Genus bounds for twisted quantum invariants

Quantum invariants, as developed by Reshetikhin and Turaev [29, 30], are topological invariants of knots and 3-manifolds built from the representation theory of quantum groups, or more generally, the theory of braided monoidal categories. The Jones polynomial is the special case when the quantum group is that associated to the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$. When the braided categories involved are semisimple, or more precisely modular, these invariants can be nicely packaged into what is called a 3-dimensional topological quantum field theory (TQFT), a mathematical notion that formalizes the various cut-and-paste properties of quantum invariants [32]. As the name suggests, these objects originate from physics, namely from Witten’s interpretation of the Jones polynomial in terms of quantum field theory [37].

When the parameter $q$ of the quantum group $U_{q}(\mathfrak{g})$ is a root of unity, the corresponding representation category becomes non-semisimple and one can find continuous families of simple modules, such as Verma modules with complex highest weights. When $\mathfrak{g}=\mathfrak{sl}_{2}$ this gives rise to a polynomial invariant, called the ADO invariant, after Akutsu-Deguchi-Ohtsuki [1]. These “non-semisimple quantum invariants” have been much less studied than their semisimple counterparts though, partially because extending these to 3-manifolds and eventually TQFTs is much more complicated and for more than 20 years they lacked a clear physical interpretation. Some of these hurdles have been overcome in the work of Geer-Patureau-Mirand and collaborators [13, 6], leading to TQFTs for ADO invariants with interesting topological features [5]. Moreover, connections between non-semisimple invariants and the physics of vertex operator algebras and logarithmic conformal field theory have recently been found [15, 7].

However, even though quantum invariants have nice cut-and-paste (TQFT) properties and physically interesting interpretations, their topological content remains mysterious. For instance, the Jones polynomial (as well as HOMFLY and Kauffman polynomials) is not clearly related to the Seifert genus of a knot, fibredness or sliceness. This is in sharp contrast to twisted Alexander polynomials (or equivalently, twisted Reidemeister torsion), which are related to all these topological properties [11] and even to hyperbolic geometry [27]. Relations between colored Jones polynomials and hyperbolic geometry or $SL(2,\mathbb{C})$-character varieties are still major open conjectures in the field.

In order to find which aspects of the theory of braided monoidal categories capture interesting topology of knot complements, a first natural question is: which “structure” behind the Alexander polynomial and Reidemeister torsion is making it capture so much topological information? The answer is very simple: these are invariants of a covering space of the knot complement $X_{K}=S^{3}\setminus K$, or in other words, invariants of a pair $(K,\rho)$ where $\rho:\pi_{1}(X_{K})\to G$ is a homomorphism into a group $G$. For instance, the Alexander polynomial comes from the covering space corresponding to the projection $\pi_{1}(X_{K})\to H_{1}(X_{K})=\mathbb{Z}$. Most applications depend on these invariants being polynomials, for which one needs $\mathbb{Z}\subset G$, in particular, $G$ is an infinite group.

It turns out that there is an extension of the theory of quantum invariants, due to Turaev [31], that produces invariants of such pairs $(K,\rho)$. This extension starts with a $G$-crossed braided monoidal category, essentially a $G$-graded category with a $G$-action and a compatible braiding, and produces invariants of tangles endowed with a representation $\rho:\pi_{1}(X_{T})\to G$, where $X_{T}=\mathbb{R}^{2}\times[0,1]\setminus T$. The $G$-action condition turns out to pose a severe restriction: a semisimple monoidal category has only finitely many tensor autoequivalences up to isomorphism [8]. In light of this, we believe it is natural to consider non-semisimple $G$-braided monoidal categories where $G$ is some infinite group.

In previous work, the first author [23] considered a special class of such categories: relative Drinfeld centers of crossed products $\text{Rep}(H)\ltimes\mathrm{Aut}(H)$ where $H$ is a finite dimensional Hopf algebra and $\mathrm{Aut}(H)$ its group of Hopf algebra automorphisms, or equivalently, $\mathrm{Aut}(H)$-twisted Drinfeld doubles ¹¹1From now on we refer to this simply as a twisted Drinfeld double as in [34]. Note that the term “twisted Drinfeld double” has another common meaning in the literature. as defined by Virelizier [34]. The reason to use such categories is that the twisted quantum invariants they define can be seen as “Fox calculus deformations” of the invariants coming from the usual Drinfeld double $D(H)$ so, in some sense, they do retain some covering space theory. Moreover, it was shown in [23, 22] that the $SL(n,\mathbb{C})$-twisted Reidemeister torsion of knot complements is obtained from the twisted Drinfeld double of an exterior algebra. Thus, one may expect that these quantum invariants contain topological information generalizing that of torsion.

