Hopf Images of Coactions

• Hopf images of coactions are defined as the minimal quantum symmetries that factor a given coaction through an inner‐faithful representation.
• They are constructed via a universal factorization method that isolates the smallest Hopf algebra or quantum subgroup underpinning effective symmetry.
• Applications include the classification of quantum symmetries in combinatorial structures and quantum principal bundles using both algebraic and analytic techniques.

A Hopf image of a coaction captures the minimal effective quantum symmetry contained in a coaction of a Hopf algebra (or a locally compact quantum group) on an algebraic, operator-algebraic, or geometric structure. This construction provides a universal factorization, isolating the smallest Hopf algebra or closed quantum subgroup through which the coaction factors, thereby providing a canonical reduction to inner-faithful symmetry. The notion of Hopf images is central to the classification of quantum symmetries and the effective symmetry reduction of quantum principal bundles, as well as to understanding quantum symmetries of combinatorial and operator-algebraic objects (Bhattacharjee, 4 Jan 2026, Bichon, 2015, Józiak et al., 2016).

1. Coactions and the Universal Property of the Hopf Image

Let $(H, \Delta, \epsilon, S)$ be a Hopf algebra over a ground field $\mathbb{k}$ of characteristic zero, and $A$ an associative unital algebra over $\mathbb{k}$. A right $H$–coaction on $A$ is a linear map $\delta: A \to A \otimes H$ satisfying

$$(\delta \otimes \operatorname{id}_H) \circ \delta = (\operatorname{id}_A \otimes \Delta) \circ \delta, \quad (\operatorname{id}_A \otimes \epsilon) \circ \delta = \operatorname{id}_A.$$

Given such a coaction $\delta$ and focusing on its symmetry content, the Hopf image $H_\delta$ is the smallest Hopf subalgebra of $H$ such that $\delta(A) \subseteq A \otimes H_\delta$. Universally, $H_\delta$ is the initial object in the category of all factorizations of $\delta$ through Hopf subalgebras, i.e., any coaction factoring $\delta$ through a subalgebra factors uniquely through $H_\delta$.

Explicitly, $$H_\delta = \bigcap_{L \subseteq H,\, \delta(A) \subseteq A \otimes L} L$$, and the restricted coaction $$\delta_{\operatorname{im}}:A \to A \otimes H_\delta$$ is inner-faithful by construction. An equivalent perspective presents $H_\delta$ as the Hopf subalgebra generated by the set of coefficients

$$\mathcal{C}_\delta = \{ (\omega \otimes \operatorname{id}) \delta(a) : a \in A,\, \omega \in A^* \} \subseteq H,$$

or as a quotient $H / I_{\operatorname{triv}}$ for the Hopf ideal $$I_{\operatorname{triv}} = \{ h \in H : \forall a \in A,\ (a_{(0)} \otimes a_{(1)})(1 \otimes h - \epsilon(h) 1 \otimes 1) = 0 \}$$ (Bhattacharjee, 4 Jan 2026).

Functoriality arises: algebra maps $\phi: A \to B$ and Hopf algebra maps $\psi: H \to K$ compatible with coactions induce morphisms between their respective Hopf images.

2. Hopf Images in Locally Compact Quantum Groups

The concept of Hopf image extends to the analytic setting of coactions of locally compact quantum groups. In this context, one works with a coaction $\Delta_\rho : A \to M(A \otimes C_0(G))$ for a $C^*$-algebra $A$, where $G$ is a locally compact quantum group in the Kustermans–Vaes framework.

A closed quantum subgroup $H \subseteq G$ and a morphism $\pi : C_0^u(G) \to C_0^u(H)$ form a Hopf image if the coaction factors as $$(\operatorname{id}_A \otimes \pi^r) \circ \Delta_\rho' = \Delta_\rho$$, and $H$ is universal with respect to this property. This is equivalent to the initial object in the category of all such subgroup factorizations (Józiak et al., 2016).

Existence and uniqueness of the Hopf image are established via the Baaj–Vaes theory: one constructs, from the associated anti-representation of the dual quantum group, a minimal Baaj–Vaes subalgebra $M_{BV}$ of $L^\infty(\widehat{G})$, which corresponds to a unique closed quantum subgroup $H$ encapsulating the effective symmetry.

The fullness or generating property of the coaction is characterized equivalently in terms of ergodicity of the induced partial coaction, density in the dual von Neumann algebra, and injectivity on restriction functors in the representation category.

3. Inner Faithfulness and Effective Quantum Symmetry

A coaction $\delta$ is called inner-faithful if $H_\delta = H$; that is, no proper Hopf subalgebra of $H$ realizes the same symmetry. The restriction of any coaction to its Hopf image yields an inner-faithful coaction, providing a canonical reduction to minimal effective symmetry.

