The Ribbon Elements of the Quantum Double of Generalized Taft–Hopf Algebra

Abstract

Let s, t be two positive integers and $\mathbb{k}$ be an algebraically closed field with char ($\mathbb{k})\nmid st$. We show that the Drinfeld double $D\left({\bigwedge }_{st,t}^{*cop}\right)$ of generalized Taft–Hopf algebra ${\bigwedge }_{st,t}^{*cop}$ has ribbon elements if and only if t is odd. Moreover, if s is even and t is odd, then $D\left({\bigwedge }_{st,t}^{*cop}\right)$ has two ribbon elements, and if both s and t are odd, then $D\left({\bigwedge }_{st,t}^{*cop}\right)$ has only one ribbon element. Finally, we compute explicitly all ribbon elements of $D\left({\bigwedge }_{st,t}^{*cop}\right)$.

1. Introduction

The representation category of a quasi-triangular Hopf algebra is a braided tensor category. The braiding structure of a quasi-triangular Hopf algebra can supply a solution to the Yang–Baxter equation. Recently, great progress has been made in the research of the quasi-triangular Hopf algebra. Drinfeld [1] constructed a quasi-triangular Hopf algebra from a finite-dimensional Hopf algebra, i.e., the quantum double (or Drinfeld double) of a Hopf algebra. Ribbon Hopf algebra is a quasi-triangular Hopf algebra with a ribbon element. The finite-dimensional ribbon Hopf algebra plays an important role in constructing invariants of three-manifolds [2]. Thus, researchers have paid much attention to the question of when a quasi-triangular Hopf algebra has ribbon structures. In [3], Chen and Yang gave a necessary and sufficient condition for the Drinfeld double of a finite-dimensional Hopf superalgebra to have a ribbon element. Kauffman and Radford [4] gave a necessary and sufficient condition for the Drinfeld double of a finite-dimensional Hopf algebra to admit a ribbon structure, and they proved that $(D\left(A_n\left(q\right)\right),\mathcal{R})$ is a ribbon Hopf algebra if and only if n is odd, where $D\left(A_n\left(q\right)\right)$ is the Drinfeld double of $n^2$-dimensional Taft algebra $A_n\left(q\right)$ and $\mathcal{R}$ is the universal $\mathcal{R}$-matrix of $D\left(A_n\left(q\right)\right)$. In [5], Benkart and Biswal computed the ribbon element of $(D\left(A_n\left(q\right)\right),\mathcal{R})$ explicitly when n is odd. In [6], Andruskiewitsch and Schneider constructed $u(\mathcal{D},0,0)$, which is a pointed Hopf algebra of the nilpotent type. In particular, if $\mathcal{D}=(G,g^s,\chi ,\mu )$, $u(\mathcal{D},0,0)$ is the generalized Taft–Hopf algebra denoted by ${\bigwedge }_{st,t}^{*cop}$. Burciu [7] provided a sufficient condition for the quantum double of $u(\mathcal{D},0,0)$ to be a ribbon Hopf algebra. If $\chi \left(g^s\right)=t$ is an odd that is coprime to three, then $D\left({\bigwedge }_{st,t}^{*cop}\right)$ is a ribbon Hopf algebra. Leduc and Ram [8] showed how the ribbon Hopf algebra structure on the Drinfeld–Jimbo quantum groups of types A, B, C, and D can be used to derive formulas giving explicit realizations of the irreducible representations of the Iwahori–Hecke algebras of type A and the Birman–Wenzl algebras. Centrone and Yasumura [9] extended the action of the n-th Taft–Hopf algebra H on $A=k\left[u\right]$ with $u^n=\beta$ to the Drinfeld double $D\left(H\right)$. This is used to show that, for each H-action on A, there is a unique left H-comodule algebra structure on A such that A is a Yetter–Drinfeld algebra over H. Montgomery and Schneider [10] characterized the action of a Taft algebra $H_n$ on finite-dimensional algebras A that satisfy that every skew derivation is inner. Farsad [11] proved that the Drinfeld double $D\left(K_n\right)$ of Nichols–Hopf algebra $K_n$ is a ribbon Hopf algebra when n is even. Chang [12] provided the explicit expression of the ribbon elements of $D\left(K_n\right)$.

In this paper, we give a sufficient and necessary condition for the Drinfeld double $D\left({\bigwedge }_{st,t}^{*cop}\right)$ of generalized Taft–Hopf algebra ${\bigwedge }_{st,t}^{*cop}$ to have a ribbon structure. The paper is organized as follows. In Section 2, we recall some definitions and notions and the structures of generalized Taft–Hopf algebra ${\bigwedge }_{st,t}^{*cop}$. In Section 3, we describe the Hopf algebra structure of ${\left({\bigwedge }_{st,t}^{*cop}\right)}^{*}$. In Section 4, we show that $(D\left({\bigwedge }_{st,t}^{*cop}\right),\mathcal{R})$ have ribbon elements if and only if t is odd. Finally, we compute all ribbon elements of $(D\left({\bigwedge }_{st,t}^{*cop}\right),\mathcal{R})$.

2. Preliminaries

Throughout, we work over an algebraically closed field $\mathbb{k}$ with char ($\mathbb{k})\nmid st$. Unless otherwise stated, all algebras and Hopf algebras are defined over $\mathbb{k}$; $dim$ and ⊗ denote $di{m}_{\mathbb{k}}$ and ${\otimes }_{\mathbb{k}}$, respectively. References [13,14,15] are basic references for the theory of Hopf algebras and quantum groups.

Let $0\ne q\in \mathbb{k}.$ For any non-negative integer n, define ${\left(n\right)}_q$ by ${\left(0\right)}_q=0$ and ${\left(n\right)}_q=1+q+\cdots +q^{n-1}$ for $n>0$. Observe that ${\left(n\right)}_q=n$ when $q=1$ and

$${\left(n\right)}_q=\frac{q^n-1}{q-1}$$

when $q\ne 1$. Define the q-factorial of n by

$$\left(n\right){!}_q=\frac{(q^n-1)(q^{n-1}-1)\dots (q-1)}{{(q-1)}^n}$$

when $n>0$ and $q\ne 1$. The q-binomial coefficients ${\left(\begin{array}{c}n\\ i\end{array}\right)}_q$ are defined inductively as follows for $0\le i\le n$:

$${\left(\begin{array}{c}n\\ 0\end{array}\right)}_q=1={\left(\begin{array}{c}n\\ n\end{array}\right)}_q,forn\ge 0,$$

$${\left(\begin{array}{c}n\\ i\end{array}\right)}_q=q^i{\left(\begin{array}{c}n-1\\ i\end{array}\right)}_q+{\left(\begin{array}{c}n-1\\ i-1\end{array}\right)}_q,for0<i<n.$$

It is well-known that ${\left(\begin{array}{c}n\\ i\end{array}\right)}_q$ is a polynomial in q with integer coefficients and with the value at $q=1$ equal to the usual binomial coefficient $\left(\begin{array}{c}n\\ i\end{array}\right)$ and that

$${\left(\begin{array}{c}n\\ i\end{array}\right)}_q=\frac{\left(n\right){!}_q}{\left(i\right){!}_q(n-i){!}_q}$$

when $(n-1){!}_q\ne 0$ and $0<i<n$.