In this work, we show that the twisted quantum invariants of [23] do indeed contain easily readable topological information about knot complements, namely, lower bounds to the Seifert genus. Moreover, we show that these invariants contain the ADO invariants, therefore giving genus bounds for these.

To state our main theorem, we briefly recall the construction of [23]. Let $H$ be a finite dimensional Hopf algebra over a field $\mathbb{K}$ such that the Drinfeld double $D(H)$ is ribbon. The twisted Drinfeld double $\underline{D(H)}$ of $H$ is a family, indexed by $\alpha\in\mathrm{Aut}(H)$, of deformations $D(H)_{\alpha}$ of the usual Drinfeld double of $H$ [34]. If $T$ is a framed, oriented, $n$-component open tangle, the twisted Drinfeld double leads to a deformation

$$Z^{\rho}_{\underline{D(H)}}(T)\in(H^{*}\otimes H)^{\otimes n}$$

of the usual universal invariant of tangles $Z_{D(H)}(T)$ (see e.g. [16]) that depends on a homomorphism $\rho:\pi_{1}(X_{T})\to\mathrm{Aut}(H)$. This invariant is essentially a special case of Turaev’s construction [31]. Now, if in addition $H$ is $\mathbb{N}^{m}$-graded, there is a canonical automorphism $\theta\in\mathrm{Aut}(H^{\prime})$, where $H^{\prime}=H\otimes_{\mathbb{K}}\mathbb{F}$ and $\mathbb{F}=\mathbb{K}[t_{1}^{\pm 1/2},\dots,t_{m}^{\pm 1/2}]$, defined by $\theta(x)=t_{1}^{k_{1}}\dots t_{m}^{k_{m}}x$ for $x\in H$ homogeneous of degree $(k_{1},\dots,k_{m})$. Then any $\rho$ as above extends to $\rho\otimes\theta:\pi_{1}(X_{T})\to\mathrm{Aut}(H^{\prime})$. This “degree twisted” extension generalizes a similar procedure in twisted Reidemeister torsion theory (e.g. [11]). We thus get an invariant

$$Z_{\underline{D(H^{\prime})}}^{\rho\otimes\theta}(T)\in(H^{\prime*}\otimes_{\mathbb{F}}H^{\prime})^{\otimes n}$$

defined for any $\rho:\pi_{1}(X_{T})\to\mathrm{Aut}^{0}_{\mathbb{N}^{m}}(H)$ (automorphisms preserving the degree and fixing the left cointegral of $H$). If $K_{o}$ is a framed, oriented, $(1,1)$-tangle whose closure is a knot $K$, then we define

$$P_{H}^{\rho\otimes\theta}(K)=\lambda\cdot\epsilon_{D(H^{\prime})}(Z_{\underline{D(H^{\prime})}}^{\rho\otimes\theta}(K_{o}))\in\mathbb{K}[t_{1}^{\pm 1},\dots,t_{m}^{\pm 1}]$$

where $\lambda\in\mathbb{F}^{\times}$ is a normalization factor making this an invariant of the unframed $(K,\rho)$. If $\rho\equiv 1$, $P_{H}^{\theta}(K)$ is a polynomial invariant of the oriented knot $K$ with no additional structure, but one still has to think that there is a canonical abelian representation $H_{1}(X_{K})\to\mathrm{Aut}(H^{\prime})$ sending the oriented meridian to $\theta$.

If $P_{H}^{\rho\otimes\theta}(K)=\sum_{\overline{k}}a_{\overline{k}}t_{1}^{k_{1}}\dots t_{m}^{k_{m}}$ where $a_{\overline{k}}\in\mathbb{K}$ and $\overline{k}=(k_{1},\dots,k_{m})\in\mathbb{Z}^{m}$, we define $\mathrm{deg}\ P_{H}^{\rho\otimes\theta}(K)$ as $\max\{k_{1}+\dots+k_{m}\ |\ a_{\overline{k}}\neq 0\}-\min\{k_{1}+\dots+k_{m}\ |\ a_{\overline{k}}\neq 0\}$. We denote $d(H)=\max\{i_{1}+\dots+i_{m}\ |\ H_{(i_{1},\dots,i_{m})}\neq 0\}$ where $H_{(i_{1},\dots,i_{m})}$ is the component of degree $(i_{1},\dots,i_{m})\in\mathbb{N}^{m}$ of $H$.

