2.1 Definition and basic properties

Recall [Reference Montgomery26, Section 3.4] that a subalgebra K of a Hopf algebra H is normal if it is invariant under the left and right adjoint actions of H; that is, for all $k\in K$ and $h\in H$ ,

\begin{equation*} (\textrm{ad}_{\ell} h)(k)= \sum h_1 k S(h_2) \in K \qquad \text{and} \qquad (\textrm{ad}_{r} h)(k)= \sum S(h_1)kh_2 \in K. \end{equation*}

The following class of Hopf algebras will be the focus of this paper.

For full proofs and references for the following see [Reference Brown and Couto3, Section 2].

2.2 Further notation

For brevity we shall henceforth record the set-up of Theorem 2.2, including $\overline{H}$ and the map $\pi$ , as

\begin{equation*}A \subseteq H \textit{ is an affine commutative-by-finite Hopf algebra.}\end{equation*}

The nilrdical N(A) of the commutative affine Hopf k-algebra A is a Hopf ideal by [Reference Brown and Couto3, Lemma 5.2(1)]. There is thus an affine algebraic k-group $G_H$ such that $A/N(A) \cong \mathcal{O}(G_H)$ [Reference Waterhouse37, 11.7]. We will often identify $G_H$ with the space ${\textrm{Maxspec}}(A)$ of maximal ideals of A and with the group of algebra homomorphisms from A to k, that is with the group-like elements $G(A^{\circ})$ of the finite dual $A^{\circ}$ . For $g \in G_H$ , we denote the corresponding members of ${\textrm{Maxspec}}(A)$ and $G(A^{\circ})$ by ${\mathfrak{m}}_g$ and $\chi_g$ .

3.1 Commutative recap

When A is a commutative affine Hopf k-algebra the structure of $A^{\circ}$ is well understood—it is cocommutative and decomposes as a smash product

(3.1) \begin{equation} A^{\circ} \; \cong \; A' \# kG,\end{equation}

by the classical structure theorem for cocommutative pointed Hopf algebras, due to Cartier–Gabriel–Kostant [Reference Montgomery26, Theorem 5.6.4(3)]. Here, A ^′ is a connected Hopf subalgebra of A, and, as discussed in Section 2.2, $G = G(A^{\circ})$ is the group of characters of A. Moreover, when k has characteristic 0, A ^′ is the enveloping algebra $U(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ of G, [Reference Montgomery26, Theorem 5.6.5]. That is, $\mathfrak{g}=(A^+/{A^+}^2)^{\ast}$ , consisting of the primitive elements of $A^{\circ}$ , the functionals vanishing at 1 and ${A^+}^2$ . Thus, by [Reference Montgomery26, Proposition 9.2.5],

(3.2) \begin{equation} A' = \{f \in A^{\circ} \;:\;f((A^+)^n) = 0, \text{ for some } n > 0 \}= U(\mathfrak{g}),\end{equation}

where the first equality holds in general and the second when k has characteristic 0. Similarly, as can easily be checked, with no condition on the characteristic of k,

(3.3) \begin{equation} kG=\lbrace f\in A^{\circ}\;:\;f({\mathfrak{m}}_1\cap \ldots\cap {\mathfrak{m}}_r)=0 , r \geq 1, {\mathfrak{m}}_i\in{\textrm{Maxspec}}(A) \rbrace,\end{equation}

the functionals vanishing on some finite intersection of maximal ideals of A.

3.2 Subspaces and quotient spaces of $H^{\circ}$

The following lemma gathers together some basic facts concerning the components $A^{\circ}$ and $\overline{H}^{\ast}$ of $H^{\circ}$ . Fix throughout the rest of the paper the Hopf embedding and the Hopf surjection

\begin{equation*}\iota\;:\;A\hookrightarrow H \; \textit{ and } \; \pi\;:\;H \longrightarrow \overline{H},\end{equation*}

and the projection of right A-modules along X,

(3.4) \begin{equation}\Pi\;:\;H = A \oplus X \twoheadrightarrow A \;:\;h = a+x \mapsto a,\end{equation}

afforded by Theorem 2.2(8) when A is semiprime or central, or H is pointed.

Proof. (1),(2) Define $T \;:\!=\; \{f \in H^{\circ} \;:\;f(A^+H) = 0\}$ . Since $A^+H$ is a Hopf ideal of H, it is clear that T is a Hopf subalgebra of $H^{\circ}$ which is isomorphic as a Hopf algebra to $\overline{H}^*$ , via the map $\pi^{\circ}\;:\;\overline{H}^{\ast} \longrightarrow H^{\circ}\;:\;\beta \mapsto \beta \circ \pi$ .

Since $HA^+ = A^+ H$ by Theorem 2.2(2), it is very easy to check that

(3.6) \begin{equation}T=\lbrace f\in H^{\circ}\;:\;f(ah)=\epsilon(a)f(h)=f(ha), \forall a\in A, \forall h\in H \rbrace.\end{equation}

To deduce normality of T, let $f\in\overline{H}^*$ and $\varphi\in H^{\circ}$ . For any $a\in A^+, h\in H$ ,

\begin{eqnarray*}(\textrm{ad}_{r} \varphi)(f) (ah) &=& \left[ \sum S^\circ(\varphi_1)f\varphi_2 \right](ah) = \sum S^\circ(\varphi_1)(a_1h_1) f(a_2h_2) \varphi_2(a_3h_3) \\&=& \sum \varphi_1(S(h_1)S(a_1)) \epsilon(a_2)f(h_2) \varphi_2(a_3h_3) = \sum \varphi(S(h_1)S(a_1)a_2h_3) f(h_2) \\&=& \epsilon(a) \sum \varphi(S(h_1)h_3) f(h_2) =0.\end{eqnarray*}

Similarly, one shows that $(\textrm{ad}_{\ell}\varphi)(ha) = 0$ for $h \in H$ and $a \in A$ , and hence T is normal by (3.6).

(3) Since $\overline{H}^{\ast}$ is a finite-dimensional and normal Hopf subalgebra of $H^{\circ}$ , $H^{\circ}$ is a free $\overline{H}^{\ast}$ -module by [Reference Schneider31, Theorem 2.1(2)].

(4),(5) Clearly, the restriction map $\iota^\circ\;:\;f \mapsto f \circ \iota$ is a Hopf algebra map whose image lies in $A^{\circ}$ .

Let $f \in A^{\circ}$ , so $f(J) = 0$ for an ideal J of A of finite codimension. Thanks to Theorem 2.2(7) $H = A \oplus X$ for a right A-submodule X of H, so the left ideal HJ of H decomposes as a right A-module,

(3.7) \begin{equation} HJ = J \oplus XJ \subseteq A \oplus X = H.\end{equation}

In view of (3.7), we can define $\widehat{f} \in H^{\ast}$ by $\widehat{f}(X) = 0$ and $\widehat{f}_{\mid A} = f$ . Since $\widehat{f}(HJ) = 0$ and $\textrm{dim}_k (H/HJ) < \infty$ , $\widehat{f} \in H^{\circ}$ by [Reference Montgomery26, Lemma 9.1.1]. By construction, $f \circ \Pi = \widehat{f}$ ; that is, $\Pi^{\circ}(f) = \widehat{f}$ . It is clear that $\iota^{\circ} \circ \Pi^{\circ} = \textrm{id}_{A^{\circ}}$ , so $\Pi^{\circ}$ is injective and $\iota^{\circ}$ surjective.

It remains to prove that $\Pi^{\circ}$ is a map of right $A^{\circ}$ -comodules. The right $A^{\circ}$ -comodule structure of $H^{\circ}$ is given by $\rho\;:\!=\;({\textrm{id}}\otimes \iota^{\circ})\Delta$ . Let $f\in A^{\circ}$ . Then for all $a\in A, h\in H$ we have

\begin{eqnarray*}\left( (\rho\circ\Pi^\circ)(f)\right)(h\otimes a) &=& \left(\rho(f\circ\Pi)\right)(h\otimes a) = \left[ \sum (f\circ\Pi)_1 \otimes (f\circ\Pi)_2\circ\iota \right](h\otimes a) \\&=& \sum (f\circ\Pi)_1(h) (f\circ\Pi)_2 (a) = (f\circ\Pi)(ha) = f(\Pi(h)a).\end{eqnarray*}

On the other hand,

\begin{equation*} \left((\Pi^\circ\otimes {\textrm{id}})\Delta(f)\right)(h\otimes a) = \left[\sum (\Pi^\circ\circ f_1) \otimes f_2\right](h\otimes a) = \sum f_1(\Pi(h)) f_2(a) = f(\Pi(h)a), \end{equation*}

so (4) is proved.

(6) $H^{\circ}$ is canonically a right and left $A^{\circ}$ -comodule algebra, the right comodule map being $\rho\;:\!=\;({\textrm{id}}\otimes\iota^{\circ})\circ\Delta$ . The right coinvariants are the maps f such that

\begin{equation*}\rho(f)=\sum f_1\otimes (f_2\circ \iota)= f\otimes \epsilon_A, \end{equation*}

so that $f(ha)=f(h)\epsilon(a)$ for all $h\in H, a\in A$ . By (3.6), these are precisely the maps in $\overline{H}^*$ . The left case is analogous.

(7) First, $(\overline{H}^{\ast})^+ H^{\circ}$ is a Hopf ideal of $H^{\circ}$ since, by (2), $\overline{H}^{\ast}$ is a normal Hopf subalgebra of $H^{\circ}$ , so [Reference Montgomery26, Section 3.4] applies. Let $f\in (\overline{H}^*)^+$ . Then $f(A^+H)=0$ by definition of $\overline{H}^{\ast}$ , and $\epsilon_{H^{\circ}}(f)=f(1)=0$ . Thus $f(A)=0$ , so that $f \in \textrm{ker }\iota^{\circ}$ . Hence $(\overline{H}^{\ast})^+ H^{\circ} \subseteq \textrm{ker }\iota^{\circ}$ , and the inclusion in (3.5) follows. For the equality in (3.5), first observe that the sums are direct by part (5). Let $f \in H^{\circ}$ . Then $(\Pi^{\circ} \circ \iota^{\circ})(f) \in \Pi^{\circ}(A^{\circ})$ , and

\begin{equation*} \iota^{\circ}(f - (\Pi^{\circ} \circ \iota^{\circ})(f)) = 0,\end{equation*}

so that $H^{\circ} = \textrm{ker }\iota^{\circ} + \Pi^{\circ}(A^{\circ})$ , as required.

3.3 Decomposition of $H^{\circ}$

Our first main result is the following. For the definition of a cleaving map, see Remark 2.3(2). Recall that Theorem 2.2(8) guarantees that under the hypotheses of the theorem below $H = A \oplus X$ for a right A-submodule X of H, and, as discussed in Section 1.1, $A \subseteq H$ satisfies $(\mathbf{{CoSplit}})$ if X can be chosen to be a coideal of H.

Proof.