Next, we use the sigma notation: for $x\in H$, H being a coalgebra,

$$\Delta \left(x\right)=\sum x_1\otimes x_2.$$

Suppose that H is a bialgebra over $\mathbb{k}$. The left and right H-module actions defined on $H^{*}$ by

$$<a\rightharpoonup p,b>=<p,ba>=<p\leftharpoonup b,a>,$$

respectively, for $a,b\in H$ and $p\in H^{*}$ give $H^{*}$ an A-bimodule structure. Likewise, the left and right $H^{*}$-modules actions on H by

$$p\rightharpoonup a=\sum a_1<p,a_2>,a\leftharpoonup p=\sum <p,a_1>a_2,$$

respectively, for $p\in H^{*}$ and $a\in H$ give H a $H^{*}$-bimodule structure.

2.1. Generalized Taft–Hopf Algebra

In this subsection, we recall the structure of generalized Taft–Hopf algebra ${\bigwedge }_{st,t}^{*cop}$.

Let $s\ge 2$, $t\ge 1$, and let $p\in \mathbb{k}$ be a primitive t-th root of unity. The generalized Taft–Hopf algebra ${\bigwedge }_{st,t}^{*cop}$ is generated as an algebra by g and x subject to the following relations:

$$g^{st}=1,x^t=0,xg=pgx.$$

The comultiplication $\Delta$, counit $\epsilon$, and antipode S are given, respectively, by

$$\Delta \left(x\right)=x\otimes g+1\otimes x,\epsilon \left(x\right)=0,S\left(x\right)=-x{g}^{-1},$$

$$\Delta \left(g\right)=g\otimes g,\epsilon \left(g\right)=1,S\left(g\right)=g^{-1}=g^{st-1}.$$

Note that dim$\left({\bigwedge }_{st,t}^{*cop}\right)=s{t}^2$, and ${\bigwedge }_{st,t}^{*cop}$ has a $\mathbb{k}$-basis $\left\{g^i{x}^j\right|0\le i\le st-1,0\le j\le t-1\}$. In case $s=1$, then ${\bigwedge }_{st,t}^{*cop}={\bigwedge }_{t,t}^{*cop}$ is the $t^2$-dimensional Taft–Hopf algebra. For this reason, ${\bigwedge }_{st,t}^{*cop}$ is called a generalized Taft algebra. For details, one can refer to [16]. In the following, we denote ${\bigwedge }_{st,t}^{*cop}$ by $\mathcal{T}$.

2.2. Ribbon Hopf Algebra

In this subsection, we recall the definition of the quasi-triangular Hopf algebra and ribbon Hopf algebra [5] (Section 3.1).

Definition 1. 

Let H be a Hopf algebra. If there exists an invertible element $\mathcal{R}\in H\otimes H$, such that

$$\begin{array}{cc}\hfill & \mathcal{R}\Delta \left(x\right)={\Delta }^{op}\left(x\right)\mathcal{R},forallx\in H,\hfill \\ \hfill & (\Delta \otimes id)\mathcal{R}={\mathcal{R}}_{13}{\mathcal{R}}_{23},\hfill \\ \hfill & (id\otimes \Delta )\mathcal{R}={\mathcal{R}}_{13}{\mathcal{R}}_{12},\hfill \end{array}$$

then H is called a quasi-triangular Hopf algebra. Here, ${\Delta }^{op}\left(x\right)$ has the tensor factors in $\Delta \left(x\right)$ interchanged, and $\mathcal{R}={\sum }_i{x}_i\otimes y_i,{\mathcal{R}}_{12}={\sum }_i{x}_i\otimes y_i\otimes 1,{\mathcal{R}}_{13}={\sum }_i{x}_i\otimes 1\otimes y_i,{\mathcal{R}}_{23}={\sum }_{i}1\otimes x_i\otimes y_i.$ Let ${\mathcal{R}}^{op}={\sum }_i{y}_i\otimes x_i.$ For instance, both Radford–Hopf algebra and generalized Taft algebra are a quasi-triangular Hopf algebra.

We assume $\mathcal{R}={\sum }_i{x}_i\otimes y_i$ as above and use the antipode S to define

$$u=\sum _{i}S\left(y_i\right)x_i\in H.$$

(1)

Then, the following expressions hold

$$ux{u}^{-1}=S^2\left(x\right)forallx\in Hand\Delta \left(u\right)={\left({\mathcal{R}}^{op}\mathcal{R}\right)}^{-1}(u\otimes u).$$

Definition 2. 

Let H be a quasi-triangular Hopf algebra. If there exists an invertible element v (the ribbon element) in the center of H such that

$$v^2=uS\left(u\right),S\left(v\right)=v,\epsilon \left(v\right)=1,\Delta \left(v\right)={\left({\mathcal{R}}^{op}\mathcal{R}\right)}^{-1}(v\otimes v),$$

(2)

where u is as in (1), then $(H,\mathcal{R},v)$ is called a ribbon Hopf algebra. For example, a$t^2$-dimensional Taft–Hopf algebra is a ribbon Hopf algebra (see [4]).

3. The Structure of ${\mathcal{T}}^{*}$

In this section, we describe the Hopf algebra structure of ${\mathcal{T}}^{*}$.

Let $\left\{\overline{g^i{x}^j}\right|0\le i\le st-1,0\le j\le t-1\}$ be the basis of Hopf algebra ${\mathcal{T}}^{*}$ such that $\overline{g^i{x}^j}\left(g^i{x}^j\right)=1$ and $\overline{g^i{x}^j}\left(g^{i^{\prime }}{x}^{j^{\prime }}\right)=0$ for $(i^{\prime },j^{\prime })\ne (i,j),0\le i,i^{\prime }\le st-1,0\le j,j^{\prime }\le t-1.$

Lemma 1. 

Let $0\le i,k\le st-1$ and $0\le j,l\le t-1$. Then,

$$\overline{g^i{x}^j}\ast \overline{g^k{x}^l}=\left\{\begin{array}{cc}0,\hfill & \mathit{if}k\ne i+j(\mathit{mod}st)\mathit{or}l+j\ge t,\hfill \\ {\left(\begin{array}{c}l+j\\ j\end{array}\right)}_p\overline{g^i{x}^{j+l}},\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Proof. 