In fact the proof shows that a similar bound holds for $Z_{\underline{D(H^{\prime})}}^{\rho\otimes\theta}(K_{o})$. As mentioned previously, the motivation for this theorem comes from the Fox calculus formulas of [22, 23]. The actual proof given here makes no mention to Fox calculus, it is instead based on an argument of Habiro [16] but generalized to include the representation $\rho\otimes\theta$, which is essential for our theorem. We illustrate the theorem directly for (even) twist knots, which have genus one, in Subsection 3.7. Note that this theorem is in contrast with the Jones polynomial whose degree coincides with the crossing number for twist knots (since they are alternating), hence the degrees are not bounded.

We now list various special cases of our theorem.

When $H=\Lambda(\mathbb{C}^{n})$ is an exterior algebra (with $\mathbb{C}^{n}$ concentrated in degree one), one has $\mathrm{Aut}^{0}_{\mathbb{N}}(H)\cong SL(n,\mathbb{C})$, $d(H)=n$ and $P_{\Lambda(\mathbb{C}^{n})}^{\rho\otimes\theta}(K)$ coincides with the $SL(n,\mathbb{C})$-twisted relative Reidemeister torsion $\tau^{\rho\otimes\theta}(X_{K},\mu)$ where $\mu\subset\partial X_{K}$ is a meridian [23]. This torsion is essentially equivalent to the twisted Alexander polynomial $\Delta_{K}^{\rho}(t)$ of Lin and Wada [19, 35]. The above theorem implies

$$\mathrm{deg}\ \tau^{\rho\otimes\theta}(X_{K},\mu)\leq 2g(K)n.$$

Since $\tau^{\rho\otimes\theta}(X_{K},\mu)=\det(t\rho(\mu)-I_{n})\tau^{\rho\otimes\theta}(X_{K})$ this is equivalent to $\mathrm{deg}\ \tau^{\rho\otimes\theta}(X_{K})\leq n(2g(K)-1)$ which is a special case of a result of Friedl and Kim [10].

Now let $H_{p}$ be the Borel part of the restricted quantum group $\overline{U}_{\zeta_{p}}(\mathfrak{sl}_{2})$ at a $2p$-th root of unity $\zeta_{p}$. Then $H_{p}$ has finite dimension, it is $\mathbb{N}$-graded (with $d(H_{p})=p-1$) and $D(H_{p})$ is ribbon, so we have a polynomial invariant $P_{H_{p}}^{\theta}(K)\in\mathbb{Z}[\zeta_{p}^{2}][t^{\pm 1}].$ On the other hand, the unrolled restricted quantum group $\overline{U}^{H}_{\zeta_{p}}(\mathfrak{sl}_{2})$ leads to the ADO invariant $ADO_{p}(K,t)\in\mathbb{Z}[\zeta_{p}^{2}][t^{\pm 1}]$ [1]. We use the normalization of the ADO invariant in which $ADO_{2}(K,t)=\Delta_{K}(t)$.

As an example, the $ADO_{4}$ invariants of the first $6$ twist knots (denoted by their usual names in knot tables) are given as follows:

$K$	$ADO_{\mathfrak{sl}_{2},4}(K,t)$
$3_{1}$	$(1-2i)+t^{-3}+it^{-2}-(1+i)t^{-1}+(1+i)t+it^{2}-t^{3}$
$4_{1}$	$7+it^{-3}-3t^{-2}-6it^{-1}+6it-3t^{2}-it^{3}$
$5_{2}$	$(-5+10i)-(2+2i)t^{-3}+(3-5i)t^{-2}+(8+4i)t^{-1}-(8+4i)t+(3-5i)t^{2}+(2+2i)t^{3}$
$6_{1}$	$(7+10i)-(2-2i)t^{-3}-(3+5i)t^{-2}+(8-4i)t^{-1}-(8-4i)t-(3+5i)t^{2}+(2-2i)t^{3}$
$7_{2}$	$(-5+4i)-(2+i)t^{-3}+(3-2i)t^{-2}+(2+4i)t^{-1}-(2+4i)t+(3-2i)t^{2}+(2+i)t^{3}$
$8_{1}$	$(13+2i)-(1-2i)t^{-3}-(6+i)t^{-2}+(1-11i)t^{-1}-(1-11i)t-(6+i)t^{2}+(1-2i)t^{3}$

