1 Introduction↩︎

The representation category \(\mathscr{Z}(G):=\operatorname{Rep}(D(G))\) of the Drinfeld double \(D(G)\) of a finite group scheme \(G\) over an algebraically closed field \(\mathbf{k}\) of characteristic \(p\ge 0\) plays a central role in the theory of finite braided tensor categories [1]–[13]. In characteristic \(p=0\), the category \(\mathscr{Z}(G)\) is a non-degenerate braided fusion category; in this semisimple setting the tensor subcategories of \(\mathscr{Z}(G)\) were classified by Naidu, Nikshych, and Witherspoon [2]. A key feature of [2] is that modular data can be used as an effective organizing principle. Namely, the \(S\)-matrix of \(\mathscr{Z}(G)\) detects when two simple objects of \(\mathscr{Z}(G)\) centralize each other, and Müger centralizers can be computed from \(S\)-matrix relations, ultimately translating the subcategory problem into explicit group-theoretic conditions involving commuting normal subgroups of \(G\).

In positive characteristic \(p>0\), \(\mathscr{Z}(G)\) is typically not semisimple. While one still has a robust notion of non-degeneracy for finite braided tensor categories (e.g. in the sense of Lyubashenko, equivalent to factorizability in the sense of Etingof–Nikshych–Ostrik by Shimizu [14]), there is in general no computable \(S\)-matrix attached to the simple objects that could play the same role as in the fusion case. Thus, the strategy of [2] does not directly extend to the non-semisimple setting.

Our approach replaces \(S\)-matrix technology with a Hopf-theoretic classification of Hopf quotient pairs of \(D(G)\), and then uses the fact that Hopf quotient pairs of \(D(G)\) encode tensor subcategories of \(\mathscr{Z}(G)\). Indeed, tensor subcategories of \(\operatorname{Rep}(H)\), for a finite dimensional Hopf algebra \(H\), correspond to equivalence classes of Hopf algebra quotient pairs of \(H\) (see, e.g. [15]), so one can study tensor subcategories of \(\mathscr{Z}(G)=\operatorname{Rep}(D(G))\) by classifying Hopf algebra quotient pairs of \(D(G)\). This perspective is particularly well-suited to positive characteristic: although \(S\)-matrices are unavailable, the braiding on \(\operatorname{Rep}(D(G))\) is still controlled by the universal \(R\)-matrix of \(D(G)\), and centralizing conditions can be tested on the Hopf algebra side via the element \(R_{21}R\) (cf. the criterion used in the proof of Theorem 12 below).

The first main result of this paper is a classification of Hopf algebra quotient pairs \((D,\theta)\) of \(D(G)\) in terms of group scheme-theoretical data. We show that every Hopf algebra quotient \(D\) of \(D(G)\) is of the form \[D(K,H,B):=\mathscr{O}(K)^{\mathrm{cop}}\#^{\tau}_{\sigma} \mathbf{k}[G/H],\] where \(K\) and \(H\) are normal subgroup schemes of \(G\) that centralize each other and \(B\colon \mathbf{k}[H]\to \mathscr{O}(K)\) is a \(G\)-equivariant Hopf algebra map, and describe the surjective Hopf algebra map \(\theta:D(G)\twoheadrightarrow D\) (Theorems 6 and 9). Moreover, \(D(K,H,B)\) is always ribbon braided, and we give explicit formulas for its \(R\)-matrix and ribbon element (Corollary 3). We also determine precise criteria for \(D(K,H,B)\) to be triangular or factorizable, yielding large families of quasitriangular Hopf algebras realized as Hopf algebra quotients of \(D(G)\).

The second main result is a complete, group scheme-theoretical description of the tensor subcategory lattice of \(\mathscr{Z}(G)\) and its centralizer theory. Using the above Hopf algebra quotient pairs classification, we show that the assignment \[(K,H,B)\mapsto \mathscr{Z}(K,H,B):=\operatorname{Rep}(D(K,H,B))\] yields a bijection between triples \((K,H,B)\) and tensor subcategories of \(\mathscr{Z}(G)\). We then compute Müger centralizers inside \(\mathscr{Z}(G)\): for each tensor subcategory \(\mathscr{Z}(K,H,B)\), its centralizer is \(\mathscr{Z}(H,K,\overline{B})\) (Theorem 12), generalizing the characteristic-\(0\) picture of [2] from modular-data methods to a purely Hopf-theoretic criterion. As consequences, we obtain explicit criteria for symmetry, non-degeneracy, and Lagrangianity of tensor subcategories of \(\mathscr{Z}(G)\) (Corollary 7).

Finally, using the geometric description of \(\mathscr{Z}(G)\) from [1], we describe the simple and projective objects in \(\mathscr{Z}(K,H,B)\) and compute their Frobenius-Perron dimensions (Theorem 13). In §5, we discuss some special cases and examples illustrating our results. In particular, our factorizability criteria produce families of non-degenerate finite braided tensor categories in positive characteristic (coming from factorizable Hopf algebra quotients of \(D(G)\)), and we exhibit examples in both the constant and connected cases.

1.1 Organization↩︎

In §2 we recall background on finite group schemes, Hopf algebras, and Drinfeld doubles. In §3 we construct and classify Hopf algebra quotient pairs of \(D(G)\) and analyze the induced (quasitriangular ribbon) structures on \(D(K,H,B)\). In §4 we apply these results to classify tensor subcategories of \(\mathscr{Z}(G)\), compute their centralizers, and derive criteria for symmetry, nondegeneracy, and Lagrangianity. We then describe the simples and projectives in \(\mathscr{Z}(K,H,B)\) using [1].

1.2 Acknowledgements↩︎

The work of S.G. was supported by Simons Foundation Award 963288.

2 Preliminaries↩︎

We work over an algebraically closed field \(\mathbf{k}\) of characteristic \(p\ge0\). We assume familiarity with the theory of finite tensor categories and finite group schemes over \(\mathbf{k}\), and refer to [16]–[18] for any unexplained notion.

2.1 Finite group schemes↩︎

A finite group scheme \(G\) over \(\mathbf{k}\) is a finite scheme over \(\mathbf{k}\) whose coordinate algebra \(\mathscr{O}(G)\) is a finite dimensional commutative Hopf algebra (see, e.g., [17], [18]), so that its group algebra \(\mathbf{k}[G]:=\mathscr{O}(G)^*\) is a finite dimensional cocommutative Hopf algebra. We set \(|G|:=\dim_{\mathbf{k}}(\mathscr{O}(G))=\dim_{\mathbf{k}}(\mathbf{k}[G])\).

Let \(G^{\circ}\) be the identity component of \(G\), and let \(G(\mathbf{k})\subseteq G\) be the subgroup of closed points of \(G\). Recall that we have a split exact sequence of finite group schemes \[\label{sesG} 1\to G^{\circ}\xrightarrow{} G \mathrel{\mathop{\rightleftarrows}^{\pi}_{\gamma}} G(\mathbf{k})\to 1.\tag{1}\]

Let \(L\) be a closed subgroup scheme of \(G\), let \[\label{iotaL} \iota=\iota_L=\iota_{L,G}:L\hookrightarrow G\tag{2}\] be the inclusion of group schemes, and let \[\label{qL} q=q_L=q_{G,L}=\iota_{L}^{\sharp}:\mathscr{O}(G)\twoheadrightarrow\mathscr{O}(L)\tag{3}\] be the corresponding surjective Hopf algebra map. Recall [19] that we can choose a section \[\label{muL} \mu=\mu_L:\mathscr{O}(L)\xrightarrow{1:1} \mathscr{O}(G)\tag{4}\] for \(q\), so that \(\mu\) is an \(\mathscr{O}(L)\)-colinear map; that is, \(\varepsilon \mu = \varepsilon\) and \[\label{mucolinear} \mu(a_1)\otimes a_2=\mu(a)_1\otimes q(\mu(a)_2);\quad\forall a\in \mathscr{O}(L),\tag{5}\] such that \(\mu(1)=1\) and \[\label{qLmuLsection} q(\mu(a))=a;\quad \forall a\in \mathscr{O}(L).\tag{6}\] We will denote the dimension of the algebra \(\mathscr{O}(G/L)=\mathscr{O}(G)^L\) by \([G:L]\).