By the coalgebra structure of $\mathcal{T}$, we have $\Delta \left(g^a{x}^b\right)=(g^a\otimes g^a){(x\otimes g+1\otimes x)}^b={\sum }_{u=0}^b{\left(\begin{array}{c}b\\ u\end{array}\right)}_p{g}^a{x}^{b-u}\otimes g^{a+b-u}{x}^u$, where $0\le a\le st-1,0\le b\le t-1$. If $0\le i$, $k\le st-1$, $0\le j$, $l\le t-1$, then $(\overline{g^i{x}^j}\ast \overline{g^k{x}^l})\left(g^a{x}^b\right)={\sum }_{u=0}^b{\left(\begin{array}{c}b\\ u\end{array}\right)}_p\overline{g^i{x}^j}\left(g^a{x}^{b-u}\right)\overline{g^k{x}^l}\left(g^{a+b-u}{x}^u\right).$ Hence, $(\overline{g^i{x}^j}\ast \overline{g^k{x}^l})\left(g^a{x}^b\right)\ne 0$ if and only if $a=i,b=l+j,j+l<tandk\equiv i+j\left(modst\right).$ Obviously, $(\overline{g^i{x}^j}\ast \overline{g^k{x}^l})\left(g^a{x}^b\right)={\left(\begin{array}{c}l+j\\ j\end{array}\right)}_p.$ □

Obviously, ${\sum }_{i=0}^{st-1}\overline{g^i}=\epsilon$ is the identity of the algebra ${\mathcal{T}}^{*}$.

Let $\omega \in \mathbb{k}$ be a primitive $st$-th root of unity with ${\omega }^s=p$. Put $\alpha ={\sum }_{i=0}^{st-1}{\omega }^i\overline{g^i}$ and $\beta ={\sum }_{i=0}^{st-1}\overline{g^{i}x}$.

Lemma 2. 

${\mathcal{T}}^{*}$ is generated as an algebra by α and β.

Proof. 

Obviously, $\alpha ,\beta \in {\mathcal{T}}^{*}.$ Let A be a subalgebra of ${\mathcal{T}}^{*}$ generated by $\alpha$ and $\beta$. It follows from Lemma 1 that ${\beta }^j=\left(j\right){!}_p(\overline{x^j}+\overline{g{x}^j}+\cdots +\overline{g^{st-1}{x}^j}),1\le j\le t-1$, and

$$\begin{array}{cc}\hfill \alpha & =\overline{1}+\omega \overline{g}+{\omega }^2\overline{g^2}+\cdots +{\omega }^{st-1}\overline{g^{st-1}},\hfill \\ \hfill {\alpha }^2& =\overline{1}+{\omega }^2\overline{g}+{\omega }^4\overline{g^2}+\cdots +{\omega }^{2(st-1)}\overline{g^{st-1}},\hfill \\ \hfill & \cdots \hfill \\ \hfill {\alpha }^{st-1}& =\overline{1}+{\omega }^{st-1}\overline{g}+{\omega }^{2(st-1)}\overline{g^2}+\cdots +{\omega }^{(st-1)(st-1)}\overline{g^{st-1}}.\hfill \end{array}$$

(3)

Then, we have

$$\begin{array}{cc}\hfill \frac{1}{\left(j\right){!}_p}\alpha {\beta }^j& =\overline{x^j}+\omega \overline{g{x}^j}+{\omega }^2\overline{g^2{x}^j}+\cdots +{\omega }^{st-1}\overline{g^{st-1}{x}^j},\hfill \\ \hfill \frac{1}{\left(j\right){!}_p}{\alpha }^2{\beta }^j& =\overline{x^j}+{\omega }^2\overline{g{x}^j}+{\omega }^4\overline{g^2{x}^j}+\cdots +{\omega }^{2(st-1)}\overline{g^{2(st-1)}{x}^j},\hfill \\ \hfill & \cdots \hfill \\ \hfill \frac{1}{\left(j\right){!}_p}{\alpha }^{st-1}{\beta }^j& =\overline{x^j}+{\omega }^{st-1}\overline{g{x}^j}+{\omega }^{2(st-1)}\overline{g^2{x}^j}+\cdots +{\omega }^{(st-1)(st-1)}\overline{g^{st-1}{x}^j},\hfill \\ \hfill \frac{1}{\left(j\right){!}_p}{\alpha }^{st}{\beta }^j& =\overline{x^j}+\overline{g{x}^j}+\overline{g^2{x}^j}+\cdots +\overline{g^{st-1}{x}^j}.\hfill \end{array}$$

The coefficient determinant of (3) is $\left|B\right|=$ $\left|\begin{array}{ccccc}1& \omega & {\omega }^2& \cdots & {\omega }^{st-1}\\ 1& {\omega }^2& {\omega }^4& \cdots & {\omega }^{2(st-1)}\\ & \cdots \\ 1& {\omega }^{st-1}& {\omega }^{2(st-1)}& \cdots & {\omega }^{(st-1)(st-1)}\end{array}\right|$ $={\prod }_{0\le i<j\le st-1}({\omega }^i-{\omega }^j)\ne 0.$

Therefore, by Cramer’s Rule, we have $\overline{g}\in A.$ Similarly, one can prove that $\overline{g^i{x}^j}\in A$ for $0\le i\le st-1,0\le j\le t-1.$ □

Proposition 1. 

In ${\mathcal{T}}^{*}$, we have

$${\alpha }^{st}=\epsilon ,{\beta }^t=0,\beta \alpha =\omega \alpha \beta .$$

Proof. 

It follows from Lemma 1 that ${\alpha }^{st}=\overline{1}+\overline{g}+\overline{g^2}+\cdots +\overline{g^{st-1}}=\epsilon$, ${\beta }^t=\left(t\right){!}_p(\overline{x^t}+\overline{g{x}^t}+\cdots +\overline{g^{st-1}{x}^t})=0$, and $\beta \alpha =\left(j\right){!}_p(\omega \overline{x}+{\omega }^2\overline{gx}+\cdots +{\omega }^{st}\overline{g^{st-1}x})=\omega \alpha \beta$. This completes the proof. □

Proposition 2. 

The comultiplication, the counit, and the antipode of ${\mathcal{T}}^{*cop}$ are given by

$$\Delta \left(\alpha \right)=\alpha \otimes \alpha ,$$

$$\Delta \left(\beta \right)=\beta \otimes 1+{\alpha }^s\otimes \beta ,$$

$$\epsilon \left(\alpha \right)=1,\epsilon \left(\beta \right)=0,$$

$$S\left(\alpha \right)={\alpha }^{st-1},S\left(\beta \right)=-{\alpha }^{-s}\beta .$$

Proof. 

