2.1 Hopf algebras and their module categories

Let H be a Hopf algebra over $\mathbb {k}$ , by which we mean a bialgebra with product $m_H$ , unit $1_H\in H$ , coproduct $\Delta _H: H\to H\otimes H$ , and counit $\varepsilon _H\colon H\to \mathbb {k}$ , that has an invertible antipode $S_H\colon H\to H$ . We use adapted Sweedler’s notation, with suppressed summation,

$$ \begin{align*}\Delta_H(h)=h_{(1)}\otimes h_{(2)}\in H\otimes H,\end{align*} $$

to denote the coproduct evaluated at $h\in H$ . We use the standard convention to denote

$$ \begin{align*}(\Delta_H\otimes {\operatorname{id}})\Delta_H(h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)}=({\operatorname{id}}\otimes\Delta_H)\Delta_H(h),\end{align*} $$

which is justified by coassociativity of $\Delta _H$ . A morphism of Hopf algebras $\varphi \colon K \to H$ is required to be a morphism of both algebras and coalgebras that commutes with the antipodes, that is, $\varphi S_K=S_H\varphi $ . We denote the $\mathbb {k}$ -dual of a finite-dimensional Hopf algebra H by $H^*$ .

We will consider the monoidal category $H\text {-}\mathbf {Mod}$ of left H-modules. The monoidal subcategory of finite-dimensional H-modules is denoted by $H\text {-}\mathbf {mod}$ . On the tensor product $V\otimes W=V\otimes _\mathbb {k} W$ of two H-modules V and W, H acts via $h\cdot (v\otimes w)=h_{(1)}\cdot v\otimes h_{(2)}\cdot w$ and the tensor unit is given by $\mathbb {k}$ with H-action via $h\cdot 1=\varepsilon (h)$ , for $h\in H$ .

2.2 Induction and restriction functors

Given a morphism of Hopf algebras $\varphi \colon K\to H$ , consider the restriction functor

$$ \begin{align*}G=\operatorname{Res}_\varphi\colon H\text{-}\mathbf{Mod}\to K\text{-}\mathbf{Mod}, \quad W\mapsto \left.W\right|{}_K. \end{align*} $$

Restriction has a left and a right adjoint functor given by

(2.1) $$ \begin{align} L=\operatorname{Ind}_\varphi(V):=H\otimes_K V, \qquad R=\operatorname{CoInd}_\varphi(V)=\operatorname{Hom}_{K}(H,V), \end{align} $$

for a K-module V. The left H-module structure on $\operatorname {CoInd}_\varphi (V)$ is

(2.2) $$ \begin{align} h\cdot f(g)=f(gh), \qquad f\in \operatorname{Hom}_{K}(H,V), ~~ h,g\in H. \end{align} $$

In this article, we will focus on the case of an inclusion $\iota \colon K\hookrightarrow H$ of Hopf algebras and denote the induction and restriction functors by

$$ \begin{align*}\operatorname{Ind}_K^H=\operatorname{Ind}_\iota\quad\dashv\quad \operatorname{Res}_K^H=\operatorname{Res}_\iota \quad\dashv\quad \operatorname{CoInd}_K^H=\operatorname{CoInd}_\iota.\end{align*} $$

2.3 Lax, oplax, and Frobenius monoidal functors

We will briefly recall the definitions of lax, oplax, strong, and Frobenius monoidal functor between monoidal categories $\mathcal {C}$ and $\mathcal {D}$ . Let $\mathcal {C}, \mathcal {D}$ denote monoidal categories which, by Mac Lane’s coherence theorem [Reference MacLane34], we can regard as strict monoidal for simplicity. A lax monoidal functor is a functor $R: \mathcal {D} \rightarrow \mathcal {C}$ together with morphisms

(2.3) $$ \begin{align} \operatorname{lax}_{X,Y}\colon RX \otimes RY \longrightarrow R(X \otimes Y), \end{align} $$

natural in $X,Y \in \mathcal {D}$ and a morphism such that the associativity constraint

(2.4) $$ \begin{align} \operatorname{lax}_{X \otimes Y, Z} (\operatorname{lax}_{X, Y} \otimes {\operatorname{id}})= \operatorname{lax}_{X, Y \otimes Z}({\operatorname{id}} \otimes \operatorname{lax}_{Y,Z}) \end{align} $$

holds for $X,Y,Z \in \mathcal {D}$ as well as the unitality constraints

(2.5)

Dually, one defines an oplax monoidal functor as a functor $L: \mathcal {D} \rightarrow \mathcal {C}$ with morphisms

(2.6) $$ \begin{align} \operatorname{oplax}_{X,Y}\colon L(X \otimes Y) \longrightarrow LX \otimes LY, \end{align} $$

natural in $X,Y$ and a morphism satisfying the conditions obtained from Equations Equations (2.4) and (2.5) by duality.

A strong monoidal functor can now be defined as a lax monoidal functor where $\operatorname {lax}_{X,Y}$ and $\operatorname {lax}_0$ are isomorphisms.

The following result is due to [Reference Kelly28] (see also [Reference Flake, Laugwitz and Posur20, Proposition 2.2] for details).

This lemma can be applied to the Hopf algebra case.

Example 2.2 For a morphism of Hopf algebras $\varphi \colon K\to H$ , we note that $G=\operatorname {Res}_\varphi $ is a strong monoidal functor and will identify $\left. (W\otimes X)\right |{}_K$ and $\left .W\right |{}_K\otimes \left. X\right |{}_K$ for H-modules $W,X$ to simplify notation. Then $R=\operatorname {CoInd}_\varphi $ and $L=\operatorname {Ind}_\varphi $ have the following lax and oplax monoidal structures, for K-modules V and U:

(2.7) $$ \begin{align} \operatorname{lax}^R_{V,U}\colon R(V)\otimes R(U)\longrightarrow R(V\otimes U),\nonumber \\ f\otimes g\mapsto \big(h\mapsto f(h_{(1)})\otimes g(h_{(2)})\big), \end{align} $$

(2.8) $$ \begin{align} \operatorname{oplax}^{L}_{V,U}\colon L(V\otimes U)\to L(V)\otimes L(U),\nonumber \\ h\otimes_K (v\otimes u)\mapsto (h_{(1)}\otimes_K v)\otimes (h_{(2)}\otimes_K u). \end{align} $$

Finally, a Frobenius monoidal functor (see, e.g., [Reference Day and Pastro10]) $F\colon \mathcal {D}\to \mathcal {C}$ is a functor equipped with a lax monoidal structure $(\operatorname {lax},\operatorname {lax}_0)$ , and an oplax monoidal structure $(\operatorname {oplax}, \operatorname {oplax}_0)$ satisfying the following two compatibility conditions, for any objects $X,Y,Z$ of $\mathcal {D}$ :

(2.9) $$ \begin{align} (\operatorname{lax}_{X,Y}\otimes {\operatorname{id}}_{F(Z)})({\operatorname{id}}_{F(X)}\otimes \operatorname{oplax}_{Y,Z}) & = \operatorname{oplax}_{X\otimes Y,Z}\operatorname{lax}_{X,Y\otimes Z} \end{align} $$

(2.10) $$ \begin{align} ({\operatorname{id}}_{F(X)}\otimes \operatorname{lax}_{Y,Z})(\operatorname{oplax}_{X,Y}\otimes {\operatorname{id}}_{F(Z)}) & = \operatorname{oplax}_{X,Y\otimes Z}\operatorname{lax}_{X\otimes Y,Z}. \end{align} $$

Recall that $\mathcal {C}$ is braided monoidal if it comes with natural isomorphisms

$$ \begin{align*}\Psi^{\mathcal{C}}_{X,Y}\colon X\otimes Y\to Y\otimes X,\end{align*} $$

called the braiding which satisfy the hexagon axioms (see, e.g., [Reference Etingof, Gelaki, Nikshych and Ostrik13, Section 8.1]). For $\mathcal {C},\mathcal {D}$ braided monoidal categories, say that a lax (or, oplax) monoidal functor $F\colon \mathcal {D}\to \mathcal {C}$ is lax braided (resp. oplax braided) if the conditions

(2.11) $$ \begin{align} \operatorname{lax}_{Y,X}\Psi^{\mathcal{C}}_{F(X),F(Y)}&=F(\Psi^{\mathcal{D}}_{X,Y})\operatorname{lax}_{X,Y}, \end{align} $$

(2.12) $$ \begin{align} \Psi^{\mathcal{C}}_{F(X),F(Y)}\operatorname{oplax}_{X,Y}&=\operatorname{oplax}_{Y,X}F(\Psi^{\mathcal{D}}_{X,Y}) \end{align} $$

hold for all objects $X,Y\in \mathcal {D}$ . A Frobenius monoidal functor between braided monoidal categories is called braided if it is both lax braided and oplax braided.

2.4 Algebraic structures in monoidal categories

Let $\mathcal {C}$ be a monoidal category. We can define an algebra in $\mathcal {C}$ as an object A in $\mathcal {C}$ together with morphisms $m_A\colon A\otimes A\to A$ , in $\mathcal {C}$ which satisfy the associativity and unitality axioms internally to $\mathcal {C}$ . Algebras in $\mathcal {C}$ form a category denoted by $\mathbf {Alg}(\mathcal {C})$ . Dually, we define coalgebras in $\mathcal {C}$ which form a category $\mathbf {CoAlg}(\mathcal {C})$ . One checks that a lax/oplax monoidal functor $F\colon \mathcal {D}\to \mathcal {C}$ induces a functor $\mathbf {Alg}(\mathcal {D})\to \mathbf {Alg}(\mathcal {C})$ /a functor $\mathbf {CoAlg}(\mathcal {D})\to \mathbf {CoAlg}(\mathcal {C})$ . The (co)multiplication and (co)units on the images of these functors are defined by

(2.13) $$ \begin{align} m_{F(A)}=F(m_A)\operatorname{lax}_{A,A}, \qquad u_{F(A)}=F(u_A)\operatorname{lax}_{0}, \end{align} $$

(2.14) $$ \begin{align} \Delta_{F(C)}=\operatorname{oplax}_{C,C}F(\Delta_C), \qquad \varepsilon_{F(C)}=\operatorname{oplax}_{0} F(\varepsilon_C), \end{align} $$

where $(A,m_A,u_A)$ is an algebra and $(C,\Delta _A,\varepsilon _A)$ is a coalgebra in $\mathcal {D}$ .

Recall that a Frobenius algebra in a monoidal category $\mathcal {C}$ is an object A in $\mathcal {C}$ which is an algebra $(A,m_A,u_A)$ and a coalgebra $(A,\Delta _A,\varepsilon _A)$ such that the compatibility conditions

(2.15) $$ \begin{align} (m_A\otimes {\operatorname{id}}_A)({\operatorname{id}}_A\otimes \Delta_A)=\Delta_A m_A=({\operatorname{id}}_A\otimes m_A)(\Delta_A\otimes {\operatorname{id}}_A) \end{align} $$

hold. A morphism of Frobenius algebras is required to commute with both the algebra and coalgebra structures. We denote the category of Frobenius algebras in $\mathcal {C}$ by $\mathbf {Frob}(\mathcal {C})$ . One checks that a Frobenius monoidal functor $F\colon \mathcal {D}\to \mathcal {C}$ induces a functor $\mathbf {Frob}(\mathcal {D})\to \mathbf {Frob}(\mathcal {C})$ that combines the induced functors on algebras and coalgebras defined in Equations (2.13) and (2.14).