1. (1) Since $X=\textrm{ker }\Pi$ is a coideal of H, $\Pi$ is a coalgebra map, and so the right $A^{\circ}$ -comodule map $\Pi^{\circ}$ of Lemma 3.1(4) is an algebra map. In particular, it is a cleaving map with convolution inverse $\Pi^{\circ}\circ S_{A^{\circ}}$ . By Remark 2.3(3), $ H^{\circ}$ is isomorphic to $\overline{H}^* \#_\sigma A^{\circ} $ for some action of $A^{\circ}$ on $\overline{H}^*$ and cocycle $\sigma$ . Since the cleaving map $\Pi^{\circ}$ is an algebra map, the cocycle $\sigma$ is trivial. By [Reference Montgomery26, Proposition 7.2.3], the isomorphism carries the structures as stated.

2. (2) Since $\gamma$ is a coalgebra map, $H=A\oplus A\gamma(\overline{H}^+)$ and $A\gamma(\overline{H}^+)$ is a coideal of H, so (1) applies. It remains to prove the action is trivial, that is $ \Pi^\circ(f')\pi^\circ(f) = \pi^\circ(f)\Pi^\circ(f'),$ for all $f\in\overline{H}^*, f'\in A^\circ$ .

Let $f\in\overline{H}^*, f'\in A^\circ, a\in A, \bar{h}\in \overline{H}$ . We have

\begin{eqnarray*}\pi^\circ(f)\Pi^\circ(f')(a\gamma(\bar{h})) &=& \sum \pi^\circ(f)(a_1\gamma(\bar{h_1}))\Pi^\circ(f')(a_2\gamma(\bar{h_2}))\\&=& \sum f\pi(a_1\gamma(\bar{h_1}))f'\Pi(a_2\gamma(\bar{h_2})) \\&=& \sum \epsilon(a_1)f\pi\gamma(\bar{h_1})f'(a_2)\epsilon(\gamma(\bar{h_2}))\\&=& \sum \epsilon(a_1)f(\bar{h_1})f'(a_2)\epsilon(\bar{h_2}) \\&=& f(\bar{h})f'(a),\end{eqnarray*}

since $\pi\circ\gamma = {\textrm{id}}$ , which follows from $\gamma$ being a right $\overline{H}$ -comodule coalgebra map. The proof for $\Pi^\circ(f')\pi^\circ(f)$ is completely analogous.

In Section 7, it is shown that Question 3.4 has a positive answer for many classes of commutative-by-finite Hopf algebras.

4.1 Definition and basic properties

Throughout this section, the hypotheses and notation of Section 2.2 remain in force. We define an analogue for the finite dual of a commutative-by-finite Hopf algebra of the irreducible component A ^′ appearing in (3.1), as follows.

Definition 4.1. Let $A \subseteq H$ be an affine commutative-by-finite Hopf k-algebra. The tangential component of $H^{\circ}$ is

\begin{equation*} W(H^{\circ}) \,\;:\!=\; \lbrace f\in H^{\circ}\;:\;f((A^+H)^n)=0, \text{ for some } n>0 \rbrace. \end{equation*}

Proof. First, $W(H^{\circ})$ is a Hopf subalgebra because $A^+H$ is a Hopf ideal of H and, for any Hopf ideal I of H,

\begin{equation*} \{ f \in H^{\circ} \;:\;f(I^n) = 0 \textit{ for some } n > 0 \} \end{equation*}

is a Hopf subalgebra of $H^\circ$ by [Reference Montgomery26, Lemma 9.2.1]. Clearly, $W(H^{\circ})$ contains $\overline{H}^*$ , by the definition of the latter subalgebra in Lemma 3.1(1). Since $\overline{H}^{\ast}$ is normal in $H^{\circ}$ by Lemma 3.1(2), it is a fortiori normal in the Hopf subalgebra $W(H^{\circ})$ .

To prove normality of $W(H^\circ)$ let $\varphi\in H^{\circ}$ and $f\in W(H^{\circ})$ , so f vanishes at $(A^+H)^n = (A^+)^n H$ for some positive integer n. Take $a_1,\ldots, a_n\in A^+$ and $h\in H$ . Then

\begin{eqnarray*}&& \left[\sum S^\circ(\varphi_1)f\varphi_2\right](a_1\ldots a_nh) \\&=& \sum S^\circ(\varphi_1)(a_{1,1}\ldots a_{n,1}h_1)f(a_{1,2}\ldots a_{n,2}h_2)\varphi_2(a_{1,3}\ldots a_{n,3}h_3) \\&=&\sum \varphi_1(S(h_1)S(a_{n,1})\ldots S(a_{1,1}))f(a_{1,2}\ldots a_{n,2}h_2)\varphi_2(a_{1,3}\ldots a_{n,3}h_3) \\&=& \sum \varphi(S(h_1)S(a_{n,1})\ldots S(a_{1,1})a_{1,3}\ldots a_{n,3}h_3)f(a_{1,2}\ldots a_{n,2}h_2) \\&=& \sum \varphi(S(h_1)S(a_{n,1})a_{n,3}\ldots S(a_{1,1})a_{1,3} h_3)f(a_{1,2}\ldots a_{n,2}h_2),\end{eqnarray*}

since A is commutative. For each i, we write $a_{i,2}=\epsilon(a_{i,2})1+a'_{\!\!i,2}$ , where $a'_{\!\!i,2}\in A^+$ . Writing [1, n] for $\lbrace 1,2,\ldots,n \rbrace$ , we now have

\begin{equation*} f(a_{1,2}\ldots a_{n,2}h_2)=\sum_{\mathcal{P}\subseteq [1,n]} \prod_{i\in\mathcal{P}}\epsilon(a_{i,2}) f\left(\prod_{i\in[1,n]\setminus\mathcal{P}}a'_{\!\!i,2}h_2\right). \end{equation*}

Therefore,

\begin{eqnarray*}&&\left[\sum S^\circ(\varphi_1)f\varphi_2\right](a_1\ldots a_n h) \\&=& \sum \sum_{\mathcal{P}\subseteq[1,n]} \varphi(S(h_1)S(a_{n,1})a_{n,3}\ldots S(a_{1,1})a_{1,3} h_3) \prod_{i\in\mathcal{P}} \epsilon(a_{i,2}) f\left(\prod_{i\in[1,n]\setminus\mathcal{P}} a'_{\!\!i,2} \, h_2\right) \\&=& \sum \sum_{\mathcal{P}\subseteq[1,n]} \varphi \left(S(h_1) \prod_{i\in\mathcal{P}} S(a_{i,1})\epsilon(a_{i,2})a_{i,3} \prod_{i\in[1,n]\setminus\mathcal{P}} S(a_{i,1})a_{i,3} \, h_3 \right) f\left(\prod_{i\in[1,n]\setminus\mathcal{P}} a'_{\!\!i,2} h_2\right) \\&=& \sum \sum_{\mathcal{P}\subseteq[1,n]} \prod_{i\in\mathcal{P}} \epsilon(a_i) \, \varphi \left(S(h_1) \prod_{i\in[1,n]\setminus\mathcal{P}} S(a_{i,1})a_{i,3} \, h_3 \right) f\left(\prod_{i\in[1,n]\setminus\mathcal{P}} a'_{\!\!i,2} h_2\right).\end{eqnarray*}

The summands where $\mathcal{P}\neq\emptyset$ vanish due to $\epsilon(a_i)=0$ for all $i\in[1,n]$ and the summands where $\mathcal{P}=\emptyset$ vanish because f is zero on $(A^+H)^n={A^+}^nH$ . The proof for the left adjoint action is similar, so normality of $W(H^{\circ})$ in $H^{\circ}$ is proved.

4.2 Decomposition of $W(H^\circ)$

To analyse the structure of $W(H^{\circ}) $ , we use the Hopf algebra surjection $\iota^{\circ}$ from $H^{\circ}$ to $A^{\circ}$ of Lemma 3.1(5), given by restriction of functionals from H to A. In the following result, we decompose $W(H^\circ)$ into the crossed product $\overline{H}^* \#_\sigma A'$ . Notice that the isomorphism (4.1) below does not require hypothesis (CoSplit), in contrast to Theorem 3.2.

Proof. (1) Since $\iota^{\circ}$ is just restriction of functionals to the subdomain A, it is clear that $\iota^{\circ}(W(H^{\circ})) \subseteq A'$ . To see that $\textrm{im } \iota^{\circ}_{\mid_{W(H^\circ)}} = A'$ , note the definition (3.2) of A ^′, and then observe that, in the proof of Lemma 3.1(4), if $f \in A' \subseteq A^{\circ}$ , then the map $\widehat{f} \in H^{\circ}$ constructed there, with $\Pi^{\circ}(f) = \widehat{f}$ , is actually in $W(H^{\circ})$ . Since $\iota^{\circ} \circ \Pi^{\circ} = {\textrm{id}}_{A^{\circ}}$ by Lemma 3.1(5), $\iota^{\circ}(\widehat{f}) = f$ and the result follows.

(2),(3): By Lemma 3.1(6), $H^{\circ}$ is a right $A^{\circ}$ -comodule algebra with $\rho=({\textrm{id}}\otimes\iota^{\circ})\Delta$ . In particular, since $W(H^{\circ})$ is a Hopf subalgebra of $H^{\circ}$ by Lemma 4.2(2), and using (1) of the present theorem,

(4.2) \begin{equation} \rho(W(H^{\circ}))\subseteq W(H^{\circ})\otimes \iota^{\circ}(W(H^{\circ})) = W(H^{\circ})\otimes A'.\end{equation}

That is, the structure of right $A^{\circ}$ -comodule of $H^{\circ}$ restricts to a structure of right A ^′-comodule on $W(H^{\circ})$ .

We have noted in the proof of (1) that the right $A^{\circ}$ -comodule map $\Pi^{\circ}\;:\;A^{\circ} \longrightarrow H^{\circ}$ maps A ^′ into $W(H^{\circ})$ . Thus, in light of (4.2), $\Pi^{\circ}_{\mid_{A'}}\;:\;A' \longrightarrow W(H^{\circ})$ is an injection of right A ^′-comodules.