Recall that the right adjoint (or conjugation) action of \(G\) on itself is given by \[\label{right32adjoint32action} {\rm ad_r}:G\times G\to G,\quad (g,f)\mapsto f^{-1}gf;\tag{7}\] that is, \({\rm ad_r}(u)(\tilde{u})=S(u_1)\tilde{u}u_2\) for \(u,\tilde{u}\in\mathbf{k}[G]\), and the algebra comorphism \[\label{coadog} {\rm ad_r}^{\sharp}:\mathscr{O}(G)\to\mathscr{O}(G)\otimes\mathscr{O}(G),\quad b\mapsto b_2\otimes S(b_1)b_3,\tag{8}\] is the right adjoint coaction of \(\mathscr{O}(G)\) on itself. Then \({\rm ad_r}^{\sharp}\) corresponds to the left coadjoint action \(\rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\) of \(\mathbf{k}[G]\) on \(\mathscr{O}(G)\), given by \[\label{left32coadjoint32action} \mathbf{k}[G]\otimes\mathscr{O}(G)\to \mathscr{O}(G),\quad u\otimes b\mapsto u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b := \left\langle S(b_1)b_3, u \right\rangle b_2.\tag{9}\] Namely, for every \(u,\tilde{u}\in \mathbf{k}[G]\) and \(b\in \mathscr{O}(G)\), we have \[\label{left32coadjoint32action2} \langle u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b,\tilde{u}\rangle=\langle b,{\rm ad_r}(u)(\tilde{u})\rangle.\tag{10}\] Clearly, if \(G\) is commutative (that is, \(\mathscr{O}(G)\) is cocommutative), then \(\rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\) is trivial.

2.2 Normal subgroup schemes↩︎

Fix a finite group scheme \(G\) over \(\mathbf{k}\) as in §2.1. Let \(H\subseteq G\) be a normal subgroup scheme; that is, \({\rm ad_r}(H\times G)\subseteq H\), so that \(G/H\) is a finite group scheme. Let \(\pi=\pi_H:G\twoheadrightarrow G/H\) be the quotient group scheme morphism. By [19], the exact sequence of Hopf algebras \[\label{gammaH} \mathbf{k}\to \mathbf{k}[H]\xrightarrow{\iota} \mathbf{k}[G]\xrightarrow{\pi} \mathbf{k}[G/H] \to \mathbf{k}\tag{11}\] is cleft. That is, viewing \(\mathbf{k}[G]\) as a right \(\mathbf{k}[G/H]\)-comodule via \[\rho=\rho_H:\mathbf{k}[G]\to \mathbf{k}[G]\otimes\mathbf{k}[G/H],\quad u\mapsto u_1\otimes\pi(u_2),\] we have \(\mathbf{k}[G]^{\text{co} \mathbf{k}[G/H]} = \mathbf{k}[H]\), and we can choose a section \[\label{gamma} \gamma=\gamma_H:\mathbf{k}[G/H]\xrightarrow{1:1} \mathbf{k}[G],\tag{12}\] which is a convolution invertible \(\mathbf{k}[G/H]\)-colinear map; that is, \(\varepsilon \gamma = \varepsilon\) and \[\label{gammacolinear} \gamma(x_1)\otimes x_2=\gamma(x)_1\otimes\pi(\gamma(x)_2);\quad\forall x\in \mathbf{k}[G/H],\tag{13}\] such that \[\pi(\gamma(x))=x;\quad\forall x\in \mathbf{k}[G/H].\]

Consider the map \(\eta=\eta_H:=\operatorname{id}\star(\gamma^{-1}\pi)\); that is, \[\label{etafromgamma} \eta: \mathbf{k}[G] \to \mathbf{k}[H],\quad u\mapsto u_1\gamma^{-1}(\pi(u_2)).\tag{14}\] Note that since \(\gamma\) and \(\gamma^{-1}\) preserve the counit, \[\pi(u_1 \gamma^{-1}(\pi(u_2))) = \pi(u_1) S(\pi(u_2)) = \varepsilon(u) = \varepsilon(\gamma^{-1}\pi(u)) = \varepsilon(u_1 \gamma^{-1}(\pi(u_2))),\] so indeed \(\eta(u) \in \mathbf{k}[G]^{\text{co} \mathbf{k}[G/H]} = \mathbf{k}[H]\).

Recall that the Hopf algebra \(\mathbf{k}[G/H]\) measures \(\mathbf{k}[H]\) via \(\cdot\), given by \[\mathbf{k}[G/H]\otimes\mathbf{k}[H]\to \mathbf{k}[H],\quad x\cdot u=\gamma(x_1)u\gamma^{-1}(x_2).\] Recall also that the \(\mathbf{k}\)-linear map \[\label{defnolsigma} \overline{\sigma}: \mathbf{k}[G/H] \otimes \mathbf{k}[G/H] \to \mathbf{k}[H],\quad \overline{\sigma}(x,y)=\gamma(x_1)\gamma(y_1)\gamma^{-1}(x_2y_2),\tag{15}\] is a \(2\)-cocycle, and the \(\mathbf{k}\)-linear map \[\label{defnoltau} \begin{align} & \overline{\tau}: \mathbf{k}[G/H] \to \mathbf{k}[H]\otimes \mathbf{k}[H],\\ & \overline{\tau}(x)= \eta^{-1}(\gamma(x)_1)_1\eta(\gamma(x)_2)\otimes \eta^{-1}(\gamma(x)_1)_2\eta(\gamma(x)_3), \end{align}\tag{16}\] is a co-cocycle. For example, to see that \(\overline{\sigma}\) is well defined, note first that \[\pi(\gamma(x_1)\gamma(y_1)\gamma^{-1}(x_2y_2)) = x_1y_1S(x_2y_2) = \varepsilon(xy).\] On the other hand, using (13 ) we have \[\begin{align} \\ & = & \varepsilon(\gamma^{-1}(\pi(\varepsilon(\gamma(x)_1) \gamma(x)_2 \varepsilon(\gamma(y)_1)\gamma(y)_2))) \\ & = & \varepsilon(\gamma^{-1}(\pi(\gamma(x)\gamma(y)))) = \varepsilon(\gamma^{-1}(xy))= \varepsilon(xy). \end{align}\] This shows that \(\gamma(x_1)\gamma(y_1)\gamma^{-1}(x_2y_2) \in \mathbf{k}[G]^{\text{co} \mathbf{k}[G/H]} = \mathbf{k}[H]\), as required.

Finally, it is well known (see e.g. [20]) that the map \[{\rm f}:\mathbf{k}[G]\to \mathbf{k}[H]\#_{\overline{\sigma}}^{\overline{\tau}} \mathbf{k}[G/H],\quad u\mapsto \eta\left(u_1\right)\# \pi\left(u_2\right),\] is a Hopf algebra isomorphism, whose inverse is given by \[{\rm f}^{-1}:\mathbf{k}[H]\#_{\overline{\sigma}}^{\overline{\tau}}\mathbf{k}[G/H]\to \mathbf{k}[G],\quad v\# x\mapsto v\gamma(x).\] For example, using (13 ) and the fact that \(\eta\) is \(\mathbf{k}[H]\)-linear, we compute \[\begin{align} \\ & = & v_1\eta\left(\gamma(x)_1\right)\otimes\varepsilon(v_2)\pi\left(\gamma(x)_2\right)= v\eta\left(\gamma(x)_1\right)\otimes\pi\left(\gamma(x)_2\right)\\ & = & v\eta\left(\gamma(x_1)\right)\otimes x_2=v\varepsilon(x_1)\otimes x_2=v\otimes x, \end{align}\] and \[\begin{align} \\ & = & u_1\gamma^{-1}(\pi(u_2))\gamma\left(\pi\left(u_3\right) \right)=u_1\varepsilon(\pi(u_2))=u, \end{align}\] so \({\rm f}\) and \({\rm f}^{-1}\) are indeed inverse of each other.