We only consider the formula of $\Delta \left(\alpha \right)$ since the proof for $\Delta \left(\beta \right)$ is similar. Suppose that $\Delta \left(\overline{g^t}\right)={\sum }_{j=0}^{st-1}{\sum }_{k=0}^{t-1}{\theta }_{j,k,j^{\prime },k^{\prime }}\overline{g^j{x}^k}\otimes \overline{g^{j^{\prime }}{x}^{k^{\prime }}}$, where ${\theta }_{j,k,j^{\prime },k^{\prime }}=\overline{g^t}\left(g^j{x}^k{g}^{j^{\prime }}{x}^{k^{\prime }}\right)$. One can prove that ${\theta }_{j,k,j^{\prime },k^{\prime }}=1$ if an only if $j+j^{\prime }\equiv t\left(modst\right),k=k^{\prime }=0$. Consequently, in ${\mathcal{T}}^{*cop}$, we have $\Delta \left(\overline{g^t}\right)={\sum }_{i+j\equiv t\left(modst\right)}\overline{g^i}\otimes \overline{g^j},$ then we have $\Delta \left(\alpha \right)=\alpha \otimes \alpha .$ Similarly, one can show the formulas of the counit $\epsilon$ and antipode S of ${\mathcal{T}}^{*cop}$. □

Definition 3 ([13] (Definition IX.4.1)). 

The quantum double $D\left(\mathcal{T}\right)$ of Hopf algebra $\mathcal{T}$ is the bicrossed product of $\mathcal{T}$ and of ${\mathcal{T}}^{*cop}.$

$$D\left(\mathcal{T}\right)={\mathcal{T}}^{*cop}\bowtie \mathcal{T}.$$

By Proposition 3 and Lemma 2, one knows that $D\left(\mathcal{T}\right)$ is generated as an algebra by $\epsilon \bowtie g$, $\epsilon \bowtie x$, $\alpha \bowtie 1$, and $\beta \bowtie 1$.

Lemma 3 ([13] (Lemma IX.4.2)).  

The multiplication, comultiplication, and counit in $D\left(\mathcal{T}\right)$ are given by

$$(f\bowtie a)(g\bowtie b)=\sum _{\left(a\right)}f(a_1\rightharpoonup g\leftharpoonup s^{-1}\left(a_3\right))\bowtie a_{2}b,$$

$$\epsilon (f\bowtie a)=\epsilon \left(a\right)f\left(1\right),$$

$$\Delta (f\bowtie a)=\sum _{\left(a\right)\left(f\right)}(f_1\bowtie a_1)(f_2\bowtie a_2),$$

$$\begin{array}{cc}\hfill S(f\bowtie a)& =\sum (S\left(a_2\right)\rightharpoonup S\left(f_1\right))\bowtie (f_2\rightharpoonup S\left(a_1\right))\hfill \\ \hfill & =\sum (S\left(f_2\right)\leftharpoonup a_1)\bowtie (S\left(a_2\right)\leftharpoonup S\left(f_1\right)),\hfill \end{array}$$

where $f,g\in {\mathcal{T}}^{*cop}$ and $a,b\in \mathcal{T},\sum f_1\left(x\right)f_2\left(y\right)=f\left(yx\right)$ for all $x,y\in \mathcal{T}$.

4. The Ribbon Elements of $\left(\mathbf{D}\right(\mathcal{T}),\mathcal{R})$

In this section, we recall some results about quasi-triangular Hopf algebras, and then, we investigate the ribbon elements of $\left(D\right(\mathcal{T}),\mathcal{R})$.

4.1. Universal R-Matrix of $D\left(\mathcal{T}\right)$

In this subsection, we determine the universal R-matrix of $D\left(\mathcal{T}\right)$.

By [13] (Lemma IX.4.2), the universal $\mathcal{R}$-matrix of the quantum double has an explicit formula:

$$\mathcal{R}=\sum _{i\in I}(1\bowtie e_i)\otimes (e^i\bowtie 1),$$

where ${\left\{e_i\right\}}_{i\in I}$ is a basis of the vector space H and ${\left\{e^i\right\}}_{i\in I}$ is its dual basis in ${\left(H^{op}\right)}^{*}={\left(H^{*}\right)}^{cop}$.

Lemma 4. 

For any $0\le i\le st-1,0\le j\le t-1$, set

$$y_{i,j}=\frac{1}{st}\frac{1}{\left(j\right){!}_p}\sum _{k=0}^{st-1}{\omega }^{-ik}{\alpha }^k{\beta }^j$$

in ${\mathcal{T}}^{*}$. Then, $y_{i,j}\left(g^{i_1}{x}^{j_1}\right)={\delta }_{i,i_1}{\delta }_{j,j_1}$ for all any $0\le i,i_1\le st-1,0\le j,j_1\le t-1$.

Proof. 

For $0\le i\le st-1$, $0\le j\le t-1$, let ${\delta }_{i,j}$ be the Kronecker symbol, then we have

$$\begin{array}{cc}\hfill & y_{i,j}\left(g^{i_1}{x}^{j_1}\right)\hfill \\ \hfill =& \frac{1}{st}\frac{1}{\left(j\right){!}_p}\left(\sum _{k=0}^{st-1}{\omega }^{-ik}{\alpha }^k{\beta }^j\right)\left(g^{i_1}{x}^{j_1}\right)\hfill \\ \hfill =& \frac{1}{st}\frac{1}{\left(j\right){!}_p}\left(j\right){!}_p({\beta }^j+{\omega }^{-1}\alpha {\beta }^j+\cdots +{\omega }^{-(st-1)i}{\alpha }^{st-1}{\beta }^j)\left(g^{i_1}{x}^{j_1}\right)\hfill \\ \hfill =& \frac{1}{st}(\sum _{v=0}^{st-1}\overline{g^v{x}^j}+{\omega }^{-i}\sum _{v=0}^{st-1}{\omega }^v\overline{g^v{x}^j}+\cdots +{\omega }^{-(st-1)i}\sum _{v=0}^{st-1}{\omega }^{(st-1)v}\overline{g^v{x}^j})\left(g^{i_1}{x}^{j_1}\right)\hfill \\ \hfill =& \frac{1}{st}(1+{\omega }^{(i_1-i)}+\cdots +{\omega }^{(i_1-i)(st-1)}){\delta }_{j,j_1}\hfill \\ \hfill =& {\delta }_{i,i_1}{\delta }_{j,j_1}.\hfill \end{array}$$