A linear map from a coalgebra to an algebra is convolution invertible if and only if its restriction to the coradical is convolution invertible, see [Reference Takeuchi35, Lemma 14] or [Reference Montgomery26, Lemma 5.2.10]. But A ^′ is connected by its definition as the irreducible component of the coalgebra $A^{\circ}$ containing $\varepsilon$ , [Reference Montgomery26, p. 78]; that is, its coradical is k. Therefore, in the language of Remark 2.3(3), $\Pi^{\circ}_{\mid_{A'}}$ is a cleaving map and by the result of Doi and Takeuchi recalled in Remark 2.3(3),

\begin{equation*} W(H^{\circ}) \; \cong \; W(H^{\circ})^{\textrm{co } \iota^{\circ}_{\mid_{W(H^\circ)}}} \#_\sigma A'. \end{equation*}

Since the A ^′-comodule structure of $W(H^{\circ})$ is the restriction of the $A^{\circ}$ -comodule structure of $H^{\circ}$ , it follows from Lemma 3.1(6) that

\begin{equation*} W(H^{\circ})^{\textrm{co } \iota^\circ_{\mid_{W(H^\circ)}}} = W(H^{\circ}) \cap (H^{\circ})^{\textrm{co }\iota^\circ} = W(H^{\circ}) \cap \overline{H}^* = \overline{H}^*. \end{equation*}

(4) Since $\textrm{char}k = 0$ , $A' = U(\mathfrak{g})$ by (3.2). It is clear from (4.1) that

\begin{equation*} W(H^{\circ})/(\overline{H}^{\ast})^+ W(H^{\circ}) \quad \cong \quad U(\mathfrak{g}). \end{equation*}

Moreover, we know from Lemma 3.1(7) that

(4.3) \begin{equation} (\overline{H}^{\ast})^+W(H^{\circ})\subseteq \textrm{ker } \iota^\circ_{\mid_{W(H^\circ)}}.\end{equation}

By (1), $\textrm{im } \iota^\circ_{\mid_{W(H^\circ)}} = U(\mathfrak{g})$ . Since A is affine, $\mathfrak{g}$ is finite dimensional, and hence $U(\mathfrak{g})$ is a noetherian algebra. In particular, $U(\mathfrak{g})$ cannot be isomorphic to a proper factor of itself, and hence (4.3) must be an equality, proving (4).

(5) This follows from (2) and Theorem 3.2(1), since the cocycle $\sigma$ in (2) is in this case trivial by the latter result. The last part is a special case of Remark 3.3(2).

Recall that, in characteristic 0, a finite dimensional Hopf algebra is semisimple if and only if cosemisimple [Reference Larson and Radford21], so the following improvement of Theorem 4.3(5) applies when $\overline{H}$ is semisimple if k has characteristic 0.

Corollary 4.4. Let $A \subseteq H$ be an affine commutative-by-finite Hopf k-algebra with A semiprime or central, or H pointed. Retain the additional notation of Theorem 4.3. Assume that $A \subseteq H$ satisfies hypothesis (CoSplit), and that $\overline{H}$ is cosemisimple. Then the action of $U(\mathfrak{g})$ on $\overline{H}^{\ast}$ is inner, so that, as algebras,

\begin{equation*} W(H^{\circ}) \; \cong \; \overline{H}^{\ast} \otimes U(\mathfrak{g}). \end{equation*}

Proof. By Theorem 4.3(5) $W(H^{\ast})$ is a smash product of algebras,

(4.4) \begin{equation} \overline{H}^{\ast} \# U(\mathfrak{g}).\end{equation}

By our hypothesis, $\overline{H}^{\ast}$ is semisimple, and so by Jacobson’s theorem, (see for example [Reference Montgomery26, Example 6.1.6, Proposition 6.2.1 and discussion following]), since k is algebraically closed, $\mathfrak{g}$ acts by inner derivations on $\overline{H}^{\ast}$ . That is, there is a copy $u(\mathfrak{g})$ of $\mathfrak{g}$ contained in $\overline{H}^{\ast}$ , (where we view $\overline{H}^{\ast}$ as a Lie algebra via the commutator bracket), such that for all $x \in \mathfrak{g}$ and $h \in \overline{H}^{\ast}$ ,

\begin{equation*} x\cdot h \; =\; [u(x),\, h]. \end{equation*}

Replacing the copy of $\mathfrak{g}$ appearing in (4.4) with the isomorphic Lie algebra

\begin{equation*}\widehat{\mathfrak{g}} \; :\!=\; \sum_{i}k(x_i - u(x_i)), \end{equation*}

where $\{x_i \}$ is a k-basis of $\mathfrak{g}$ , we see that

\begin{equation*} W(H^{\circ}) \; = \; \overline{H}^{\ast} \otimes U(\widehat{\mathfrak{g}}), \end{equation*}

as claimed.

5.1 Orbits in ${\textrm{Maxspec}}(A)$

Recall the description (3.3) in Section 3.1 of the Hopf subalgebra kG of $A^{\circ}$ . In this section, we extend this viewpoint to H, defining a subcoalgebra $\widehat{kG_H}$ of $H^{\circ}$ which is a Hopf subalgebra under mild additional hypotheses and which reduces to $kG_H$ when $H = A$ . We saw in Theorem 2.2(3) that the left adjoint action of H on A factors over the ideal $A^+ H$ , so that A is a left $\overline{H}$ -module algebra. An ideal I of A is said to be $\overline{H}$ -stable if

\begin{equation*} (\textrm{ad}_{\ell}h)(a)\in I, \; \text{for all } h\in\overline{H}, a\in I. \end{equation*}

Since the extension $A\subseteq H$ is faithfully flat by Theorem 2.2(4), I is $\overline{H}$ -stable if and only if $HI=IH$ .

Definition 5.1. Let T be a Hopf algebra and R a left T-module algebra, with V a subspace of R. The T-core of V is the subspace

\begin{equation*} V^{(T)} \; = \; \{ v \in V \;:\;t\cdot v \in V, \, \forall t \in T \}.\end{equation*}

Note that if V is a right or left ideal of R in Definition 5.1 then so is $V^{(T)}$ . The crucial parts (2) and (3) of the following result are essentially due to Skryabin [Reference Skryabin33, Theorems 1.1, 1.3]. For the proofs of Proposition 5.2 and Corollary 5.4, see [Reference Brown and Couto3, Proposition 4.4].

Following still Skryabin, we can use Proposition 5.2(2) to extend the fundamental idea that a group of automorphisms of the algebra A defines orbits in ${\textrm{Maxspec}} (A)$ , to the present setting where the group is replaced by a Hopf algebra.

It’s clear that $\sim^{(\overline{H})}$ is an equivalence relation; but it is far from obvious, although it follows easily from Proposition 5.2(2), that the $\overline{H}$ -orbits have the following simple description.

5.2 Orbital semisimplicity

The results in Section 5.1 lead naturally to:

Note that, in view of Corollary 5.4(1), $A \subseteq H$ is orbitally semisimple if and only if

(5.1) \begin{equation} {\mathfrak{m}}^{(\overline{H})} = \bigcap_{{\mathfrak{m}}'\in\mathcal{O}_{\mathfrak{m}}} {\mathfrak{m}}'.\end{equation}

Since we have used the left adjoint action in Definition 5.5, one might choose to call the above property left orbital semisimplicity; but in fact it is shown in [Reference Brown and Couto3, Lemma 4.10(2)] that $A \subseteq H$ is left orbitally semisimple if and only if it is right orbitally semisimple.

We know no examples of affine commutative-by-finite Hopf algebras that are not orbitally semisimple. When A is central in H orbital semsimplicity is trivial; the next result lists other positive cases. In characteristic 0, there are overlaps between these, thanks to the result of Etingof and Walton [Reference Etingof and Walton15] implying that the action of $\overline{H}$ factors through a group algebra when $\overline{H}$ is semisimple and cosemisimple and A is a domain.

We discuss particular families of examples of affine commutative-by-finite Hopf algebras, and their status with respect to orbital semisimplicity, in Section 7.

5.3 The character component of $H^{\circ}$

Here is the definition, we have been working towards:

Here are some basic properties of these spaces.

Proof. (1) This follows from the definition of $\sim^{(\overline{H})}$ .

(2) If $g, h \in G_H$ with $\mathcal{O}_{{\mathfrak{m}}_g} \neq \mathcal{O}_{{\mathfrak{m}}_h}$ , then the ideals ${\mathfrak{m}}_g^{(\overline{H})}$ and ${\mathfrak{m}}_h^{(\overline{H})}$ of A are comaximal, by Corollary 5.4(1). The same is thus true of the ideals $H{\mathfrak{m}}_g^{(\overline{H})}$ and $H{\mathfrak{m}}_h^{(\overline{H})}$ of H. Therefore, if $g_1, \ldots , g_r \in G_H$ are representatives of distinct $\sim^{(\overline{H})}$ -orbits, the Chinese Remainder Theorem shows that

\begin{equation*} \left. H \middle/ \bigcap_{i=1}^r H{\mathfrak{m}}_{g_i}^{(\overline{H})} \right. \cong \bigoplus_{i=1}^r H/H{\mathfrak{m}}_{g_i}^{(\overline{H})}. \end{equation*}

Taking duals on both sides proves (2).

(3),(4): Let $g \in G_H$ . That $\widehat{g}$ is a subcoalgebra of $H^{\circ}$ is simply a special case of the elementary fact that, for an ideal I of finite codimension in an algebra R, $(R/I)^{\ast}$ is a subcoalgebra of $R^{\circ}$ . So (3) and (4) follow from this observation together with (2).

(5) The first inclusion is clear. For the second, note first that

(5.5) \begin{equation} \Delta (H{\mathfrak{m}}_g) = \Delta(H)\Delta({\mathfrak{m}}_g) \subseteq H{\mathfrak{m}}_{1_{G_H}} \otimes H + H \otimes H{\mathfrak{m}}_g,\end{equation}

by the definition of the coproduct in $A/N(A) = \mathcal{O}(G_H)$ , combined with the fact that N(A) is a Hopf ideal of A, [Reference Brown and Couto3, Lemma 5.2(1)]. Let $f \in \overline{H}^{\ast}$ . Then, using (5.5),

\begin{equation*} (f\Pi^{\circ}(\chi_g)) (H{\mathfrak{m}}_g) \subseteq f(H{\mathfrak{m}}_{1_{G_H}})\Pi^{\circ}(\chi_g)(H) + f(H)\Pi^{\circ}(\chi_g)(H{\mathfrak{m}}_g) = 0.\end{equation*}

So $\overline{H}^{\ast}\Pi^{\circ}(\chi_g) \subseteq (H/H{\mathfrak{m}}_g)^{\ast}$ . The proof of the remaining inclusion is similar.

(6) Let $g \in G_H$ . To prove that $S^{\circ}(\widehat{g})=\widehat{g^{-1}}$ it suffices to show that $S({\mathfrak{m}}_g^{(\overline{H})}) = {\mathfrak{m}}_{g^{-1}}^{(\overline{H})}$ . It is easy to see that

\begin{equation*} {\mathfrak{m}}_g^{(\overline{H})}=\textrm{r-Ann}_A(H/{\mathfrak{m}}_gH)=\textrm{l-Ann}_A(H/H{\mathfrak{m}}_g). \end{equation*}

Since S is an algebra anti-isomorphism of H with $S(A)=A$ , it follows that

\begin{equation*} S({\mathfrak{m}}_g^{(\overline{H})}) = \textrm{l-Ann}_A(H/HS({\mathfrak{m}}_g)) = \textrm{l-Ann}_A(H/H{\mathfrak{m}}_{g^{-1}}). \end{equation*}

This proves (6).

(7) The final equality is clear from the definitions, and this also shows that $\overline{H}^{\ast} \subseteq \widehat{kG_H}$ . Thus, by Lemma 4.2(2), $\overline{H}^{\ast}\subseteq W(H^{\circ}) \cap \widehat{kG_H}$ .