Recall that an algebra A in a braided monoidal category $\mathcal {C}$ with braiding $\Psi $ is commutative if $m_A\Psi _{A,A}=m_A$ and a coalgebra C is cocommutative if $\Psi _{C,C}\Delta _C=\Delta _C$ . A commutative Frobenius algebra is a Frobenius algebra which is commutative as an algebra and note that this implies cocommutativity as a coalgebra.

One checks that a braided lax (resp. braided oplax) monoidal functor preserves commutative algebras (resp. cocommutative coalgebras). Hence, a braided Frobenius monoidal functor preserves commutative Frobenius algebras.

2.5 The Drinfeld center, Yetter–Drinfeld modules, and the Drinfeld double

For a monoidal category $\mathcal {C}$ , the Drinfeld center $\mathcal {Z}(\mathcal {C})$ is a braided monoidal category (see [Reference Joyal and Street25, Definition 3] and [Reference Majid35, Example 3.4]). In this article, we will describe the Drinfeld center $\mathcal {Z}(H\text {-}\mathbf {Mod})$ by the equivalent category ${\vphantom {\mathbf {YD}}}^{H}_{H}{\mathbf {YD}}$ of Yetter–Drinfeld modules (or YD modules for short) over a Hopf algebra H (see [Reference Majid36, Example 9.1.8], [Reference Yetter52]). This category ${\vphantom {\mathbf {YD}}}^{H}_{H}{\mathbf {YD}}$ consists of objects V which are both left H-modules and left H-comodules with the coaction

$$ \begin{align*}\delta\colon V\to H\otimes V, \qquad v\mapsto v^{(-1)}\otimes v^{(0)},\end{align*} $$

satisfying the Yetter–Drinfeld condition

(2.16) $$ \begin{align} \delta(h v)=h_{(1)}v^{(-1)}S(h_{(3)})\otimes h_{(2)}v^{(0)}, \end{align} $$

for $h\in H, v\in V$ . The tensor product $V\otimes W$ of two YD modules is simply the tensor product of the underlying H-modules and H-comodules via $\Delta _H$ and $m_H$ , respectively. The braiding $\Psi _{V,W}\colon V\otimes W\to W\otimes V$ of two YD modules is given by

(2.17) $$ \begin{align} \Psi_{V,W} \colon V\otimes W\to W\otimes V, \qquad v\otimes w \mapsto (v^{(-1)}\cdot w)\otimes v^{(0)}, \end{align} $$

for any $v\in V,w\in W$ . It shall be noted that, when H is finite-dimensional, the category of Yetter–Drinfeld modules is equivalent to modules over a Hopf algebra, called the Drinfeld double of H (see, e.g., [Reference Etingof, Gelaki, Nikshych and Ostrik13, Section 7.14]). In this article, we will work with the YD module description of the center which is more convenient, especially when working with infinite-dimensional Hopf algebras.

3.1 Isomorphisms of induction and coinduction functors

Let $K\subset H$ be an inclusion of Hopf algebras (from now on called an extension of Hopf algebras).

We first recall when induction and coinduction along $K\subset H$ are isomorphic based on maps called Frobenius morphisms (see [Reference Kassel27, Reference Menini and Năstăsescu38, Reference Nakayama and Tsuzuku41]).

We adopt the following terminology from [Reference Fischman, Montgomery and Schneider18, Reference Schneider45].

If H is finite projective over K, Frobenius morphisms are in bijection with natural isomorphisms $\theta \colon \operatorname {Ind}_K^H\stackrel {\sim }{\longrightarrow } \operatorname {CoInd}_K^H$ . Indeed, a Frobenius morphism $\operatorname {tr}\colon H\to K$ gives a natural isomorphism $\theta \colon \operatorname {Ind}_K^H\to \operatorname {CoInd}_K^H$ with component maps

(3.1) $$ \begin{align} \theta_V(h\otimes v)=\big(g\mapsto \operatorname{tr}(gh)v \big). \end{align} $$

Not all extensions of Hopf algebras admit a Frobenius morphism, for instance, the inclusion of the group algebra $\mathbb {k} \mathsf {C}_\ell $ of a cyclic group of order $\ell>2$ into the Taft algebra $T_\ell (q)$ , associated with an $\ell $ -th root of unity, does not, as shown in [Reference Flake, Laugwitz and Posur21, Example 5.7].

For Frobenius extensions of Hopf algebras, it is useful to fix the following dual sets of generators for H (see [Reference Flake, Laugwitz and Posur21, Lemma 5.8] and cf. [Reference Fischman, Montgomery and Schneider18, 1.3. Proposition]).

Corollary 3.5 If $K\subset H$ is a Frobenius extension with Frobenius morphism $\operatorname {tr}$ , then there exist $n\geq 1$ and generators $h_1,\ldots ,h_n$ of H as a left K-module and $\delta _1,\ldots , \delta _n$ of H as a right K-module such that, for all $h\in H$ ,

(3.2) $$ \begin{align} \sum_{i=1}^n \delta_i\otimes_K h_i h &= \sum_{i=1}^n h\delta_i\otimes_K h_i, \end{align} $$

(3.3) $$ \begin{align} h =\sum_{i=1}^n \operatorname{tr}(h\delta_i)h_i&= \sum_{i=1}^n \delta_i\operatorname{tr}(h_i h). \end{align} $$

Moreover, the inverse of $\theta _V$ from Equation (3.1) is given by

(3.4) $$ \begin{align} \theta_V^{-1}\colon \operatorname{CoInd}_K^H(V)\to \operatorname{Ind}_K^H(V), \quad f\mapsto \sum_{i=1}^n\delta_i\otimes_K f(h_i). \end{align} $$

For a Frobenius extension $K\subset H$ , we can give $F:=\operatorname {Ind}_K^H$ the structure of a left and right adjoint to $G:=\operatorname {Res}_K^H$ . The units and counits of these adjunctions are

(3.5) $$ \begin{align} \operatorname{unit}^{G\dashv F}_W&\colon W\to FG(W), &w&\mapsto \sum_{i=1}^r \delta_i\otimes_K h_iw, \end{align} $$

(3.6) $$ \begin{align} \operatorname{counit}^{G\dashv F}_V&\colon GF(V)\to V, & h\otimes_K v& \mapsto \operatorname{tr}(h)v, \end{align} $$

(3.7) $$ \begin{align} \operatorname{unit}^{F\dashv G}_V&\colon V\to GF(V), &v&\mapsto 1\otimes_K v, \end{align} $$

(3.8) $$ \begin{align} \operatorname{counit}^{F\dashv G}_W&\colon FG(W)\to W, &h\otimes_K w&\mapsto hw, \end{align} $$

for any K-module V and any H-module W. Moreover, the functor F is lax and oplax monoidal with the lax and oplax monoidal structures

(3.9) $$ \begin{align} \operatorname{lax}_{V,U}\colon F(V)\otimes F(U)&\to F(V\otimes U),\nonumber \\ \quad (h\otimes_K v)\otimes (g\otimes_K u)&\mapsto \sum_i \delta_i\otimes_K (\operatorname{tr}((h_i)_{(1)}h)v\otimes \operatorname{tr}((h_i)_{(2)}g)u), \end{align} $$

(3.10) $$ \begin{align} \operatorname{oplax}_{V,U}\colon F(V\otimes U)&\to F(V)\otimes F(U),\nonumber \\ \quad h\otimes_K (v\otimes u)&\mapsto (h_{(1)}\otimes_K v)\otimes (h_{(2)}\otimes_K u), \end{align} $$

(3.11)

(3.12)

The following natural isomorphisms, called projection formula morphisms, will be important later. The general categorical aspects of the theory behind these morphisms are described in general in [Reference Flake, Laugwitz and Posur20] (see also [Reference Bruguières, Lack and Virelizier4, Reference Fausk, Hu and May17, Reference Flake, Laugwitz and Posur21]).

Definition 3.6 Given a Frobenius extension $K\subset H$ , we define the left projection formula morphisms to be the following natural transformations:

(3.13) $$ \begin{align} \mathrm{lproj}_{{W},{V}}^{F, G\dashv F}\colon W\otimes F(V)&\to F(G(W)\otimes V),\nonumber \\ w\otimes (h\otimes_K v)&\mapsto \sum_{i=1}^r \delta_i\otimes_K ((h_i)_{(1)}w\otimes \operatorname{tr}((h_i)_{(2)}h)v), \end{align} $$

(3.14) $$ \begin{align} \mathrm{lproj}_{{W},{V}}^{F, F\dashv G}\colon F(G(W)\otimes V)&\to W\otimes F(V),\nonumber \\h\otimes_K (w\otimes v)&\mapsto h_{(1)}w\otimes (h_{(2)}\otimes_K v), \end{align} $$

and the right projection formula morphisms to be

(3.15) $$ \begin{align} \mathrm{rproj}_{{V},{W}}^{F, G\dashv F}\colon F(V)\otimes W&\to F(V\otimes G(W)),\nonumber \\ (h\otimes_K v)\otimes w&\mapsto \sum_{i=1}^r \delta_i\otimes_K (\operatorname{tr}((h_i)_{(1)}h)v\otimes (h_i)_{(2)}w), \end{align} $$

(3.16) $$ \begin{align} \mathrm{rproj}_{{V},{W}}^{F, F\dashv G}\colon F(V\otimes G(W))&\to F(V)\otimes W,\nonumber \\ h\otimes_K (v \otimes w) &\mapsto (h_{(1)}\otimes_K v)\otimes h_{(2)}w. \end{align} $$

We note that the pair $\mathrm {lproj}^{F, G\dashv F}$ and $\mathrm {rproj}^{F, G\dashv F}$ comes from the adjunction $G\dashv F$ , where F is right adjoint to G while the other pair comes from the adjunction $F\dashv G$ , where F is the left adjoint. This is indicated in the superscripts.

Proof This follows from the results of [Reference Flake, Laugwitz and Posur20, Section 7]. The invertibility of the projection formula morphisms can also be checked directly using the inverse antipode. For instance, one checks that

(3.17) $$ \begin{align} (\mathrm{lproj}^{F,F\dashv G}_{{W},{V}})^{-1}\colon W\otimes F(V)&\to F(G(W)\otimes V),\nonumber \\ w\otimes (h\otimes_K v)&\mapsto h_{(2)}\otimes_K( S^{-1}(h_{(1)})w\otimes v), \end{align} $$

(3.18) $$ \begin{align} (\mathrm{rproj}^{F,F\dashv G}_{{V},{W}})^{-1}\colon F(V)\otimes W&\to F(V\otimes G(W)),\nonumber \\ (h\otimes_K v)\otimes w &\mapsto h_{(1)}\otimes_K( v\otimes S(h_{(2)})w). \end{align} $$

Formulas for the inverses for the other two morphisms can be obtained using Corollary 3.5 but we will not require them.