These all have degree $\leq 6$, since $g(K)=1$ for these knots, this matches with our theorem. Formulas for ADO invariants of double twist knots (genus 1) and torus knots $T_{(2,2k+1)}$ (genus $k$) are given in [3]. All these satisfy the genus bound above.

More generally, for a simple Lie algebra $\mathfrak{g}$ of rank $m$ and a primitive root of unity $\zeta_{p}^{2}$ of order $p$ (say, $p$ coprime to the determinant of the Cartan matrix), our construction provides a polynomial invariant $P_{\mathfrak{g},p}(K)\in\mathbb{Z}[\zeta_{p}][t_{1}^{\pm 1},\dots,t_{m}^{\pm 1}]$ that gives a lower bound to the genus. For instance

$$\mathrm{deg}\ P_{\mathfrak{sl}_{N+1},p}(K)\leq 2g(K)(p-1)\frac{1}{6}N(N+1)(N+2),$$

which generalizes the above bound for ADO to all $N$. The invariant $P_{\mathfrak{g},p}$ is obtained by letting $H$ be the Borel part of the corresponding restricted quantum group (which is $\mathbb{N}^{m}$-graded) in our construction. We expect that this invariant recovers other “ADO-like” invariants defined in the literature, e.g. [17], [4], thus implying genus bounds for all of them. For instance, the $\mathfrak{sl}_{3}$-ADO invariant at $p=2$ (so $\zeta_{p}=i$) defined in [17] is given as follows for the above twist knots:

Our genus bound implies $\mathrm{deg}\ P_{\mathfrak{sl}_{3},2}(K)\leq 8g(K)$, which is satisfied by the above polynomials. This will be studied in a separate paper.

In [25], Ohtsuki proved a genus bound for the 2-loop expansion of the Kontsevich integral, which is a 2-variable knot polynomial $\Theta_{K}(x,y)$. No genus bound seems to be known for higher-loop expansions. The colored Jones polynomial is a specialization of the Kontsevich integral, so Ohtsuki’s theorem implies a genus bound for the 2-loop part of this expansion as well. However, this is a limited bound and no further results seem to be known for colored Jones polynomials. Note that Ohtsuki’s invariant satisfies $\Theta_{\overline{K}}(x,y)=-\Theta_{K}(x,y)$ so it vanishes on amphichiral knots, on the other hand, ADO invariants do not vanish on amphichiral knots. Now, the family of colored Jones polynomials turns out to be equivalent to the family of $ADO_{p}$ polynomials (for varying $p$) by a theorem of Willetts [36]. Thus, our theorem shows that non-semisimple quantum knot invariants contain more transparent topological information than their semisimple counterparts.

In the case of $\mathfrak{sl}_{2}$, Bar-Natan and the second author studied the $n$-loop polynomial using $h$-adic Hopf algebra techniques [2]. This gives another proof of Ohtsuki’s genus bound. This approach has the advantage of being polynomial-time computable but a genus bound was only given in the 2-loop case. Note that Seifert genus is in NP and co-NP [18].

Finally, we note that there already exist topological invariants detecting the Seifert genus. On the one hand, a theorem of Friedl and Vidussi [12] states that there is always some $n$ and a representation $\rho:\pi_{1}(X_{K})\to U(n)$ for which $\mathrm{deg}\ \tau^{\rho\otimes\theta}(X_{K},\mu)=2g(K)n$. The authors do not know whether there is an algorithm to find such a $\rho$ though. On the other hand, knot Floer homology, which categorifies the Alexander polynomial, detects the Seifert genus [26]. It is an interesting question whether one can actually detect the Seifert genus of a knot using Hopf algebra/representation theoretic techniques, e.g. by keeping $\rho\equiv 1$ and varying $H$ in our theorem.