2.3 The Drinfeld double of a finite group scheme↩︎

Fix a finite group scheme \(G\) over \(\mathbf{k}\) as in §2.1. Recall that the Drinfeld double \[D(G)=\mathscr{O}(G)^{{\rm cop}} \bowtie \mathbf{k}[G]\] of \(\mathbf{k}[G]\) is the tensor coalgebra \(\mathscr{O}(G)^{{\rm cop}} \otimes \mathbf{k}[G]\), equipped with the product rule \[(b \bowtie u)(b' \bowtie u') = b(u_1 \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b') \bowtie u_2u',\] where \(\rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\) is defined in (9 ). Recall that \(D(G)\) is an involutive Hopf algebra, such that the natural maps \[\mathbf{k}[G] \hookrightarrow D(G),\quad {\rm and}\quad \mathscr{O}(G)^{{\rm cop}} \hookrightarrow D(G)\] are Hopf algebra inclusions. Since for every \(u \in \mathbf{k}[G]\) and \(b \in \mathscr{O}(G)\), \[\begin{align} \\ & = & (u_1 \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b) \bowtie u_2S(u_3)= (u_1 \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b) \bowtie \varepsilon(u_2) = (u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b) \bowtie 1, \end{align}\] it follows that \(\mathscr{O}(G)\) is a normal Hopf subalgebra of \(D(G)\), so we have an exact sequence of Hopf algebras \[\mathbf{k}\rightarrow \mathscr{O}(G) \rightarrow D(G) \rightarrow D(G)/(D(G)\mathscr{O}(G)^+) \rightarrow \mathbf{k}.\] Since the map \[\label{qkg} D(G)/(D(G)\mathscr{O}(G)^+) \to \mathbf{k}[G],\quad [b \bowtie u]\mapsto \varepsilon(b)u,\tag{17}\] is a Hopf algebra isomorphism, we have an exact sequence of Hopf algebras \[\mathbf{k}\rightarrow \mathscr{O}(G) \rightarrow D(G) \rightarrow \mathbf{k}[G] \rightarrow \mathbf{k}.\]

Finally, let \(\mathcal{B}\) be a basis for \(\mathbf{k}[G]\), and let \(\{\delta_u\mid u\in\mathcal{B}\}\) be the dual basis for \(\mathscr{O}(G)\). Recall that the element \[\label{rmatrix} R:=\sum_{u \in \mathcal{B}} (1\bowtie u) \otimes (\delta_u\bowtie 1)\tag{18}\] is an \(R\)-matrix for \(D(G)\), and \((D(G),R)\) is factorizable and ribbon, with ribbon element \[\label{ribbon} V:=\sum_{u \in \mathcal{B}} S(\delta_u)\bowtie u.\tag{19}\]

2.4 The representation category of \(D(G)\)↩︎

Fix a finite group scheme \(G\) over \(\mathbf{k}\) as in §2.1. Let \({\rm ad_{\ell}}\) be the left adjoint action of \(G\) on itself, defined by \[{\rm ad_{\ell}}(u)(\tilde{u})=u_1\tilde{u}S(u_2);\quad \forall u,\tilde{u}\in \mathbf{k}[G].\] Let \({\rm C}:=G/{\rm ad_{\ell}}\) be the finite quotient scheme of conjugacy orbits in \(G\) [1], [11], and let \({\rm p}:G\twoheadrightarrow {\rm C}:=G/{\rm ad_{\ell}}\) be the quotient scheme morphism [17]. Then \({\rm p}^{\sharp}:\mathscr{O}({\rm C})\xrightarrow{1:1} \mathscr{O}(G)\) is an injective algebra map, and \[\text{im}({\rm p}^{\sharp})=\mathscr{O}(G)^{{\rm co}G}:=\{b\in \mathscr{O}(G)\mid {\rm ad_{\ell}}^{\sharp}(b)=1\otimes b\}\] is a subalgebra of \(\mathscr{O}(G)\), so that \(\mathscr{O}({\rm C})=\mathscr{O}(G)^{{\rm co}G}\) via \({\rm p}^{\sharp}\).

Given a closed point \(g\in G(\mathbf{k})\), consider the scheme morphism \[\label{adg} {\rm ad}_g:G\to G,\quad f\mapsto {\rm ad_{\ell}}(g,f)=fgf^{-1},\tag{20}\] and let \[\label{Cg} C_g:=\text{im}({\rm ad}_g)\subset G\tag{21}\] be the conjugacy orbit of \(g\) [1], [11]. Then \[\mathbf{k}[C_g]:=\{{\rm ad_{\ell}}(u)(g)\mid u\in \mathbf{k}[G]\}\] is a subcoalgebra of \(\mathbf{k}[G]\), \(C_g(\mathbf{k})\subset G(\mathbf{k})\) is the conjugacy class of \(g\) in \(G(\mathbf{k})\), and \({\rm C}(\mathbf{k})=\{C_g\mid g\in G(\mathbf{k})\}\). Also, let \[\label{Gg} G_g:=\{f\in G\mid {\rm ad}_g(f)=g\}\tag{22}\] be the centralizer of \(g\) in \(G\); it is a subgroup scheme of \(G\), so that \[\mathbf{k}[G_g]:=\{u\in \mathbf{k}[G]\mid ug=gu\}\] is a Hopf subalgebra of \(\mathbf{k}[G]\). The scheme morphism \({\rm ad}_g\) (20 ) factors through the finite quotient scheme \(G/G_g\), and induces a canonical scheme isomorphism \[\label{iotag} i_g:G/G_g\xrightarrow{\cong} C_g\tag{23}\] (see [17] for more details). In other words, the map \[i_g^{\sharp}:\mathscr{O}(C_g)\to \mathscr{O}(G/G_g)=\mathscr{O}(G)^{G_g}\subseteq \mathscr{O}(G),\quad \langle i_g^{\sharp}(c),u\rangle=\langle c,u_1gS(u_2)\rangle,\] is an algebra isomorphism.

Recall that \(\mathscr{O}(G/G_g)\subseteq \mathscr{O}(G)\) is a left coideal subalgebra; that is, \[\Delta\left(\mathscr{O}(G/G_g)\right)\subset \mathscr{O}(G)\otimes\mathscr{O}(G/G_g).\] Let \(q_g:=q_{G_g}:\mathscr{O}(G)\twoheadrightarrow \mathscr{O}(G_g)\) be as in (3 ), and consider the Hopf-Galois extension \[0\to \mathscr{O}(G/G_g)\hookrightarrow \mathscr{O}(G)\xrightarrow{q_g}\mathscr{O}(G_g)\to 0,\] where we view \(\mathscr{O}(G)\) as a right \(\mathscr{O}(G_g)\)-comodule via \[\rho_g:\mathscr{O}(G)\to \mathscr{O}(G)\otimes\mathscr{O}(G_g),\quad b\mapsto b_1\otimes q_g(b_2),\] so that \(\mathscr{O}(G/G_g)=\mathscr{O}(G)^{\text{co} \mathscr{O}(G_g)}=\{b\in \mathscr{O}(G)\mid b_1\otimes q_g(b_2)=b\otimes 1\}\subseteq \mathscr{O}(G)\). Let \[\mu_g:=\mu_{G_g}:\mathscr{O}(G_g)\xrightarrow{1:1}\mathscr{O}(G)\] be a section for \(q_g\) as in (4 ), and let \[\alpha_g:\mathscr{O}(G)\twoheadrightarrow\mathscr{O}(G/G_g),\quad b\mapsto b_1\mu_g^{-1}(q_g(b_2)),\] be a retraction for the inclusion map \(\mathscr{O}(G/G_g)\subseteq \mathscr{O}(G)\); it is \(\mathscr{O}(G/G_g)\)-linear. Recall that \(\mu_g\) is a convolution invertible \(\mathscr{O}(G/G_g)\)-colinear map; that is, \(\varepsilon \mu_g = \varepsilon\) and \[\label{mu-gcolinear} \mu_g(d_1)\otimes d_2=\mu_g(d)_1\otimes q_g(\mu_g(d)_2);\quad\forall d\in \mathscr{O}(G_g),\tag{24}\] such that for every \(d\in \mathscr{O}(G_g)\), we have \[\label{mu-greta-g} q_g(\mu_g(d))=d\quad and \quad\alpha_g(\mu_g(d))=\varepsilon(d).\tag{25}\]

