}(G)$ defined by (5.18) in [1].

Example 5.7**.**

The quasi-Hopf algebra $D^{\omega}(G)$ is ribbon.

Proof.

We have $D^{\omega}(G)=D^{\omega}(k[G])$ and $k[G]$ is unimodular. Indeed, $t=\sum_{g\in G}g$ is a left and right integral in $k[G]$. ∎

Example 5.8**.**

Suppose that ${\rm dim}_{k}(H)\not=0$ in $k$ (this happens for instance when $k$ is of characteristic zero). Then $\nu=u(\beta\#1_{H})$ is a ribbon element for $D^{\omega}(H)$.

Proof.

Since $S^{2}={\rm Id}_{H}$, by the trace formula proved in [22, Proposition 2 (c)] we get that $H$ is semisimple, and so unimodular, too. Hence $\nu=u(\beta\#1_{H})$ is a ribbon element for $D^{\omega}(H)$. ∎