Hopf images and inner faithful representations

Introduction

The aim of this paper is to provide an axiomatization and systematic study of the concept of Hopf image of a Hopf algebra representation, as well as the related concept of inner faithful representation. These notions appeared, under various degrees of generality, in a number of independent investigations: vertex models and related quantum groups , , , locally compact quantum groups and their outer actions , . These were used extensively in the recent paper in order to study the quantum symmetries of Hadamard matrices and of the corresponding subfactors.

The leading idea is that we want to translate the notion of faithful representation of a discrete group at a Hopf algebra level. Let $\Gamma$ be a group, let $A$ be a $k$-algebra ($k$ is a field) and consider a representation

There are two possible notions of faithfulness for $\pi$:

faithfulness of $\pi$ as an algebra map, in which case we simply say that $\pi$ is faithful;

faithfulness of the induced group morphism $\pi_{|\Gamma}:\Gamma\longrightarrow A^{\times}$.

It is clear that the first notion is much more restrictive than the second one, and choice of one of these notions as the good one for faithfulness depends on whether one is interested by the algebra $k[\Gamma]$ or rather by the group $\Gamma$ itself.

It is not difficult so see that $\pi_{|\Gamma}$ is faithful if and only if ${\rm Ker}(\pi)\subset k[\Gamma]$ does not contain any non-zero Hopf ideal. This simple observation leads to a notion of inner faithful representation for arbitrary Hopf algebras: if $H$ a Hopf algebra, we say that a representation

is inner faithful is ${\rm Ker}(\pi)$ does not contain any non-zero Hopf ideal. Of course $H$ is viewed as the group algebra of a discrete quantum group, and faithfulness refers to this discrete quantum group.

If the representation $\pi$ is not inner faithful, there is however a minimal Hopf algebra $H_{\pi}$ that factorizes $\pi$, that we call the Hopf image of $\pi$ (for $H=k[\pi]$, we have $k[\Gamma]\pi=k[\pi(\Gamma)]$). The exact universal property of the Hopf image is stated in Section 2. The Hopf image measures how much $\pi$ fails to be faithful in the discrete quantum group sense. In the situation of , the Hopf image of the representation of the quantum permutation algebra associated to a complex Hadamard matrix measures the complexity of the quantum invariants of the Hadamard matrix.

A natural concept arising from these considerations is the notion of inner linear Hopf algebra: we say that a Hopf algebra is inner linear if it admits a finite-dimensional inner faithful representation. Therefore the problem of inner linearity for Hopf algebras is a generalization of the celebrated linearity problem for discrete groups. More generally, we believe that the study of Hopf images leads to interesting new questions and problems in Hopf algebra theory, as well as in group theory through the study of group duals.

The paper is devoted to a general study of the notion of Hopf image, culminating in a Tannakian formulation, together with the study of several key examples.

Notations and conventions. We work in general over a fixed field $k$. We assume that the reader has some familiarity with Hopf algebras, for which the textbooks or are convenient. Our terminology and notation are the standard ones: in particular, for a Hopf algebra, $\Delta$, $\varepsilon$ and $S$ denote the comultiplication, counit and antipode, respectively.

Ackowledgements. We wish to thank Rupert Yu for helpful discussions on algebraic groups.

Construction of the Hopf image and basic properties

In this section we give the precise formulation of the concept of Hopf image, prove its existence and study of some of its basic properties. The case of Hopf $*$-algebras is also considered at the end of the section.

First, let us give a precise formulation of the notion of Hopf image of a representation. Let $H$ be a Hopf algebra over a field $k$. As usual, a representation of $H$ on an algebra $A$ is an algebra morphism $\pi:H\longrightarrow A$.

Let us say that a factorization of $\pi$ is a triple $(L,q,\varphi)$ where $L$ is a Hopf algebra, $q:H\longrightarrow L$ is a surjective Hopf algebra map and $\varphi:L\longrightarrow A$ is a representation, with the decomposition $\pi=\varphi\circ q$. We define in a straightforward manner the category of factorizations of $\pi$, and the Hopf image of $\pi$ is defined to be the final object in this category (hence we can also say that this is a minimal factorization).

The above proof uses the Hopf ideal $I_{\pi}$, the largest Hopf ideal contained in ${\rm Ker}(\pi)$. This Hopf ideal is constucted in a very abstract manner, and it is useful for several purposes to have a more concrete description of $I_{\pi}$, that we give now.

To any element $g\in F$, we associate an algebra $A^{g}$, defined inductively on the lenght of $g$ as follows. We put $A^{1}=k$, $A^{\alpha_{k}}=A$ if $k$ is even and $A^{\alpha_{k}}=A^{\rm op}$ if $k$ is odd. Now for $g,h\in F$ with $l(g)>1$ and $l(h)>1$, we put $A^{gh}=A^{g}\otimes A^{h}$.

Now we associate an algebra morphism $\pi^{g}:H\longrightarrow A^{g}$ to any $g\in F$, again by induction on the length of $g$. We put $\pi^{1}=\varepsilon$, $\pi^{\alpha_{k}}=\pi\circ S^{k}$, and for $g,h\in F$ with $l(g)>1$ and $l(h)>1$, we put $\pi^{gh}=(\pi^{g}\otimes\pi^{h})\circ\Delta$.

Let $\pi:H\longrightarrow A$ be a representation and let $I_{\pi}$ be the largest Hopf ideal contained in ${\rm Ker}(\pi)$. We have

The proof of Proposition 2.2 uses several lemmas. We put $J_{\pi}=\bigcap_{g\in F}{\rm Ker}(\pi^{g})\subset H$. By construction $J_{\pi}$ is an ideal in $H$, and we wish to prove now that $J_{\pi}$ is a Hopf ideal. The following result ensures that it is a coideal.