We begin in Section 2 with some Hopf algebra preliminaries. Here we recall some definitions from Hopf $G$-coalgebras, we define twisted Drinfeld doubles and study ribbon elements in these. In Section 3 we define the invariant $Z^{\rho}_{\underline{D(H)}}(T)$, the polynomial invariant $P_{H}^{\rho\otimes\theta}(K)$, and we prove some properties of these invariants. We illustrate our main theorem with the case of twist knots in Subsection 3.7. In Section 4 we prove our genus bound (Theorem 1). Finally, in Section 5, we study the case where the Hopf algebra is the $H_{p}$ above and prove Theorem 2.

For basic definitions on Hopf algebras, see e.g. [28]. We denote multiplication, coproduct, unit, counit and antipode of a Hopf algebra $H$ over $\mathbb{K}$ by $m,\Delta,1,\epsilon,S$ respectively. We will employ Sweedler’s notation for the coproduct of a Hopf algebra, that is,

$$\Delta(x)=x_{(1)}\otimes x_{(2)},(\Delta\otimes\text{id})\Delta(x)=x_{(1)}\otimes x_{(3)}\otimes x_{(3)},$$

etc. The dual $H^{*}$ is also a Hopf algebra with

$$\langle p\cdot q,x\rangle=\langle p,x_{(1)}\rangle\langle q,x_{(2)}\rangle$$

for each $p,q\in H^{*}$ and $x\in H$. Here $\langle\ ,\ \rangle$ denotes the usual pairing and $\langle p,x\rangle=\langle x,p\rangle$ for any $p\in H^{*},x\in H$. We denote by $\mathrm{Aut}(H)$ the group of Hopf algebra automorphisms of $H$ and $\mathrm{Aut}^{0}(H)$ the subgroup of $\mathrm{Aut}(H)$ fixing a non-zero cointegral (either left or right, see Subsection 2.4 below for definitions).

A Hopf algebra $H$ is $\mathbb{N}^{m}$-graded if $H=\bigoplus_{I\in\mathbb{N}^{m}}H_{I}$ with $m(H_{I}\otimes H_{J})\subset H_{I+J}$, $\Delta(H_{N})\subset\sum_{I+J=N}H_{I}\otimes H_{J}$ and $S(H_{I})\subset H_{I}$ for each $I,J,N\in\mathbb{N}^{m}$. We denote by $\mathrm{Aut}^{0}_{\mathbb{N}^{m}}(H)$ the subgroup of automorphisms of $\mathrm{Aut}^{0}(H)$ preserving the $\mathbb{N}^{m}$-degree.

A Hopf $G$-coalgebra is a family $\underline{H}=\{H_{\alpha}\}_{\alpha\in G}$ where each $H_{\alpha}$ is an algebra with unit $1_{\alpha}$ together with a family of algebra morphisms $\Delta_{\alpha_{1},\alpha_{2}}:H_{\alpha_{1}\alpha_{2}}\to H_{\alpha_{1}}\otimes H_{\alpha_{2}}$ for each $\alpha_{1},\alpha_{2}\in G$, an algebra morphism $\epsilon:H_{1}\to\mathbb{K}$ and algebra antiautomorphisms $S_{\alpha}:H_{\alpha}\to H_{\alpha^{-1}}$ for each $\alpha$ satisfying graded versions of the Hopf algebra axioms, see [31, 33] for more details. Note that $H=H_{1}$ is a Hopf algebra in the usual sense (with the coproduct $\Delta_{1,1}$ and antipode $S_{1}$).

A Hopf $G$-coalgebra is said to be crossed if it comes with a family $\varphi=\{\varphi_{\alpha,\beta}\}_{\alpha,\beta\in G}$ of algebra isomorphisms $\varphi_{\alpha,\beta}:H_{\alpha}\to H_{\beta\alpha\beta^{-1}}$, simply denoted $\varphi_{\beta}$, preserving the Hopf structure and satisfying $\varphi_{\beta_{2}}\varphi_{\beta_{1}}=\varphi_{\beta_{2}\beta_{1}}$.