3.2 The augmented algebra associated with an extension of Hopf algebras

Let $K\subset H$ be an extension of Hopf algebras. In this section, we introduce the associated augmented algebra $\overline {H}^*$ following [Reference Fischman, Montgomery and Schneider18, Reference Schneider45]. To do this, we first recall the definition of integrals of augmented algebras.

It follows from the definition that the space of right integrals forms a left A-submodule of A. In particular, if the space of right integrals is one-dimensional, this gives rise to an algebra map $\alpha _A\colon A\to \mathbb {k}$ determined by the left multiplication

(3.19) $$ \begin{align} a \Lambda = \alpha_A(a)\Lambda, \qquad \forall a\in A, \end{align} $$

for any nonzero right integral $\Lambda $ of A. Note that $\alpha _A$ is independent of the choice of $\Lambda $ . The following lemma shows that it detects unimodularity.

Proof To prove Part (i), note that $K^+\subset K$ is a complement to the space $\mathbb {k} 1_K$ as a $\mathbb {k}$ -vector space. Thus, any $k\in K$ decomposes as $k=k'+\varepsilon (k)1$ with $k'\in K^+$ and hence

$$ \begin{align*}\overline{hk}=\overline{hk'}+\varepsilon(k)\overline{h}=\varepsilon(k)\overline{h}.\end{align*} $$

To prove Part (ii), assume H is generated by a finite set of elements $\{h_i\}_i$ over K. Then for every $h\in H$ , there are elements $\{k_i\}_i$ in K such that $h=\sum _i h_i k_i$ , and

$$ \begin{align*}\overline{h} = \sum_i \overline{h_i k_i} = \sum_i \varepsilon(k_i) \overline{h_i},\end{align*} $$

by Part (i). Hence, $H/HK^+$ is finite-dimensional.

3.3 Frobenius extensions of right integral type

In this section, we recall extensions of Hopf algebras of right integral type and their main properties (see [Reference Fischman, Montgomery and Schneider18, 1.10. Definition]).

We recall some results from [Reference Fischman, Montgomery and Schneider18].

The following argument is taken from [Reference Fischman, Montgomery and Schneider18, 1.8. Corollary].

Proof The immediate implications are (b) $\Rightarrow $ (c) $\Rightarrow $ (a), so it remains to prove (a) $\Rightarrow $ (b).

Assume $K\subset H$ is a $\beta $ -Frobenius extension and a Frobenius extension. Then we have an isomorphism $\phi \colon H\to H^{\beta }$ of H-K-bimodules, where the right K-action on $H^{\beta }$ is twisted by $\beta $ . Set $u:=\phi (1)$ . We compute

$$ \begin{align*}u\phi^{-1}(1) = \phi^{-1}(\phi(1)) = 1, \end{align*} $$

in particular, $\varepsilon (u)\neq 0$ . We also compute for all $k\in K,$

$$ \begin{align*}k u = \phi(k) = u \beta(k) , \end{align*} $$

which implies $\varepsilon \beta =\varepsilon $ on K. Finally, for all $k\in K$ ,

$$ \begin{align*}\varepsilon\beta(k) = \chi(k_{(1)}) \varepsilon(k_{(2)}) = \chi(k). \end{align*} $$

We have shown $\chi =\varepsilon \beta =\varepsilon $ .

We recall the following key result obtained from [Reference Schneider45, 3.3. Theorem].

A condition on H that implies that any extension $K\subset H$ is of right integral type is cocommutativity of the coradical $H_0$ of H (see, e.g., [Reference Montgomery39]).

A Hopf algebra is pointed if every simple subcoalgebra is one-dimensional.

3.4 Normal bases and extensions of right integral type

In the following, we recall classes of extensions of Hopf algebras which are of right integral type. The next definition follows [Reference Fischman, Montgomery and Schneider18, 4.1. Definition].

In particular, the existence of a right normal basis for H over K implies that H is free as a right K-module. In particular, H is faithfully flat over K. Moreover, [Reference Fischman, Montgomery and Schneider18, 4.8. Theorem] proves the following result.

Combining this result with [Reference Schneider45, 2.4. Theorem] yields the following result.

3.5 Criteria for Frobenius extensions of Hopf algebras

We include some sufficient criteria to obtain Frobenius extensions of Hopf algebras. The following proposition describes the case where the Hopf algebras are finite-dimensional.

Remark 3.28 If $K\subset H$ is an inclusion of finite-dimensional Hopf algebras, the relative modular function is given by

(3.22) $$ \begin{align}\chi=(\alpha_H)|_K * \alpha_K^{-1},\end{align} $$

for the convolution inverse $\alpha _K^{-1}=\alpha _K S_K$ of the distinguished grouplike of $K^*$ [Reference Fischman, Montgomery and Schneider18, 1.6. Definition], where $S_K$ is the antipode of K.

To provide another sufficient criterion for Frobenius extensions, recall the concept of a normal Hopf subalgebra [Reference Montgomery39, 3.4.1 Definition], that is, a Hopf subalgebra $K\subset H$ such that K is an H-H-subbimodule with respect to the adjoint action. In this case, $HK^+=K^+H$ and hence $HK^+$ is a Hopf ideal and $\overline {H}$ a Hopf algebra. For example, if K is contained in the center $Z(H)$ of H, then K is a normal Hopf subalgebra.

Proof By [Reference Fischman, Montgomery and Schneider18, 1.11. Example], since K is a normal Hopf subalgebra such that $\overline {H}$ is finite-dimensional by Lemma 3.12(ii), it follows that $K\subset H$ is an extension of right integral type. As H is faithfully flat over K, by Theorem 3.18, we know that H is finite projective over K and that, to show that $K\subset H$ is a Frobenius extension, it remains to check that $\chi =\varepsilon _K$ . Since $\overline {H}=H/HK^+$ is a finite-dimensional Hopf algebra, $\overline {H}^*$ has a unique nonzero right integral $\lambda $ up to scalars. Moreover, using that $\overline {hk}=\overline {h}\varepsilon _K(k)$ , for $k\in K$ , we compute

$$ \begin{align*} \lambda(\overline{k} \overline{h}) =\lambda(\overline{h_{(1)}S(h_{(2)}) k h_{(3)}}) &=\lambda(\overline{h_{(1)}})\varepsilon_K(S(h_{(2)})k h_{(3)})\\ &=\lambda(\overline{h})\varepsilon_K(k) =\chi(k) \lambda(\overline{h}). \end{align*} $$

This shows that $\chi =\varepsilon _K$ , and $K\subset H$ is a Frobenius extension by Theorem 3.18.

The statement of Corollary 3.30 simplifies when H is finite-dimensional. In this case, any normal Hopf subalgebra $K\subset H$ gives a Frobenius extension since H is free over K.

4.1 Conditions for induction functors to be Frobenius monoidal

We will now prove that the induction functor $\operatorname {Ind}_K^H$ , for a Frobenius extension $K\subset H$ of Hopf algebras of right integral type, is a Frobenius monoidal functor together with the lax and oplax monoidal structures from Equations (3.9) and (3.10).

We equip H with the regular right H-coaction. Then, K has a right H-coaction given by extending the regular K-coaction via the inclusion $K\subset H$ . We recall the following results from [Reference Flake, Laugwitz and Posur21, Proposition 5.13 and Corollary 5.14].

Theorem 4.2 Let $K\subset H$ be a Frobenius extension of Hopf algebras of right integral type (see Definition 3.13). Then

$$ \begin{align*}\operatorname{Ind}_K^H\colon K\text{-}\mathbf{Mod}\to H\text{-}\mathbf{Mod}\end{align*} $$

is a Frobenius monoidal functor.

This functor restricts to finite-dimensional modules, $\operatorname {Ind}_K^H\colon K\text {-}\mathbf {mod}\to H\text {-}\mathbf {mod}$ .

Proof Assume that $K\subset H$ is a Frobenius extension. Then H is finite projective over K. Now, [Reference Schneider45, 1.8. Corollary] shows that H is faithfully flat over K. Thus, Theorem 3.18 implies that $K\subset H$ admits a Frobenius morphism of the form

$$ \begin{align*}\operatorname{tr}(h)=\lambda(\overline{h_{(1)}}) h_{(2)}.\end{align*} $$

Such a Frobenius morphism is clearly a morphism of right H-comodules by coassociatity of the coproduct of $\Delta (h)=h_{(1)}\otimes h_{(2)}$ of H. Thus, the claim follows from Proposition 4.1. As H is, in particular, finite over K, it is clear that induction restricts to the subcategories of finite-dimensional modules.

The following corollaries of Theorem 4.2 present specific assumptions that ensure that $\operatorname {Ind}_K^H$ is a Frobenius monoidal functor based on the results of Sections 3.3 and 3.4.

4.2 Frobenius monoidal functors of Yetter–Drinfeld modules

In this section, we will introduce a class of Frobenius extensions of Hopf algebras $K\subset H$ , called central Frobenius extensions, for which the functor $\operatorname {Ind}_K^H$ extends to a braided Frobenius monoidal functor on categories of YD modules denoted by $\mathcal {Z}(\operatorname {Ind}_K^H)$ .

The condition that $\operatorname {tr}\colon H\to K$ is a morphism of H-H-bicomodules amounts to

(4.1) $$ \begin{align} h_{(1)}\otimes \operatorname{tr}(h_{(2)})\otimes h_{(3)}=\operatorname{tr}(h)_{(1)}\otimes \operatorname{tr}(h)_{(2)}\otimes \operatorname{tr}(h)_{(3)}\quad \in \quad H\otimes K\otimes H, \end{align} $$

for all $h\in H$ , where we identify $K\otimes K\otimes K$ as a subspace of $H\otimes K\otimes H$ . Note that if a Frobenius morphism $\operatorname {tr}$ with this property exists, then it is unique up to a nonzero scalar by Proposition 4.1(3).

The concept of a central Frobenius extension is justified by the following result.

Theorem 4.9 [Reference Flake, Laugwitz and Posur21, Proposition 5.19 and Corollary 5.20]

A Frobenius extension $K\subset H$ is a central Frobenius extension if and only if the natural transformations $\mathrm {lproj}^{F, F\dashv G}$ and $\mathrm {lproj}^{F, G\dashv F}$ as well as $\mathrm {rproj}^{F, F\dashv G}$ and $\mathrm {rproj}^{F, G\dashv F}$ from Definition 3.6 are mutual inverses. In this case,

(4.2) $$ \begin{align} \mathcal{Z}(\operatorname{Ind}_K^H)\colon {\vphantom{\mathbf{YD}}}^{K}_{K}{\mathbf{YD}}\to {\vphantom{\mathbf{YD}}}^{H}_{H}{\mathbf{YD}},\quad (V,\delta)\mapsto (\operatorname{Ind}_K^H(V),\delta^{\operatorname{Ind}_K^H(V)}),\qquad \text{where}\nonumber \\ \delta^{\operatorname{Ind}_K^H(V)}(h\otimes v)=h_{(1)}v^{(-1)}S(h_{(3)})\otimes h_{(2)}\otimes v^{(0)}, \end{align} $$

for $v\in V, h\in H$ , is a braided Frobenius monoidal functor. Its lax and oplax monoidal structure are given in Equations (3.9) and (3.10), that is,for YD modules $(V,\delta ^V)$ , $(W,\delta ^W)$ over K,

(4.3) $$ \begin{align} \operatorname{lax}_{(V,\delta^V),(W,\delta^W)}=\operatorname{lax}_{V,W}, \qquad \operatorname{oplax}_{(V,\delta^V),(W,\delta^W)}=\operatorname{oplax}_{V,W}. \end{align} $$

More generally, we can prove that Frobenius extensions of cocommutative Hopf algebras or quasitriangular Hopf algebras, where the inclusion preserves the R-matrix, are central Frobenius extensions. Recall that $1\otimes 1$ is an R-matrix for a cocommutative Hopf algebra.