For each \(C\in {\rm C}(\mathbf{k})\) (21 ), let \(\pi_C:C\twoheadrightarrow C(\mathbf{k})\) and \(\gamma_C:C(\mathbf{k})\xrightarrow{1:1} C\) be the scheme morphisms obtained from \(\pi:G\twoheadrightarrow G(\mathbf{k})\) and \(\gamma:G(\mathbf{k})\xrightarrow{1:1} G\) (1 ) by restriction, and define the \(\mathscr{O}(G)\)-linear algebra maps \[\begin{gather} \chi_{C}:=\operatorname{id}\otimes\gamma_{C}^{\sharp}:\mathscr{O}(G^{\circ})\otimes\mathscr{O}(C)\twoheadrightarrow \mathscr{O}(G^{\circ})\otimes\mathscr{O}(C(\mathbf{k})),\,\,\,{\rm and}\\ \nu_{C}:=\operatorname{id}\otimes\pi_{C}^{\sharp}:\mathscr{O}(G^{\circ})\otimes\mathscr{O}(C(\mathbf{k}))\xrightarrow{1:1}\mathscr{O}(G^{\circ})\otimes\mathscr{O}(C). \end{gather}\] (See [1], [11] for more details.)

Now let \(\mathscr{Z}(G):=\operatorname{Rep}(D(G))\) be the category of finite dimensional \(\mathbf{k}\)-representations of \(D(G)\); it is a finite non-degenerate ribbon braided tensor category. Recall [9] that there is a canonical tensor equivalence \[\label{the32center32is32gequiv} {\rm Coh}(G)^{G}\simeq \mathscr{Z}(G),\tag{26}\] where \({\rm Coh}(G)^{G}\) is the category of \(G\)-equivariant sheaves on \(G\) with respect to the right conjugation action of \(G\) on itself (7 ). That is, an object of \({\rm Coh}(G)^{G}\) is an object \(X\in {\rm Coh}(G)\) equipped with a right \(\mathscr{O}(G)\)-comodule structure \(\rho:X\to X\otimes\mathscr{O}(G)\), such that \(\rho(a\cdot x)={\rm ad_r}^{\sharp}(a)\cdot \rho(x)\). In particular, \(\mathscr{Z}(G)\) is a group scheme-theoretical category [9].

For each \(C\in {\rm C}(\mathbf{k})\), let \(\mathscr{Z}(G)_{C}\subset \mathscr{Z}(G)\) be the full Abelian subcategory consisting of all objects annihilated by the defining ideal \(\mathscr{I}(C)\subset \mathscr{O}(G)\) of \(C\), and let \(\overline{\mathscr{Z}(G)_{C}}\) be the Serre closure of \(\mathscr{Z}(G)_{C}\) inside \(\mathscr{Z}(G)\); that is, \(\overline{\mathscr{Z}(G)_{C}}\) is the full Abelian subcategory of \(\mathscr{Z}(G)\) consisting of all objects whose composition factors belong to \(\mathscr{Z}(G)_{C}\).

3 Hopf algebra quotients of \(D(G)\)↩︎

Fix a finite group scheme \(G\) over \(\mathbf{k}\) as in §2.1.

3.1 Construction of Hopf quotients of \(D(G)\)↩︎

Let \(K\) be a normal subgroup scheme of \(G\). Let \(q_K: \mathscr{O}(G) \twoheadrightarrow \mathscr{O}(K)\) be the corresponding surjective Hopf algebra map, and \[\label{mu} \mu_K:\mathscr{O}(K)\xrightarrow{1:1} \mathscr{O}(G)\tag{28}\] be a section for \(q_{K}\) as in (3 )-(4 ). Since \(K\) is normal in \(G\), the Hopf subalgebra \(\mathscr{O}(G/K)\) of \(\mathscr{O}(G)\) is stable under the left coadjoint \(\mathbf{k}[G]\)-action (9 ), so we obtain a well defined action \(*\) of \(\mathbf{k}[G]\) on \(\mathscr{O}(K)\), given by \[\label{themeasureofog} \mathbf{k}[G] \otimes \mathscr{O}(K) \to \mathscr{O}(K),\quad u\otimes a\mapsto u * a:= q_K(u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\mu_K(a)),\tag{29}\] which is independent of the choice of \(\mu_K\). Indeed, if \(\mu_K':\mathscr{O}(K)\to \mathscr{O}(G)\) is another section of \(q_K\), then \(\mu_K'(a)-\mu_K(a)\in \mathscr{O}(G/K)^+\mathscr{O}(G)\) for every \(a\in \mathscr{O}(K)\). Since the Hopf ideal \(\mathscr{O}(G/K)^+\mathscr{O}(G)\) is stable under the left coadjoint \(\mathbf{k}[G]\)-action (9 ), we have \(u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup(\mu_K'(a)-\mu_K(a)) \in \mathscr{O}(G/K)^+\mathscr{O}(G)\) for all \(u \in \mathbf{k}[G]\). Hence, \[q_K (u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\mu_K'(a)) = q_K (u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\mu_K(a)),\] so \(*\) (29 ) is independent of the choice of \(\mu_K\), as claimed.

In particular, we have \[\label{piyaction} u * q_K(b):= q_K(u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b);\quad \forall u\in \mathbf{k}[G],\, b\in \mathscr{O}(G).\tag{30}\]

Suppose now that \(H\) is another normal subgroup scheme of \(G\), so that we have an exact sequence of Hopf algebras \[\mathbf{k}\xrightarrow{\iota_H} \mathbf{k}[H]\to \mathbf{k}[G]\xrightarrow{\pi_H} \mathbf{k}[G/H] \to \mathbf{k},\] and choose a section \(\gamma_H\) as in (12 ). Assume further that \(H\) and \(K\) centralize each other; that is, \(vw=wv\) for every \(v\in \mathbf{k}[H]\) and \(w \in \mathbf{k}[K]\). Equivalently, \[{\rm ad_{r}}(v)(w)=S(v_1)wv_2 = \varepsilon(v)w;\quad\forall v\in \mathbf{k}[H],\,w \in \mathbf{k}[K].\] Then for any \(v \in \mathbf{k}[H]\) and \(b \in \mathscr{O}(G)\), we have for every \(w \in \mathbf{k}[K]\), \[\begin{align} \\ & = & {\left\langle b, {\rm ad_{r}}(v)(\iota_K(w)) \right\rangle} = {\left\langle b, \varepsilon(v)\iota_K(w) \right\rangle}={\left\langle \varepsilon(v)q_K(b),w \right\rangle}, \end{align}\] so \(q_K(v \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b) = \varepsilon(v)q_K(b)\). Thus, (29 ) induces a well defined \(\mathbf{k}\)-linear map \(\cdot\), given by \[\label{themeasure} \mathbf{k}[G/H] \otimes \mathscr{O}(K) \to \mathscr{O}(K),\quad x \otimes a\mapsto x\cdot a:=\gamma_H(x)*a =q_K(\gamma_H(x)\rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\mu_K(a)),\tag{31}\] which is independent of the choice of \(\gamma_H\) (as well as \(\mu_K\)). Indeed, if \(\gamma_H':\mathbf{k}[G/H]\to \mathbf{k}[G]\) is another section of \(\pi_H\), then for each \(x\in \mathbf{k}[G/H]\), we have \(\gamma_H'(x)-\gamma_H(x)\in \mathbf{k}[G]\mathbf{k}[H]^+\). But for any \(uv\), \(u\in \mathbf{k}[G]\) and \(v\in \mathbf{k}[H]^+\), we have that \[(uv)*a=u*(v*a)=u*(q_K(v \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup\mu_K(a)))=u*(\varepsilon(v)q_K(\mu_K(a)))=0\] for every \(a\in \mathscr{O}(K)\). Therefore, for every \(a\in \mathscr{O}(K)\), \[\gamma_H'(x)*a = (\gamma_H(x)+\gamma_H'(x)-\gamma_H(x))*a = \gamma_H(x)*a,\] so \(\cdot\) (31 ) is independent of the choice of \(\gamma_H\), as claimed.