A crossed Hopf $G$-coalgebra $(\underline{H},\varphi)$ is quasi-triangular if it comes with a family $R=\{R_{\alpha,\beta}\}_{\alpha,\beta\in G}$ of invertible elements in $H_{\alpha}\otimes H_{\beta}$, satisfying

1. (1)

   $R_{\alpha,\beta}$ is $\varphi$-invariant, that is, $(\varphi_{\gamma}\otimes\varphi_{\gamma})(R_{\alpha,\beta})=R_{\gamma\alpha\gamma^{-1},\gamma\beta\gamma^{-1}}$,

2. (2)

   $R_{\alpha,\beta}\cdot\Delta_{\alpha,\beta}(x)=(\tau[(\varphi_{\alpha^{-1}}\otimes\text{id}_{H_{\alpha}})\Delta_{\alpha\beta\alpha^{-1},\alpha}(x)])\cdot R_{\alpha,\beta}$ where $\tau$ denotes permutation of two factors (with a sign in the super-case),

3. (3)

   $(\text{id}_{H_{\alpha}}\otimes\Delta_{\beta,\gamma})(R_{\alpha,\beta\gamma})=(R_{\alpha,\gamma})_{1\beta 3}\cdot(R_{\alpha,\beta})_{12\gamma}$,

4. (4)

   $(\Delta_{\alpha,\beta}\otimes\text{id}_{H_{\gamma}})(R_{\alpha\beta,\gamma})=((\text{id}_{H_{\alpha}}\otimes\varphi_{\beta^{-1}})(R_{\alpha,\beta\gamma\beta^{-1}}))_{1\beta 3}\cdot(R_{\beta,\gamma})_{\alpha 23}$

for each $\alpha,\beta,\gamma\in G$, where we used the notation $X_{12\gamma}=x\otimes y\otimes 1_{\gamma}$ for any $X=\sum x\otimes y\in H_{\alpha}\otimes H_{\beta}$ and similarly for $Y_{\alpha 23},Z_{1\beta 3}$ ($Y\in H_{\beta}\otimes H_{\gamma},Z\in H_{\alpha}\otimes H_{\gamma}$). If $(\underline{H},\varphi,R)$ is quasi-triangular, the graded Drinfeld element is $u=\{u_{\alpha}\}_{\alpha\in G}$ defined by

$$u_{\alpha}=m_{\alpha}(S_{\alpha^{-1}}\varphi_{\alpha}\otimes\text{id}_{H_{\alpha}})\tau_{\alpha,\alpha^{-1}}(R_{\alpha,\alpha^{-1}})\in H_{\alpha}.$$

A quasi-triangular Hopf $G$-coalgebra $(\underline{H},\varphi,R)$ is $G$-ribbon if it comes with a family $v=\{v_{\alpha}\}_{\alpha\in G}$ of invertible elements $v_{\alpha}\in H_{\alpha}$ satisfying, for each $\alpha,\beta\in G$:

1. (1)

   $\Delta_{\alpha,\beta}(v_{\alpha\beta})=(v_{\alpha}\otimes v_{\beta})(\tau_{\beta,\alpha}[\varphi_{\alpha^{-1}}\otimes\text{id}(R_{\alpha\beta\alpha^{-1},\alpha})])R_{\alpha,\beta}$,

2. (2)

   $S_{\alpha}(v_{\alpha})=v_{\alpha^{-1}}$,

3. (3)

   $\varphi_{\alpha}(x)=v_{\alpha}^{-1}xv_{\alpha}$ for all $x\in H_{\alpha}$,

4. (4)

   $\varphi_{\beta}(v_{\alpha})=v_{\beta\alpha\beta^{-1}}$.

If $\boldsymbol{g}_{\alpha}=v_{\alpha}u_{\alpha}$, we call $\boldsymbol{g}=\{\boldsymbol{g}_{\alpha}\}_{\alpha\in G}$ the $G$-pivot of $\underline{H}$, this satisfies $S_{\alpha^{-1}}S_{\alpha}(x)=\boldsymbol{g}_{\alpha}x\boldsymbol{g}_{\alpha}^{-1}$ for each $x\in H_{\alpha},\alpha\in G$ and is group-like in the sense that $\Delta_{\alpha,\beta}(\boldsymbol{g}_{\alpha\beta})=\boldsymbol{g}_{\alpha}\otimes\boldsymbol{g}_{\beta}$.

We now define the twisted Drinfeld double of a Hopf algebra $H$, which is a quasi-triangular Hopf $\mathrm{Aut}(H)$-coalgebra extending the Drinfeld double construction. This was introduced by Virelizier in [34], here we follow the conventions of [23] (see remark 2.1 below).