We conclude this section with a detailed example of a Frobenius extension of infinite-dimensional Hopf algebras where $\operatorname {tr}$ is only a morphism of right H-comodules but not of left H-comodules so that $\operatorname {Ind}_K^H$ is Frobenius monoidal but Theorem 4.9 does not apply. A finite-dimensional example was given in [Reference Flake, Laugwitz and Posur21, Example 5.25].

Example 4.14 Let $\epsilon $ be a primitive $\ell $ -th root of unity, with $\ell>1$ , and consider the Hopf algebra

$$ \begin{align*}B=\mathbb {k} \langle g,x\rangle/(gx-\epsilon xg),\end{align*} $$

which can be seen as an infinite-dimensional analog of the Taft algebra (which, additionally, quotients out the relations $x^\ell =0$ and $g^\ell =1$ ). The coproduct, counit, and antipode are defined by the following formulas on generators:

$$ \begin{align*} \Delta(g)=g\otimes g, \quad \Delta(x)=x\otimes 1+g\otimes x, \quad \varepsilon(g)=1, \quad \varepsilon(x)=0, \\ S(g)=g^{-1}, \quad S(x)=-g^{-1}x. \end{align*} $$

Inductively, the coproduct is given by

(4.4) $$ \begin{align} \Delta(g^ix^j)=\sum_{a=0}^j \binom{j}{a}_\epsilon \epsilon^{i(a-j)}g^{i+j-a}x^a\otimes g^ix^{j-a}, \end{align} $$

where the $\epsilon $ -binomial coefficients are

$$ \begin{align*}\binom{j}{a}_\epsilon=\frac{[j]_\epsilon!}{[a]_\epsilon![j-a]_\epsilon!},\end{align*} $$

for $[n]_\epsilon =1+\epsilon +\cdots +\epsilon ^{n-1}$ and $[n]_\epsilon !=[n]_\epsilon [n-1]_\epsilon \dots [1]_\epsilon $ . One checks that the subalgebra K generated by $g^{\pm \ell }$ and $x^\ell $ is central. Thus, $K \subset B$ is a Frobenius extension by Corollary 3.30.

A basis for $\overline {B}$ is given by $\left \lbrace \overline {g^ix^j}\mid 0\leq i,j\leq \ell -1\right \rbrace $ and one checks that a right integral is

$$ \begin{align*}\lambda\colon \overline{B}\to \mathbb {k}, \quad \overline{g^ix^j}\mapsto \delta_{i,0}\delta_{j,\ell-1},\end{align*} $$

where $\delta _{i,j}$ denotes the Kronecker delta. Equation (3.21) yields the Frobenius morphism

$$ \begin{align*}\operatorname{tr}\colon B\to K, \quad g^ix^j\mapsto \delta_{i,0}\delta_{j,\ell-1},\end{align*} $$

for all $0\leq i,j\leq \ell -1$ . We compute, using the formula for the coproduct (4.4), that

$$ \begin{align*} (g^ix^j)_{(1)}\otimes \operatorname{tr}((g^ix^j)_{(2)}) =g^{\ell-1}\otimes 1 \neq 1\otimes 1=\operatorname{tr}(g^ix^j)_{(1)}\otimes \operatorname{tr}(g^ix^j)_{(2)}. \end{align*} $$

Hence, $\operatorname {tr}$ is not a morphism of left H-comodules. As $\operatorname {tr}$ is a morphism of right H-comodules, and hence unique up to a scalar by Proposition 4.1(3), there cannot be a Frobenius morphism that is a morphism of H-H-bicomodules. Hence, $K\subset B$ is not a central Frobenius extension.

4.3 Unimodularity and semisimplicity conditions for central Frobenius extensions

We now prove the second main theorem of the article that a Frobenius extension $K\subset H$ of finite-dimensional Hopf algebras is a central Frobenius extension if $K^*$ and $H^*$ are unimodular (Theorem 4.21). As a special case, when H is semisimple and co-semisimple, any Hopf subalgebra $K\subset H$ gives a central Frobenius extension (Corollary 4.24).

We first characterize central Frobenius extensions of Hopf algebras of right integral type.

Proof Assume $K\subset H$ is a central Frobenius extension. Then by Theorem 3.18, $\chi =\varepsilon _K$ and a Frobenius morphism is given by

$$ \begin{align*}\operatorname{tr}(h)=\lambda(\overline{h_{(1)}}) h_{(2)} .\end{align*} $$

It is clearly a morphism of right H-comodules and, hence, unique (up to a nonzero scalar) with this property. As $K\subset H$ is a central Frobenius extension, this Frobenius morphism $\operatorname {tr}$ is also a morphism of left H-comodules, and we have

$$ \begin{align*} h_{(1)}\otimes \lambda(\overline{h_{(2)}}) h_{(3)} =h_{(1)}\otimes \operatorname{tr}(h_{(2)}) &=\operatorname{tr}(h)_{(1)}\otimes \operatorname{tr}(h)_{(2)}\\ &=\operatorname{tr}(h_{(1)})\otimes h_{(2)} =\lambda(\overline{h_{(1)}}) h_{(2)}\otimes h_{(3)}, \end{align*} $$

where the third equality uses that $\operatorname {tr}$ is a morphism of right H-comodules. Applying ${\operatorname {id}}_H\otimes \varepsilon _H$ to this equality, we obtain Equation (4.5) from the counit axioms.

Conversely, assume $\chi =\varepsilon _K$ and Equation (4.5). Then $K\subset H$ is a Frobenius extension by Theorem 3.18 with Frobenius morphism $\operatorname {tr}$ as above and Equation (4.5) implies that

$$ \begin{align*}h_{(1)}\otimes \operatorname{tr}(h_{(2)})=\operatorname{tr}(h_{(1)})\otimes h_{(2)}.\end{align*} $$

Hence, $K\subset H$ is a central Frobenius extension.

Example 4.17 If $H=\mathbb {k} \mathsf {G}$ is a group acting on a Lie algebra $\mathfrak {g}$ by Lie algebra automorphisms and $\mathsf {K}$ is a finite-index subgroup, we can form the semidirect product $\mathbb {k} \mathsf {K} \ltimes U(\mathfrak {g}$ ) (cf. [Reference Montgomery39 Section 5.6]). If $\operatorname {char} \mathbb {k}=0$ , any pointed cocommutative Hopf algebra is of this form. Now,

$$ \begin{align*}\mathbb {k} \mathsf{K}\ltimes U(\mathfrak{g})=K\subset H=\mathbb {k} \mathsf{G}\ltimes U(\mathfrak{g})\end{align*} $$

is a finite extension of right integral type by Corollary 3.20 since H is cocommutative. Here, $\overline {H}^*=\mathbb {k} (\mathsf {G}/\mathsf {K})$ is the algebra of functions on the quotient $\mathsf {G}$ -set $\mathsf {G}/\mathsf {K}$ . A right integral is given by the indicator function $\delta _{\mathsf {K}}$ of the coset $\mathsf {K}$ . For $k\in \mathsf {K},g\in \mathsf {G}$ , and $u\in \mathfrak {g}$ ,

$$ \begin{align*}\delta_{\mathsf{K}}(\overline{kh})=\delta_{\mathsf{K}}(\overline{h})=\varepsilon_K(k)\delta_{\mathsf{K}}(\overline{h}), \quad \delta_{\mathsf{K}}(\overline{ku})=0=\varepsilon_K(u)\delta_{\mathsf{K}}(\overline{u}).\end{align*} $$

This shows that $\chi =\varepsilon _K$ and hence $K\subset H$ is a Frobenius extension. As H is cocommutative, $K\subset H$ is a central Frobenius extension by Corollary 4.16.

Next, we derive the following necessary condition for central Frobenius extensions.

As the following example shows, the converse of Proposition 4.18 is false in the sense that there are Frobenius extensions $K\subset H$ such that $\overline {H}^*$ is unimodular that are not central.

Example 4.19 Let $u_\epsilon (\mathfrak {sl}_2)$ be the small quantum group of $\mathfrak {sl}_2$ associated with a root of unity $\epsilon $ of order $\ell \geq 2$ . Consider the Frobenius extension $\mathbb {k} \mathsf {C}_\ell \subset u_\epsilon (\mathfrak {sl}_2)$ from [Reference Flake, Laugwitz and Posur21, Examples 5.16 and 5.25]. The quotient map $u_\epsilon (\mathfrak {sl}_2)\to \overline {u_\epsilon (\mathfrak {sl}_2)}$ identifies $f^je^ik^m$ and $f^je^i$ and,

$$ \begin{align*}\overline{H}\cong \mathbb {k}\left\lbrace f^je^i\mid 0\leq i,j\leq \ell-1\right\rbrace,\end{align*} $$

as a $\mathbb {k}$ -vector space. Using the coproduct formulas from [Reference Kassel27, Proposition VII.1.3], we compute the coproduct $\underline {\Delta }$ of $\overline {H}$ to be

$$ \begin{align*}\underline{\Delta}(f^je^i)=\sum_{r=0}^i\sum_{s=0}^j\epsilon^{r(i-r)+s(j-s)-2(i-r)(j-s)}\binom{i}{r}_\epsilon\binom{j}{s}_\epsilon f^se^{i-r}\otimes f^{j-s}e^r.\end{align*} $$

This coalgebra is not a bialgebra as $\mathbb {k} \mathsf {C}_\ell $ is not a normal Hopf subalgebra. With the right integral $\lambda (f^je^i)=\delta _{i,\ell -1}\delta _{j,\ell -1}$ for $\overline {u_\epsilon (\mathfrak {sl}_2)}^*$ , we compute that

$$ \begin{align*}(\lambda\otimes {\operatorname{id}})\underline{\Delta}(f^je^i)=\delta_{i,\ell-1}\delta_{j,\ell-1}\epsilon^{0}\binom{\ell-1}{0}_\epsilon\binom{\ell-1}{\ell-1}_\epsilon f^0e^0=({\operatorname{id}}\otimes \lambda)\underline{\Delta}(f^je^i).\end{align*} $$

Thus, $\overline {u_\epsilon (\mathfrak {sl}_2)}^*$ is unimodular. However, it was shown in [Reference Flake, Laugwitz and Posur21, Example 5.16] that the associated Frobenius morphism $\operatorname {tr}$ is not a morphism of $u_\epsilon (\mathfrak {sl}_2)$ -bicomodules. Thus, $\mathbb {k} \mathsf {C}_\ell \subset u_\epsilon (\mathfrak {sl}_2)$ is not a central Frobenius extension.