In particular, we have \[\label{piyactiononq} \pi_H(u)\cdot a=u*a;\quad \forall u\in \mathbf{k}[G],\, a\in \mathscr{O}(K),\tag{32}\] and by (30 ), we have \[\label{piyactiononq0} \pi_H(u)\cdot q_K(b)=u*q_K(b)= q_K(u \rightharpoonup\mathrel{\mspace{-15mu}}\rightharpoonup b);\quad \forall u\in \mathbf{k}[G],\, b\in \mathscr{O}(G).\tag{33}\]

Now suppose in addition that we have a Hopf algebra map \[\label{B} B : \mathbf{k}[H] \to \mathscr{O}(K),\tag{35}\] which is \(G\)-equivariant, in the sense that \[\label{Binv} \pi_H(u)\cdot B(v)=u*B(v)=B({\rm ad_{\ell}}(u)(v));\quad \forall u \in \mathbf{k}[G],\,v \in \mathbf{k}[H].\tag{36}\]

Now let \(\eta_H\), \(\overline{\sigma}\) and \(\overline{\tau}\) be as in (14 ), (15 ) and (16 ), respectively, and define the \(\mathbf{k}\)-linear maps \[\sigma:=B\overline{\sigma}\quad\text{and}\quad\tau:=B^{\otimes 2}\overline{\tau};\] that is, \[\label{defnsigma} \sigma : \mathbf{k}[G/H] \otimes \mathbf{k}[G/H] \to \mathscr{O}(K),\quad \sigma(x,y)=B\left(\gamma_H(x_1)\gamma_H(y_1)\gamma_H^{-1}(x_2y_2)\right),\tag{39}\] and \[\label{defntau} \begin{align} & \tau : \mathbf{k}[G/H] \to \mathscr{O}(K)\otimes \mathscr{O}(K),\\ & \tau(x)=(B\otimes B) \left\{ \eta^{-1}_H(\gamma_H(x)_1)_1\eta_H(\gamma_H(x)_2)\otimes \eta^{-1}_H(\gamma_H(x)_1)_2\eta_H(\gamma_H(x)_3) \right\}, \end{align}\tag{40}\] and write \(\tau(x)=\tau(x)^1\otimes \tau(x)^2\). Note that \(\tau(x)^1\otimes \tau(x)^2=\tau(x)^2\otimes \tau(x)^1\), by cocommutativity of \(\mathbf{k}[G]\).

3.2 Classification of Hopf quotient pairs of \(D(G)\)↩︎

A Hopf quotient pair of \(D(G)\) is a pair \((D,\varphi)\) where \(\varphi:D(G)\twoheadrightarrow D\) is a surjective Hopf algebra map. We will say that two Hopf quotient pairs \((D_1,\varphi_1)\) and \((D_2,\varphi_2)\) are equivalent if there is a Hopf algebra isomorphism \(f:D_1\xrightarrow{\cong} D_2\), such that \(\varphi_2=f\varphi_1\).

Fix two Hopf quotient pairs \((D(K,H,B),\theta)\) and \((D\left(K',H',B'\right),\theta')\) of \(D(G)\), and define the Hopf algebra map \[\label{phibbprime} \begin{align} & \beta _{B,B'}:\mathbf{k}[K\cap K']\to \mathscr{O}(H\cap H')\\ & w\mapsto \left(\iota^{\sharp}_{H\cap H',H}B^*\iota_{K\cap K',K}(w_1)\right)\left(\iota^{\sharp}_{H\cap H',H'}\overline{B'}\iota_{K\cap K',K'}(w_2)\right); \end{align}\tag{45}\] that is, \(\beta_{B,B'} =\left(\iota^{\sharp}_{H\cap H',H}B^*\iota_{K\cap K',K}\right)\star\left(\iota^{\sharp}_{H\cap H',H'}\overline{B'}\iota_{K\cap K',K'}\right)\).

Let \(L\subseteq K\cap K'\) be the subgroup scheme, such that \[{\rm Ker}(\beta_{B,B'})=\mathbf{k}[L]^+\mathbf{k}[K\cap K'],\] and define the \(G\)-equivariant Hopf algebra map \[\label{psibbprime} \begin{align} & \mathbf{B}_{B,B'}:\mathbf{k}[L]\to \mathscr{O}(HH'),\\ & w\mapsto \left(\mu_{H,HH'}B^*\iota_{L,K}(w_1)\right)\left(\mu_{H',HH'}\overline{B'}\iota_{L,K'}(w_2)\right); \end{align}\tag{46}\] that is, \(\mathbf{B}_{B,B'}=\left(\mu_{H,HH'}B^*\iota_{L,K}\right)\star\left(\mu_{H',HH'}\overline{B'}\iota_{L,K'}\right)\). Set \[\label{Bint} \mathbf{B}:=\mathbf{B}_{B,B'}^*:\mathbf{k}[HH']\to \mathscr{O}(L),\tag{47}\] and let \(\left(D\left(L,HH',\mathbf{B}\right),\Theta\right)\) be the corresponding Hopf quotient pair of \(D(G)\).

3.3 The quasitriangular ribbon structure on \(D(K,H,B)\)↩︎

Fix bases \(\mathcal{B}_K\subseteq \mathcal{B}\) for \(\mathbf{k}[K]\) and \(\mathbf{k}[G]\), respectively, and let \(\{\delta_u\mid u\in \mathcal{B}\}\) be the dual basis for \(\mathscr{O}(G)\). It is clear that \(\{q_K(\delta_u)\mid u\in \mathcal{B}_K\}\) is the dual basis of \(\mathcal{B}_K\) for \(\mathscr{O}(K)\), and that for \(u\in \mathcal{B}_K\), \(\langle q_K(\delta_u),v\rangle\ne 0\) if and only if \(v\in \mathcal{B}_K\).

4 Tensor subcategories of \(\operatorname{Rep}(D(G))\)↩︎

Fix a finite group scheme \(G\) over \(\mathbf{k}\) as in §2.1.

4.1 Construction and classification↩︎

Let \(\mathscr{Z}(G):=\operatorname{Rep}(D(G))\) as in §2.4. For each triple \((K,H,B)\) as in Theorem 6, set \[\mathscr{Z}(K,H,B):=\operatorname{Rep}(D(K,H,B)).\] Then using the surjective Hopf algebra map \(\theta:D(G)\twoheadrightarrow D(K,H,B)\), we can (and will) view \(\mathscr{Z}(K,H,B)\) as a tensor subcategory of \(\mathscr{Z}(G)\).

Theorem 11 yields a classification of certain finite braided group scheme-theoretical categories [9].

4.2 The Müger centralizer of \(\mathscr{Z}(K,H,B)\)↩︎

For each Hopf algebra \(D(K,H,B)\), recall the Hopf algebra \(D(H,K,\overline{B})\) from Proposition 8(6).

Recall (see, e.g., [16]) that a finite braided tensor category is called non-degenerate if its Müger center is trivial.

