Inner-Faithful Quantum Symmetry

• Inner-Faithful Quantum Symmetry is a concept describing Hopf algebra coactions that do not factor through any proper subalgebra, ensuring maximum effective action.
• It underpins quantum principal bundles by facilitating canonical reductions and classifying bundles through minimal effective quantum symmetry.
• Key examples include the coproduct in Hopf algebras and coactions on quantum groups that exhibit inner-faithfulness, highlighting its practical value in quantum symmetry research.

Inner-faithful quantum symmetry characterizes quantum symmetries encoded by Hopf algebra coactions that are non-degenerate in the sense that they do not factor through any proper Hopf subalgebra. This maximal non-degeneracy ensures that all available quantum symmetries act "effectively" on the underlying algebraic or geometric structure. The concept plays a pivotal role in the structure theory of quantum principal bundles, where one seeks to classify bundles up to effective symmetry and identify canonical reductions with minimal quantum symmetry.

1. Hopf Algebra Coactions and Matrix Coefficients

Let $H$ be a Hopf algebra with coproduct $\Delta$, counit $\varepsilon$, and bijective antipode $S$. Given an associative unital algebra $A$, a right $H$-coaction is a map

$$\delta: A \rightarrow A \otimes H,\qquad \delta(a) = a_{(0)} \otimes a_{(1)}$$

satisfying the coassociativity and counitality axioms: \begin{align*} (\delta \otimes \mathrm{id}_H) \circ \delta &= (\mathrm{id}_A \otimes \Delta) \circ \delta,\ (\mathrm{id}_A \otimes \varepsilon) \circ \delta &= \mathrm{id}_A. \end{align*} Matrix coefficients associated with the coaction are elements of the form $(\omega \otimes \mathrm{id})\delta(a)$ for $a \in A$, $\omega \in A^*$.

2. Hopf Images and Their Universal Property

A factorization of the coaction $\delta: A \to A \otimes H$ is a triple $(L, \iota_L, \delta_L)$, where $L \subseteq H$ is a Hopf subalgebra, $\iota_L : L \hookrightarrow H$ is the inclusion, and $\delta_L : A \to A \otimes L$ is a right $L$-coaction such that $$\delta = (\mathrm{id}_A \otimes \iota_L) \circ \delta_L$$.

The Hopf image $H_\delta$ of $\delta$ is the initial object in the category of its factorizations: it is the smallest Hopf subalgebra of $H$ through which $\delta$ factors. Explicitly,

$$H_\delta = \bigcap_{\substack{L \subseteq H\text{ Hopf}\\delta(A) \subseteq A \otimes L}} L,$$

or equivalently, the Hopf subalgebra generated by all matrix coefficients $(\omega \otimes \mathrm{id})\delta(a)$. The induced coaction $\delta_{\mathrm{im}}: A \to A \otimes H_\delta$ satisfies the universal property that any other factorization factors uniquely through it via a Hopf algebra map.

3. Inner-Faithful Coactions and Inner-Faithful Quantum Symmetry

A coaction $\delta: A \to A \otimes H$ is called inner-faithful when $H_\delta = H$, that is, $\delta$ does not factor through any proper Hopf subalgebra. Equivalent characterizations include:

• There is no proper Hopf subalgebra $L \subsetneq H$ with $\delta(A) \subseteq A \otimes L$.
• Every factorization of $\delta$ is given by the identity inclusion on $H$.

Moreover, for any coaction $\delta$, the induced coaction $\delta_{\mathrm{im}}$ on the Hopf image $H_\delta$ is inner-faithful and is universal with respect to this property. Thus, inner-faithful quantum symmetry is the setting in which the quantum group acts maximally effectively on the algebra or bundle.

4. Quantum Principal Bundles and Hopf-Image Reduction

A quantum principal $H$-bundle is a structure consisting of a right $H$-comodule algebra $(A, \delta)$, a right-covariant first-order differential calculus $\Omega^1(A)$, such that:

1. The canonical map $\mathrm{can}: A \otimes_B A \to A \otimes H$ ($B = A^{\mathrm{co}H}$) is bijective (Hopf–Galois condition).
2. The defining right ideal for the differential calculus is stable under the right adjoint coaction $\mathrm{Ad}_R$ of $H$.

Given $(A, \Omega^1(A), H, \delta)$, let $(H_\delta, \iota, \delta_{\mathrm{im}})$ be its Hopf image. To construct the associated reduction, form the largest two-sided ideal

$$I = \sum\{J \subseteq A \mid J \text{ ideal and } \delta_{\mathrm{im}}(J) \subseteq J \otimes H_\delta\}$$

and set $A_0 = A / I$, with canonical projection $\pi: A \twoheadrightarrow A_0$. The coaction $\delta_{\mathrm{im}}$ descends to a well-defined right coaction $\bar{\delta}_{\mathrm{im}}$ on $A_0$, making $$(A_0, \Omega^1(A_0), H_\delta, \bar{\delta}_{\mathrm{im}})$$ a quantum principal $H_\delta$-bundle with inner-faithful quantum symmetry. This process is termed the Hopf-image reduction.

5. Rigidity and Minimality of Effective Quantum Symmetry

For cosemisimple $H$, the Hopf-image reduction yields a bundle whose quantum symmetry is minimal in the following sense: given any other quantum principal $K$-bundle on the same total algebra $A_0$, with the same coinvariants and inner-faithful coaction, there exists a unique injective Hopf algebra map $\iota: H_\delta \hookrightarrow K$ intertwining the comodule structure. Thus, $H_\delta$ captures the minimal effective symmetry group compatible with the bundle structure. This rigidity result formalizes the notion of the smallest quantum symmetry still acting effectively on the reduced space.

6. Classification up to Effective Quantum Symmetry and Functoriality

The assignment taking a quantum principal bundle to its Hopf-image reduction extends to a functor

$$\mathcal{R} : \mathsf{QPB}_{\mathrm{cosemi}} \to \mathsf{QPB}_{\mathrm{inner,cosemi}}$$

where $\mathsf{QPB}_{\mathrm{cosemi}}$ is the category of quantum principal bundles with cosemisimple Hopf algebras, and morphisms given by pairs $(\phi, \psi)$ of algebra and Hopf maps compatible with coactions and calculus. The fiber over a reduced bundle consists of all bundles having the same Hopf-image reduction up to isomorphism. As a result, the classification of quantum principal bundles up to effective quantum symmetry is equivalent to the classification of isomorphism classes of inner-faithful quantum principal bundles. This connects the structure theory of quantum bundles with the universal properties of Hopf images.

7. Illustrative Examples

Two principal examples clarify the concepts of Hopf images and inner-faithful quantum symmetry:

Example	Description	Inner-Faithfulness
Levi-subgroup coaction	$G$ a complex semisimple Lie group, $L_S \subset G$ a Levi subgroup, with $$\pi: \mathcal{O}_q(G) \twoheadrightarrow \mathcal{O}_q(L_S)$$. The natural coaction $$\delta = (\mathrm{id} \otimes \pi)\circ \Delta_{\mathcal{O}_q(G)}$$ on $\mathcal{O}_q(G)$ is inner-faithful: no proper Hopf subalgebra carries the full image.	Yes
Regular coaction	For any Hopf algebra $H$, the coproduct $\Delta: H \to H \otimes H$ is inner-faithful: it cannot factor through any proper subalgebra.	Yes

These examples illustrate that both natural quantum group actions (arising from morphology in algebraic groups) and canonical structures on the Hopf algebra itself exhibit inner-faithful quantum symmetry (Bhattacharjee, 4 Jan 2026).