Example 4.20 Continuing Example 4.14, we note that $\underline {\Delta }(\overline {g^ix^j})$ is computed using the same formula (4.4), for $0\leq i,j\leq \ell -1$ , replacing $g^ax^b$ by $\overline {g^ax^b}$ for all $0\leq a,b\leq \ell -1$ . Thus, we check that $\overline {B}^*$ is not unimodular since

$$ \begin{align*}(\lambda\otimes {\operatorname{id}})\Delta(\overline{g^ix^j})=\overline{1}\otimes \overline{1} \neq \overline{g^{\ell-1}}\otimes \overline{1}=({\operatorname{id}}\otimes \lambda)\Delta(\overline{g^ix^j}).\end{align*} $$

This provides another way to see that $K\subset B$ is not a central Frobenius extension.

Proof By Proposition 3.26, a right integral for $\overline {H}^*$ is given by the formula

$$ \begin{align*}\lambda_{\overline{H}}(\overline{h})=\lambda_H(h\Lambda_K),\end{align*} $$

where $\lambda _H$ is a right integral for $H^*$ and $\Lambda _K$ is a left integral for K. Since $K^*$ is unimodular, the distinguished grouplike element $g_K=1$ and we find that

(4.6) $$ \begin{align} (\Lambda_K)_{(2)}\otimes S^{-1}((\Lambda_K)_{(1)})=(\Lambda_K)_{(1)}\otimes S((\Lambda_K)_{(2)}), \end{align} $$

for all $k\in K$ by [Reference Radford44, Theorem 10.5.4]. Denoting $\lambda =\lambda _H$ , we compute

$$ \begin{align*} \lambda_{\overline{H}}(\overline{h_{(1)}}) h_{(2)}&=\lambda(h_{(1)}\Lambda_K) h_{(2)}\\ &=\lambda((h(\Lambda_K)_{(1)})_{(1)}) (h(\Lambda_K)_{(1)})_{(2)}S((\Lambda_K)_{(2)})\\ &=\lambda((h(\Lambda_K)_{(2)})_{(1)}) (h(\Lambda_K)_{(2)})_{(2)}S^{-1}((\Lambda_K)_{(1)})\\ &=(h(\Lambda_K)_{(2)})_{(1)}S^{-1}((\Lambda_K)_{(1)}) \lambda((h(\Lambda_K)_{(2)})_{(2)})\\ &=h_{(1)}(\Lambda_K)_{(2)}S^{-1}((\Lambda_K)_{(1)}) \lambda(h_{(2)}(\Lambda_K)_{(3)})\\ &=h_{(1)} \lambda(h_{(2)}\Lambda_K)=h_{(1)} \lambda_{\overline{H}}(\overline{h_{(2)}}), \end{align*} $$

for any $h\in H$ . Here, the second equality uses that for any $h,g\in H$ ,

$$ \begin{align*}h_{(1)}g\otimes h_{(2)}=(hg_{(1)})_{(1)}\otimes (hg_{(1)})_{(2)}S(g_{(2)}),\end{align*} $$

followed by applying Equation (4.6) in the third equality. The fourth equality uses that $\lambda (h_{(1)}) h_{(2)}=h_{(1)} \lambda (h_{(2)}),$ for all $h\in H$ , by unimodularity of $H^*$ , followed by the bialgebra axiom in the fifth, and $h_{(2)}S^{-1}(h_{(1)})=\varepsilon _H(h)1_H$ and the counit axiom in the sixth equality. Thus, by Proposition 4.15, $K\subset H$ is a central Frobenius extension. The last conclusion follows from Theorem 4.9.

The following example shows that even if $H^*$ and K are unimodular (K even semisimple), $K\subset H$ might not be a (central) Frobenius extension of Hopf algebras.

Example 4.23 Let $\mathfrak {g}$ be a restricted Lie algebra over a field $\mathbb {k}$ of characteristic p with p-map $\mathfrak {g}\to \mathfrak {g}, x\mapsto x^{[p]}$ . Denote its (finite-dimensional) restricted universal enveloping algebra by $u(\mathfrak {g})$ (see, e.g., [Reference Montgomery39, Section 2.3]). Now, by cocommutativity, $u(\mathfrak {g})^*$ is unimodular.

Consider a restricted Lie subalgebra $\mathfrak {k}\subset \mathfrak {g}$ which is abelian and such that $\mathfrak {k}^{[p]}=\mathfrak {k}$ . Then $u(\mathfrak {k})\subset u(\mathfrak {g})$ is a commutative Hopf subalgebra. By [Reference Hochschild22], $u(\mathfrak {k})$ is semisimple and hence unimodular. However, $u(\mathfrak {k})\subset u(\mathfrak {g})$ might not be a Frobenius extension of Hopf algebras (unless, say, $\mathfrak {k}\subset \mathfrak {g}$ is normal, that is, closed under the adjoint action). A counterexample is given by the two-dimensional restricted Lie algebra

$$ \begin{align*}\mathfrak{g}=\mathbb {k} x\oplus \mathbb {k} t, \quad [t,x]=x, \quad t^{[p]}=t, \quad x^{[p]}=0 ,\quad\mathfrak{k}=\mathbb {k} t\subset\mathfrak{g}.\end{align*} $$

A left (but not right) integral for $u(\mathfrak {g})$ is $(t^{p-1}-1)x^{p-1}$ and a left integral for $u(\mathfrak {k})$ is $t^{p-1}-1$ . Thus,

$$ \begin{align*}\alpha_{u(\mathfrak{k})}(t)=0\neq 1=\alpha_{u(\mathfrak{g})}(t),\end{align*} $$

see [Reference Farnsteiner16], and we do not have a Frobenius extension by the converse implication in Proposition 3.26(ii). However, considering instead the Hopf subalgebra $u(\mathfrak {h})$ , with $\mathfrak {h}=\mathbb {k} x$ ,

$$ \begin{align*}\alpha_{u(\mathfrak{h})}(x)=0=\alpha_{u(\mathfrak{g})}(x),\end{align*} $$

which yields a Frobenius extension of Hopf algebras, and by Theorem 4.21, a central Frobenius extension. Note that $u(\mathfrak {h})$ is not semisimple.

5.1 Drinfeld doubles of unimodular Hopf algebras

Next, we apply our results to the Drinfeld double $\operatorname {Drin}(H)$ of a unimodular Hopf algebra H. We use the presentation of $\operatorname {Drin}(H)$ of [Reference Radford44, Section 13.1].

Proof Factorizable Hopf algebras are unimodular [Reference Radford44, Proposition 12.4.2]. Hence, the Drinfeld double of a finite-dimensional Hopf algebra H is unimodular as it is factorizable [Reference Radford44, Theorem 3.2.1]. Hence, $\alpha _{\operatorname {Drin}(H)}=\varepsilon $ . As H is also unimodular by assumption, we have that $\alpha _H=\varepsilon $ . Thus, $\chi =(\alpha _{\operatorname {Drin}(H)})|_H *\alpha _H = \varepsilon $ so that $H\subset \operatorname {Drin}(H)$ is a Frobenius extension.

As shown in [Reference Fischman, Montgomery and Schneider18, 1.9. Example], a right integral $\lambda $ for $\overline {\operatorname {Drin}(H)}^*\cong H^{\mathrm {op}}$ corresponds to a left integral $\Lambda _H$ of H and we may write

$$ \begin{align*}\lambda_{\overline{\operatorname{Drin}(H)}}(f\otimes h)=\varepsilon(h)f(\Lambda_H).\end{align*} $$

We now compute, for $f\in H^*$ and $h\in H$ ,

$$ \begin{align*} (\lambda_{\overline{\operatorname{Drin}(H)}}\otimes {\operatorname{id}}_{\operatorname{Drin}(H)})\Delta_{\operatorname{Drin}(H)}(f\otimes h)&= \lambda_{\overline{\operatorname{Drin}(H)}}(f_{(2)}\otimes h_{(1)})\otimes (f_{(1)}\otimes h_{(2)})\\ &=f_{(2)}(\Lambda_H) f_{(1)}\otimes h\\ &=f_{(1)}(\Lambda_H) f_{(2)}\otimes h\\ &=({\operatorname{id}}_{\operatorname{Drin}(H)}\otimes \lambda_{\overline{\operatorname{Drin}(H)}})\Delta_{\operatorname{Drin}(H)}(f\otimes h), \end{align*} $$

where the second to last equality uses that $H\cong H^{**}$ is unimodular. Thus, $H\subset \operatorname {Drin}(H)$ is a central Frobenius extension by Proposition 4.15.

In particular, the induction functor

$$ \begin{align*}\operatorname{Ind}_H^{\operatorname{Drin}(H)}\colon H\text{-}\mathbf{Mod}\to \operatorname{Drin}(H)\text{-}\mathbf{Mod}\simeq {\vphantom{\mathbf{YD}}}^{H}_{H}{\mathbf{YD}},\end{align*} $$

which is left and right adjoint to the forgetful functor $F\colon {\vphantom {\mathbf {YD}}}^{H}_{H}{\mathbf {YD}}\to H\text {-}\mathbf {Mod}$ , is a Frobenius monoidal functor which extends to Drinfeld centers as a braided Frobenius monoidal functor.

5.2 Quantum groups at roots of unity

In this section, we apply the results of Section 4.1 to obtain Frobenius monoidal functors to categories of representations of quantum groups at roots of unity.

We let $U_\epsilon (\mathfrak {g})$ denote the De Concini–Kac–Procesi form of the quantized enveloping algebra, or, quantum group, associated with a fixed Cartan datum of a semisimple Lie algebra $\mathfrak {g}$ and specialized to a root of unity $\epsilon \in \mathbb{C}$ of odd order $\ell $ Footnote ² (see [Reference Chari and Pressley6, Section 9.1] or [Reference De Concini and Kac11]). We let $E_i, F_i, K_i$ denote the usual generators of these quantum groups.

We will also consider the small quantum group $u_\epsilon (\mathfrak {g})$ , defined as the quotient of $U_\epsilon (\mathfrak {g})$ by the ideal generated by $E_\alpha ^\ell , F_\alpha ^\ell , K_i^\ell $ , for positive roots $\alpha $ of $\mathfrak {g}$ . The finite-dimensional Hopf algebra $u_\epsilon (\mathfrak {g})$ is factorizable [Reference Lyubashenko33, Corollary A.3.3] and hence unimodular (see, e.g., [Reference Radford44, Proposition 12.4.2]). We refer to [Reference De Renzi, Geer and Patureau-Mirand12, Section 2.1] for explicit formulas of the braiding, ribbon structure, and integrals for $u_\epsilon (\mathfrak {g})$ and its dual.