□

By Lemma 4, one can easily know that the $\mathcal{R}$-matrix of $D\left(\mathcal{T}\right)$ is

$$\begin{array}{cc}\hfill \mathcal{R}& =\sum _{i,j}(1\bowtie g^i{x}^j)\otimes (y_{i,j}\bowtie 1)\hfill \\ \hfill & =\frac{1}{st}\sum _{i,j,k}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}(1\bowtie g^i{x}^j)\otimes ({\alpha }^k{\beta }^j\bowtie 1).\hfill \end{array}$$

4.2. The Existence of Ribbon Elements

In this subsection, we review some facts about the integral and quasi-ribbon element for a finite-dimensional Hopf algebra H. For $h\in H$ and $\alpha$ in the dual space $H^{*}$, we define

$$\langle \alpha ,h\rangle :=\alpha \left(h\right)\in \mathbb{k}.$$

The following results on the integral can be found in [14] (Chapter 2):

• A left integral element in H is an element t in H such that $ht=\epsilon \left(h\right)t,\forall h\in H.$ A right integral element in H is an element $t^{\prime }$ in H such that $t^{\prime }h=\epsilon \left(h\right)t^{\prime },\forall h\in H.$

${\int }_H^l$ denotes the subspace of left integrals in H, and ${\int }_H^r$ denotes the subspace of right integrals in H. H is called unimodular if ${\int }_H^l={\int }_H^r$.

• Let H be a finite-dimensional Hopf algebra. Then, we have the following:

  (1)

  ${\int }_H^l$ and ${\int }_H^r$ are each one-dimensional.

  (2)

  The antipode S of H is bijective, and $S\left({\int }_H^l\right)={\int }_H^r,S\left({\int }_H^r\right)={\int }_H^l$.

• Suppose $t\in {\int }_H^l$ and $T\in {\int }_{H^{*}}^r$. Notice that the left integrals for H form a one-dimensional ideal of H. Hence, there is a unique $\tilde{\alpha }\in G\left(H^{*}\right)$ such that $th=\langle \tilde{\alpha },h\rangle t$ for all $h\in H$. The condition that H is unimodular is equivalent to $\tilde{\alpha }=\epsilon$.

Likewise, there is a unique $\tilde{g}\in H$ such that $qT=\langle q,\tilde{g}\rangle T$, for all $q\in H^{*}$. We call $\tilde{\alpha }$ and $\tilde{g}$ the distinguished group-like elements of $H^{*}$ and H, respectively.

As above, assume the R-matrix is $\mathcal{R}={\sum }_i{x}_i\otimes y_i$, and define

$$\begin{array}{c}\hfill g_{\tilde{\alpha }}=\sum _i{x}_i\tilde{\alpha }\left(y_i\right),and{h}_{\tilde{\alpha }}=g_{\tilde{\alpha }}{\tilde{g}}^{-1},\end{array}$$

(4)

where $\tilde{\alpha }$ is the distinguished group-like element of $H^{*}$ and $\tilde{g}$ is the distinguished group-like element of H.

A quasi-ribbon element of Hopf algebra H is an element satisfying all the ribbon conditions in (2), except for the requirement that it is central. Our approach to finding an explicit formula for the ribbon element of $D\left(\mathcal{T}\right)$ is to use the following results from quasi-ribbon elements.

Theorem 1 ([4] (Theorem 1)). 

Let $(H,\mathcal{R})$ be a finite-dimensional quasi-triangular Hopf algebra over a field $\mathbb{k}$. Suppose $h_{\tilde{\alpha }}^{\prime }$ is any element of H such that ${\left(h_{\tilde{\alpha }}^{\prime }\right)}^2=h_{\tilde{\alpha }}$, i.e., $h_{\tilde{\alpha }}^{\prime }$ is any square root of the element $h_{\tilde{\alpha }}$ in (4). Then, $v=u{h}_{\tilde{\alpha }}^{\prime }$ is a quasi-ribbon element, where u is as in (1). Moreover, $v=u{h}_{\tilde{\alpha }}^{\prime }$ is a ribbon element of $(H,\mathcal{R})$ if and only if $S^2\left(a\right)={\left(h_{\tilde{\alpha }}^{\prime }\right)}^{-1}a{h}_{\tilde{\alpha }}^{\prime }$ for all $a\in H$.

Theorem 2 ([4] (Theorem 3)). 

Suppose that H is a finite-dimensional Hopf algebra with antipode S over a field $\mathbb{k}$. Let $\tilde{g}$ and $\tilde{\alpha }$ be the distinguished group-like elements of H and $H^{*}$, respectively. Then, we have the following:

(1) 

$\left(D\right(H),\mathcal{R})$ has a quasi-ribbon element if and only if there are $h\in G\left(H\right)$ and $\gamma \in G\left(H^{*}\right)$ such that $h^2=\tilde{g}$ and ${\gamma }^2=\tilde{\alpha }$.

(2) 

$\left(D\right(H),\mathcal{R})$ has a ribbon element if and only if there are $h\in G\left(H\right)$ and $\gamma \in G\left(H^{*}\right)$ as in part (1) such that

$$S^2\left(y\right)=h(\gamma \rightharpoonup y\leftharpoonup {\gamma }^{-1})h^{-1}$$

for all $y\in H$.

Corollary 1 ([4] (Proposition 3)). 

Let $\mathcal{S}$ be the antipode of $D\left(H\right)$, and set $\mathcal{U}=\sum \mathcal{S}\left(y_i\right)x_i$. Then,

$$(\gamma ,h)\mapsto \mathcal{U}({\gamma }^{-1}\otimes h^{-1})$$

defines a one-to-one correspondence between those pairs $(\gamma ,h)\in G\left(H^{*}\right)\times G\left(H\right)$ such that $h^2=\tilde{g}$ and ${\gamma }^2=\tilde{\alpha }$ and the quasi-ribbon elements of $\left(D\right(H),\mathcal{R}).$ The ribbon elements correspond to those pairs $(\gamma ,h)$, which further satisfy $S^2\left(y\right)=h(\gamma \rightharpoonup y\leftharpoonup {\gamma }^{-1})h^{-1}$, for all $y\in H$.

By [17], $D\left(\mathcal{T}\right)$ is unimodular, so that ${\alpha }_D={\epsilon }_D$, where ${\alpha }_D$ and ${\epsilon }_D$ are the distinguished group-like element and the counit of ${\left(D\left(\mathcal{T}\right)\right)}^{*}$, respectively.

Lemma 5. 