4.3 The simples and projectives of \(\mathscr{Z}(K,H,B)\)↩︎

Retain the notation from §2.4.

Fix a triple \(\left(K,H,B\right)\) as in Theorem 6. Recall that \(K\subseteq G\) is a normal subgroup scheme, so \(C_g\subset K\) for every \(g\in K(\mathbf{k})\), and let \({\rm C}_K(\mathbf{k}):=\{C_g\mid g\in K(\mathbf{k})\}\subseteq {\rm C}(\mathbf{k})\).

Fix \(C_g\in {\rm C}_K(\mathbf{k})\), with representative \(g\in C_g(\mathbf{k})\), and let \((M,\rho_M)\in {\rm Corep}(\mathscr{O}(G_g))\). Recall that for \(m\in M\), we write \(\rho_M(m)=m^{(-1)}\otimes m^{(0)}\in \mathscr{O}(G_g)\otimes M\). Let \[\begin{align} & \rho_M^g:\mathscr{O}(C_g)\otimes M\to \mathscr{O}(C_g)\otimes M\otimes \mathscr{O}(G),\\ & \rho_M^g(c\otimes m)= \left(i_g^{-1}\right)^{\sharp}\alpha_{g}\left(i_g^{\sharp}(c)_1\mu_g\left(m^{(-1)}\right)_1 \right)\otimes m^{(0)}\otimes i_g^{\sharp}(c)_2\mu_g\left(m^{(-1)}\right)_2, \end{align}\] and define \(\mathbf{F}_{C_g}(M,\rho_M)=\left(\mathscr{O}(C_g)\otimes M,\rho_M^g\right)\in \mathscr{Z}(G)_{C_g}\). In particular, the action of \(u\in\mathbf{k}[G]\) on \(c\otimes m\in \mathscr{O}(C_g)\otimes M\) is given by \[\label{actGonCg} \begin{align} & u\cdot (c\otimes m)=\left\langle u,i_g^{\sharp}(c)_2\mu_g\left(m^{(-1)}\right)_2\right\rangle \left(i_g^{-1}\right)^{\sharp}\alpha_{g}\left(i_g^{\sharp}(c)_1\mu_g\left(m^{(-1)}\right)_1 \right)\otimes m^{(0)}. \end{align}\tag{48}\] Note that since \(g\in K\), \(H\subseteq G\) is normal, and \(H\) centralizes \(K\), it follows that \(H\subseteq G_g\) is a normal subgroup scheme, and \(q_{G,H}=q_{G_g,H}\circ q_{G,G_g}\).

From the \(G\)-equivariant Hopf algebra map \(B:\mathbf{k}[H]\to \mathscr{O}(K)\), we obtain the map \[\label{charBg} B_g:\mathbf{k}[H]\to \mathbf{k},\quad v\mapsto \langle B(v),g\rangle,\tag{49}\] which is a \(G_g\)-invariant character since for every \(u\in \mathbf{k}[G_g]\) and \(v\in \mathbf{k}[H]\), \[B_g({\rm ad_{\ell}}(S(u))(v))=\langle B({\rm ad_{\ell}}(S(u))(v)),g\rangle=\langle B(v),{\rm ad_{\ell}}(u)(g)\rangle=\langle B(v),g\rangle=B_g(v).\] We also obtain the Drinfeld twist \(\psi_g\) for \(\mathscr{O}(G_g/H)\) (i.e., a Hopf \(2\)-cocycle for \(\mathbf{k}[G_g/H]\)), given by \[\label{twistJg} \psi_g(x,y)=\langle \sigma(x,y),g\rangle ;\quad\forall x,y\in \mathbf{k}[G_g/H].\tag{50}\] Let \(\mathbf{k}^{\psi_g}[G_g/H]=(\mathscr{O}(G_g/H)_{\psi_g})^*\) be the associated twisted group algebra, and let \(\cdot\) denote its multiplication.

The injective coalgebra homomorphism \(p_g^*:\mathscr{O}(G_g/H)_{\psi_g}\xrightarrow{1:1}\mathscr{O}(G_g)\) induces an Abelian embedding \({\rm Corep}(\mathscr{O}(G_g/H)_{\psi_g})\subseteq {\rm Corep}(\mathscr{O}(G_g))\), so that \({\rm Corep}(\mathscr{O}(G_g/H)_{\psi_g})\) consists of all \(G_g\)-modules \(M\) for which \[\label{hinv} m\cdot \iota_{H,G_g}(v)=B_g(v)m;\quad \forall v\in \mathbf{k}[H],\,m\in M.\tag{51}\]

Now for each \(C_g\in {\rm C}_K(\mathbf{k})\), consider the full Abelian subcategory \[\widetilde{\mathscr{Z}}(K,H,B)_{C_g}:=\mathbf{F}_{C_g}({\rm Corep}(\mathscr{O}(G_g/H)_{\psi_g}))\subseteq \mathscr{Z}(G)_{C_g},\] and let \[\label{Dg} \widetilde{\mathscr{Z}}(K,H,B):=\bigoplus_{C_g\in {\rm C}_K(\mathbf{k})}\overline{\widetilde{\mathscr{Z}}(K,H,B)_{C_g}}\subseteq \mathscr{Z}(G).\tag{52}\]

5 Some special cases and examples↩︎

5.1 Canonical examples↩︎

The following are some canonical Hopf quotients of \(D(G)\) and their corresponding braided subcategories of \(\mathscr{Z}(G)\):

(1) We have \[D(1,G,1)=\mathbf{k},\quad D(G,1,1)=D(G),\quad D(1,1,1)=\mathbf{k}[G].\] Thus, we have \[\mathscr{Z}(1,G,1)={\rm Vect},\quad \mathscr{Z}(1,G,1)'=\mathscr{Z}(G,1,1)=\mathscr{Z}(G),\quad \mathscr{Z}(1,1,1)=\operatorname{Rep}(G).\]

(2) For every normal subgroup scheme \(K\subseteq G\), we have \(D(1,K,1)=\mathbf{k}[G/K]\), and \(D(K,1,1)=\mathscr{O}(K)^{{\rm cop}}\# \mathbf{k}[G]\) is a tensor product coalgebra and smash product algebra. In particular, \(D(1,G^{\circ},1)=\mathbf{k}[G(\mathbf{k})]\), and \(D(G^{\circ},1,1)=\mathscr{O}(G^{\circ})^{{\rm cop}}\# \mathbf{k}[G]\).

(3) For every normal subgroup scheme \(K\subseteq G\), a central subgroup scheme \(H\subseteq G\), and a \(G\)-equivariant group scheme morphism \(B:K\to H^{\vee}\), the Hopf algebra \[D(K,H,B)=\mathscr{O}(K)\#_{\sigma}^{\tau} \mathbf{k}[G/H]\] is a tensor product coalgebra and smash product algebra, and by Proposition 8(7), \[D(H,K,\overline{B})=\mathbf{k}[H^{\vee}]\#_{\sigma}^{\tau} \mathbf{k}[G/K]=\mathbf{k}[\widetilde{G}],\] where \(\widetilde{G}\) is the group scheme extension of \(H^{\vee}\) by \(G/K\) corresponding to \[\sigma\in Z^2(H^{\vee},G/K),\quad \tau\in Z^1(H^{\vee},G/K),\] with trivial action. Thus, we have \[\mathscr{Z}(K,H,B)=\operatorname{Rep}\left(\mathscr{O}(K)\#_{\sigma}^{\tau} \mathbf{k}[G/H]\right),\] and \(\mathscr{Z}(K,H,B)'=\mathscr{Z}(H,K,\overline{B})=\operatorname{Rep}(\widetilde{G})\).