For quantum principal $H$-bundles $(A, \Omega^1(A), H, \delta)$, under cosemisimplicity, the Hopf image reduction produces a quotient $A_0 = A / I$ and an induced coaction $\bar{\delta}$ that is automatically inner-faithful. This leads to a classification of quantum principal bundles up to effective symmetry, where $H_\delta$ is the unique minimal symmetry acting effectively on the reduced total space. Any other reduction to an inner-faithful coaction is essentially equivalent, via a unique injective Hopf algebra morphism (Bhattacharjee, 4 Jan 2026).

4. Explicit Constructions and Classification in Group-Theoretic and Smash Coproduct Settings

For algebraic settings involving finite groups and smash coproducts, the Hopf image provides an explicit classification of quantum symmetries. Given an action of a finite group $H$ on another finite group $G$, the smash coproduct $k[G] \# k^H$ admits coactions whose Hopf images are determined using so-called quotient data: triples $(G', N, \Phi)$ with $G' \subseteq H$, $N \triangleleft G$ normal and $G'$-stable, and a morphism $\Phi: N \to (kG')^\times$ subject to specific compatibility conditions. Every Hopf algebra quotient of $k[G] \# k^H$ is isomorphic to a twisted smash product $k[G/N] \#_\Phi k^{G'}$ for some unique quotient datum (Bichon, 2015).

This framework enables concrete computation of Hopf images and supports the classification of quantum symmetry groups, especially for combinatorial and operator-algebraic structures like quantum permutations of finite sets and deformations of quantum group symmetries.

5. Applications to Quantum Principal Bundles

Hopf image reduction plays a central role in the geometry of quantum principal bundles. Given a quantum principal $H$-bundle equipped with a right-covariant first-order differential calculus and assuming cosemisimplicity of $H$, every such bundle admits a canonical reduction to a quantum principal $H_\delta$-bundle with inner-faithful symmetry.

Formally, with $A$ as total space and $B = A^{\operatorname{co}H}$ as base, one constructs $A_0 = A / I$, where $I$ is the largest $H_\delta$-stable ideal, and obtains a reduced bundle $(A_0, \Omega^1(A_0), H_\delta, \bar{\delta})$. The coaction $\bar{\delta}$ is inner-faithful, yielding a rigidity result: $H_\delta$ is the minimal quantum symmetry acting effectively. Any morphism of quantum principal bundles descends functorially to the level of their Hopf image reductions (Bhattacharjee, 4 Jan 2026).

6. Examples and Representation-Theoretic Aspects

Several prototypical examples illustrate the Hopf image construction:

• Coproduct coaction: For $H$ a Hopf algebra, the coproduct $\delta = \Delta : H \to H \otimes H$ is inner-faithful; thus, its Hopf image is $H$,
• Levi-subgroup coaction: For $\mathcal{O}_q(G) \to \mathcal{O}_q(L_S)$ the canonical quotient map of quantized function algebras of a semisimple group $G$ and Levi subgroup $L_S$, the coaction is inner-faithful,
• Finite group action: For $A$ a $G$-graded algebra and $H = k[G]$, the Hopf image reproduces the group algebra of the effective subgroup of $G$ appearing in the grading (Bhattacharjee, 4 Jan 2026),
• Representation-theoretic picture: In the locally compact setting, the notion of generating morphism is translated via functorial restriction of representations and intertwiner conditions, providing equivalence between Hopf image fullness and generation by families of closed subgroups (Józiak et al., 2016).

These examples collectively demonstrate that the Hopf image formalism isolates the effective quantum symmetry acting via a coaction, admitting both algebraic and analytic instantiations across the theory of quantum groups.

7. Significance, Algorithms, and Further Directions

The classification of Hopf images has key implications for symmetry reduction in noncommutative geometry, algorithmic computation of quantum symmetries, and the understanding of deformation and quotient theory in Hopf algebras and quantum groups (Bichon, 2015). The universality and functoriality of the Hopf image yield robust tools for reducing redundant symmetries, classifying quantum principal bundles up to effective actions, and analyzing representation categories of quantum symmetries. Applications extend to the study of quantum permutations, quantum symmetry groups of combinatorial and operator-algebraic objects (such as complex Hadamard matrices), and the description of deformations and semisimple/cosemisimple Hopf quotients in characteristic zero.

The operator-algebraic generalization and the unification of partial action, representation category, and Tannaka-type approaches further enhance the applicability of the Hopf image paradigm in analytic quantum group theory and its connections with noncommutative geometry and quantum topology (Józiak et al., 2016).