(1) 

${\sum }_{i=0}^{st-1}{\omega }^{(t-1)i}{\alpha }^i{\beta }^{t-1}$ is a right integral in ${\mathcal{T}}^{*}$, and the distinguished group-like element of $\mathcal{T}$ is $g^{1-t}$.

(2) 

${\sum }_{i=0}^{st-1}{g}^i{x}^{t-1}$ is a left integral in $\mathcal{T}$, and the distinguished group-like element of ${\mathcal{T}}^{*}$ is ${\alpha }^{-s}$.

Proof. 

Let $\lambda ={\sum }_{\begin{array}{c}0\le i\le st-1\\ 0\le j\le t-1\end{array}}{a}_{ij}{\alpha }^i{\beta }^j.$ We use the definition of a right integral of ${\mathcal{T}}^{*}$:

$$\lambda \in {\int }_{{\mathcal{T}}^{*}}^r\iff \lambda h^{*}=\epsilon \left(h^{*}\right)\lambda ,$$

for all $h^{*}\in {\mathcal{T}}^{*}.$ Let $h^{*}=\beta$; we have

$$\sum _{\begin{array}{c}0\le i\le st-1\\ 0\le j\le t-1\end{array}}{a}_{ij}{\alpha }^i{\beta }^{j+1}=0.$$

Since $\left\{{\alpha }^i{\beta }^k\right|0\le i\le st-1,0\le j\le t-1\}$ is the basis of ${\mathcal{T}}^{*}$, and ${\beta }^t=0$, we obtain

$$\lambda =\sum _{i=0}^{st-1}{a}_{i,t-1}{\alpha }^i{\beta }^{t-1}.$$

Let $h^{*}=\alpha$. We have

$$\sum _{i=0}^{st-1}{a}_{i,t-1}{\omega }^{t-1}{\alpha }^{i+1}{\beta }^{t-1}=\sum _{i=0}^{st-1}{a}_{i,t-1}{\alpha }^i{\beta }^{t-1},$$

Then, $a_{i,t-1}{\omega }^{t-1}=a_{i+1,t-1}$. Moreover, ${\int }_{{\mathcal{T}}^{*}}^r$ is one-dimensional. Therefore, $\lambda ={\sum }_{i=0}^{st-1}{\omega }^{(t-1)i}{\alpha }^i{\beta }^{t-1}.$

Let $\tilde{g}={\sum }_{\begin{array}{c}0\le k\le st-1\\ 0\le l\le t-1\end{array}}{b}_{kl}{g}^k{x}^l$ be the distinguished group-like element of $\mathcal{T}$. Then, we have

$$h^{*}\lambda =h^{*}\left(\tilde{g}\right)\lambda .$$

Let $h^{*}={\alpha }^j$. We have

$$\sum _{\begin{array}{c}0\le i\le st-1\\ 0\le j\le t-1\end{array}}{\omega }^{(t-1)i}{\alpha }^{i+j}{\beta }^{t-1}=\langle \sum _{\begin{array}{c}0\le i\le st-1\\ 0\le j\le t-1\end{array}}{\omega }^{ij}\overline{g^i},\sum _{\begin{array}{c}0\le k\le st-1\\ 0\le l\le t-1\end{array}}{b}_{kl}{g}^k{x}^l\rangle \sum _{i=0}^{st-1}{\omega }^{(t-1)i}{\alpha }^i{\beta }^{t-1}.$$

Thus, ${\sum }_{k=0}^{st-1}{\omega }^{kj}{b}_{k0}={\omega }^{(1-t)j},0\le j\le st-1.$ By Cramer’s Rule and the Vandermonde determinant, we have

$$b_{k0}=\left\{\begin{array}{cc}1,\hfill & k=st+1-t,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.,where0\le k\le st-1.$$

Let $h^{*}={\alpha }^j{\beta }^z$, $0\le j\le st-1,1\le z\le t-1$. We have

$$\langle \left(z\right){!}_p\sum _{k=0}^{st-1}{\omega }^{jk}\overline{g^k{x}^z},\sum _{\begin{array}{c}0\le k\le st-1\\ 0\le l\le t-1\end{array}}{b}_{kl}{g}^k{x}^l\rangle \sum _{i=0}^{st-1}{\omega }^{(t-1)i}{\alpha }^i{\beta }^{t-1}=0.$$

Since $\left\{{\alpha }^i{\beta }^k\right|0\le i\le st-1,0\le j\le t-1\}$ is the basis of ${\mathcal{T}}^{*},$ we have

$$\langle \left(z\right){!}_p\sum _{k=0}^{st-1}{\omega }^{jk}\overline{g^k{x}^z},\sum _{\begin{array}{c}0\le k\le st-1\\ 0\le l\le t-1\end{array}}{b}_{kl}{g}^k{x}^l\rangle =\left(z\right){!}_p\sum _{k=0}^{st-1}{\omega }^{jk}{b}_{kz}=0.$$

By Cramer’s Rule and the Vandermonde determinant, we have

$$b_{kl}=0,0\le k\le st-1,1\le l\le t-1.$$

Consequently, $\tilde{g}=g^{1-t}.$ Similarly, we can prove part (2). □

Theorem 3. 

(1) 

$\left(D\right(\mathcal{T}),\mathcal{R})$ has quasi-ribbon elements if and only if t is odd.

(2) 

$\left(D\right(\mathcal{T}),\mathcal{R})$ has unique ribbon element if and only if both s and t are odd.

(3) 

$\left(D\right(\mathcal{T}),\mathcal{R})$ has two ribbon elements if and only if s is even and t is odd.

Proof. 

(1)

By part (1) of Theorem 2, $\left(D\right(\mathcal{T}),\mathcal{R})$ has a quasi-ribbon element if and only if there exist $h=g^j\in G\left(\mathcal{T}\right)$, $\gamma ={\alpha }^k\in G\left({\mathcal{T}}^{*}\right),wherej,k\in \mathbb{Z}$, such that ${\left(g^j\right)}^2=g^{1-t}$, ${\alpha }^{2k}={\alpha }^{-s}$, which implies $st\mid (2j+t-1)$ and $st\mid (2k+s).$ Since, when t is even, $2j+t-1$ is odd, $st$ is even, which contradicts $st\mid (2j+t-1)$. However, when t is odd, no matter whether s is even or odd, there exist $j=\frac{2st+1-t}{2},k=\frac{st-s}{2}\in \mathbb{Z}$ such that $st\mid (2j+t-1),st\mid (2k+s)$. Thus, one knows that $\left(D\right(\mathcal{T}),\mathcal{R})$ has quasi-ribbon elements if and only if t is odd.