For the reverse inclusion, suppose $f \in H^{\circ}$ with f vanishing on $(A^+H)^n={A^+}^nH$ and also on $I \;:\!=\; \bigcap_{i=1}^r {\mathfrak{m}}_{g_i}^{(\overline{H})}H$ , for some $n,r \geq 1$ and distinct $g_1,\ldots,g_r\in G_H/\sim^{(\overline{H})}$ . We claim that

(5.6) \begin{equation} A^+H \subseteq I + {A^+}^nH.\end{equation}

If $g_i\neq 1_{G_H}$ for all $i = 1, \ldots , r$ , then I and ${A^+}^nH$ are comaximal by the discussion in the proof of (2), so that (5.6) is clear. Suppose on the other hand that, say, $g_1 = 1_{G_H}$ . Then $I + {A^+}^nH \subseteq A^+ H$ , and the left A-module $A^+H/(I + {A^+}^n H)$ is annihilated by both of the comaximal ideals ${A^+}^{n-1}$ and by $\bigcap_{i=2}^r{\mathfrak{m}}_{g_i}^{(\overline{H})}$ . Hence $A^+H/(I + {A^+}^n H) = \{0\}$ and again (5.6) is proved. Thus (5.6) holds in all cases and so $f \in \overline{H}^{\ast}$ , as required.

(8) Assume that A is a domain. Localise A at the maximal ideal ${\mathfrak{m}}_g$ , and consider the left $A_{{\mathfrak{m}}_g}$ -module $H_{{\mathfrak{m}}_g} \;:\!=\; A_{{\mathfrak{m}}_g} \otimes_A H$ . Since H is a finitely generated projective A-module by Theorem 2.2(6), and $A_{{\mathfrak{m}}_g}$ is local, $H_{{\mathfrak{m}}_g} $ is a free $A_{{\mathfrak{m}}_g}$ -module of finite rank r. Let Q denote the quotient field of A. Since H is a torsion free A-module,

\begin{equation*} r \; = \; \textrm{dim}_Q (Q \otimes_{A_{{\mathfrak{m}}_g}} H_{{\mathfrak{m}}_g}) \; = \; \textrm{dim}_Q (Q \otimes_A H),\end{equation*}

so r is constant as g varies through $G_H$ . Taking $g = 1_{G_H}$ and applying Nakayama’s lemma,

(5.7) \begin{equation} r = \textrm{dim}_k (H/{\mathfrak{m}}_{1_{G_H}}H) = \textrm{dim}_k (\overline{H}),\end{equation}

proving the second equality in (5.2). The first equality is a consequence of the second, using the antipode. Since the algebra $H/{\mathfrak{m}}_{g}^{(\overline{H})}H$ is artinian, it is isomorphic as an $A/{\mathfrak{m}}_g^{(\overline{H})}$ -module to the factor $H_{{\mathfrak{m}}_g}/{\mathfrak{m}}_{g}^{(\overline{H})}H_{{\mathfrak{m}}_g}$ of $H_{{\mathfrak{m}}_g}$ . But the latter is a free $A/{\mathfrak{m}}_{g}^{(\overline{H})}$ -module of rank r. Combining this with (5.7) proves the equality in (5.3), and the inequality is then given by [Reference Brown and Couto3, Theorem 6.1(2)]. If H is prime then A is a domain and the inequality (5.4) follows from [Reference Brown and Couto3, Corollary 6.2 and its proof].

5.4 Subalgebras and subcoalgebras of $\widehat{kG_H}$

In order to better control the structure of $\widehat{kG_H}$ , we need to impose on the inclusion $A \subseteq H$ one or both of the additional hypotheses $(\mathbf{{CoSplit}})$ and $(\mathbf{{OrbSemi}})$ defined respectively in Section 1.1 and Definition 5.5. First we assume $(\mathbf{{CoSplit}})$ in Proposition 5.9, then $(\mathbf{{OrbSemi}})$ in Proposition 5.10, before combining these in Theorem 5.13.

Proposition 5.9. Let $A \subseteq H$ be an affine commutative-by-finite Hopf k-algebra satisfying $(\mathbf{{CoSplit}})$ . Assume also that A is semiprime or central, or that H is pointed. Keep the notation introduced in Sections 5.3 and 2.2 and in Lemma 3.1. Define

\begin{equation*}\widetilde{G_H} = \{ \Pi^{\circ} (\chi_g) \;:\;g \in G_H\} \subseteq H^{\circ}.\end{equation*}

1. (1) $\widetilde{G_H}$ is a subgroup of the group of units of $H^{\circ}$ , and the algebra injection $\Pi^{\circ}\;:\;A^{\circ} \to H^{\circ}$ restricts to an isomorphism of groups from the group-likes $G_H$ of $A^{\circ}$ to $\widetilde{G_H}$ .

2. (2) The subalgebra of $H^{\circ}$ generated by $\widetilde{G_H}$ is the group algebra $k\widetilde{G_H}$ , and $\Pi^{\circ}$ restricts to an algebra isomorphism from $kG_H$ to $k\widetilde{G_H}$ .

3. (3) For all $g \in G_H$ , $\overline{H}^{\ast}\Pi^{\circ}(\chi_g) = \Pi^{\circ}(\chi_g)\overline{H}^{\ast}$ is free of rank 1 as an $\overline{H}^{\ast}$ -bimodule. Hence $\langle \overline{H}^{\ast}, \widetilde{G_H} \rangle$ is a skew group algebra $\overline{H}^{\ast} \# \widetilde{G_H}$ ; that is,

   \begin{equation*} \langle \overline{H}^{\ast}, \widetilde{G_H} \rangle \; = \; \overline{H}^{\ast} \# \widetilde{G_H} \; = \; \oplus_{g \in G_H} \overline{H}^{\ast}\Pi^{\circ}(\chi_g) \end{equation*}
   is a subalgebra of $H^{\circ}$ contained in the subcoalgebra $\widehat{kG_H}$ .

4. (4) There are inclusions

   \begin{equation*} \widehat{kG_H} = \bigoplus_{g \in G_H/\sim^{(\overline{H})}} \widehat{g} \supseteq \bigoplus_{g \in G_H/\sim^{(\overline{H})}}\left(\bigoplus_{h\sim^{(\overline{H})} g} (H/H\mathfrak{m}_h)^{\ast}\right) \supseteq \bigoplus_{g \in G_H/\sim^{(\overline{H})}}\left(\bigoplus_{h\sim^{(\overline{H})} g} \overline{H}^{\ast}\Pi^{\circ}(\chi_h)\right) = \overline{H}^{\ast} \# \widetilde{G_H}. \end{equation*}
   If A is a domain then the second inclusion is an equality.

Proof. (1),(2) The $(\mathbf{{CoSplit}})$ hypothesis ensures that $\Pi$ is a coalgebra map, so $\Pi^\circ$ is an algebra homomorphism, and is an injection by Lemma 3.1(5). Thus $\Pi^{\circ}$ maps $G_H$ isomorphically to a subgroup $\widetilde{G_H}$ of the group of units of $H^{\circ}$ , and

\begin{equation*} k\widetilde{G_H} = \Pi^{\circ}(kG_H) \cong kG_H.\end{equation*}

(3) By Remark 3.3(2), the action of $kG_H\subset A^\circ$ on $\overline{H}^*$ is

\begin{equation*} \chi_g\cdot f = \Pi^\circ(\chi_g) f \Pi^\circ(\chi_{g^{-1}}) \in \overline{H}^*, \end{equation*}

for any $f\in \overline{H}^*, g\in G_H$ , hence $\Pi^\circ(\chi_g)f\in \overline{H}^*\Pi^\circ(\chi_g)$ , proving $\Pi^\circ(\chi_g)\overline{H}^* \subseteq \overline{H}^*\Pi^\circ(\chi_g)$ and the converse is proved similarly. Both these $\overline{H}^{\ast}$ -modules are free of rank 1 since $\Pi^{\circ}(G_H)$ consists of units of $H^{\circ}$ . From Theorem 3.2(1) and its proof, we know that $H^{\circ}$ is a free left and right $\overline{H}^{\ast}$ -module with any vector space basis of $\Pi^{\circ}(A^{\circ})$ as free basis. In particular, $\{\Pi^{\circ}(\chi_g) \;:\;g \in G_H \}$ spans a free left and right $\overline{H}^{\circ}$ -submodule of $\widehat{kG_H}$ .

(4) The first equality and first inclusion are immediate from Definition 5.7 and Lemma 5.8(5). The second inclusion follows from Lemma 5.8(5) and the third equality is part (3) of the present lemma. If A is a domain the second inclusion is seen to be an equality by comparing vector space dimensions using Lemma 5.8(8).

We examine next the effect of the $(\mathbf{{OrbSemi}})$ hypothesis on $\widehat{kG_H}$ .

Proof. Note that (3) is a special case of (2), since $\overline{H}^{\ast} = \widehat{1_{G_H}}$ . By Lemma 5.8(3),(6), (1) will follow if $\widehat{kG_H}$ is closed under multiplication. Moreover, by Lemma 5.8(2), closure under multiplication will follow at once from (2). We claim that to prove (5.8) amounts to showing that

(5.9) \begin{equation}\Delta\left( \bigcap_{g'\sim^{(\overline{H})} g, h'\sim^{(\overline{H})} h} {\mathfrak{m}}_{g'h'} \right)\subseteq {\mathfrak{m}}_g^{(\overline{H})}\otimes A + A\otimes {\mathfrak{m}}_h^{(\overline{H})} =\;:\;J.\end{equation}

For, the flatness of H over A together with the Chinese remainder theorem implies that $\bigcap_{g',h'} (H{\mathfrak{m}}_{g'h'}^{(\overline{H})}) = H\left(\bigcap_{g',h'} {\mathfrak{m}}_{g'h'}^{(\overline{H})}\right)$ . Hence, given (5.9),

(5.10) \begin{equation} \Delta\left(\bigcap_{g'\sim^{({\overline{H}})} g, h'\sim^{({\overline{H}})} h} H{\mathfrak{m}}_{g'h'}^{(\overline{H})}\right) \subseteq H{\mathfrak{m}}_g^{(\overline{H})}\otimes H + H \otimes H{\mathfrak{m}}_h^{(\overline{H})}.\end{equation}

Consider now $\beta\gamma $ for $\beta \in\widehat{g}$ and $\gamma \in\widehat{h}$ . It follows from (5.10) that

\begin{equation*} \beta \gamma \left( \bigcap_{g'\sim^{({\overline{H}})} g, h'\sim^{({\overline{H}})} h} H{\mathfrak{m}}_{g'h'}^{(\overline{H})} \right) \subseteq \beta (H{\mathfrak{m}}_g^{(\overline{H})})\gamma (H) + \beta (H)\gamma (H{\mathfrak{m}}_h^{(\overline{H})}) = 0,\end{equation*}

proving part (2) of the proposition.