It is well-known that $U_\epsilon (\mathfrak {g})$ contains a large central Hopf subalgebra $Z_0$ , which can be identified with the coordinate Hopf algebra $\mathcal {O}_{\mathsf {H}}$ of an affine algebraic group $\mathsf {H}$ , giving an extension of Hopf algebras $\iota \colon \mathcal {O}_{\mathsf {H}}\hookrightarrow U_\epsilon (\mathfrak {g})$ . Being generated by grouplike and skew-primitive elements, $U_\epsilon (\mathfrak {g})$ is a pointed Hopf algebra and $U_\epsilon (\mathfrak {g})$ is free and finite over $\mathcal {O}_{\mathsf {H}}$ . Thus, by Corollary 3.21, the extension $\mathcal {O}_{\mathsf {H}}\subset U_\epsilon (\mathfrak {g})$ is of right integral type and since $\mathcal {O}_{\mathsf {H}}\subset Z(H)$ , $\mathcal {O}_{\mathsf {H}}$ is a normal Hopf subalgebra. Thus, we obtain the following result from Corollary 3.30.

Corollary 5.3 The induction functor

$$ \begin{align*}\operatorname{Ind}_{\iota}\colon \mathcal{O}_{\mathsf{H}}\text{-}\mathbf{Mod}\to U_\epsilon(\mathfrak{g})\text{-}\mathbf{Mod}\end{align*} $$

is a Frobenius monoidal functor.

Because $\mathcal {O}_{\mathsf {H}}$ is a central subalgebra, we have that the ideal $U_\epsilon (\mathfrak {g})\mathcal {O}_{\mathsf {H}}^+$ is two-sided. This makes $\overline {U_\epsilon (\mathfrak {g})}$ a Hopf algebra which can be identified with $u_\epsilon (\mathfrak {g})$ . Thus, as a right integral, we can use the integral $\lambda $ of $u_\epsilon (\mathfrak {g})^*$ from [Reference De Renzi, Geer and Patureau-Mirand12, Section 2.1] for which, by Corollary 4.6(ii),

$$ \begin{align*}\operatorname{tr}(h)=\lambda(\pi(h_{(1)}))\otimes h_{(2)}\end{align*} $$

is a Frobenius morphism, where $\pi \colon U_\epsilon (\mathfrak {g})\to u_\epsilon (\mathfrak {g})$ is the quotient morphism.

Remark 5.4 The functor $\operatorname {Ind}_{\mathcal {O}_{\mathsf {H}}}^{U_\epsilon (\mathfrak {g})}$ was shown to induce an oplax monoidal functor $\mathcal {Z}(\operatorname {Ind}_\iota )$ on the Drinfeld centers in [Reference Flake, Laugwitz and Posur20, Corollary 8.8(ii)]. We note that $\mathcal {O}_{\mathsf {H}}\subset U_\epsilon (\mathfrak {g})$ is not a central Frobenius extension of Hopf algebras. Indeed, $\operatorname {tr}$ is not a morphism of H-H-bicomodules by Proposition 4.15, since

$$ \begin{align*}\operatorname{tr}(h)=\lambda(\pi(h_{(1)}))\otimes h_{(2)}\neq h_{(1)}\otimes \lambda(\pi(h_{(2)})).\end{align*} $$

To verify this, consider the unique monomial

$$ \begin{align*}X=F_1^{\ell-1}\ldots F_r^{\ell-1}K_{2\rho} E_1^{\ell-1}\ldots E_r^{\ell-1},\end{align*} $$

on which the integral is nonzero, the coproduct takes the form

$$ \begin{align*} \Delta(X) \quad\in\quad &X\otimes K_{2\rho} K_{1}^{\ell-1}\ldots K_r^{\ell-1}+K_{2\rho}K_{1}^{1-\ell}\ldots K_r^{1-\ell}\otimes X \\ &+\ker(\lambda\pi)\otimes U_\epsilon(\mathfrak{g})+U_\epsilon(\mathfrak{g})\otimes \ker(\lambda\pi). \end{align*} $$

Thus, applying $\lambda (\pi (h_{(1)}))\otimes h_{(2)}$ and $h_{(1)}\otimes \lambda (\pi (h_{(2)}))$ yields

$$ \begin{align*}K_{2\rho} K_{1}^{\ell-1}\ldots K_r^{\ell-1}\quad \text{and}\quad K_{2\rho}K_{1}^{1-\ell}\ldots K_r^{1-\ell}, \quad\text{respectively}.\end{align*} $$

Multiplying with $K_{2\rho }^{-1}$ yields distinct elements in the group algebra generated by the $K_i$ , unless $\ell =2$ , which contradicts our assumption that $\ell $ is odd. Thus, we do not obtain a Frobenius monoidal functor $\mathcal {Z}(\operatorname {Ind}_{\mathcal {O}_{\mathsf {H}}}^{U_\epsilon (\mathfrak {g})})$ from Theorem 4.9.

5.3 Hopf subalgebras of small quantum groups

Recall that the small quantum group, for a root of unity of odd order $\ell>2$ , has a triangular decomposition

$$ \begin{align*}u_\epsilon(\mathfrak{g})=u_\epsilon(\mathfrak{n}_-)\otimes \mathbb{C}\mathsf{K}\otimes u_\epsilon(\mathfrak{n}_+),\end{align*} $$

as a $\mathbb{C}$ -vector space, where $\mathsf {K}$ is the group generated by the $K_i$ , which is isomorphic to $\mathbb{Z}_\ell ^r$ , and $u_\epsilon (\mathfrak {n}_-)$ and $u_\epsilon (\mathfrak {n}_+)$ are the subalgebras generated by the $F_i$ and the $E_i$ , respectively.

We will now consider triples $(\Sigma , I_+, I_-)$ defining a Hopf subalgebra of $u_\epsilon (\mathfrak {g})$ . Here, $\Sigma \subset \mathsf {K}$ is a subgroup and $I_\pm $ are subsets of $\{1,\ldots , r\}$ which are identified with subsets of the simple roots of $\mathfrak {g}$ . We require that $K_i\in \Sigma $ if $i\in I_+\cup I_-$ . It is easy to verify that the subalgebra generated by the $E_i$ , for $i\in I_+$ , $F_j$ , for $j\in I_-$ , and $g\in \Sigma $ generate a Hopf subalgebra of $u_\epsilon (\mathfrak {g})$ which we denote by $u_\epsilon (\Sigma , I_+, I_-)$ . By [Reference Müller40, Theorem 6.3], see also [Reference Andruskiewitsch and García1, Corollary 1.13], all possible Hopf subalgebras of $u_\epsilon (\mathfrak {g})$ are obtained this way.

We will now determine those Hopf subalgebras of $u_\epsilon (\mathfrak {g})$ that lead to Frobenius extensions.

Proof of Proposition 5.7

Consider any triple $(\Sigma ,I_+,I_-)$ as above. Denote $u'=u_\epsilon (\Sigma , I_+, I_-)$ and $u=u_\epsilon (\mathfrak {g})$ . For $I=I_+\cap I_-$ , we have a sequence of extensions of Hopf algebras

$$ \begin{align*}u_\epsilon(\Sigma, I,I)\subset u_\epsilon(\Sigma, I_+,I)\subset u'\subset u.\end{align*} $$

We first claim that $u_\epsilon (\Sigma , I,I)$ is unimodular. Indeed, restricting $\Sigma $ to the subgroup generated by those $K_i$ with $i\in I$ we obtain a quantum group $u_\epsilon (\mathfrak {h})$ , where $\mathfrak {h}$ is obtained by restricting the Cartan datum to the subset I. As $\mathfrak {h}$ is again a semisimple Lie algebra, the Hopf algebra $u_\epsilon (\mathfrak {h})$ is unimodular. Further, arguing as in the proof of Corollary 4.24, a nonzero left integral $\Lambda $ for $u_\epsilon (\Sigma , I,I)$ is obtained as a product $\Lambda _2=\Lambda _1\Lambda '$ , where $\Lambda _1$ is a left integral for $u_\epsilon (\mathfrak {h})$ and $\Lambda '\in \mathbb{C} \Sigma $ (as otherwise $\Lambda _2=0$ ). We now check that for $K_j$ , with $j\notin I$ ,

$$ \begin{align*}\Lambda_2K_j=\Lambda_1\Lambda'K_j=\Lambda_1K_j\Lambda'=K_j\Lambda_1\Lambda'=K_j\Lambda_2=\Lambda_2.\end{align*} $$

The second equality uses that $\Lambda '\in \mathbb{C} \Sigma $ , which is commutative. The third equality uses the form of $\Lambda _1$ , see [Reference Lyubashenko33], in which for any $E_i$ , the corresponding $F_i$ appears with the same power in $\Lambda _1$ , and that $E_iK_j=\epsilon ^{-i\cdot j}K_jE_i$ and $F_iK_j=\epsilon ^{i\cdot j}F_iK_j$ . The last equality uses the assumption that $\Lambda _2$ is a left integral for $u_\epsilon (\Sigma , I,I)$ . This shows that $u_\epsilon (\Sigma , I,I)$ is unimodular.

Next, we claim that $u_\epsilon (\Sigma , I_+,I)$ is not unimodular unless $I_+=I$ . Assume $\Lambda _3=\Lambda _2\Lambda "$ is a nonzero left integral for $u_\epsilon (\Sigma , I_+,I)$ , with $\Lambda _2$ as in the previous paragraph. Given $j\in I_+\setminus I$ , we use the triangular decomposition of u to see that $\Lambda "\in \mathbb{C}\Sigma \prod _{k\in I_+\setminus I}E_k^{\ell -1}$ . Thus, $\Lambda _3E_j=0=\varepsilon (E_j)\Lambda _3$ for any $j\in I_+\setminus I$ . It also follows that

$$ \begin{align*}\Lambda_3K_j=\epsilon^{\sum_{k\in I_+\setminus I}(1-\ell)j\cdot k}\Lambda_2K_j\Lambda"=\epsilon^{\sum_{k\in I_+\setminus I}j\cdot k}\Lambda_3,\end{align*} $$

using that $\Lambda _2K_j=\Lambda _2$ as $\Lambda _2$ is a left and right integral for $u_\epsilon (\Sigma , I,I)$ . Now, if $\epsilon ^{\sum _{k\in I_+\setminus I}j\cdot k}=1$ , for all $j\in I_+\setminus I$ , then the restriction of the Cartan matrix to the subset $I_+\setminus I$ has only zero row sums when viewed as a matrix with entries in $\mathbb{Z}_{\ell }$ . However, Lemma 5.9(i) below shows that this is impossible. Hence, $\Lambda _3$ is not a right integral and $u_\epsilon (\Sigma , I_+,I)$ is not unimodular.

Finally, again assuming $\ell>3$ , we claim that $u'=u_\epsilon (\Sigma , I_+,I_-)$ is unimodular if and only if $I_-=I_+=I$ . Let $\Lambda _4=\Lambda _3\Lambda "'$ be a nonzero left integral for $u'$ with $\Lambda _3$ a left integral for $u_\epsilon (\Sigma , I_+,I)$ as in the previous paragraph. Take $F_m$ with $m\in I_-\setminus I$ . A similar calculation as in the previous paragraph shows that

$$ \begin{align*}\Lambda_4K_m=\epsilon^{-\sum_{n\in I_-\setminus I}m\cdot n}\Lambda_3K_m \Lambda"'=\epsilon^{\sum_{k\in I_+\setminus I}m\cdot k-\sum_{n\in I_-\setminus I}m\cdot n}\Lambda_4.\end{align*} $$

Similarly, we have, for $j\in I_+\setminus I$ , that

$$ \begin{align*}\Lambda_4K_j=\epsilon^{\sum_{k\in I_+\setminus I}j\cdot k-\sum_{n\in I_-\setminus I}j\cdot n}\Lambda_4.\end{align*} $$

Note that $(I_+\setminus I)\cap (I_-\setminus I)$ is empty. Hence, $\Lambda _4$ is a right integral if and only if the square matrix $d=(d_{a,b})$ , with entries in $\mathbb{Z}_\ell $ and rows and columns indexed by $a,b\in (I_+\setminus I)\cup (I_-\setminus I)$ , defined by

$$ \begin{align*}d_{a,b}=\begin{cases} a\cdot b, &\text{if } b \text{ in } (I_+\setminus I),\\ -a\cdot b, &\text{if } b \text{ in } (I_-\setminus I), \end{cases} \end{align*} $$

has only zero row sums. This is not possible as shown in Lemma 5.9(ii) below, since, by assumption, $\ell>3$ is odd.