Let $C_{\pi}$ be the linear subspace in $H^{*}$ generated by the elements

Then $C_{\pi}$ is a subalgebra of $H^{*}$, and $J_{\pi}=C_{\pi}^{\perp}$. In particular $J_{\pi}$ is a coideal in $H$.

For $\psi,\phi\in H^{*}$ and $g,h\in F$, we have

and since $\varepsilon\in C_{\pi}$, we conclude that $C_{\pi}$ is a subalgebra of $H^{*}$. It is clear that $C_{\pi}^{\perp}=I_{\pi}$, and we conclude that $J_{\pi}$ is a coideal by Proposition 1.4.6 in . ∎

For all $g\in F$, there exists a linear isomorphism $R_{g}:A^{\tau(g)}\longrightarrow A^{g}$ such that $\pi^{g}\circ S=R_{g}\circ\pi^{\tau(g)}$. In particular $S(J_{\pi})\subset J_{\pi}$, and $J_{\pi}$ is a Hopf ideal in $H$.

We prove the result by induction on $l(g)$. For $l(g)=0$ or $l(g)=1$, this follows immediately from the definitions. So assume that $l(g)>1$, so that $g=hh^{\prime}$, with $l(g)>l(h)\geq 1$ and $l(g)>l(h^{\prime})\geq 1$. For $x$ in $H$, we have

The following lemma finishes the proof of Proposition 2.2.

Let $\pi:H\longrightarrow A$ be an algebra map, let $q:H\longrightarrow L$ be a Hopf algebra map, and let $\varphi:L\longrightarrow A$ be an algebra map with $\pi=\varphi\circ q$. Then ${\rm Ker}(q)\subset J_{\pi}$, and hence any Hopf ideal contained in ${\rm Ker}(\pi)$ is contained in $J_{\pi}$.

with the same notation for $\varphi$ as the one for $\pi$. We show this by induction on $l(g)$. This is true if $l(g)=0$ since $q$ is a coalgebra map and if $l(g)=1$, we have $\pi^{\alpha_{k}}=\pi\circ S^{k}=\varphi\circ q\circ S^{k}=\varphi\circ S^{k}\circ q=\varphi^{\alpha_{k}}\circ q$. Assume now that $l(g)>1$, so that $g=hh^{\prime}$ with $l(g)>l(h)\geq 1$ and $l(g)>l(h^{\prime})\geq 1$, and that the result is proved for elements of lenght $<l(g)$. Then

and this proves our assertion by induction. Hence we have ${\rm Ker}(q)\subset J_{\pi}$. Any Hopf ideal is the kernel of a Hopf algebra map, and hence the last assertion follows. ∎

The notion of Hopf image considered here is in general different from the one given in , Definition 1.2.0, which refers to the category of Hopf algebras. Our definition of Hopf image uses the largest category of algebras.

2. Inner faithful representations

The following definition was already given in the introduction.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$. We say that $\pi$ is inner faithful if ${\rm Ker}(\pi)$ does not contain any non-zero Hopf ideal.

We have several equivalent formulations for inner faithfulness.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$. The following assertions are equivalent.

The Hopf algebra morphism $p:H\longrightarrow H_{\pi}$ in Theorem 2.1 is an isomorphism.

If $(L,q,\varphi)$ is any factorization of $\pi$, then $q$ is an isomorphism.

The equivalence between these assertions follows from the previous considerations and Proposition 2.2. Under a special assumption, we also have another equivalent criterion for inner faithfulness.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$ such that ${\rm Ker}(\pi)$ is $S$-stable: $S({\rm Ker}(\pi))\subset{\rm Ker}(\pi)$. Then $\pi$ is inner faithful if and only if $(0)$ is the only bi-ideal contained in ${\rm Ker}(\pi)$.

We also have a characterization of the Hopf image using inner faithfulness, which can be useful in some contexts (for example see the proof of Proposition 3.3).

When considering Hopf algebra maps, Hopf images are usual images and inner faithfulness is equivalent to faithfulness.

Let $H$ and $A$ be Hopf algebras and $\pi:H\longrightarrow A$ be a Hopf algebra map. Then the Hopf image of $\pi$ is $\pi(H)$, and $\pi$ is inner faithful if and only if it is faithful.

The first assertion is immediate, as well as the second one, since ${\rm Ker}(\pi)$ is a Hopf ideal. The following result is also immediate using the universal property of the Hopf image.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$ and let $\theta:A\longrightarrow B$ be an algebra isomorphism. Then we have a Hopf algebra isomorphism $H_{\theta\circ\pi}\simeq H_{\pi}$. In particular $\theta\circ\pi$ is inner faithful if and only if $\pi$ is inner faithful.

Of course one cannot expect that an inner faithful representation $H\longrightarrow A$ will transmit all the algebra properties of the algebra $A$ to the algebra $H$. This is true however for commutativity.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on a commutative algebra $A$. Then the Hopf image $H_{\pi}$ is a commutative algebra, and hence if $\pi$ is inner faithful, then $H$ is also commutative.

Since $A$ is commutative, the ideal of $H$ generated by the commutators of elements of $H$ is contained in ${\rm Ker}(\pi)$. But the commutator ideal is also a Hopf ideal, so is contained in $I_{\pi}$, and hence $H_{\pi}$ is commutative. ∎

3. Hopf ∗*-algebras.

Let us begin by recalling the appropriate language. First, a Hopf $*$-algebra is a Hopf algebra $H$ which is a $*$-algebra and such that the comultiplication $\Delta:H\longrightarrow H\otimes H$ is a $*$-algebra map. It follows that the counit is a $*$-morphism and that the antipode is bijective, and its inverse satisfies $S^{-1}(x)=S(x^{*})^{*}$. A $*$-representation of $H$ on a $*$-algebra is a $*$-algebra map $H\longrightarrow A$.

The formulation of the problem of the existence of a Hopf $*$-image is the same as in the introduction, adding “$*$” where needed. In this framework, Theorem 2.1 has the following form.

