---
title: Inner-Faithful Quantum Symmetry
url: https://www.emergentmind.com/topics/inner-faithful-quantum-symmetry
type: topic
---

# Inner-Faithful Quantum Symmetry

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 [2601.01520].

Source: https://www.emergentmind.com/topics/inner-faithful-quantum-symmetry