For each $\alpha\in\mathrm{Aut}(H)$ we define an algebra $D(H)_{\alpha}$ as follows: $D(H)_{\alpha}=H^{*}\otimes H$ as a vector space and we define the product by

$$(p\otimes a)\cdot_{\alpha}(q\otimes b)=\langle a_{(1)},q_{(3)}\rangle\langle S^{-1}\alpha^{-1}(a_{(3)}),q_{(1)}\rangle p\cdot q_{(2)}\otimes a_{(2)}\cdot b$$

where $p,q\in H^{*},a,b\in H$. This is an associative algebra with unit $1_{\alpha}=\epsilon_{H}\otimes 1_{H}$. For each $\alpha,\beta\in\mathrm{Aut}(H)$ we define a coproduct $\Delta_{\alpha,\beta}:D(H)_{\alpha\beta}\to D(H)_{\alpha}\otimes D(H)_{\beta}$ by

$$\Delta_{\alpha,\beta}(p\otimes a)=(p_{(2)}\otimes a_{(1)})\otimes(p_{(1)}\otimes\alpha^{-1}(a_{(2)})).$$

We will now establish under which conditions the twisted Drinfeld double has a graded ribbon element. For this, we briefly recall what happens for the usual Drinfeld double, for more details see [28].

Recall first that a left cointegral of a Hopf algebra $H$ is an element $\Lambda_{l}\in H$ such that $x\Lambda_{l}=\epsilon(x)\Lambda_{l}$ for each $x\in H$. Dually, a right integral is $\lambda_{r}\in H^{*}$ such that $\lambda_{r}(x_{(1)})x_{(2)}=\lambda_{r}(x)1_{H}$ for each $x\in H$. As a consequence of uniqueness of integrals/cointegrals, there are unique group-likes $\boldsymbol{\alpha}\in G(H^{*}),\boldsymbol{a}\in G(H)$ satisfying

$$x_{(1)}\lambda_{r}(x_{(2)})=\lambda_{r}(x)\boldsymbol{a},\hskip 28.45274pt\Lambda_{l}x=\boldsymbol{\alpha}(x)\Lambda_{l}$$

for all $x\in H$. These are the distinguished group-likes of $H$.

Now, a theorem of Kauffman and Radford states that the Drinfeld double $D(H)$ is ribbon if and only if there are group-likes $\boldsymbol{b}\in G(H),\boldsymbol{\beta}\in G(H^{*})$ such that $\boldsymbol{b}^{2}=\boldsymbol{a},\boldsymbol{\beta}^{2}=\boldsymbol{\alpha}$ and

$$S^{2}=\text{ad}_{\boldsymbol{\beta}^{-1}}\circ\text{ad}_{\boldsymbol{b}}.$$

In such a case, the ribbon element is $v=u^{-1}(\boldsymbol{\beta}\otimes\boldsymbol{b})$ where $u$ is the Drinfeld element of $D(H)$.

We can now give sufficient conditions for $\underline{D(H)}$ to be ribbon (or rather, a restriction to a certain subgroup $G\subset\mathrm{Aut}(H)$). Let’s suppose that $H$ satisfies the above condition making $D(H)$ ribbon. Let $G\subset\mathrm{Aut}(H)$ be a subgroup fixing such $\boldsymbol{b},\boldsymbol{\beta}$. Let $r_{H}:\mathrm{Aut}(H)\to\mathbb{K}^{\times}$ be the homomorphism characterized by $\alpha(\Lambda_{l})=r_{H}(\alpha)\Lambda_{l}$ (this is defined by uniqueness of cointegrals). Then, as shown in [23, Prop. 2.4], $\underline{D(H)}|_{G}$ is $G$-ribbon if and only if the homomorphism $r_{H}|_{G}:G\to\mathbb{K}^{\times}$ has a square root. In such a case, the $G$-ribbon element is given by

$$v_{\alpha}=\sqrt{r_{H}(\alpha)}^{-1}(\text{id}_{H^{*}}\otimes\alpha)(v)$$

for any $\alpha\in G$, where $v$ is the ribbon element of $D(H)$. The $G$-pivot is given by

$$\boldsymbol{g}_{\alpha}=\sqrt{r_{H}(\alpha)}^{-1}(\boldsymbol{\beta}\otimes\boldsymbol{b}).$$

The square root condition is immediately satisfied if $G=\text{Ker}\,(r_{H})=\mathrm{Aut}^{0}(H)$.