Proof First, observe that if the statements hold for all indecomposable Cartan matrices then they hold for decomposable Cartan matrices.

To prove Part (i), note that the statement is easily verified for Dynkin diagrams of rank $1$ or $2$ . Next, every Dynkin diagram for rank $r\geq 3$ has at least two endpoints with just a single other vertex connected to it. Using a suitable ordering of simple roots, this vertex corresponds to a row $(2,-1,0,\dots ,0)$ in the Cartan matrix, whose row sum is nonzero modulo any $\ell \neq 1$ .

To prove Part (ii), multiply each column of the roots from a subset J of the simple roots I by $-1$ . For rank $1$ , the statement is clear since $\ell \neq 2$ . For rank $2$ , we check all three cases separately. The Cartan matrices are given by

$$ \begin{align*} \begin{pmatrix} 2\eta_1 & -\eta_2\\ -\eta_1 & 2\eta_2 \end{pmatrix}, \qquad \begin{pmatrix} 2\eta_1 & -\eta_2\\ -2\eta_1 & 2\eta_2 \end{pmatrix},\qquad \begin{pmatrix} 2\eta_1 & -\eta_2\\ -3\eta_1 & 2\eta_2 \end{pmatrix}, \qquad \eta_1,\eta_2\in \{\pm 1\}. \end{align*} $$

For the second two matrices, it is not possible to choose $\eta _1,\eta _2$ such that both row sums are zero modulo any $\ell \neq 0$ . For the first matrix, this is only possible when $\eta _1=-\eta _2$ and $\ell =3$ . However, we assume that $\ell>3$ so at least one row sum is nonzero modulo $\ell $ in all cases.

For rank $r\geq 2$ , we can again find an endpoint whose row in the Cartan matrix is $(2,-1,0,\dots ,0)$ . The corresponding row sum is now given by $2\eta _1-\eta _2$ , for some $\eta _1,\eta _2\in \{\pm 1\}$ . This row sum can only be zero if $\eta _1=-\eta _2$ and $\ell =3$ . As $\ell>3$ by assumption, this case is ruled out as well.

Example 5.11 Let $\epsilon $ be a primitive third root of unity and consider the Hopf subalgebra $u'$ of $u_\epsilon (\mathfrak {sl}_3)$ generated by $K_1,K_2, E_1, F_2$ only. Let $\Lambda _K$ be a nonzero left and right integral for the Cartan group algebra. Then

$$ \begin{align*}\Lambda=\Lambda_KE_1^2F_2^2\end{align*} $$

is a left and right integral for $u'$ . Thus, $u'\subset u_\epsilon (\mathfrak {sl}_3)$ is a Frobenius extension. This case is excluded in Proposition 5.7 by assuming that $\ell>3$ .

5.4 The Kac–Paljutkin algebra

In this section, we study an eight-dimensional Hopf algebra $H_8$ which gives a central Frobenius extension $K=\mathbb {k} \mathsf {K}\subset H_8$ , where $\mathsf {K}=\mathsf {C}_2\times \mathsf {C}_2$ is the Klein four-group and $\operatorname {char} \mathbb {k}=0$ .

Definition 5.15 (The Kac–Paljutkin algebra)

The Kac–Paljutkin algebra $H_8$ of [Reference Masuoka37] is defined as

$$ \begin{align*}H_8=\mathbb {k} \langle x,y,z ~|~ x^2=y^2=1,~z^2=\frac{1}{2}(1+x+y-xy),~xy=yx,~zx=yz,~zy=xz \rangle,\end{align*} $$

and becomes a Hopf algebra with coproduct determined by

$$ \begin{align*}\Delta(x)=x\otimes x, \quad \Delta(y)=y\otimes y,\quad \Delta(z)=\frac{1}{2}(1\otimes 1+1\otimes x+y\otimes 1-y\otimes x)(z\otimes z).\end{align*} $$

Note that $\varepsilon (x)=\varepsilon (y)=1$ , $\varepsilon (z)=0$ determine the counit.

A basis for $H_8$ is given by $\{ 1, x, y, z,xy, xz, yz, xyz\}$ . The Hopf algebra $H_8$ is a semisimple Hopf algebra that is neither commutative nor cocommutative. The group algebra of the Klein four group $K=\mathbb {k}\mathsf {K} =\mathbb {k} \langle x,y\rangle $ is a Hopf subalgebra. As H is semisimple, the extension $K\subset H$ is a Frobenius extension by Corollary 4.7.

Note that $\ker \varepsilon _K$ has basis $1-x,1-y,1-xy$ and thus $\overline {H_8}$ has basis $1,z$ (where we use elements from $H_8$ to denote their cosets in $\overline {H_8}$ ), and both elements are grouplike. We verify that the augmented algebra

$$ \begin{align*}\overline{H_8}^*\cong\mathbb {k}[a]/(a^2-1)\cong \mathbb {k} \mathsf{C}_2,\end{align*} $$

a group algebra. Here, $1=\delta _1+\delta _z$ and $a=\delta _1-\delta _z$ in the dual basis $\{\delta _1,\delta _z\}$ to $\{1,z\}$ . An integral is given by $1+a$ which, when viewed as a function on $\overline {H_8}$ , corresponds to $2\delta _1$ . Thus, by Equation (3.21), we have the Frobenius morphism determined by

(5.1) $$ \begin{align}\operatorname{tr}\colon H_8\to K, \quad 1\mapsto 2, \quad z\mapsto 0.\end{align} $$

Our results imply, as $H_8$ is semisimple, that $K\subset H_8$ is a central Frobenius extension (see Corollary 4.24). Thus, we obtain a Frobenius monoidal functor

$$ \begin{align*}\mathcal{Z}(\operatorname{Ind}_K^{H_8})\colon {\vphantom{\mathbf{YD}}}^{K}_{K}{\mathbf{YD}}\to {\vphantom{\mathbf{YD}}}^{H_8}_{H_8}{\mathbf{YD}}. \end{align*} $$

By Equations (3.9) and (3.10), the lax and oplax monoidal structures are given by

Moreover, this Frobenius monoidal functor is separable, that is, for any $V,U$ in ${\vphantom {\mathbf {YD}}}^{K}_{K}{\mathbf {YD}}$ ,

$$ \begin{align*}\operatorname{lax}_{V,U}\operatorname{oplax}_{V,U}={\operatorname{id}}_{F(V\otimes U)}.\end{align*} $$

Lemma 5.16 The functor

$$ \begin{align*}\mathcal{Z}(\operatorname{Ind}_{K}^{H_8})\colon {\vphantom{\mathbf{YD}}}^{K}_{K}{\mathbf{YD}}\to {\vphantom{\mathbf{YD}}}^{H_8}_{H_8}{\mathbf{YD}}\end{align*} $$

is a separable braided Frobenius monoidal functor.

Assume now that $\mathbb {k}$ is algebraically closed and recall that $\operatorname {char} \mathbb {k}=0$ . The simple YD modules over the Kac–Paljutkin algebra $H_8$ were explicitly described in [Reference Hu and Zhang23, Reference Shi46]. With the notation of [Reference Shi46], there are five non-isomorphic simple $H_8$ -modules

$$ \begin{align*}V_1(b), \quad b\in \left\lbrace \pm1, \pm i\right\rbrace, \quad V_2,\end{align*} $$

where on the one-dimensional simples $V_1(b)$ , z acts by b and $x,y$ act by $b^2$ . For the two-dimensional simple $V_2=\mathbb {k}\left \lbrace v_1,v_2\right \rbrace $ , we have

$$ \begin{align*}zv_1=v_1,\quad zv_2=-v_2, \quad xv_1=-v_2, \quad yv_1=v_2.\end{align*} $$

The simple K-modules are one-dimensional and denoted by $\mathbb {k}_{\epsilon _x,\epsilon _y}$ , where x, y act by $\epsilon _x,\epsilon _y\in \left \lbrace \pm 1\right \rbrace $ , respectively. We compute for $\operatorname {Ind}=\operatorname {Ind}_K^{H_8}$ that

$$ \begin{align*} \operatorname{Ind}(\mathbb {k}_{1,1})&\cong V_1(1)\oplus V_1(-1), \\ \operatorname{Ind}(\mathbb {k}_{1,-1})&\cong \operatorname{Ind}(\mathbb {k}_{-1,1})\cong V_2,\\ \operatorname{Ind}(\mathbb {k}_{-1,-1})&\cong V_1(i)\oplus V_1(-i). \end{align*} $$

Thus, every simple $H_8$ -module is a summand of an object in the image of $\operatorname {Ind}$ .

We observe that a YD module $(V,\delta _V)$ over $H_8$ appears in the closure of the image of $\mathcal {Z}(\operatorname {Ind}_K^{H_8})$ under direct summands if and only if its coaction $\delta _V$ factors through $K\otimes V$ , for $K=\mathbb {k} \mathsf {K}$ . In the list of simple YD modules over $H_8$ from [Reference Shi46, Theorem 3.13], we can identify the images of the simple YD modules over K under $\mathcal {Z}(\operatorname {Ind})=\mathcal {Z}(\operatorname {Ind}_K^{H_8})$ as follows:

$$ \begin{align*} \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,1}^1)&\cong M\left\langle 1,1\right\rangle\oplus M\left\langle -1,1\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,1}^x)&\cong M\left\langle (x,y)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,1}^{xy})&\cong M\left\langle 1,xy\right\rangle\oplus M\left\langle -1,xy\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,1}^y)&\cong M\left\langle (x,y)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,-1}^x)&\cong M\left\langle i,x\right\rangle\oplus M\left\langle -i,x\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,-1}^1)&\cong M\left\langle (1,xy)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,-1}^{y})&\cong M\left\langle i,y\right\rangle\oplus M\left\langle -i,y\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,-1}^{xy})&\cong M\left\langle (1,xy)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,-1}^{1})&\cong M\left\langle (1,y))\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,-1}^{x})&\cong M\left\langle (x,1)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,-1}^{y})&\cong M\left\langle (y,xy))\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{1,-1}^{xy})&\cong M\left\langle (xy,x)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,1}^{y})&\cong M\left\langle (1,y))\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,1}^{1})&\cong M\left\langle (x,1)\right\rangle,\\ \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,1}^{xy})&\cong M\left\langle (y,xy))\right\rangle, & \mathcal{Z}(\operatorname{Ind})(\mathbb {k}_{-1,1}^{x})&\cong M\left\langle (xy,x)\right\rangle. \end{align*} $$

This list shows that all eight one-dimensional simples

$$ \begin{align*}M\left\langle a,b\right\rangle, \text{ with } \left\langle a,b\right\rangle=\left\langle \pm 1,1\right\rangle,\left\langle \pm 1,xy\right\rangle,\left\langle \pm i,x\right\rangle,\left\langle \pm i,y\right\rangle,\end{align*} $$

and a family of six two-dimensional simples

$$ \begin{align*}M(\left\langle c,d\right\rangle), \text{ with } \left\langle c,d\right\rangle=\left\langle x,y\right\rangle,\left\langle 1,xy\right\rangle,\left\langle x,1\right\rangle,\left\langle 1,y\right\rangle,\left\langle xy,x\right\rangle,\left\langle y,xy\right\rangle,\end{align*} $$

appear as summands. There are eight further simple YD modules over $H_8$ with coactions that do not factor over the group algebra K and that do not appear in the image of $\mathcal {Z}(\operatorname {Ind})$ .