The details are left to the reader (to show that $\bigcap_{g\in F^{+}}{\rm Ker}(\pi^{g})$ is $*$-stable, one uses the formula $S\circ*=*\circ S^{-1})$. ∎

We have used the same notation $H_{\pi}$ for the Hopf $*$-image of a $*$-representation $\pi:H\longrightarrow A$ and its Hopf image. In general it seems possible that two notions might not coincide, although we do not have an explicit example. This should cause no confusion: unless specifically notified, when we have such a $*$-representation, the notation $H_{\pi}$ will always denote the Hopf $*$-image.

We will say that a $*$-representation $\pi:H\longrightarrow A$ of a Hopf $*$-algebra $H$ on an $*$-algebra $A$ is inner faithful if ${\rm Ker}(\pi)$ does not contain any non zero Hopf $*$-ideal (similarly to Proposition 2.8 there are several equivalent characterizations). Again it seems to be possible that such a $*$-representation be inner faithful as a $*$-representation, but not as a representation. However the two notions coincide when $S^{2}={\rm id}_{H}$ (or more generally if some power of $S^{2}$ is an inner automorphism).

The proof of Theorem 2.14 also shows that Hopf images exist in the category of Hopf algebras having a bijective antipode (more precisely, the Hopf ideal $I^{+}_{\pi}$ is the largest Hopf ideal with $S(I_{\pi}^{+})=I_{\pi}^{+}$ contained in ${\rm Ker}(\pi)$), and the Hopf image in this category coincides with the Hopf $*$-image. Note that the construction of Theorem 2.1 does not give the Hopf image in the category of Hopf algebras with bijective antipode, since there exist quotients of Hopf algebras with bijective antipode that do not have a bijective antipode: see e.g. .

We now turn to compact Hopf algebras: these are the Hopf $*$-algebras having all their finite-dimensional comodules equivalent to unitary ones (see , these are called CQG algebras there), and are the algebras of representative functions on compact quantum groups.

Let $\pi:H\longrightarrow A$ be a $*$-representation of a compact Hopf algebra $H$ on a $*$-algebra $A$. Then the Hopf $*$-image $H_{\pi}$ is a compact Hopf algebra, and hence Hopf $*$-images exist in the category of compact Hopf algebras.

It is clear from Theorem 27, Section 11.3 in , that a quotient of a compact Hopf algebra is again a compact Hopf algebra. Thus since we have a surjective Hopf $*$-algebra map $H\longrightarrow H_{\pi}$, we conclude that $H_{\pi}$ is a compact Hopf algebra. ∎

Classical examples: group algebras and Lie algebras

In this section we get back to the motivating examples for the notion of inner faithfulness: group algebras and universal enveloping algebras of Lie algebras.

The following result is announced in the introduction. Its origin goes back to Proposition 2.2 in .

Let $\Gamma$ be group, let $A$ be an algebra and let $\pi:k[\Gamma]\longrightarrow A$ be an algebra map. Then we have

In particular $\pi$ is inner faithful if and only if the group morphism $\pi_{|\Gamma}:\Gamma\longrightarrow A^{\times}$ is injective (faithful).

Let $(L,q,\varphi)$ be a factorization of $\pi$. Then $L$ is the group algebra $k[q(\Gamma)]$, and we have a group factorization

Hence we have $q(\Gamma)\subset\pi(\Gamma)$, and this induces the appropriate Hopf algebra map $k[q(\Gamma)]=L\longrightarrow k[\pi(\Gamma)]$. ∎

The above result motivates the following definition.

A Hopf algebra $H$ is said to be inner linear if there exists an inner faithful representation $\pi:H\longrightarrow A$ into a finite-dimensional algebra $A$.

Indeed the group algebra $k[\Gamma]$ is inner linear if and only the group $\Gamma$ is linear (over $k$). It is clear that a Hopf algebra is inner linear if and only if it contains an ideal of finite codimension that does not contain non-zero Hopf ideals.

2. Lie algebras

Let $\mathfrak{g}$ be a Lie algebra, and let $U(\mathfrak{g})$ be its universal enveloping algebra. Algebra maps $U(\mathfrak{g})\longrightarrow A$ correspond exactly to Lie algebra maps $\mathfrak{g}\longrightarrow\mathfrak{gl}(A)$, where $\mathfrak{gl}(A)$ is the Lie algebra associated to $A$: as a vector space $\mathfrak{gl}(A)=A$ and the Lie bracket is defined by $[a,b]=ab-ba$, $\forall a,b\in A$. The Hopf image is described as follows in characteristic zero.

Let $\mathfrak{g}$ be a Lie algebra over a characteristic zero field, let $A$ be an algebra and let $\pi:U(\mathfrak{g})\longrightarrow A$ be an algebra map. Then we have

In particular $\pi$ is inner faithful if and only if the Lie algebra map $\pi_{|\mathfrak{g}}:\mathfrak{g}\longrightarrow\mathfrak{gl}(A)$ is injective (faithful). Thus the Hopf algebra $U(\mathfrak{g})$ is inner linear if and only if $\mathfrak{g}$ is finite-dimensional.

Let us first show if $\pi_{|\mathfrak{g}}:\mathfrak{g}\longrightarrow\mathfrak{gl}(A)$ is injective, then $\pi:U(\mathfrak{g})\longrightarrow A$ is inner faithful. The space of primitive elements $\mathcal{P}(U(\mathfrak{g}))$ equals $\mathfrak{g}$ by the characteristic zero assumption and $1$ is the only group-like in $U(\mathfrak{g})$ (see e.g. Proposition 5.5.3 in ). Thus the canonical map $p:U(\mathfrak{g})\longrightarrow U(\mathfrak{g})_{\pi}$ is injective on primitive elements, and since the Hopf algebra $U(\mathfrak{g})$ is pointed, we use Corollary 5.4.7 in to conclude that $p$ is injective, and hence $\pi$ is inner faithful.