Let us prove (5.9). By the definition of the coproduct of A,

\begin{eqnarray*} \Delta\left( \bigcap_{g'\sim^{({\overline{H}})} g, h'\sim^{({\overline{H}})} h} {\mathfrak{m}}_{g'h'} \right) &\subseteq& \bigcap_{g'\sim^{({\overline{H}})} g, h'\sim^{({\overline{H}})} h} \Delta\left( {\mathfrak{m}}_{g'h'}\right)\\ &\subseteq& \bigcap_{g'\sim^{({\overline{H}})} g, h'\sim^{({\overline{H}})} h} ({\mathfrak{m}}_{g'}\otimes A + A\otimes{\mathfrak{m}}_{h'}) =\;:\;I.\end{eqnarray*}

Thus (5.9) will follow if it can be shown that $I = J$ . It is clear that $J \subseteq I$ . To prove equality, let $r=|\mathcal{O}_{{\mathfrak{m}}_g}|$ and $s=|\mathcal{O}_{{\mathfrak{m}}_h}|$ . Then it follows from $(\mathbf{{OrbSemi}})$ that

\begin{equation*} \textrm{dim}_k((A \otimes A)/J) = rs = \textrm{dim}_k((A \otimes A)/I) .\end{equation*}

This completes the proof of (5.9).

(4) Since H is orbitally semisimple, $\left| \mathcal{O}_{{\mathfrak{m}}_g} \right| = \textrm{dim}_k (A/ {\mathfrak{m}}_g^{(\overline{H})})$ by (5.1), hence (4) is (5.3).

(5) Since H is orbitally semisimple, $H/H{\mathfrak{m}}_g^{(\overline{H})} = \bigoplus_{h \sim^{(\overline{H})} g} H/H{\mathfrak{m}}_h$ , so this follows from the definition of $\widehat{g}$ .

Remark 5.11. The formula (5.8) cannot be further simplified. To see this, consider the group algebra $H \;:\!=\; kD$ of the dihedral group as in Remark 3.3(1). Recall that $A = \; k[b^{\pm 1}] \; = \; \mathcal{O}(k^{\times}),$ so, $G_H = k^{\times}$ . Now $\overline{H} = k\langle a \rangle$ , so the $\overline{H}$ -orbits in $\textrm{Maxspec}(A)$ are

(5.11) \begin{equation} \{ \mathfrak{m}_g, \mathfrak{m}_{g^{-1}} \, \;:\;\, g \in k^{\times} \},\end{equation}

and $\overline{H}^{\ast} \cong \overline{H}.$ Clearly H satisfies both $(\mathbf{{CoSplit}})$ with $X \;:\!=\; (a-1)kA$ , and $(\mathbf{{OrbSemi}})$ , since $\overline{H}$ is a group algebra. Therefore, by Propositions 5.9(4) and 5.10(5), and using (5.11),

(5.12) \begin{equation} \widehat{g} \; = \; (k\Pi^\circ(\chi_g) + k\Pi^\circ(\chi_{g^{-1}}))\otimes kC_2,\end{equation}

and $\widehat{1}=1\otimes kC_2$ and $\widehat{-1}=k\Pi^\circ(\chi_{-1})\otimes kC_2$ . Hence, for $g,h\in k^\times\setminus\lbrace \pm 1 \rbrace$ ,

\begin{equation*} \widehat{g}\widehat{h} = \left[ k\Pi^\circ(\chi_{gh}) + k\Pi^\circ(\chi_{g^{-1}h}) + k\Pi^\circ(\chi_{gh^{-1}}) + k\Pi^\circ(\chi_{g^{-1}h^{-1}}) \right]\otimes kC_2 = \widehat{gh}+\widehat{g^{-1}h}. \end{equation*}

Thus the inclusion of (5.8) is an equality in this instance.

5.5 Decomposition of $\widehat{kG_H}$

Before uniting the conclusions of Propositions 5.9 and 5.10 in Theorem 5.13, we need one more lemma. Recall from Section 2.2 and Lemma 3.1 the Hopf surjection $\iota^{\circ} \;:\;H^{\circ} \longrightarrow A^{\circ}$ , so that $H^{\circ}$ is a right $A^{\circ}$ -comodule algebra via the map $\rho \;:\;= ({\textrm{id}} \otimes \iota^{\circ})\circ \Delta$ .

Proof.

1. (1) Let $g \in G_H$ and take $f\in\widehat{g}$ , so that $\left.f\right|_A({\mathfrak{m}}_g^{(\overline{H})}) =0$ . Since $A \subseteq H$ satisfies $(\mathbf{{OrbSemi}})$ the Chinese remainder theorem implies that

   \begin{equation*} \iota^{\circ}(f) = \left.f\right|_A\in \left(A/{\mathfrak{m}}_g^{(\overline{H})}\right)^{\ast}\cong \bigoplus_{h\sim^{({\overline{H}})} g}(A/{\mathfrak{m}}_h)^{\ast} = \bigoplus_{h\sim^{({\overline{H}})} g} k\chi_h. \end{equation*}
   Conversely, let $\alpha=\sum_{h\sim^{({\overline{H}})} g} \lambda_h\chi_h \in A^{\circ}$ for some $\lambda_h\in k$ . Then $\alpha=\iota^\circ(\Pi^\circ(\alpha))$ by Lemma 3.1(5). Recall that $H=A\oplus X$ as right A-modules by (1.1) in Section 1.2. Thus
   \begin{equation*} \Pi^\circ(\alpha)(H{\mathfrak{m}}_g^{(\overline{H})}) = \Pi^\circ(\alpha)({\mathfrak{m}}_g^{(\overline{H})}\oplus X{\mathfrak{m}}_g^{(\overline{H})}))= \alpha({\mathfrak{m}}_g^{(\overline{H})}) = \sum_{h\sim^{(\overline{H})} g} \lambda_h\chi_h({\mathfrak{m}}_g^{(\overline{H})})=0, \end{equation*}
   as ${\mathfrak{m}}_g^{(\overline{H})}\subseteq {\mathfrak{m}}_h$ for every $h\sim^{(\overline{H})} g$ . Thus $\Pi^\circ(\alpha)\in\widehat{g}$ and $\alpha\in\iota^\circ(\widehat{g})$ , proving (1).

2. (2) Let $f \in H^{\circ}$ and suppose $\rho(f)=f\otimes \sum_{h\sim^{(\overline{H})} g} \lambda_h \chi_h$ for some $\lambda_h\in k$ . Then

   \begin{eqnarray*}f(H{\mathfrak{m}}_g^{(\overline{H})}) &=& \sum f_1(H) f_2({\mathfrak{m}}_g^{(\overline{H})}) = \sum f_1(H) \left.f_2\right|_A ({\mathfrak{m}}_g^{(\overline{H})}) \\&=& \rho(f)(H\otimes {\mathfrak{m}}_g^{(\overline{H})}) = f(H)\left( \sum_{h\sim^{({\overline{H}})} g} \lambda_h \chi_h({\mathfrak{m}}_g^{(\overline{H})}) \right) =0,\end{eqnarray*}
   as ${\mathfrak{m}}_g^{(\overline{H})}\subseteq {\mathfrak{m}}_h$ for every $h\sim^{(\overline{H})} g$ . Hence $f\in \widehat{g}$ .

3. (3) That the left side contains the right is a special case of (2). For the reverse inclusion, let $f\in\overline{H}^*=\widehat{1_G}$ . By part (1)

   \begin{equation*}\rho(f)\in ({\textrm{id}} \otimes \iota^{\circ})(\overline{H}^* \otimes \overline{H}^*) = \overline{H}^*\otimes k \varepsilon_A, \end{equation*}
   say $\rho(f)=f'\otimes \varepsilon_A$ for some $f'\in\overline{H}^*$ . By the counit axiom of right comodules, $f=\varepsilon_A(1_A)f'=f'$ , hence $\rho(f)=f\otimes 1_{A^\circ}$ , proving (3).

We can now show that when $A \subseteq H$ satisfies $(\mathbf{{OrbSemi}})$ $\widehat{kG_H}$ decomposes as a crossed or smash product whenever $H^{\circ}$ does, generalising the presence of the group algebra $kG_H$ in the dual of a commutative semiprime Hopf algebra.

Proof.

1. (1) Suppose that $H^\circ\cong \overline{H}^* \#_\sigma A^\circ$ . By [Reference Doi and Takeuchi14] (see Remark 2.3(4)), there exists a convolution invertible right $A^\circ$ -comodule map $\gamma:A^\circ\to H^\circ$ such that the isomorphism is given by $f\# \varphi\mapsto f\gamma(\varphi)$ . The core of this proof is showing that

   (5.15) \begin{equation} \widehat{g}=\bigoplus_{h\sim^{(\overline{H})} g} \overline{H}^*\gamma(\chi_h). \end{equation}
   We first claim that $\gamma(\chi_h)\in\widehat{g}$ . Since $\gamma:A^\circ\to H^\circ$ is a right $A^\circ$ -comodule map and $\chi_h$ is a grouplike element of $A^{\circ}$ , then
   \begin{equation*} \rho\gamma(\chi_h) = (\gamma\otimes {\textrm{id}})\Delta_{A}(\chi_h) = \gamma(\chi_h)\otimes \chi_h \end{equation*}
   and the claim follows from Lemma 5.12(2). Therefore, by Proposition 5.10(3)
   \begin{equation*} \overline{H}^*\gamma(\chi_h) \subseteq \overline{H}^*\widehat{g}\subseteq \widehat{g}, \end{equation*}
   proving the inclusion of the right of (5.15) in the left. For the reverse inclusion, let $f\in\widehat{g}$ . By the second sentence of the proof, $f=\sum_{i=1}^n f_i\gamma(u_i\#\chi_{g_i})$ , where $f_i\in\overline{H}^*, u_i\in U(\mathfrak{g}), g_i\in G_H$ . Since $\rho$ is an algebra homomorphism and $\gamma:A^\circ\to H^\circ$ is a right $A^\circ$ -comodule map, and since $\chi_{g_i} \in A^{\circ}$ is grouplike and $\rho(f_i)=f_i\otimes 1$ by Lemma 5.12(2) we have
   \begin{eqnarray*}\rho(f) &=& \sum_{i=1}^n \rho(f_i)\rho\gamma(u_i\#\chi_{g_i}) = \sum_{i=1}^n \rho(f_i) (\gamma\otimes {\textrm{id}})\Delta(u_i\#\chi_{g_i}) \\&=& \sum_{i=1}^n \sum_{(u_i)} f_i\gamma(u_{i,1}\#\chi_{g_i})\otimes (u_{i,2}\#\chi_{g_i}).\end{eqnarray*}
   But by Lemma 5.12(1)
   \begin{equation*} \rho(f)\in \widehat{g}\otimes\iota^\circ(\widehat{g})=\bigoplus_{h\sim^{(\overline{H})} g} \widehat{g}\otimes \chi_h.\end{equation*}
   Moreover, since each $u_i$ is a polynomial on primitive elements, we must have $g_i\sim^{(\overline{H})} g$ for every i, and each $u_i=1$ . Hence f has the required form, and (5.15) follows. Therefore,
   \begin{equation*} \widehat{kG_H} = \overline{H}^*\gamma(kG_H) \cong \overline{H}^*\otimes kG_H \end{equation*}
   as vector spaces, left $\overline{H}^*$ -modules and right $kG_H$ -comodules, since by the calculations above $\rho(\widehat{kG_H})\subseteq \widehat{kG_H}\otimes kG_H$ , that is the $A^\circ$ -comodule structure of $H^\circ$ restricts to a $kG_H$ -comodule structure for $\widehat{kG_H}$ . In particular, by Lemma 3.1
   \begin{equation*} \widehat{kG_H}^{co\, kG_H} = \widehat{kG_H}\cap (H^\circ)^{co\,A^\circ} = \widehat{kG_H}\cap\overline{H}^* = \overline{H}^*. \end{equation*}
   Also, since $\gamma\;:\;A^\circ\to H^\circ$ is convolution invertible, then so is its restriction to the coradical $\left.\gamma\right|_{kG_H}:\;kG_H\to \widehat{kG_H}$ , [Reference Takeuchi35, Lemma 14]. Therefore,
   \begin{equation*} \widehat{kG_H} \cong \overline{H}^* \#_\tau kG_H \end{equation*}
   and by inspecting the formulae for the cocyles $\sigma$ and $\tau$ one easily sees that $\tau=\left.\sigma\right|_{kG_H\otimes kG_H}$ .