(2)

By part (2) of Theorem 2, $\left(D\right(\mathcal{T}),\mathcal{R})$ has ribbon elements if and only if there exist $\gamma ={\alpha }^k\in G\left({\mathcal{T}}^{*}\right)$, $h=g^j\in G\left(\mathcal{T}\right)$, which satisfy

$$\begin{array}{cc}\hfill & h^2=g^{1-t},{\gamma }^2={\alpha }^{-s},\hfill \\ \hfill & S^2\left(x\right)=h(\gamma \rightharpoonup x\leftharpoonup {\gamma }^{-1})h^{-1},\hfill \\ & S^2\left(g\right)=h(\gamma \rightharpoonup g\leftharpoonup {\gamma }^{-1})h^{-1},\hfill \end{array}$$

(5)

where x and g are the generators of $\mathcal{T}$, $k,j\in \mathbb{Z}$ and $\gamma ,h$ are given in part $\left(1\right)$.

It follows $S^2\left(x\right)={\omega }^{-s}x$, $h(\gamma \rightharpoonup x\leftharpoonup {\gamma }^{-1})h^{-1}=g^j\left({\omega }^{k}x\right)g^{-j}={\omega }^{k-sj}x$, $S^2\left(g\right)=g$, $h(\gamma \rightharpoonup g\leftharpoonup {\gamma }^{-1})h^{-1}=g^{j}g{g}^{-j}=g$ that $\left(D\right(\mathcal{T}),\mathcal{R})$ has ribbon elements if and only if there exist pairs $(\gamma ,h)=({\alpha }^k,g^j)$ such that

$$\begin{array}{cc}\hfill & g^{2j}=g^{1-t},{\alpha }^{2k}={\alpha }^{-s},\hfill \\ \hfill & {\omega }^{-s}x={\omega }^{k-sj}x.\hfill \end{array}$$

Since the order of $\alpha$ is $st$ and the order of g is $st$, $\left(D\right(\mathcal{T}),\mathcal{R})$ has ribbon elements if and only if there exist pairs $(\gamma ,h)=({\alpha }^k,g^j)$ such that

$$st\mid (2j+t-1),st\mid (2k+s),st\mid (ks-sj).$$

⇔

$$j=\frac{stu+1-t}{2}\in \mathbb{Z},k=\frac{stv-s}{2}\in \mathbb{Z},v-su=2w-1,$$

(6)

where $u,v,w\in \mathbb{Z}.$

By part $\left(1\right)$, one knows that, if $\left(D\right(\mathcal{T}),\mathcal{R})$ has ribbon elements, then t must be odd. If v is odd, then ${\alpha }^k={\alpha }^{\frac{stv-s}{2}}={\alpha }^{\frac{s(t-1)}{2}}$; if v is even, ${\alpha }^k={\alpha }^{\frac{stv-s}{2}}$, $k=\frac{stv-s}{2}\in \mathbb{Z}$, which implies that s must be even. If u is odd, $j=\frac{stu+1-t}{2}\in \mathbb{Z}$, which implies s must be even, then we have $g^j=g^{\frac{stu+1-t}{2}}=g^{\frac{st+1-t}{2}}$; if u is even, then $g^j=g^{\frac{stu+1-t}{2}}=g^{\frac{1-t}{2}}$. If both t and s are odd, u is even, v is odd or u is odd, and v is even since $v-su=2w-1$. Moreover, $j=\frac{stu+1-t}{2}\in \mathbb{Z}$ and $k=\frac{stv-s}{2}\in \mathbb{Z}$ imply that u is even and v is odd, then there exists a unique pair $(\gamma ,h)=({\alpha }^{\frac{s(t-1)}{2}},g^{\frac{1-t}{2}})$ satisfying (5). If t is odd and s is even, v must be odd since $v-su=2w-1$. In this case, no matter whether p is odd or even, $j=\frac{stu+1-t}{2}\in \mathbb{Z}$ and $k=\frac{stv-s}{2}\in \mathbb{Z}$. Thus, there exist two pairs $({\gamma }_1,h_1)=({\alpha }^{\frac{s(t-1)}{2}},g^{\frac{1-t}{2}})$ and $({\gamma }_2,h_2)=({\alpha }^{\frac{s(t-1)}{2}},g^{\frac{(s-1)t+1}{2}})$ satisfying (5). Consequently, by Corollary 1, $\left(D\right(\mathcal{T}),\mathcal{R})$ has a unique ribbon element if and only if both s and t are odd; $\left(D\right(\mathcal{T}),\mathcal{R})$ has two ribbon elements if and only if s is even and t is odd. □

4.3. Computation of the Ribbon Elements of $D\left(\mathcal{T}\right)$

Throughout this subsection, assume t is an odd integer. Notice that $\tilde{\alpha }={\alpha }^{-s}$ and $\tilde{g}=g^{1-t}$ are the distinguished group-like elements in ${\mathcal{T}}^{*}$ and $\mathcal{T}$, respectively. By the description about the distinguished group-like element of the Drinfeld double of a finite-dimensional quasi-triangular Hopf algebra in [17], the distinguished group-like element in $D\left(\mathcal{T}\right)$ is $\tilde{\alpha }\bowtie \tilde{g}$.

Recall the universal R-matrix of $D\left(\mathcal{T}\right)$ given in Section 4.1:

$$\begin{array}{cc}\hfill \mathcal{R}& =\frac{1}{st}\sum _{i,j,k}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}(1\bowtie g^i{x}^j)\otimes ({\alpha }^k{\beta }^j\bowtie 1).\hfill \end{array}$$

(7)

Theorem 4. 

Assume that t is an odd integer:

(1) 

When s is odd, the unique ribbon element in $D\left(\mathcal{T}\right)$ is

$$v=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}}),$$

$$whereu=\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}{(-1)}^j\frac{1}{\left(j\right){!}_p}{\omega }^{-(i+j)k-\frac{j(j-1)s}{2}}({\alpha }^{-sj-k}{\beta }^j\bowtie g^i{x}^j).$$

(2) 

When s is even, the ribbon elements in $D\left(\mathcal{T}\right)$ are

$$v_1=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}}),v_2=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t(s+1)-1}{2}}),$$

$$whereu=\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}{(-1)}^j\frac{1}{\left(j\right){!}_p}{\omega }^{-(i+j)k-\frac{j(j-1)s}{2}}({\alpha }^{-sj-k}{\beta }^j\bowtie g^i{x}^j).$$

Proof. 