In general we have a Hopf algebra factorization

The algebra map on the right is inner faithful by the previous discussion, and hence by Proposition 2.9 we have the announced isomorphism $U(\mathfrak{g})_{\pi}\simeq U(\pi(\mathfrak{g}))\simeq U(\mathfrak{g}/{\rm Ker}(\pi_{|\mathfrak{g}}))$. Finally if $\pi$ is inner faithful, it induces an isomorphism $U(\mathfrak{g})\simeq U(\pi(\mathfrak{g}))$, which induces an isomorphism between the Lie algebras of primitive elements, and hence between $\mathfrak{g}$ and $\pi(\mathfrak{g})$: $\pi_{|\mathfrak{g}}$ is injective.

If $U(\mathfrak{g})$ is inner linear, then $\mathfrak{g}$ is finite-dimensional by the previous discussion. The converse follows from Ado’s Theorem: a finite dimensional Lie algebra $\mathfrak{g}$ has a faithful finite-dimensional representation $\mathfrak{g}\longrightarrow\mathfrak{gl}_{n}(k)$ for some $n$. ∎

In fact the main argument we have used for the proof this proposition, Corollary 5.4.7 in , a result due independently to Takeuchi and Radford, is valid for arbitrary pointed Hopf algebras. We use it in a similar manner in the next section to get an inner faithfulness criterion for representations of arbitrary pointed Hopf algebras.

Pointed Hopf algebras

After the motivating examples of group algebras and enveloping algebras of Lie algebras, the next natural step is the study of pointed Hopf algebras. These have been much studied in recent years (see e.g. for quantized enveloping algebras of Lie algebras and for the finite-dimensional case). In this section we study the inner faithfulness of a representation of a pointed Hopf algebra. The criterion that we give unifies those given in the previous section.

Recall that a Hopf algebra $H$ is said to be pointed if all its simple comodules are one-dimensional, hence corresponding to group-like elements. The group of group-like elements of $H$ is denoted by Gr$(H)$. An element $x\in H$ is said to be $(g,h)$-primitive for $g,h\in{\rm Gr}(H)$ if

The space of $(g,h)$-primitive elements is denoted $\mathcal{P}_{g,h}(H)$.

Let $\pi:H\longrightarrow A$ be a representation of a pointed Hopf algebra $H$ on an algebra $A$. The following assertions are equivalent.

$\forall g\in{\rm Gr}(H)$, the restriction $\pi_{|\mathcal{P}_{g,1}(H)}:\mathcal{P}_{g,1}(H)\longrightarrow A$ is injective.

$\forall g\in{\rm Gr}(H)$, the restriction $\pi_{|\mathcal{P}_{1,g}(H)}:\mathcal{P}_{1,g}(H)\longrightarrow A$ is injective.

We begin by showing that $\pi$ is inner faithful if and only if for all $g,h\in{\rm Gr}(H)$, the restriction $\pi_{|\mathcal{P}_{g,h}(H)}:\mathcal{P}_{g,h}(H)\longrightarrow A$ is injective. Assume that $\pi$ is inner faithful. Let $x\in\mathcal{P}_{g,h}(H)$ be such that $\pi(x)=0$ and let $I$ be the ideal of $H$ generated by $x$. It is clear from the identities

that $I$ is a Hopf ideal, and since $I$ is contained in ${\rm Ker}(\pi)$, we get $I=0$ and $x=0$.

Conversely, assume that $\pi$ is injective on each space of primitives. Then so is $p:H\longrightarrow H_{\pi}$, and hence by Corollary 5.4.7 in , we conclude that $p$ is an isomorphism and that $\pi$ is inner faithful.

For $x\in\mathcal{P}_{g,h}(H)$, we have $g^{-1}x\in\mathcal{P}_{1,g^{-1}h}(H)$ and $h^{-1}x\in\mathcal{P}_{h^{-1}g,1}(H)$. Hence if (2) or (3) holds, then $\pi$ is injective when restricted to each space of primitives, and the previous discussion shows that $\pi$ is inner faithful. ∎

It is a direct computation to check that the conditions of Theorem 4.1 are fulfilled, and hence $\pi$ is inner faithful. If $q$ is a root of unity, then $\pi$ is not injective on the group-like elements and hence is not inner faithful. ∎

Function algebras

In this section we study Hopf images for various function algebras: polynomial functions on algebraic groups and representative functions on compact groups. The idea for the computation of the Hopf image goes back to , but we are a little bit more general here. These simple examples are already interesting for testing the possibility of generalizing representation theoretic properties of discrete group algebras to arbitrary Hopf algebras.

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic zero and let $\pi:\mathcal{O}(G)\longrightarrow A$ be a representation on an algebra $A$. Assume that the algebra $\pi(A)$ is reduced, so that $\pi(A)\simeq\mathcal{O}(X)$ for an affine algebraic set $X$ and that the algebra map $\mathcal{O}(G)\longrightarrow\pi(A)\simeq\mathcal{O}(X)$ is induced by a polynomial map $\nu:X\longrightarrow G$. Then $\mathcal{O}(G)_{\pi}\simeq\mathcal{O}(\overline{\langle\nu(X)\rangle})$, where $\overline{\langle\nu(X)\rangle}$ is the Zariski closure in $G$ of the subgroup generated by $\nu(X)$.