5.2 Direct products↩︎

Let \(G:=K\times H\), and let \(B:\mathbf{k}[H]\xrightarrow{}\mathscr{O}(K)\) be any \(G\)-equivariant Hopf algebra map. Then \[\mathscr{Z}(K\times 1,1\times H,B)\subseteq \mathscr{Z}(G)\] is a non-degenerate braided subcategory. Note that since \(\sigma\) and \(\tau\) are trivial by Remark 5, it follows that \[\mathscr{Z}(K\times 1,1\times H,B)\cong \mathscr{Z}(K)\] as braided categories (regardless of the choice of \(B\)).

5.3 Constant groups↩︎

Assume that \(G\) is constant; that is, \(G=G(\mathbf{k})\). For \(g\in G\), let \(\delta_g\in \mathscr{O}(G)\) be the delta function at \(g\). The elements \(\{\delta_g\}_{g\in G}\) are pairwise orthogonal idempotents, hence any \(\mathscr{O}(G)\)-module \(V\) decomposes as \[V = \bigoplus_{g\in G} V_g, \qquad V_g:=\delta_gV.\] Recall that in this case a \(D(G)\)-module is a \(G\)-graded vector space \(V=\bigoplus_{g\in G}V_g\), together with an action of \(G\), such that \(x\cdot V_g \subseteq V_{xgx^{-1}}\) for every \(x,g\in G\).

Now fix a triple \((K,H,B)\) and let \(\theta:D(G)\twoheadrightarrow D(K,H,B)\) be the quotient map as in Theorem 6. Since in this case \(\gamma_H\) is a coalgebra map, we have that \[D(K,H,B)=\mathscr{O}(K)^{{\rm cop}}\#_{\sigma}\mathbf{k}[G/H]\] is a tensor product coalgebra by Proposition 8(4).

If \(V\in\operatorname{Rep}(D(K,H,B))=\mathscr{Z}(K,H,B)\), and we inflate \(V\) along \(\theta\) to a \(D(G)\)-module, then the \(\mathscr{O}(G)\)-action factors through \(q_K:\mathscr{O}(G)\twoheadrightarrow\mathscr{O}(K)\). In particular, \(q_K(\delta_g)=0\) for \(g\notin K\), so \(V_g=\delta_gV=0\) unless \(g\in K\). Thus, \(V\) is supported on \(K\). Moreover, for \(h\in H\), we have \(\pi_H(h)=1\) and \(\eta_H(h)=h\), hence \[\theta(1\bowtie h)=B(h)\# 1\in \mathscr{O}(K)\#_{\sigma}\mathbf{k}[G/H],\] so on each homogeneous component \(V_k\), \(k\in K\), the element \(h\) acts by the scalar \(B(h)(k)\). Equivalently, setting \[\beta:H\times K\to \mathbf{k}^\times,\qquad \beta(h,k):=B(h)(k),\] we see that \(\mathscr{Z}(K,H,B)\) consists precisely of \(G\)-equivariant sheaves on \(G\) supported on \(K\) for which the action of \(H\) on the fibre over \(k\in K\) is given by \(\beta(h,k)\), so in the fusion case, \(\mathscr{Z}(K,H,B)\) is the category \(\mathcal{S}(K,H,B)\) from [2].

It follows from Theorem 13, similarly to [11], that we have a direct sum decomposition of Abelian categories \[\mathscr{Z}(K,H,B)=\bigoplus_{C_g\in {\rm C}_K}\mathscr{Z}(K,H,B)_{C_g}\simeq \bigoplus_{C_g\in {\rm C}_K}{\rm Corep}(\mathscr{O}(G_g/H)_{\psi_g}),\] given by \(\mathbf{F}_{C_g}\), \(C_g\in {\rm C}_K\), and for every simple \(M\in {\rm Corep}(\mathscr{O}(G_g/H)_{\psi_g})\), we have \[P_{\mathscr{Z}(K,H,B)}\left(\mathbf{F}_{C_g}(M)\right)= \mathbf{F}_{C_g}(P_{(G_g/H,\psi_g)}(M))= \left(\mathscr{O}(C_g)\otimes P_{(G_g/H,\psi_g)}(M),\rho_{P_{(G_g/H,\psi_g)}(M)}^g\right),\] where \(\mathscr{O}(K)\) acts on the first factor.

5.4 Connected groups↩︎

Assume that \(G\) is connected; that is, \(G(\mathbf{k})=1\) (e.g., \(G\) is the finite group scheme associated to a finite dimensional restricted \(p\)-Lie algebra \(\mathfrak{g}\)). Then it follows from Theorem 13, similarly to [11], that for each triple \(\left(K,H,B\right)\) as in Theorem 6, we have \(\mathscr{Z}(K,H,B)=\overline{{\rm Rep}\left(G/H\right)}\), and for every simple \(M\in \operatorname{Rep}\left(G/H\right)\), we have \[P_{\mathscr{Z}(K,H,B)}\left(\mathbf{F}(M)\right)\cong \left(\mathscr{O}(K)\otimes P_{G/H}(M),\operatorname{id}\otimes\rho_{P_{G/H}(M)}\right),\] where \(\mathscr{O}(K)\) acts on the first factor.

5.5 Trivial \(B\)↩︎

Set \(\mathscr{Z}(K,H):=\mathscr{Z}(K,H,1)\). Since by definition, an object \(X\) in \({\rm Coh}(K)^{G/H}\) is an \(\mathscr{O}(K)\)-module in the category \(\operatorname{Rep}(G/H)\), it follows from Proposition 8(5) that we have a natural embedding of tensor categories \[{\rm Coh}(K)^{G/H}\xrightarrow{1:1} \mathscr{Z}(K,H).\] Since both categories have the same Frobenius-Perron dimension, it follows that we have an equivalence of tensor categories \[\mathscr{Z}(K,H)\simeq {\rm Coh}(K)^{G/H}.\]

Now consider \(K\) as a subgroup scheme of \(K\times G/H\) via the group scheme embedding \[\Delta_{K,H}:K\to K\times G/H,\quad k\mapsto (k,\pi_H(k)).\] Then, similarly to the proof that \(\mathscr{Z}(G)\simeq {\rm Coh}(G)^{G}\simeq \mathscr{C}(G\times G,\Delta_{G,1}(G))\) [1], we have an equivalence of tensor categories \[{\rm Coh}(K)^{G/H}\simeq \mathscr{C}(K\times G/H,\Delta_{K,H}(K)),\] so \(\mathscr{Z}(K,H)\) is group scheme-theoretical in the sense of [9]. In particular, by [13], it follows that \(\mathscr{Z}(K,H)\) has the finite generation property, and [9] implies the classification of exact indecomposable module categories over \(\mathscr{Z}(K,H)\).

5.6 Commutative groups↩︎

Assume that \(A\subseteq G\) is a normal commutative subgroup scheme (e.g. \(A\) is contained in the center of \(G\)). Then \(D(A,A,1)=\mathbf{k}[A^{\vee}\times G/A]\) is a triangular Hopf algebra, where \(A^{\vee}\) is the Cartier dual of \(A\), and \[\mathscr{Z}(A,A,1)=\operatorname{Rep}\left(A^{\vee}\times G/A\right)\subseteq \mathscr{Z}(G)\] is a Lagrangian subcategory.

More generally, if \(B:A\to A^{\vee}\) is any \(G\)-equivariant group scheme morphism, then \((D(A,A,B),R_B)=(\mathbf{k}[A^{\vee}]\#_{\sigma}^{\tau} \mathbf{k}[G/A],R_B)=(\mathbf{k}[\widetilde{G}],R_B)\) is a quasitriangular Hopf algebra (see Proposition 8(7)), and by Corollary 7(3), the braided subcategory \[\mathscr{Z}(A,A,B)=\operatorname{Rep}\left(\mathbf{k}[A^{\vee}]\#_{\sigma}^{\tau} \mathbf{k}[G/A],R_B\right)=\operatorname{Rep}(\widetilde{G},R_B)\subseteq \mathscr{Z}(G)\] is Lagrangian if and only if \(B=\overline{B}:=B^{\vee}S\).