As an application, we can now construct commutative Frobenius algebras in ${\vphantom {\mathbf {YD}}}^{H_8}_{H_8}{\mathbf {YD}}$ .

Example 5.17 Let be the trivial Frobenius algebra. Then is the two-dimensional Frobenius algebra $\overline {H_8}^*=\mathbb {k}[a]/(a^2-1)$ in $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ , where $1$ , a correspond to $\tfrac {1}{2}(1+z)\otimes _K 1$ , $\tfrac {1}{2}(1-z)\otimes _K 1$ , respectively. The coproduct is given by

using that $\frac {1}{2}(1_A+a)=1$ , and $ \frac {1}{2}(1_A-a)=z.$ Moreover,

The counit for the Frobenius algebra is thus given by $\epsilon _A(1_A)=1$ and $\epsilon _A(a)=0.$ The $H_8$ -Yetter–Drinfeld module structure on A consists of the trivial $H_8$ -coaction and the $H_8$ -action

$$ \begin{align*}z\cdot a=-a, \quad x\cdot a=y\cdot a=a.\end{align*} $$

5.5 Rigid Frobenius algebras in the Drinfeld center of the Kac–Paljutkin algebra

As an application of our results, we conclude by constructing Frobenius algebras in $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ whose categories of local modules are modular fusion categories. We now assume that $\operatorname {char}\mathbb {k}=0$ and $\mathbb {k}$ is algebraically closed.

An étale algebra in an abelian tensor category $\mathcal {C}$ is a commutative separable algebra [Reference Davydov, Müger, Nikshych and Ostrik8, Definition 3.1]. We say that A is connected if . Any connected étale algebra in a fusion category is a Frobenius algebra (see, e.g., [Reference Laugwitz and Walton32, Proposition 3.11]). Connected étale algebras in $\mathcal {Z}(\mathbb {k} \mathsf {G}\text {-}\mathbf {mod})$ , for a finite group $\mathsf {G}$ , were classified in [Reference Davydov7]. These are described by a normal subgroup $\mathsf {N}\triangleleft \mathsf {H}$ , for a subgroup $\mathsf {H}\subset \mathsf {G}$ and cocycle data (see [Reference Laugwitz and Walton32, Section 6.2] for details).

We now apply this classification to $\mathsf {K}=\mathsf {C}_2\times \mathsf {C}_2$ . In case $\mathsf {N}=\{1\},$ we only get the trivial algebra in $\mathcal {Z}(\mathbb {k} \mathsf {K}\text {-}\mathbf {mod})$ and its image in $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ was considered in Example 5.17. If $\mathsf {N}=\mathsf {H}$ , the categories of local modules are trivial (see [Reference Davydov and Simmons9] or [Reference Laugwitz and Walton32, Corollary 6.18(3) and (4)]). We will not compute the images of these so-called Lagrangian subalgebras under $\mathcal {Z}(\operatorname {Ind}_{\mathbb {k} \mathsf {K}}^{H_8})$ here. Hence, we will consider the cases

$$ \begin{align*}\mathsf{H}=\mathsf{K}=\mathsf{C}_2\times \mathsf{C}_2\quad \text{ and }\quad\mathsf{N}=\left\langle n\right\rangle\cong \mathsf{C}_2, \quad \text{ where } n=x, y, \text{ or } xy.\end{align*} $$

The triples $\{(n,\epsilon _x,\epsilon _y):n\in \{x,y,xy\},\epsilon _x,\epsilon _y\in \{\pm 1\}\}$ parameterize a discrete family of 12 non-isomorphic two-dimensional étale algebras $A_{n,\epsilon _x,\epsilon _y}$ in $\mathcal {Z}(\mathbb {k} \mathsf {K}\text {-}\mathbf {mod})$ . We use the basis $\left \lbrace e_1,e_n\right \rbrace $ , with $\deg e_1=1$ and $\deg e_n=n$ . The action is given by $x\cdot e_m=\epsilon _x e_m$ , $y\cdot e_m=\epsilon _x e_m$ , for some $\epsilon _x,\epsilon _y\in \left \lbrace \pm 1\right \rbrace $ , where $m=1,n$ . These algebras are Frobenius algebras with coproduct and counit

$$ \begin{align*}\Delta(e_m)=e_1\otimes e_{m}+e_n\otimes e_{nm},\end{align*} $$

for $m=1,n$ , by [Reference Laugwitz and Walton32, Proposition 6.12].

Our results imply that these algebras induce four-dimensional commutative algebras in $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ . We denote these algebras by

$$ \begin{align*}B_{n,\epsilon_x,\epsilon_y}=\mathcal{Z}(\operatorname{Ind}_{\mathbb {k} \mathsf{K}}^{H_8})(A_{n,\epsilon_x,\epsilon_y}).\end{align*} $$

A basis for each of these algebras is given by

$$ \begin{align*}\{ b_{1m}:=1\otimes e_m,\quad b_{zm}:=z\otimes e_m~|~ m=1,n \}.\end{align*} $$

Recall the definition of a rigid Frobenius algebra in a ribbon category $\mathcal {D}$ [Reference Laugwitz and Walton32, Definition 3.8].Footnote ⁴ A rigid Frobenius algebra A is, equivalently, a connected, commutative, special Frobenius algebra, that is,

for . The categories of internal modules $\mathbf {Mod}_{\mathcal {D}}\text {-}A$ over such an algebra A provide a source of tensor categories and their centers are given by the categories $\mathbf {Mod}^{\mathrm {loc}}_{\mathcal {D}}\text {-}A$ of so-called local modules [Reference Kirillov and Ostrik29, Reference Laugwitz and Walton32, Reference Schneider45].

The Drinfeld center $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ of the category of finite-dimensional $H_8$ -modules is a ribbon category. Indeed, $H_8$ is a spherical Hopf algebra as it is involutory, that is, $S^2={\operatorname {id}}_H$ for the antipode S [Reference Barrett and Westbury3, Example 3.2]. Thus, the ribbon structure on the center of $H_8\text {-}\mathbf {mod}$ is obtained from the spherical structure of $H_8\text {-}\mathbf {mod}$ via [Reference Shimizu49]. As $\operatorname {char} \mathbb {k} =0$ , the category $H_8\text {-}\mathbf {mod}$ is fusion and so is its Drinfeld center [Reference Etingof, Nikshych and Ostrik14, Theorem 2.15].

Corollary 5.19 The algebras $B=B_{n,\epsilon _x,\epsilon _y}$ are rigid Frobenius algebras in $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ . In particular, the categories $\mathbf {Mod}_{H_8\text {-}\mathbf {mod}}\text {-}B$ are fusion categories and their Drinfeld centers

$$ \begin{align*}\mathcal{Z}(\mathbf{Mod}_{H_8\text{-}\mathbf{mod}}\text{-}B)\simeq\mathbf{Mod}^{\mathrm{loc}}_{\mathcal{Z}(H_8\text{-}\mathbf{mod})}\text{-}B\end{align*} $$

are modular fusion categories.

Proof As , our explicit calculations after Lemma 5.16 show that $B=\mathcal {Z}(\operatorname {Ind}_K^{H_8})(A_{n,\epsilon _x,\epsilon _y})$ only contains a single copy of the tensor unit and is, hence, connected. It remains to show that B is special. By definition, this amounts to the composition $m_B\Delta _B$ of coproduct and product being a nonzero multiple of the identity. We know by [Reference Laugwitz and Walton32, Theorem 6.14] that for the algebras $A=A_{n,\epsilon _x,\epsilon _y}$ ,

$$ \begin{align*}m_A\Delta_A=|\mathsf{N}|{\operatorname{id}}_{A}=2{\operatorname{id}}_A.\end{align*} $$

As, by Lemma 5.16, $\operatorname {lax}_{A,A}\operatorname {oplax}_{A,A}={\operatorname {id}}$ , it follows that

$$ \begin{align*}m_B\Delta_B&=\operatorname{Ind}(m_A)\operatorname{lax}_{A,A}\operatorname{oplax}_{A,A}\operatorname{Ind}(\Delta_A)=2{\operatorname{id}}_{\operatorname{Ind}(A)}=2{\operatorname{id}}_B,\\ \varepsilon_Bu_B&=\operatorname{oplax}_0 (b_{11}+b_{z1})=\varepsilon_{H_8}(1)+\varepsilon_{H_8}(z)=1, \end{align*} $$

showing that B is a special Frobenius algebra.

The non-degenerate braided tensor categories $\mathbf {Mod}^{\mathrm {loc}}_{\mathcal {Z}(H_8\text {-}\mathbf {mod})}\text {-}B$ inherit the ribbon structure of $\mathcal {Z}(H_8\text {-}\mathbf {mod})$ by [Reference Kirillov and Ostrik29] and are, hence, modular tensor categories.

5.6 Further questions

Here, we collect some open questions and directions not investigated in this article and [Reference Flake, Laugwitz and Posur21].Footnote ⁵

For fusion categories of the form $\mathcal {C}=H\text {-}\mathbf {mod}$ , for a finite-dimensional semisimple Hopf algebra H over $\mathbb {k}$ , with $\operatorname {char} \mathbb {k}=0$ , it follows from Corollary 4.24 and Theorem 4.9 that Question 5.20 is answered in the affirmative for tensor functors induced by extensions of Hopf algebras. The question can be asked, specifically, for categories of modules over semisimple quasi-Hopf algebras.

Another interesting question may be the explicit computation of the lax and oplax monoidal structures for the functor in Corollaries 5.3 and 5.12 and the resulting Frobenius algebras in $U_\epsilon (\mathfrak {g})\text {-}\mathbf {Mod}$ and $u_\epsilon (\mathfrak {g})\text {-}\mathbf {Mod}$ , respectively.

Finally, concerning the commutative Frobenius algebras $B=B_{m,\epsilon _x,\epsilon _y}$ from Section 5.4, it would be interesting to identify their tensor categories of modules and their modular categories of local modules from Corollary 5.19 among known modular fusion categories, for example, as Drinfeld centers of finite-dimensional modules over (quasi-)Hopf algebras.