By the assumption and Proposition 2.12 we may assume that $\pi:\mathcal{O}(G)\longrightarrow\mathcal{O}(X)$ is induced by an injective polynomial map $\nu:X\longrightarrow G$. The injections $X\rightarrow\overline{\langle\nu(X)\rangle}\subset G$ yield a factorization of $\pi$. Now let $(L,q,\varphi)$ be a factorization of $\pi$. Then $L$ is finitely generated and is reduced by Cartier’s theorem ($k$ has characteristic zero), hence we can assume that $L=\mathcal{O}(H)$ for a linear algebraic group $H$, and that $q$ and $\rho$ are induced by polynomial maps $X\rightarrow H\subset G$ whose composition is $\nu$. Hence $\overline{\langle\nu(X)\rangle}\subset H$, and this gives the required Hopf algebra map $\mathcal{O}(H)\longrightarrow\mathcal{O}(\overline{\langle\nu(X)\rangle})$. ∎

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic zero and let $g_{1},\ldots,g_{n}\in G$. The algebra map

has $\mathcal{O}(\overline{\langle g_{1},\ldots,g_{n}\rangle})$ as Hopf image and is inner faithful if and only if $G=\overline{\langle g_{1},\ldots,g_{n}\rangle}$.

The compact group case is worked out similarly. Let $G$ be a compact group. The Hopf $*$-algebra of representative functions on $G$ is denoted by $\mathcal{R}(G)$: this a dense $*$-subalgebra of C$(G)$ by the Peter-Weyl theorem, and moreover C$(G)$ is the enveloping ${\rm C}^{*}$-algebra of $\mathcal{R}(G)$.

Let $G$ be a compact group and let $\pi:\mathcal{R}(G)\longrightarrow A$ be a $*$-representation on a ${\rm C^{*}}$-algebra $A$. Extend $\pi$ to a $*$-algebra map $\pi^{+}:C(G)\longrightarrow A$, and let $X$ be the spectrum of $\pi^{+}(C(G))$, so that $\pi^{+}(C(G))\simeq C(X)$ and that the induced ${\rm C^{*}}$-algebra map $C(G)\longrightarrow\pi^{+}(C(G))\simeq C(X)$ comes from a continuous map $\nu:X\longrightarrow G$. Then $\mathcal{R}(G)_{\pi}\simeq\mathcal{R}(\overline{\langle\nu(X)\rangle})$, where $\overline{\langle\nu(X)\rangle}$ is the closure in $G$ of the subgroup generated by $\nu(X)$.

The proof is essentially the same as the one of the previous proposition, because a quotient of a compact Hopf algebra is itself compact, and by the Hopf algebra version of the Tannaka-Krein theorem, a commutative compact Hopf algebra is the algebra of representative functions on a unique compact group. ∎

We end the section by showing that the simple example of function algebras shows that it is not possible to deduce inner faithfulness of a representation of a cosemisimple Hopf algebra by only studying its restriction to characters.

Let us assume that $k$ is algebraically closed. Recall that a cosemisimple Hopf algebra is a Hopf algebra $H$ whose category of comodules is semisimple. This is equivalent to say that $H$ has a Peter-Weyl decomposition

where $\Lambda$ is the set of simple $H$-comodules and for $\lambda\in\Lambda$, $H_{\lambda}$ is the comatrix coalgebra of corresponding coefficients. Let $d_{\lambda}$ be the dimension of the simple $H$-comodule corresponding to $\lambda$, and let $\chi_{\lambda}\in H_{\lambda}$ be the corresponding character.

The group algebra case corresponds to when $d_{\lambda}=1$ for any $\lambda$, and then the characters correspond to the group-like elements. Since for group algebras inner faithfulness can be detected by only studying the restriction of a representation to group-like elements, it would be natural to hope that in the general case, the restriction to characters would lead to the same information. The following simple example shows that this is not true.

The elements $\tau_{1}$, $\tau_{2}$ and $\tau_{3}$ are all conjugate in $S_{4}$, so this representation is not injective on characters. Thus an inner faithful representation is not necessarily injective on characters.

is injective on characters (as one easily checks on the character table of $S_{4}$) but is not inner faithful since the elements $\tau_{1}$ and $(1,2,3)$ do not generate $S_{4}$.

As a conclusion, it seems that there is no link between the inner faithfulness of a representation and its injectivity on characters.

Twistings

In this section we study the behaviour of Hopf images under various deformation procedures such as Drinfeld’s twisting, or the dual operation, often called 2-cocycle deformation, that we will call here cotwisting for simplicity.

We begin with the twisting operation, in the sense of Drinfeld . More exactly the definitions we use are taken or adapted from .

Let $H$ be a Hopf algebra and let $\Omega$ be an invertible element in $H\otimes H$. Consider the linear maps $\delta_{\Omega},\Delta_{\Omega}:H\longrightarrow H\otimes H$ defined respectively by

We say that $\Omega$ is a twist on $H$ if $(H,\delta_{\Omega},\varepsilon)$ is a coalgebra. The element $u=m\circ({\rm id_{H}}\otimes S)(\Omega)$ is then invertible in $H$, and $H_{\Omega}=(H,m,u,\Delta_{\Omega},\varepsilon,S_{u})$ is a Hopf algebra, where $S_{u}:H\longrightarrow H$ is defined by $S_{u}(x)=uS(x)u^{-1}$.

We say that $\Omega$ is a pseudo-twist on $H$ if $(\varepsilon\otimes{\rm id})(\Omega)=1=({\rm id}\otimes\varepsilon)(\Omega)$, if $(H,\Delta_{\Omega},\varepsilon)$ is a coalgebra, and if there exists an invertible element $u\in H$ such that $S_{u}:H\longrightarrow H$, defined by $S_{u}(x)=uS(x)u^{-1}$, is an antipode for the bialgebra $(H,m,u,\Delta_{\Omega},\varepsilon)$, so that $H_{\Omega}=(H,m,u,\Delta_{\Omega},\varepsilon,S_{u})$ is a Hopf algebra.

A Hopf algebra having the form $H_{\Omega}$ for some twist (resp. pseudo-twist) $\Omega$ on $H$ is said to be a twist (resp. pseudo-twist) of $H$.

The following lemma gives some basic properties of twisting, probably well-known. The direct verification is left to the reader.

Let $\Omega$ be a pseudo-twist on a Hopf algebra $H$.