2. (2) If X can be chosen to be a coideal, then $H^\circ\cong \overline{H}^* \# A^\circ$ by Theorem 3.2(1), hence the cocycle $\sigma$ is trivial and $\widehat{kG_H}$ decomposes as a smash product. The action of $kG_H$ on $\overline{H}^*$ is a special case of Remark 3.3(2).

7.1 Enveloping algebras of Lie algebras in positive characteristic

Assume that k has positive characteristic p. Let $\mathfrak{g}$ be a k-Lie algebra with basis $\{x_1, \ldots , x_n\}$ . Then $H \;:\!=\; U(\mathfrak{g})$ is a free module of finite rank over a central polynomial Hopf subalgebra

\begin{equation*}A\;:\!=\;k[y_1,\ldots,y_n],\end{equation*}

where each $y_i$ is a p-polynomial in $x_i$ and hence is primitive; see [Reference Jacobson18, Proposition 1]. When $\mathfrak{g}$ is restricted, with restriction map $x\mapsto x^{[p]}$ , one takes $y_i \;:\!=\; x_i^p - x_i^{[p]}$ for all i, and in this case the Hopf quotient $\overline{H}$ is the restricted enveloping algebra of $\mathfrak{g}$ , usually denoted by $u^{[p]}(\mathfrak{g})$ , with $\textrm{dim}_k (u^{[p]}(\mathfrak{g})) = p^n$ . In the non-restricted case, it is still the case by [Reference Jacobson18, Proposition 2] that $U(\mathfrak{g})$ is a free A-module, with basis

(7.1) \begin{equation} \{x_1^{j_1} \cdots x_n^{j_n} \;:\;0 \leq j_i < p^{d_i} \},\end{equation}

where $y_i$ is a linear combination of $x_{i}^{p^j}$ for $j = 0, \ldots , d_i$ . Thus, in the general non-restricted case,

\begin{equation*} \textrm{dim}_k(\overline{H}) \; = \; p^{\sum_i d_i}.\end{equation*}

Since A is central $(\mathbf{{OrbSemi}})$ is trivially true for $H = U(\mathfrak{g})$ . Moreover it is clear from (7.1) that, as an A-module complement to A in H, we can choose

\begin{equation*} X \; :\!=\; \sum Ax_1^{j_1} \cdots x_n^{j_n}, \; 0 \leq j_i < p^{d_i}, \; \textit{ not all } j_i = 0.\end{equation*}

Thus X is a coideal of H and so $(\mathbf{{CoSplit}})$ is also valid. We can therefore apply Theorem 6.1(1). Note that

\begin{equation*}k[y_1, \ldots , y_n]\; \cong \; \mathcal{O}((k,+)^n),\end{equation*}

and that $H^{\circ}$ is commutative since H is cocommutative, so that the actions in the smash products composing $H^{\circ}$ are all trivial. We conclude that, as algebras,

\begin{equation*} U(\mathfrak{g})^{\circ} \; \cong \; A' \otimes \overline{H}^{\ast} \otimes k(k,+)^n, \end{equation*}

with $\overline{H}^{\ast}$ , $W(H^{\circ}) \, \;:\!=\; \, A' \otimes \overline{H}^{\ast}$ and $\widehat{k(k,+)^n}\, \;:\!=\; \, \overline{H}^{\ast} \otimes k(k,+)^n$ each being a Hopf subalgebra of $U(\mathfrak{g})^{\circ}$ . Here, as an algebra, A ^′ is a divided powers algebra in n variables,

\begin{equation*} A' \; = \; k[f^{(m)}_1, \ldots , f^{(m)}_n \;:\;m \geq 0],\end{equation*}

as defined and discussed at [Reference Montgomery26, Examples 5.6.8 and 9.1.7]. Modulo the Hopf ideal $(\overline{H}^{\ast})^+ A'$ of $W(H^{\circ})$ the coproducts of the elements of A ^′ take their classical cocommutative forms. Similarly, the elements of $(k,+)^n$ are group-like modulo $(\overline{H}^{\ast})^+k(k,+)^n$ .

7.2 Group algebras of finitely generated abelian-by-finite groups

Let G be a finitely generated group with an abelian normal subgroup N of finite index. For convenience, we may as well choose N to be torsion free, and hence free abelian of rank n, say. With $H \;:\!=\; kG$ and $A\;:\!=\; kN$ , $\overline{H} = k(G/N)$ . Clearly H is a free A-module with basis a set of coset representatives of N in G, and we can choose as a complement to A in H the free A-submodule X with basis the non-trivial coset representatives. Thus X is a subcoalgebra of H, so $(\mathbf{{CoSplit}})$ holds, and $(\mathbf{{OrbSemi}})$ holds since the adjoint action on A is by the group $G/N$ . Thus once again Theorem 6.1(1) applies, and again the smash products occurring are trivial because $H^{\circ}$ is commutative. We deduce that, as algebras,

\begin{equation*} kG^{\circ} \; \cong \; k[z_1, \ldots , z_n] \otimes \overline{H}^{\ast} \otimes k((k^{\times})^n), \end{equation*}

with

\begin{equation*}\overline{H}^{\ast} \cong k^{\oplus|G/N|}\end{equation*}

and $\overline{H}^{\ast}$ , $W(H^{\circ}) \,\;:\!=\; \, k[z_1, \ldots , z_n] \otimes \overline{H}^{\ast}$ and $\widehat{k((k^{\times})^n)} \, \;:\!=\; \, \overline{H}^{\ast} \otimes k((k^{\times})^n)$ each being a Hopf subalgebra of $kG^{\circ}$ . As in Section 7.1, the elements $z_i$ are primitive modulo $(\overline{H}^{\ast})^+k[z_1, \ldots , z_n]$ and the elements of $(k^{\times})^n$ are group-like modulo $(\overline{H}^{\ast})^+k((k^{\times})^n)$ .

7.3 Quantized coordinate rings at a root of unity

In this subsection and the next, we assume that $k = \mathbb{C}$ and that $\ell$ is an odd positive integer with $\ell \geq 3$ , and prime to 3 if $\mathfrak{g}$ contains a factor of type $G_2$ . The quantized coordinate ring $\mathcal{O}_q(G)$ of a connected, simply connected, semisimple Lie group G is a noetherian Hopf algebra, [Reference De Concini and Lyubashenko9, Sections 4.1 and 6.1]. If $q=\epsilon$ is a primitive $\ell$ th root of unity, $\mathcal{O}_\epsilon(G)$ is a finite module over a central Hopf subalgebra isomorphic to $\mathcal{O}(G)$ , [Reference De Concini and Lyubashenko9, Proposition 6.4], [Reference Brown and Goodearl4, Theorem III.7.2]. Thus

\begin{equation*} A \; :\!=\; \mathcal{O}(G) \; \subseteq \; \mathcal{O}_{\epsilon}(G) =\;:\;H \end{equation*}

is an affine commutative-by-finite Hopf algebra which satisfies $(\mathbf{{OrbSemi}})$ because of the centrality of A. In fact, the extension $\mathcal{O}(G)\subseteq \mathcal{O}_\epsilon(G)$ is cleft in the sense of Remark 2.3(3), as is noted in [Reference Andruskiewitsch and Andrés-Garcia1, Remark 2.18(b)]), with a cleaving map which is a coalgebra map, as shown in the proof of [Reference Andruskiewitsch and Andrés-Garcia1, Proposition 2.8(c)]. Thus, $(\mathbf{{CoSplit}})$ is satisfied. The finite-dimensional Hopf quotient $\overline{H} \;:\!=\;\mathcal{O}_\epsilon(G)/\mathcal{O}(G)^+\mathcal{O}_\epsilon(G)$ is the restricted quantized coordinate ring, sometimes denoted by $o_\epsilon(G)$ ; its vector space dimension is $\ell^{\textrm{dim}(G)}$ . Each of $u_{\epsilon}(\mathfrak{g})$ , (defined in Section 7.4), and $o_{\epsilon}(G)$ is the Hopf dual of the other by [Reference Brown and Goodearl4, Theorem III.7.10]. Thus Theorem 3.2(2) applies, yielding the isomorphism of algebras, left $u_\epsilon(\mathfrak{g})$ -modules and right $U(\mathfrak{g})\#kG$ -comodules

(7.2) \begin{equation} \mathcal{O}_\epsilon(G)^\circ \; \cong \; u_\epsilon(\mathfrak{g})\otimes (U(\mathfrak{g})\# kG),\end{equation}

where $\mathfrak{g}=\textrm{Lie } G$ denotes the Lie algebra of G and $u_{\epsilon}(\mathfrak{g})$ is $\overline{U_{\epsilon}(\mathfrak{g})}$ , as defined in Section 7.4. By Theorems 6.1(1), 4.3(5), and 5.13(2), $\mathcal{O}_{\epsilon}(G)$ contains the Hopf subalgebras $u_\epsilon(\mathfrak{g})$ ,

\begin{equation*}W(H^{\circ}) \; = \; u_\epsilon(\mathfrak{g})\otimes U(\mathfrak{g})\end{equation*}

and

\begin{equation*} \widehat{kG_H} \;= \; u_\epsilon(\mathfrak{g}) \otimes kG, \end{equation*}

and the tensorand $U(\mathfrak{g})\# kG$ on the right of (7.2) is isomorphic to $\mathcal{O}(G)^{\circ}$ as an algebra.