Suppose that $G$ is an abelian group with an homomorphism $\phi:G\to\mathrm{Aut}(H)$ and consider $\underline{D(H)}|_{G}$ as defined above. Then $G$ acts on each $D(H)_{\alpha}$ by $\beta\mapsto\varphi_{\phi(\beta)}|_{D(H)_{\alpha}}$ and setting $A_{\alpha}=\mathbb{K}[G]\ltimes D(H)_{\alpha}$ we get another Hopf $G$-coalgebra $\underline{A}=\{A_{\alpha}\}_{\alpha\in G}$. The Hopf structure is extended to $\underline{A}$ by declaring the elements of $G$ to be group-likes. In what follows we denote $\beta\in G$ by $\varphi_{\beta}$ when considered as an element of $A_{\alpha}$ so that $\varphi_{\beta}x=\varphi_{\phi(\beta)}(x)\varphi_{\beta}$ in $A_{\alpha}$ for each $x\in D(H)_{\alpha}$. It is easy to see that

$$R_{\alpha,\beta}^{A}=(1\otimes\varphi_{\alpha})R_{\alpha,\beta}\in A_{\alpha}\otimes A_{\beta}$$

is an $R$-matrix for $\underline{A}$ in (almost) the usual sense, that is, it satisfies

$$R^{A}_{\alpha,\beta}\cdot\Delta^{A}_{\alpha,\beta}(x)=\Delta^{A,op}_{\beta,\alpha}(x)\cdot R^{A}_{\alpha,\beta}.$$

In other words, the category $\mathcal{C}=\coprod_{\alpha\in G}\text{Rep}(A_{\alpha})$ is a braided category in the usual sense. It is not difficult to see that the Drinfeld element of $\underline{A}$ is determined by $u^{A}_{\alpha}=u_{\alpha}\varphi_{\alpha}^{-1}$ where $u=\{u_{\alpha}\}$ is the Drinfeld element of $\underline{D(H)}$. If $v=\{v_{\alpha}\}$ is a ribbon element in $\underline{D(H)}$, then $\underline{A}$ is ribbon with $v^{A}_{\alpha}=v_{\alpha}\varphi_{\alpha}$. It follows that the pivot of $\underline{A}$ is the same as in $\underline{D(H)}$.

Recall that a super vector space is a vector space $V$ with a mod 2 grading, that is, a decomposition $V=V_{0}\oplus V_{1}$. The category of super vector spaces is symmetric with $\tau_{V,W}(v\otimes w)=(-1)^{|v||w|}w\otimes v$ where $|v|,|w|$ denotes the mod 2 degree of homogeneous elements $v\in V,w\in W$. A super Hopf algebra is a Hopf algebra $(H,m,1,\Delta,\epsilon,S)$ in the category of super vector spaces. This amounts to the same axioms as for a Hopf algebra, except that the coproduct $\Delta$ satisfies

$$\Delta\circ m=(m\otimes m)\circ(\text{id}\otimes\tau_{H,H}\otimes\text{id})\circ(\Delta\otimes\Delta).$$

More generally, we can talk about super Hopf $G$-coalgebras, now it is $\Delta_{\alpha,\beta}$ that satisfies the above property (with $\tau_{H_{\beta},H_{\alpha}}$ in place of $\tau_{H,H}$). In the super case, $\mathrm{Aut}(H)$ denotes the automorphisms preserving the mod 2 degree. The twisted Drinfeld double of a super Hopf algebra $H$ is defined as before, but with some additional signs:

	$\displaystyle(p\otimes a)\cdot_{\alpha}(q\otimes b)$	$\displaystyle=(-1)^{|a_{(1)}|+|q_{(1)}|+|a_{(2)}||q_{(2)}|+|a_{(1)}||q_{(2)}|+|a_{(2)}||q_{(3)}|}$
		$\displaystyle\langle a_{(1)},q_{(3)}\rangle\langle S^{-1}\alpha^{-1}(a_{(3)}),q_{(1)}\rangle p\cdot q_{(2)}\otimes a_{(2)}\cdot b.$