Let $f:H\longrightarrow L$ be a surjective Hopf algebra map. Then $f^{*}(\Omega)=(f\otimes f)(\Omega)$ is also a pseudo-twist for $L$.

The Hopf ideals in $H$ are exactly the Hopf ideals in $H_{\Omega}$.

Let $H$ be a Hopf algebra, let $\Omega$ be a pseudo-twist on $H$ and let $\pi:H\longrightarrow A$ be a representation on an algebra $A$, that we also view as a representation $\pi:H_{\Omega}\longrightarrow A$. We have $(H_{\Omega})_{\pi}=(H_{\pi})_{p^{*}(\Omega)}$, where $p:H\longrightarrow H_{\pi}$ is the canonical projection, and $\pi$ is inner faithful as a representation of $H$ if and only if it is inner faithful as a representation of $H_{\Omega}$.

Since by the previous lemma the Hopf ideals of $H$ and of $H_{\Omega}$ are the same, this is also true for the Hopf ideals contained in ${\rm Ker}(\pi)$, and the largest one is the same. So the defining Hopf ideal of $(H_{\Omega})_{\pi}$ is the Hopf ideal $I_{\pi}$ constructed in Section 2, so that $(H_{\Omega})_{\pi}=H_{\Omega}/I_{\pi}$, and since $(H/I_{\pi})_{p^{*}}(\Omega)=H_{\Omega}/I_{\pi}$, we have the claimed result. The last assertion is also immediate after the previous discussion. ∎

It follows that the twist of an inner linear Hopf algebra is still inner linear. The reader will find several examples of twisted Hopf algebras in , for example, for which the above theorem furnishes inner faithful representations of the new Hopf algebra from the old one. Some of the Hopf algebras considered in Section 9 are also obtained by twisting.

2. Cotwisting

We now deal with the dual operation to twisting, that we call cotwisting. The situation is more complicated here, because we deform the product rather than the coproduct, and so a representation of the given Hopf algebra is not a representation of the deformed one.

Let us recall the basic vocabulary, which is dual to the one of the previous subsection. We only consider the cotwist case.

Let $H=H$ be a Hopf algebra. We use Sweedler’s notation $\Delta(x)=x_{1}\otimes x_{2}$. Recall (see e.g. ) that a cotwist (=2-cocycle) is a convolution invertible linear map $\sigma:H\otimes H\longrightarrow{\rm C}$ satisfying

and $\sigma(x,1)=\sigma(1,x)=\varepsilon(x)$, for $x,y,z\in H$.

Following and , we associate various algebras to a cotwist. First consider the algebra ${}_{\sigma}\!H$. As a vector space we have ${}_{\sigma}\!H=H$ and the product of ${}_{\sigma}H$ is defined to be

where an element $x\in H$ is denoted $\{x\}$, when viewed as an element of ${}_{\sigma}\!H$.

We also have the algebra $H_{\sigma^{-1}}$, where $\sigma^{-1}$ denotes the convolution inverse of $\sigma$. As a vector space we have $H_{\sigma^{-1}}=H$ and the product of $H_{\sigma^{-1}}$ is defined to be

where an element $x\in H$ is denoted $\langle x\rangle$, when viewed as an element of $H_{\sigma^{-1}}$. The cocycle condition ensures that ${}_{\sigma}\!H$ and $H_{\sigma^{-1}}$ are associative algebras with $1$ as a unit.

Finally we have the Hopf algebra $H^{\sigma}={{}_{\sigma}\!H}\!_{\sigma^{-1}}$. As a coalgebra $H^{\sigma}=H$. The product of $H^{\sigma}$ is defined to be

where an element $x\in H$ is denoted $[x]$, when viewed as an element of $H^{\sigma}$, and we have the following formula for the antipode of $H^{\sigma}$:

A Hopf algebra having the form $H^{\sigma}$ for some cotwist $\sigma$ on $H$ is said to be a cotwist of $H$.

Let $H$ be a Hopf algebra and let $\sigma:H\otimes H\to k$ be a cotwist induced by a finite-dimensional quotient Hopf algebra of $H$. Assume that $S^{2}$ is an inner automorphism of $H$. If $H$ is inner linear, then $H^{\sigma}$ is also inner linear.

The proof will be a consequence of the following technical result.

Let $\theta:H\longrightarrow A\otimes L$ be an algebra map, where $H,L$ are Hopf algebras and $A$ is an algebra, and let $\varphi:L\longrightarrow B$ be an inner faithful representation of $L$ on an algebra $B$. Assume that there exists $\psi\in A^{*}$ such that $(\psi\otimes{\rm id}_{L})\circ\theta:H\longrightarrow L$ is an injective coalgebra map. Assume moreover that one of the following conditions holds.

$S_{L}\circ((\psi\otimes{\rm id}_{L})\circ\theta)=((\psi\otimes{\rm id}_{L})\circ\theta)\circ S_{H}$.

$S_{L}({\rm Ker}(\varphi))\subset{\rm Ker}(\varphi)$.

Then the representation $({\rm id}_{A}\otimes\varphi)\circ\theta:H\longrightarrow A\otimes B$ is inner faithful.

Let $I\subset{\rm Ker}(({\rm id}_{A}\otimes\varphi)\circ\theta)$ be a Hopf ideal. Let $J=(\psi\otimes{\rm id}_{L})(\theta(I))\subset L$: this is a coideal since $(\psi\otimes{\rm id}_{L})\circ\theta$ is a coalgebra map. We have

and hence $J\subset{\rm Ker}(\varphi)$. Let $J^{+}$ be the ideal of $L$ generated by $J$: we have $J^{+}\subset{\rm Ker}(\varphi)$ and $J^{+}$ is a bi-ideal in $L$.

Assume that condition (1) holds. Then $J$ is $S_{L}$-stable and hence so is $J^{+}$, which is a Hopf ideal. Since $\varphi$ is inner faithful, we have $J^{+}=J=(0)$, and we conclude that $I=(0)$ by the injectivity of $(\psi\otimes{\rm id}_{L})\circ\theta$.