7.4 Quantized enveloping algebras at a root of unity

The quantized enveloping algebra $U_q(\mathfrak{g})$ of a semisimple finite-dimensional Lie algebra $\mathfrak{g}$ is a noetherian Hopf algebra [Reference De Concini, Procesi, (Zampieri and D’Agnolo10, Section 9.1], [Reference Chari and Pressley7, Section 9.1A]. When $q=\epsilon$ is a primitive $\ell$ th root of unity $U_\epsilon(\mathfrak{g})$ is a free module of rank $\ell^{\dim_k(\mathfrak{g})}$ over a central Hopf domain A [Reference De Concini, Procesi, (Zampieri and D’Agnolo10, Corollary and Theorem 19.1], [Reference Brown and Goodearl4, Theorem III.6.2]. The Hopf subalgebra A is the coordinate ring of a certain solvable Poisson algebraic group T with $\textrm{dim}T = \textrm{dim}_k(\mathfrak{g})$ ; see [Reference De Concini, Procesi, (Zampieri and D’Agnolo10] or [Reference Brown and Goodearl4, III.6.5]. The finite-dimensional Hopf quotient $\overline{H}=U_\epsilon(\mathfrak{g})/A^+U_\epsilon(\mathfrak{g})$ of $U_\epsilon(\mathfrak{g})$ is the restricted quantized enveloping algebra, denoted by $u_\epsilon(\mathfrak{g})$ .

Since A is central in $U_\epsilon(\mathfrak{g})$ $(\mathbf{{OrbSemi}})$ holds, but we do not know whether $(\mathbf{{CoSplit}})$ is valid. However, for the two easiest cases—that is when $\mathfrak{g}$ is $\mathfrak{sl}_2$ or $\mathfrak{sl}_3$ - $(\mathbf{{CoSplit}})$ has been confirmed by hand, as follows.

For $U_\epsilon(\mathfrak{sl}_2)$ with its standard generators $\{E,F, K^{\pm 1}\}$ , $A = k[E^{\ell}, F^{\ell}, K^{\pm \ell}]$ and one may check that the A-submodule X of $U_\epsilon(\mathfrak{sl}_2(k))$ generated by

\begin{equation*} \lbrace F^rK^sE^t\;:\;0\leq r,s,t<l \text{ and } r,t \text{ not both zero} \rbrace \cup \lbrace K^s-1 \;:\;1\leq s<l \rbrace \end{equation*}

is a coideal of $U_\epsilon(\mathfrak{sl}_2(k))$ . When $\mathfrak{g}$ is $\mathfrak{sl}_2$ , the group T is a semidirect product, $T = (k,+)^2\rtimes k^{\times}$ ; see e.g. [Reference Brown and Goodearl4, Section III.6.5]. Let $\mathfrak{t}$ denote its Lie algebra. Since $u_\epsilon(\mathfrak{sl}_2)^{\ast} = o_\epsilon(SL(2))$ by [Reference Brown and Goodearl4, Theorem III.7.10], we deduce from Theorem 6.1 that

\begin{equation*} U_\epsilon(\mathfrak{sl}_2)^\circ \; \cong \; (o_\epsilon(SL_2)\#U(\mathfrak{t}))\#kT ,\end{equation*}

with Hopf subalgebras $o_\epsilon(SL_2)$ , $W(U_\epsilon(\mathfrak{sl}_2)^{\circ})\, \;:\!=\; \, o_\epsilon(SL_2)\#U(\mathfrak{t})$ and $\widehat{kT} \, \;:\!=\; \, o_\epsilon(SL_2)\#kT$ .

For $ U_\epsilon(\mathfrak{sl}_3)$ , the Hopf centre A is $\mathcal{O}(S)$ for a Poisson algebraic group S which, using [Reference Brown and Goodearl4, III.6.5], is a semidirect product $(k,+)^6 \rtimes (k^{\times})^2$ , whose Lie algebra $\mathfrak{s}$ has basis $\lbrace e_1,e_2,e_3,f_1,f_2,f_3,k_1,k_2 \rbrace,$ with non-zero brackets

\begin{equation*} [e_1,e_2]=e_3, [f_1,f_2]=-f_3, [e_1,k_1]=-e_1, [e_2,k_2]=-e_2, \end{equation*}

\begin{equation*} [e_3,k_i]=-e_3, [f_1,k_1]=-f_1, [f_2,k_2]=-f_2, [f_3,k_i]=-f_3. \end{equation*}

An explicit coalgebra A-module complement to A in $U_\epsilon(\mathfrak{sl}_3)$ has been described as a result of extensive calculations in [Reference Couto11, p. 131 and Appendix A.1], so that $A \subseteq U_\epsilon(\mathfrak{sl}_3)$ satisfies $(\mathbf{{CoSplit}})$ . Therefore, applying Theorem 6.1, we obtain

\begin{equation*} U_\epsilon(\mathfrak{sl}_3)^\circ \; \cong\; (o_\epsilon(SL_3)\#U(\mathfrak{s}))\# kS,\end{equation*}

with Hopf subalgebras $o_\epsilon(SL_3)$ ,

\begin{equation*} W(U_\epsilon(\mathfrak{sl}_3)^{\circ}) \; \cong \; o_\epsilon(SL_3)\#U(\mathfrak{s})\end{equation*}

and

\begin{equation*} \widehat{kS} \; \cong \; o_\epsilon(SL_3)\# kS.\end{equation*}

In view of these two cases, it seems reasonable to ask.

7.5 Prime regular affine Hopf algebras of Gelfand–Kirillov dimension 1

Here, “regular” means “having finite global dimension,” necessarily then equal to 1. These Hopf k-algebras were classified when k is an algebraically closed field of characteristic 0 by Wu et al. in [Reference Ding12], building on [Reference Brown and Zhang5] and [Reference Lu, Wu and Zhang24]; see also the survey article [Reference Brown and Zhang6]. By a fundamental result of Small et al. [Reference Small, Stafford and Warfield34], a semiprime affine algebra of GK-dimension one is a finite module over its centre. But in fact more is true for these Hopf algebras—they are all commutative-by-finite. This can be checked on a case-by-case basis, as we now briefly outline. For k algebraically closed of characteristic 0, there are two finite families and three infinite families, as follows.

1. (I) The commutative algebras k[x] and $k[x^{\pm 1}]$ .

2. (II) The unique noncommutative cocommutative example, the group algebra $H = kD$ of the infinite dihedral group D, defined and discussed in Remarks 3.3(1).

3. (III) The infinite dimensional Taft algebras

   \begin{equation*}T(n,t,q)\; :\!=\;\; k\langle g,x\;:\;g^n=1, xg=qgx \rangle,\end{equation*}
   where $n \in \mathbb{Z}_{\geq 2},$ $1 \leq t \leq n$ and q is a primitive nth root of 1 in k, with g group-like and $\Delta (x) = x \otimes1 + g^t \otimes x$ . With $n^{\prime} \;:\!=\;n/\gcd(n,t)$ , $A \;:\!=\;k[x^{n^{\prime}}]$ is a commutative normal Hopf subalgebra.

4. (IV) The generalised Liu algebras B(n,w,q), where n and w are positive integers and q is a primitive nth root of 1. Here,

   \begin{equation*} B(n,w,q) \; :\!=\;\; k\langle x^{\pm 1}, g^{\pm 1}, y\,:\, x \textit{ central},\, yg=qgy, \, g^n=x^w=1-y^n \rangle, \end{equation*}
   with x and g group-like and $\Delta(y) = y \otimes 1 + g\otimes y$ . One can show that $A\;:\!=\;k[x^{\pm 1}]$ is a central Hopf subalgebra over which B(n,w,q) is free of rank $n^2$ .

5. (V) Let m and d be positive integers with $(1+m)d$ even, and let q be a primitive $2m^{\textrm{th}}$ root of 1 in k. The Hopf algebras D(m,d,q) are defined in [Reference Ding12, Section 4.1]. D(m,d,q) is finitely generated over the normal commutative Hopf subalgebra $A\;:\!=\;k[x^{\pm 1}]$ , [Reference Ding12, (4.7)].

The above algebras are all free over the listed normal commutative Hopf subalgebras. Families (I)–(IV) are pointed and decompose as crossed products

\begin{equation*} H\; \cong \; A\#_\sigma \overline{H};\end{equation*}

but D(m,d,q) is not pointed, [Reference Ding12, Proposition 4.9]. Applying Theorems 3.2, 4.3, and 5.13 to these families yields the following information about their Hopf duals, (omitting the algebras in (I) and (II), which are already covered by Sections 3.1 and 7.2). In the proposition, for coprime integers s and m with $1 \leq s < m$ and a primitive $m^{\textrm{th}}$ root of unity $q \in k$ , $T_f(m,s,q)$ denotes the finite dimensional Taft algebra in these parameters; that is, $T_f(m,s,q) \;:\!=\; T(m,s,q)/\langle x^m \rangle$ . For a positive integer m, $C_m$ denotes the cyclic group of order m.

For reasons of space, we omit some details in the proofs for (III) and (IV), and omit the entire proof of (V). Full proofs can be found in [Reference Couto11] and [Reference Jahn19]; detailed references are listed in Remarks 7.1.

Proof. (III) Let $A = k[x^{n^{\prime}}]$ as noted above. Thus

\begin{equation*}\overline{H} \; :\!=\; H/ x^{n^{\prime}}H \; = \; k\langle \overline{x}, \overline{g}\, \;:\;\, \overline{g}^n = 1,\, \overline{x}^{n^{\prime}} = 0, \, \overline{x}\overline{g} = q\overline{g}\overline{x} \rangle.\end{equation*}

Thus, as a right A-module $H = A \oplus X$ , where X is the free right A-module on basis $\mathcal{P}$ , where

\begin{equation*}\mathcal{P} \; :\!=\; \lbrace g^ix^j \;:\;0\leq i<n, 1\leq j<n^{\prime} \rbrace \cup \lbrace g^i-1 \;:\;1\leq i<n \rbrace.\end{equation*}

One checks that X is a coideal of H, so that H satisfies $(\mathbf{{CoSplit}})$ . Moreover, it is clear that, for $ 0\leq i<n,\, 0 \leq j<n^{\prime}$ , the map

\begin{equation*} \gamma\;:\;\overline{H}\longrightarrow H\;:\;\overline{g}^i\overline{x}^j \mapsto g^i x^j \end{equation*}

is a cleaving map of coalgebras, so that Theorem 3.2(2) applies, yielding the isomorphism of algebras

(7.3) \begin{equation} H^{\circ} \; \cong \; \overline{H}^{\ast} \otimes A^{\circ}.\end{equation}

It is now straightforward to calculate that, as an algebra,

\begin{equation*} \overline{H} \; \cong \; T_f(n^{\prime},t^{\prime},q^d) \#_{\tau} kC_d, \end{equation*}

where $\tau$ is a cocycle, while as coalgebra

\begin{equation*} \overline{H} \; \cong \; T_f(n^{\prime},t^{\prime},q^d) \otimes kC_d.\end{equation*}

Since $\textrm{gcd}(n^{\prime},t^{\prime}) = 1$ , $T_f(n^{\prime},t^{\prime},q^d)$ is self-dual by [Reference Radford28, Exercise 7.4.3], as also is $kC_d$ , so we deduce that, as an algebra,