(1)

We adopt the previous conventions and set $g_{{\alpha }_D}=g_{{\epsilon }_D}$ (which holds as $D\left(\mathcal{T}\right)$ is unimodular). By (4) and (7), we have

$$\begin{array}{cc}\hfill g_{{\epsilon }_D}& =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}\epsilon ({\alpha }^k{\beta }^j\bowtie 1)(1\bowtie g^i{x}^j)\hfill \\ \hfill & =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}\epsilon \left(1\right){\alpha }^k{\beta }^j\left(1\right)(1\bowtie g^i{x}^j).\hfill \end{array}$$

Since ${\beta }^j\left(1\right)=0$ when $j\ne 0$ and ${\epsilon }_D$ is an algebra homomorphism, only the terms with $j=0$ survive, and therefore,

$$\begin{array}{cc}\hfill g_{{\epsilon }_D}& =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}{\omega }^{-ik}(1\bowtie g^i)\hfill \\ \hfill & =\frac{1}{st}\sum _{i=0}^{st-1}\left(\sum _{k=0}^{st-1}{\omega }^{-ik}\right)(1\bowtie g^i).\hfill \end{array}$$

Observe that

$$\sum _{k=0}^{st-1}{\omega }^{-ik}=\frac{1-{\left({\omega }^{-i}\right)}^{st}}{1-{\omega }^{-i}}=0$$

unless $i=0$, in which case ${\sum }_{k=0}^{st-1}{\omega }^{-ik}=st$. Therefore, $g_{{\epsilon }_D}=1_{D\left(E\right)}.$

By the discussion above, the distinguished group-like element in $D\left(\mathcal{T}\right)\simeq {\mathcal{T}}^{*}\otimes \mathcal{T}$ is $\widehat{g}={\alpha }^{-s}\bowtie g^{1-t}$.

By (4), $h_{{\epsilon }_D}=g_{{\epsilon }_D}{\left(\widehat{g}\right)}^{-1}={({\alpha }^{-s}\bowtie g^{1-t})}^{-1}={\alpha }^s\bowtie g^{t-1}.$ When s and t are both odd, the square root $h_{{\epsilon }_D}^{\prime }$ of $h_{{\epsilon }_D}$ is unique, because $h_{{\epsilon }_D}$, and therefore, $h_{{\epsilon }_D}^{\prime }$ has odd order. Thus,

$$h_{{\epsilon }_D}^{\prime }={\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}},$$

$$v=u{h}_{{\epsilon }_D}^{\prime }=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}}).$$

By Theorem 1, the quasi-ribbon element v is the unique ribbon element of $D\left(\mathcal{T}\right)$.

(2)

When s is even and t is odd, $h_{{\epsilon }_D}$ has four square roots:

$$h_{{\epsilon }_{D1}}^{\prime }={\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}},h_{{\epsilon }_{D2}}^{\prime }={\alpha }^{\frac{s}{2}}\bowtie g^{\frac{t-1}{2}},$$

$$h_{{\epsilon }_{D3}}^{\prime }={\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t(s+1)-1}{2}},h_{{\epsilon }_{D4}}^{\prime }={\alpha }^{\frac{s}{2}}\bowtie g^{\frac{t(s+1)-1}{2}}.$$

By Theorem 1,

$$S^2(\epsilon \bowtie g)={\left(h_{{\epsilon }_{Di}}^{\prime }\right)}^{-1}(\epsilon \bowtie g)h_{{\epsilon }_{Di}}^{\prime },S^2(\epsilon \bowtie x)={\left(h_{{\epsilon }_{Di}}^{\prime }\right)}^{-1}(\epsilon \bowtie x)h_{{\epsilon }_{Di}}^{\prime },$$

$$S^2(\alpha \bowtie 1)={\left(h_{{\epsilon }_{Di}}^{\prime }\right)}^{-1}(\alpha \bowtie 1)h_{{\epsilon }_{Di}}^{\prime },S^2(\beta \bowtie 1)={\left(h_{{\epsilon }_{Di}}^{\prime }\right)}^{-1}(\beta \bowtie 1)h_{{\epsilon }_{Di}}^{\prime },$$

where $i=1,3$ and $\epsilon \bowtie g,\epsilon \bowtie x,\alpha \bowtie 1,\beta \bowtie 1$ are the generators of $D\left(\mathcal{T}\right)$.

Therefore, quasi-ribbon elements

$$v_1=u{h}_{{\epsilon }_{D1}}^{\prime }=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}})and{v}_2=u{h}_{{\epsilon }_{D3}}^{\prime }=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t(s+1)-1}{2}})$$

are the ribbon elements of $D\left(\mathcal{T}\right)$.

It remains to show that u has the expression in (1). Recall that $u={\sum }_{i}S\left(y_i\right)x_i$, where

$$\mathcal{R}=\frac{1}{st}\sum _{i,j,k}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}(1\bowtie g^i{x}^j)\otimes ({\alpha }^k{\beta }^j\bowtie 1).$$

Therefore,

$$\begin{array}{cc}\hfill u& =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}S({\alpha }^k{\beta }^j\bowtie 1)(1\bowtie g^i{x}^j)\hfill \\ \hfill & =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}\frac{1}{\left(j\right){!}_p}{\omega }^{-ik}\left({(-{\alpha }^{-s}\beta )}^j{\left({\alpha }^{st-1}\right)}^k\right)\bowtie 1)(1\bowtie g^i{x}^j)\hfill \\ \hfill & =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}{(-1)}^j\frac{1}{\left(j\right){!}_p}{\omega }^{-ik-jk-\frac{j(j-1)s}{2}}({\alpha }^{-sj-k}{\beta }^j\bowtie 1)(1\bowtie g^i{x}^j)\hfill \\ \hfill & =\frac{1}{st}\sum _{\begin{array}{c}0\le i,k\le st-1\\ 0\le j\le t-1\end{array}}{(-1)}^j\frac{1}{\left(j\right){!}_p}{\omega }^{-(i+j)k-\frac{j(j-1)s}{2}}({\alpha }^{-sj-k}{\beta }^j\bowtie g^i{x}^j).\hfill \end{array}$$

Finally, we know that, when s is odd, $v=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}})$ is the unique ribbon element of $D\left(\mathcal{T}\right)$. When s is even, $v_1=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t-1}{2}})and{v}_2=u({\alpha }^{\frac{s(t+1)}{2}}\bowtie g^{\frac{t(s+1)-1}{2}})$ are the ribbon elements of $D\left(\mathcal{T}\right)$. □

Remark 1. 

The generalized Taft algebra is a special rank-one pointed Hopf algebra of the nilpotent type. We have provided a necessary and sufficient condition for the quantum double of generalized Taft algebra ${\bigwedge }_{st,t}^{*cop}$ to be a ribbon Hopf algebra. Besides, we computed all ribbon elements of $D\left({\bigwedge }_{st,t}^{*cop}\right)$. Further research is required to obtain the necessary and sufficient condition for the quantum double of all pointed Hopf algebras of the nilpotent type.