Assume now that condition (2) holds. Then $J^{+}=(0)=J$ by Proposition 2.8, and $I=(0)$, again by the injectivity of $(\psi\otimes{\rm id}_{L})\circ\theta$. ∎

By the assumption, there is a Hopf algebra surjection $\pi:H\to K$ onto a finite dimensional Hopf algebra $K$ and a cotwist $\tau:K\otimes K\to k$ such that $\sigma=\tau_{\pi}$. As noted in , we have an injective algebra map

Consider the linear map $\psi=\varepsilon\otimes\varepsilon:{{}_{\tau}\!K}\otimes K_{\tau^{-1}}\longrightarrow k$. We have $(\psi\otimes{\rm id})\circ\theta={\rm id}$, and hence is a coalgebra map. Let $\varphi_{0}:H\longrightarrow B$ be an inner faithful finite-dimensional representation, and let $\varphi:H\longrightarrow B\times B^{\rm op}$, $x\longmapsto(\varphi_{0}(x),\varphi_{0}(S(x)))$. It is clear that $\varphi$ is still inner faithful, and the second condition in the previous theorem is satisfied since $S^{2}$ is inner. Hence the previous theorem ensures that the representation $({\rm id}\otimes\varphi)\circ\theta:H\longrightarrow{{}_{\tau}\!K}\otimes K_{\tau^{-1}}\otimes(B\times B^{\rm op})$ is inner faithful, and we are done since ${{}_{\tau}\!K}\otimes K_{\tau^{-1}}$ is finite-dimensional. ∎

Tensor and free product of representations

The Hopf image does not behave well with respect to the tensor or free product: the Hopf image of a tensor or free product is not necessarily the tensor or free product of the Hopf images. Here is what can be said in general.

Let $H$ and $L$ be Hopf algebras and let $\pi:H\longrightarrow A$ and $\varphi:L\longrightarrow B$ be algebra maps. Then there are surjective Hopf algebra maps

We consider the Hopf algebra factorizations

and the universal property of the Hopf image yields the announced Hopf algebra maps, which are clearly surjective. ∎

The morphisms in the proposition are not injective in general, as the following example shows.

For the tensor product, a faithfulness assumption on one of the algebra morphisms enables one to describe the Hopf image easily. We begin with a lemma. We use the notation of Section 2.

Let $H$ and $L$ be Hopf algebras and let $\pi:H\longrightarrow A$ and $\varphi:L\longrightarrow B$ be algebra maps. Then

We first show that for any $g\in F$, there exists a linear isomorphism $T_{g}:A^{g}\otimes B^{g}\longrightarrow(A\otimes B)^{g}$ such that $(\pi\otimes\varphi)^{g}=T_{g}\circ(\pi^{g}\otimes\varphi^{g})$. We prove this by induction on $l(g)$. This is clear if $l(g)\leq 1$, so we assume that $g=hh^{\prime}$ with $l(g)>l(h)\geq 1$ and $l(g)>l(h^{\prime})\geq 1$. Let $x\in H$ and $y\in L$. We have, using the induction assumption,

and hence we have the desired result. Thus for $g\in F$, we have ${\rm Ker}((\pi\otimes\varphi)^{g})={\rm Ker}(\pi^{g}\otimes\varphi^{g})$, and we have our result. ∎

Let $H$ and $L$ be Hopf algebras and let $\pi:H\longrightarrow A$ and $\varphi:L\longrightarrow B$ be algebra maps. Assume that $\pi$ is faithful and that the antipode of $H$ is injective. Then $(H\otimes L)_{\pi\otimes\varphi}\cong H\otimes L_{\varphi}$. In particular if $\pi$ is faithful and $\varphi$ is inner faithful, then $\pi\otimes\varphi$ is inner faithful.

It is sufficent to show that $I_{\pi\otimes\varphi}=H\otimes I_{\varphi}$, since $(H\otimes L)/(H\otimes I_{\varphi})\simeq H\otimes(L/I_{\varphi})$. For any $g\in F$, the representation $\pi^{g}$ is faithful since $\Delta$ and $S$ are injective, so ${\rm Ker}(\pi^{g}\otimes\varphi^{g})=H\otimes{\rm Ker}(\varphi^{g})$. By the previous lemma we have

and we have our result. The last assertion is immediate. ∎

Of course the assumption of the injectivity of the antipode is satisfied in the most interesting cases. See for an example of a Hopf algebra having a non-injective antipode.

The result in the previous proposition naturally leads to the following question.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$. Does there exist a condition on $\pi$, weaker than faithfulness, that ensures that for any inner faithful representation $\varphi:L\longrightarrow B$ of a Hopf algebra $L$ on an algebra $B$, then the representation $\pi\otimes\varphi$ is inner faithful?

If a representation $\pi$ satisfies to the hypothetic condition of Question 7.6, then in particular $\pi\otimes\pi$ will be inner faithful. It seems to be simpler to study first representations with this weaker property, and this leads to the following definition.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$. We say that $\pi$ is projectively inner faithful if $\pi\otimes\pi$ is inner faithful.

Once again the terminology is motivated by the discrete group case: a representation $\pi:k[\Gamma]\longrightarrow A$ is projectively inner faithful if and only if the associated group morphism $\Gamma\longrightarrow A^{\times}/k^{*}$ is faithful. When $A={\rm End}(V)$, this means that $\Gamma$ embeds into the projective linear group ${\rm PGL}(V)$.

The example of function algebras shows that it is in general difficult to decide when an inner faithful representation is projectively inner faithful. This specific example leads to the following definition. This is a purely discrete group theoretic definition, but of course we have in mind discrete groups embedded as dense subgroups of algebraic groups.