(7.4) \begin{equation} \overline{H}^{\ast} \; \cong \; kC_d \otimes T_f(n^{\prime},t^{\prime},q^d).\end{equation}

Combining (7.3), (7.4) and the fact that $A^{\circ} = k[f] \otimes k(k,+)$ , we deduce the algebra isomorphism in (III), with $\overline{H}^{\ast}$ a Hopf subalgebra by Lemma 3.1(1). Moreover $A \subseteq H$ also satisfies $(\mathbf{{OrbSemi}})$ since the action of $\overline{H}$ on A factors through the group algebra $k\langle\overline{g}\rangle$ . Thus the claims regarding $W(H^{\circ})$ and $\widehat{kG_H}$ follow, respectively, from Theorems 4.3(5) and 5.13(2)

(IV) Let $H = B(n,w,q)$ , so $H = \bigoplus_{0 \leq i,j < n}y^ig^jA$ is a free A-module, and

(7.5) \begin{equation} \overline{H} \; = \; H/A^+H \; \cong \; T_f(n,1,q).\end{equation}

As a complement X to A in H choose the A-module generated by

\begin{equation*} \lbrace g^iy^j \;:\;0\leq i<n, 1\leq j<n \rbrace \cup \lbrace g^i-1\;:\;1\leq i<n \rbrace.\end{equation*}

One checks routinely that X is a coideal of H. Therefore $(\mathbf{{CoSplit}})$ holds, and Theorem 3.2(1) implies that

\begin{equation*} H^{\circ} \; \cong \; \overline{H}^{\ast} \# A^{\circ} \; \cong \; T_f(n,1,q) \# (k[f] \otimes k(k^{\times})),\end{equation*}

by (7.5) and the fact that $T_f(n,1,q)$ is self-dual by [Reference Radford28, Exercise 7.3.4].

Since A is central $A \subseteq H$ satisfies $(\mathbf{{OrbSemi}})$ , so Theorem 6.1(2) confirms the claims regarding $W(H^{\circ})$ and $\widehat{kG_H}$ .

(V) The fact that D(m,d,q) satisfies $(\mathbf{{CoSplit}})$ and $(\mathbf{{OrbSemi}})$ is proved in [Reference Couto11, Section 2.2.5, pp. 52--55 and Proposition 3.1.13], and the structure of its Hopf dual is described in [Reference Couto11, Corollary 4.4.6(V) and Appendix A.2].

7.6 Noetherian PI Hopf domains of Gelfand-Kirillov dimension two

Let k be algebraically closed of characteristic 0 and let H be a noetherian Hopf k-algebra domain with ${\textrm{GK dim}} (H) = 2$ . Such Hopf algebras were classified in [Reference Goodearl and Zhang17] under the additional assumption that H has an infinite dimensional commutative factor, or equivalently by [Reference Goodearl and Zhang17, Proposition 3.8(c)] that $ {\textrm{Ext}}_H^1({_Hk},{_Hk})\neq 0.$

The algebras in this classification which satisfy a polynomial identity (PI) are all easily seen to be commutative-by-finite, as we now indicate. Omitting the 2 group algebras and the rank 2 polynomial algebra, since these have already been discussed, the remaining PI families are as follows, using the numbering scheme of [Reference Goodearl and Zhang17]:

1. (III) Hopf algebras $A(\ell, n,q)$ , where $\ell$ and n are integers with $\ell >1$ and $n > 0$ , and q is a primitive $\ell^{\textrm{th}}$ root of 1 in k. As an algebra

   \begin{equation*} A(\ell, n , q) \; :\!=\; k\langle x^{\pm 1},y\;:\;xy = qyx \rangle, \end{equation*}
   the localised quantum plane, with x group-like and $\Delta (y) = y \otimes 1 + x^n \otimes y$ . These algebras are free of finite rank over the normal commutative Hopf subalgebra $A\;:\!=\;k [x^{\pm \ell}, y^{\ell'} ]$ , where $\ell' \;:\!=\; \ell/ \gcd (n, \ell)$ .

2. (IV) Hopf algebras $B(n, p_0, \dots , p_s,q)$ , where $s \geq 2$ , $n,p_0,\ldots , p_s$ are positive integers with $p_0 \mid n$ and $\{p_i \;:\;i \geq 1 \}$ strictly increasing and pairwise relatively prime, and q is a primitive $\ell^{\textrm{th}}$ root of 1 where $\ell \;:\!=\; (n/p_0)p_1 \ldots p_s$ . Then $B(n, p_0, \dots , p_s,q)$ is the subalgebra of the localised quantum plane from (III) generated by $\{x^{\pm 1}, y^{m_i} \;:\;1 \leq i \leq s \}$ , where $m_i \;:\!=\; \Pi_{j \neq i} p_j.$ This is a Hopf subalgebra of $A(\ell,n,q)$ , with x group-like and $\Delta(y^{m_i}) = y\otimes 1 + x^{m_i n} \otimes y$ . One can easily check that $A \;:\!=\; k[ y^{p_1 \cdots p_s}, x^{\pm \ell} ]$ is a normal commutative Hopf subalgebra over which $B(n,p_0,\ldots , p_s,q)$ is a finite module.

We proceed to describe the Hopf duals of these algebras. Once again we leave some calculations to the reader in order to save space; details of the omitted arguments can be found in [Reference Jahn19] or [Reference Couto11].

Proof. (III) As noted above, $A\;:\!=\;k [x^{\pm \ell}, y^{\ell'} ]$ is a normal commutative Hopf subalgebra of H. It is clear that H is a free right A-module on the basis $\{y^i x^j \;:\;0 \leq i \leq \ell', \, 0 \leq j < \ell \}$ . By a straightforward analysis detailed in [Reference Couto11, Example 1.1.21 and Section 2.2.6], as algebras

\begin{equation*} \overline{H} \;:\!=\; H/A^+ H \; \cong \; T_f(\ell', n^{\prime}, q^{-d}) \#_{\sigma} kC_d ,\end{equation*}

for a cocycle $\sigma$ which is in general non-trivial; and, as coalgebras,

(7.6) \begin{equation} \overline{H} \; \cong \; T_f(\ell', n^{\prime}, q^{-d}) \otimes kC_d .\end{equation}

Since $\textrm{gcd}(n^{\prime},\ell') = 1$ , both tensorands in (7.6) are self-dual (using [Reference Radford28, Exercise 7.3.4] for the left-hand one), and hence

(7.7) \begin{equation} \overline{H}^{\ast} \; \cong \; kC_d \otimes T_f(\ell', n^{\prime}, q^{-d})\end{equation}

as algebras. Moreover $A \cong \mathcal{O}((k,+)\times k^{\times})$ , so that, as Hopf algebras,

(7.8) \begin{equation} A^{\circ} \; \cong \; k[f,f'] \otimes k((k,+) \times k^{\times}),\end{equation}

as recalled in Section 3.1.

Let X be the right A-submodule of H generated by $ \lbrace y^ix^j \;:\;1\leq i<\ell', 0\leq j<\ell \rbrace \cup \lbrace x^j-1\;:\;1\leq j<\ell \rbrace$ . Clearly $H = A \oplus X$ , and it is routine to check that X is a coideal of H. Therefore, $A \subseteq H$ satisfies $(\mathbf{{CoSplit}})$ .

Thus Theorem 3.2(1) applies, and with (7.7) and (7.8) it yields the isomorphism of algebras

\begin{eqnarray*} H^{\circ} \; &\cong& \; \overline{H}^{\ast}\# A^{\circ}\\&\cong& \; (kC_d \otimes T_f(\ell', n^{\prime}, q^{-d})) \# (k[f,f'] \otimes k((k,+)\times k^{\times})),\end{eqnarray*}

with Hopf subalgebra $ (kC_d \otimes T_f(\ell', n^{\prime}, q^{-d}))$ .

It is routine to check that the adjoint action of $\overline{H}$ on A factors through the group algebra $kC_{\ell} = k\langle \overline{x} \rangle$ , so that $A \subseteq H$ satisfies $(\mathbf{{OrbSemi}})$ . Therefore, Theorem 6.1(2) yields the remaining parts of (III).

(IV) With $A = k[y^{p_1 \cdots p_s}, x^{\pm \ell}]$ , a commutative normal Hopf subalgebra of H, one can check that H is a free right A-module on the basis $\{ y_1^{i_1}\ldots y_s^{i_s}x^j \;:\;0\leq j<\ell, 0\leq i_t < p_t \} $ . The structure of the quotient Hopf algebra $\overline{H} \;:\!=\; H/A^+ H$ has been analysed in [Reference Couto11, Lemma 2.2.5]. As an algebra it is an iterated crossed product whose detailed description we do not need here; but as a coalgebra it decomposes as

\begin{equation*} \overline{H} \; \cong \; T_f(p_1,p_0, \xi_1) \otimes T_f(p_2, p_0p_1, \xi_2)\otimes \cdots \otimes T_f(p_s,p_0p_1 \cdots p_{s-1}, \xi_s) \otimes kC_{n/p_0}.\end{equation*}

Since the above tensorands are all self-dual because the $p_i$ are mutually coprime [Reference Radford28, Exercise 7.4.3], we deduce that, as an algebra,

(7.9) \begin{equation} \overline{H}^{\ast} \; \cong \; T_f(p_1,p_0, \xi_1) \otimes T_f(p_2, p_0p_1, \xi_2)\otimes \cdots \otimes T_f(p_s,p_0p_1 \cdots p_{s-1}, \xi_s) \otimes kC_{n/p_0}.\end{equation}

Just as in (III),

(7.10) \begin{equation} A^{\circ} \; \cong \; k[f,f'] \otimes k((k,+) \times k^{\times}).\end{equation}

Define X to be the right A-module generated by

\begin{equation*} \mathcal{B} \; :\!=\; \lbrace y_1^{i_1}\ldots y_s^{i_s}x^j \;:\;0\leq j<\ell, 0\leq i_t<p_t, \text{ some } i_t\geq 1 \rbrace \cup \lbrace x^j-1\;:\;1\leq j<\ell \rbrace.\end{equation*}

Then X is a free A-module on the basis $\mathcal{B}$ with $H = A \oplus X$ . It is routine to confirm that X is a coideal of H, so that $A \subseteq H$ satisfies $(\mathbf{{CoSplit}})$ . Therefore, Theorem 3.2(1) applies implying that

(7.11) \begin{equation} H^{\circ} \; \cong \; \overline{H}^{\ast} \#A^{\circ}.\end{equation}

Combining (7.9), (7.10) and (7.11) gives the first two isomorphisms in (IV).

Finally, note that the adjoint action of each of the elements $y_i$ on A is trivial. Therefore, the action of $\overline{H}$ on A factors through the group algebra $kC_{\ell} = k\langle \overline{x}\rangle$ . Hence, $A \subseteq H$ satisfies $(\mathbf{{OrbSemi}})$ , and so the final two isomorphisms in (IV) follow from Theorem 6.1(2).