Assume further that \(G\) is commutative. Then for any group scheme morphism \(B:A\xrightarrow{}A^{\vee}\), we have the braided subcategory \[\mathscr{Z}(A,A,B)=\operatorname{Rep}\left(\mathbf{k}[A^{\vee}]\#_{\sigma}^{\tau} \mathbf{k}[G/A],R_B\right)=\operatorname{Rep}(\widetilde{G},R_B)\subseteq \mathscr{Z}(G);\] by Corollary 7(3), it is Lagrangian if and only if \(B=\overline{B}\).

Assume even further that \(G\) is self dual. Then for each group scheme morphism \(B:G\xrightarrow{}G^{\vee}\), by Corollary 7(2), the braided subcategory \[\mathscr{Z}(G,G,B)=\operatorname{Rep}\left(\mathbf{k}[G^{\vee}],R_B\right)\subseteq \mathscr{Z}(G)\] is non-degenerate if and only if \(B\star B^{\vee}:G\xrightarrow{\cong }G^{\vee}\) is a group scheme isomorphism.

For example, let \(G:=\mathbb{G}_{a,1}\). In §6.2 we show that \[\mathscr{Z}(\mathbb{G}_{a,1},\mathbb{G}_{a,1},B_{\lambda})=\operatorname{Rep}\left(\mathbf{k}[\mathbb{G}_{a,1}^{\vee}],R_{\lambda}\right)\subseteq \mathscr{Z}(\mathbb{G}_{a,1})\] is an infinite family of non-equivalent non-degenerate braided subcategories of Frobenius-Perron dimension \(p\), parametrized by \(\lambda\in \mathbf{k}^{\times}\), where \(R_{\lambda}\) is given in (54 ).

6 Appendix↩︎

6.1 The group scheme \(\mathbb{G}_{a,r}\)↩︎

Let \(\mathbb{G}_{a,r}\) denote the \(r\)-th Frobenius kernel of \(\mathbb{G}_a\). Recall that \(\mathscr{O}(\mathbb{G}_{a,r}) = \mathbf{k}[t]/(t^{p^r})\) with \(t\) primitive. Fix the basis \(\{t^i\}_{0\le i\le p^r-1}\) for \(\mathscr{O}(\mathbb{G}_{a,r})\) and denote by \(\{\delta_i\}_{0\le i\le p^r-1}\) the dual basis. The multiplication in \(\mathbf{k}[\mathbb{G}_{a,r}]\) is determined by \[\delta_m\delta_n=\begin{cases} \binom{m+n}{m}\delta_{m+n},& m+n<p^r\\ 0,& \text{otherwise}. \end{cases}\] The unit is \(\delta_0\), and one has \[\Delta(\delta_n)=\sum_{a+b=n}\delta_a\otimes\delta_b,\quad \varepsilon(\delta_n)=\delta_{n,0},\quad S(\delta_n)=(-1)^n\delta_n.\] It is well-known that the map \[F_r: \mathbf{k}[X_0,\ldots,X_{r-1}]/(X_0^p,\ldots,X_{r-1}^p) \to \mathbf{k}[\mathbb{G}_{a,r}],\quad X_i\mapsto \delta_{p^i},\] is an algebra isomorphism with inverse given by \[F_r^{-1}:\mathbf{k}[\mathbb{G}_{a,r}]\to \mathbf{k}[X_0,\ldots,X_{r-1}]/(X_0^p,\ldots,X_{r-1}^p),\quad \delta_n\mapsto \frac{1}{a_{0}! \cdots a_{r-1}!} X_0^{a_0} \cdots X_{r-1}^{a_{r-1}},\] where \(n = a_0 + \cdots + a_{r-1}p^{r-1}\) is the \(p\)-adic expansion of \(n\).

6.2 The center \(\mathscr{Z}(\mathbb{G}_{a,1})\)↩︎

Since \(F_1^{-1}:\mathbf{k}[\mathbb{G}_{a,1}] \xrightarrow{\cong }\mathscr{O}(\mathbb{G}_{a,1})\) is a Hopf algebra isomorphism, it follows that Hopf algebra maps \(\mathbf{k}[\mathbb{G}_{a,1}] \to \mathscr{O}(\mathbb{G}_{a,1})\) are parameterized by elements \(\lambda \in \mathbf{k}\), where \[\label{bz} B_{\lambda}: \mathbf{k}[\mathbb{G}_{a,1}] \to \mathscr{O}(\mathbb{G}_{a,1}),\quad \delta_n\mapsto \frac{\lambda^n}{n!}t^n.\tag{53}\] This is an isomorphism if and only if \(\lambda \neq 0\). Note that \(B_{\lambda}=\overline{B_{\lambda}}\) if and only if \(\lambda=0\) or \(p=2\) (since \(B_{\lambda}=B_{\lambda}^*\)).

It now follows from Corollary 3 in a straightforward manner that for each \(\lambda\in\mathbf{k}\), \[\label{rz1} R_{\lambda}:=\sum_{i=0}^{p-1}\frac{\lambda^i}{i!}(t^i\otimes t^i)\tag{54}\] is an \(R\)-matrix for \(D(\mathbb{G}_{a,1},\mathbb{G}_{a,1},B_{\lambda})=\mathbf{k}[\mathbb{G}_{a,1}^{\vee}]\). By Corollary 7(2), \((\mathbf{k}[\mathbb{G}_{a,1}^{\vee}],R_{\lambda})\) is factorizable if and only if \(\lambda \neq 0\); that is, \(R_{\lambda}\ne 1\otimes 1\).

6.3 The center \(\mathscr{Z}(\mathbb{G}_{a,2})\)↩︎

By Theorem 6, since \(\mathbb{G}_{a,2}\) is commutative, Hopf quotient pairs of \(D(\mathbb{G}_{a,2})\) are indexed by \((K,H,B)\), where \(K\) and \(H\) are subgroup schemes of \(\mathbb{G}_{a,2}\), and \(B : \mathbf{k}[H] \to \mathscr{O}(K)\) is a Hopf algebra map. We will now compute the Hopf quotient pairs for \(K = H = \mathbb{G}_{a,1}\), up to equivalence, following the construction laid out in §3.

We have \[q:=q_{\mathbb{G}_{a,1}} : \mathscr{O}(\mathbb{G}_{a,2}) \twoheadrightarrow \mathscr{O}(\mathbb{G}_{a,1}),\quad t\mapsto t,\] with section \[\mu:=\mu_{\mathbb{G}_{a,1}} : \mathscr{O}(\mathbb{G}_{a,1}) \xrightarrow{1:1} \mathscr{O}(\mathbb{G}_{a,2}),\quad t\mapsto t.\]

Next consider the exact sequence of Hopf algebras \[\mathbf{k} \to \mathbf{k}[\mathbb{G}_{a,1}] \to \mathbf{k}[\mathbb{G}_{a,2}] \xrightarrow{\pi:=\pi_{\mathbb{G}_{a,1}}} \mathbf{k}[\mathbb{G}_{a,2}/\mathbb{G}_{a,1}] \to \mathbf{k}.\]

Recall the definitions of \(\overline{\sigma}, \overline{\tau}\) and \(\sigma, \tau\) given in (15 ), (16 ) and (39 ), (40 ).

Finally, it follows from Corollary 3 in a straightforward manner that \[\label{rz2} R_{\lambda}:=\sum_{i=0}^{p-1}\frac{\lambda^i}{i!}(t^i\#1)\otimes(t^i\#1)\tag{55}\] is an \(R\)-matrix for \(D(\mathbb{G}_{a,1},\mathbb{G}_{a,1},B_{\lambda})= \mathbf{k}[\mathbb{G}_{a,1}^{\vee}]\#^{\tau} \mathbf{k}[\mathbb{G}_{a,2}/\mathbb{G}_{a,1}]=\mathbf{k}[\widetilde{\mathbb{G}_{a,1}}]\) for each \(\lambda\in\mathbf{k}\).

References↩︎