Let $\Gamma$ be a discrete group and let $\{s_{1},\ldots,s_{n}\}$ be a family of generators of $\Gamma$. We say that $\{s_{1},\ldots,s_{n}\}$ is a family of projective generators of $\Gamma$ if the group $\Gamma\times\Gamma$ is generated by the elements $(s_{i},s_{j})$, $1\leq i,j\leq n$.

Of course $\{s_{1},\ldots,s_{n}\}$ is a family of projective generators if $1\in\{s_{1},\ldots s_{n}\}$. It would be interesting to have more examples and to characterize the family of projective generators of a group.

We end the section by noting that there is no analogous result to Proposition 7.4. in the free product case, as shown by the following example.

Tannaka duality

Tannaka duality studies the interplays between a Hopf algebra and its category of comodules. In this section we formulate some Tannaka type results for Hopf images. These results are used in , in the study of quantum permutation groups associated to complex Hadamard matrices.

Let $H$ be a Hopf algebra and let $U,V$ be $H$-comodules. The coaction on $U$ is denoted $\alpha_{U}:U\longrightarrow U\otimes H$. The set of $H$-comodule morphisms from $U$ to $V$ is denoted ${\rm Hom}_{H}(U,V)$. If $f:H\longrightarrow L$ is a Hopf algebra map, then $f$ induces natural $L$-comodule structures on $U$ and $V$ (the resulting comodules still being denoted $U$ and $V$ if no confusion arises), with

We now introduce a key space for the computation of Hom spaces of a Hopf image.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$, and let $U,V$ be some $H$-comodules. The space of $\pi$-morphisms from $U$ to $V$ is defined by

The idea is that the space of $\pi$-morphisms is a more concrete one than the space of $H_{\pi}$-comodules (at least if the algebra $A$ is a more concrete one than $H$), and hence should be easier to describe. In fact it contains all the necessary information.

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$. Let $U$ and $V$ be finite dimensional $H$-comodules and let $(L,q,\varphi)$ be a factorization of $\pi$. Then we have

The first three inclusions on the left arise from the Hopf algebra maps $H\rightarrow L\rightarrow H_{\pi}$. Let $f\in{\rm Hom}_{H_{\pi}}(U,V)$. Then we have, with the notations of Theorem 2.1:

Hence $f\in{\rm Hom}(U_{\pi},V_{\pi})$ and ${\rm Hom}_{H_{\pi}}(U,V)\subset{\rm Hom}(U_{\pi},V_{\pi})$. Assume conversely that $f\in{\rm Hom}(U_{\pi},V_{\pi})$, and let $e_{1},\ldots,e_{m}$ and $e^{\prime}_{1},\ldots,e^{\prime}_{m}$ be respective bases of $U$ and $V$ with

Let $(\lambda_{ij})\in M_{m,n}(k)$ be such that $f(e_{i})=\sum_{j}\lambda_{ji}e^{\prime}_{j}$. Then since $f\in{\rm Hom}(U_{\pi},V_{\pi})$, we have for $1\leq i\leq n$, $1\leq l\leq m$:

Let $I$ be the ideal generated by the elements $P_{li}$. We have

and hence that $I$ is a bi-ideal. Multiplying $P_{il}$ on the left by $S(v_{rl})$ and on the right by $S(u_{is})$ and summing over $i$ and $l$ gives

Thus $I$ is a Hopf ideal and hence $I\subset I_{\pi}={\rm Ker}(p)$. Thus we have, for $1\leq i\leq n$, $1\leq l\leq m$:

This exactly means that $f\in{\rm Hom}_{H_{\pi}}(U,V)$, and we are done. ∎

Let $\pi:H\longrightarrow A$ be a representation of a Hopf algebra $H$ on an algebra $A$. If $\pi$ is inner faithful, then we have,

The converse of the corollary is not true in general. To see this, assume that $k$ has characteristic zero and consider $H=\mathcal{O}({\rm SL}_{2}(k))$ and $L=\mathcal{O}(B)$, with $B$ being the Borel subgroup of ${\rm SL}_{2}(k)$ consisting of triangular matices. The restriction map $\mathcal{O}({\rm SL}_{2}(k))\longrightarrow\mathcal{O}(B)$ is not inner faithful because it is not faithful, and has $\mathcal{O}(B)$ as Hopf image. One easily sees that for the irreducible representations of ${\rm SL}_{2}(k)$ (the symmetric powers of the fundamental representation), one has ${\rm Hom}_{{\rm SL}_{2}(k)}(U,V)={\rm Hom}_{B}(U,V)$, and hence by the cosemisimplicity of $\mathcal{O}({\rm SL}_{2}(k))$, this is true for any representation of ${\rm SL}_{2}(k)$.

However we have a converse to the corollary if both $H$ and the Hopf image are cosemisimple.

Let $\pi:H\longrightarrow A$ be a representation of a cosemisimple Hopf algebra $H$ on an algebra $A$. Assume that the Hopf image $H_{\pi}$ is cosemisimple. Then $\pi$ is inner faithful if and only if

for any simple $H$-comodules $U$ and $V$.

The $\Rightarrow$ part is the previous corollary. Conversely, let us assume that

for any simple $H$-comodules $U$ and $V$. Then the canonical projection $p:H\longrightarrow H_{\pi}$ induces an injection from the set of simple $H$-comodules to the set of simple $H_{\pi}$-comodules, and using the respective Peter-Weyl decompositions of $H$ and $H_{\pi}$, we see that $p$ is injective, and hence is an isomorphism. ∎

It seems to be difficult in general to ensure that the Hopf image is cosemisimple. However this is automatically true when one works in the category of compact Hopf algebras (in the sense of Section 2), and hence we get the following interesting characterization of inner faithfulness for $*$-representations.

Let $\pi:H\longrightarrow A$ be a $*$-representation of a compact Hopf algebra $H$ on a $*$-algebra $A$. Then $\pi$ is inner faithful if and only if