Linear Duality on Action Bialgebroids

• Linear Duality on Action Bialgebroids demonstrates that dualization commutes with forming action bialgebroids, preserving smash product structures via explicit isomorphisms.
• The construction relies on categorical equivalences and finite projectivity conditions to transfer braided-commutative and Yetter–Drinfeld algebra structures.
• Implications extend to quantization, Lie bialgebroid crossed modules, and quantum groupoid theory, reinforcing duality in noncommutative geometry.

Linear duality on action bialgebroids concerns the commutation properties between the dualization functor and the process of forming action (or smash product, or scalar extension) bialgebroids. This phenomenon is central in both classical and quantum groupoid theory, with substantial implications for categorical duality, Yetter–Drinfeld modules, and quantization via Drinfeld functors. The duality has a precise algebraic formulation across bialgebroids and their braided and crossed module counterparts, manifesting a deep compatibility between algebraic structures and their linear duals.

1. Foundations: Action Bialgebroids, Duals, and Yetter–Drinfeld Algebras

Let $(H, A, s, t, \Delta_H, \epsilon_H)$ be a left $A$-bialgebroid, i.e., $H$ is an $A$-ring and $A$-coring with suitable compatibility. An action (or smash product, or scalar extension) bialgebroid arises when an algebra $R$ is embedded into the category of left $H$-modules and left $H$-comodules, specifically as a braided-commutative Yetter–Drinfeld algebra.

For $R$ a braided-commutative Yetter–Drinfeld algebra in ${}_H$, the smash product

$R \# H := R \otimes_A H$

inherits a left $R$-bialgebroid structure with the following explicit operations:

• Multiplication: $$(r \otimes_A h) \cdot (r' \otimes_A h') = r \cdot (h_{(1)} \triangleright r') \otimes_A h_{(2)} h'$$
• Source/Target: $s_R(r) = r \otimes_A 1_H$, $t_R(r) = r_{[0]} \otimes_A r_{[1]}$ (with notation $r \mapsto r_{[0]} \otimes r_{[1]}$ for the $H$-coaction)
• Coproduct: $$\Delta_{R\#H}(r \otimes_A h) = (r \otimes_A h_{(1)}) \otimes_R (1_R \otimes_A h_{(2)})$$
• Counit: $$\epsilon_{R\#H}(r \otimes_A h) = r \cdot \epsilon_H(h)$$

The linear dual of a finitely generated projective $A$-module $H$ is $H_* := \operatorname{Hom}_A({}_2 H, A)$, which itself is a right $A$-bialgebroid with dual structure maps. Notably, the Yetter–Drinfeld and braided-commutative structures can be transported between $H$ and $H_*$ via canonical monoidal and braided equivalences among module and comodule categories.

2. Main Duality Theorem: Commutation of Duality with Action Bialgebroids

If $R$ is a braided-commutative Yetter–Drinfeld algebra over a left $A$-bialgebroid $H$ with the finite projectivity assumption on ${}_2 H$, the following duality results hold (Chemla et al., 11 Nov 2025):

• $R$ is also a braided-commutative Yetter–Drinfeld algebra over the dual bialgebroid $H_*$.
• There is a canonical isomorphism of right $R$-bialgebroids:

$(R \# H)^* \cong H_* \# R$

where $(R \# H)^*$ denotes the linear dual over the base $R$, and $H_* \# R$ is the action bialgebroid formed for $H_*$ acting on $R$. In general, for smash products $A \#_H B$:

$(A \#_H B)^* \cong A^* \#_{H^*} B^*$

provided the requisite finite projectivity of $A$ and $B$ as (bi)modules.

The equivalence is obtained via explicit “matrix-element” isomorphisms, most notably the map

$$\eta: \operatorname{Hom}_R(R \otimes_A H, R) \to H_* \otimes_A R, \qquad f \mapsto \sum_i e^i \otimes f(1 \otimes e_i),$$

with $\{e_i\},\{e^i\}$ a dual basis of $H$ over $A$. This intertwines the smash product ring structures and dual corings, and verifies compatibility of multiplication, comultiplication, and source/target maps in both constructions.

3. Categorical Structures and Equivalences

Commutation of linear duality with the action bialgebroid is understood through monoidal and braided equivalences between four pivotal categories:

$${}_H \simeq \text{Comod-}H^* \simeq \text{Mod-}H_* \simeq {}^{H_*}_{H_*}$$

where $H^*$ and $H_*$ are left and right dual bialgebroids. Crucially, Yetter–Drinfeld structures and braided commutativity transfer across these equivalences, ensuring that the property of being a braided-commutative monoid is preserved under dualization.

The proof leverages the structure of (bi)modules and (co)modules, and the functoriality of tensor products, capitalizing on transformations such as

$\eta(fg) = \eta(f) \eta(g)$

for the dual product, and compatibility with the dual coproduct when paired with $R \otimes_A H$ elements.

4. Examples: Group Actions, Lie Bialgebroids, and Quantum Duality

Finite group case: For $G$ a finite group acting by automorphisms on a commutative $k$-algebra $R$, $H = kG$ is a Hopf algebra and $R$ forms a Yetter–Drinfeld $kG$-algebra. The smash product $R \# kG$ is a Hopf algebroid, and its linear dual is canonically $(kG)^* \# R$, corresponding to functions on $G$ smashed with $R$.

Lie bialgebroid setting (Lang et al., 2019): Action algebroids of the form $A = M \ltimes \mathfrak{g}$ and $X = M \times V$ (for a Lie algebra $\mathfrak{g}$ acting on a vector space $V$) yield action bialgebroids whose duals correspond to matched pairs of Lie algebroid crossed modules, characterized as co-quadratic Manin triples $(K, P, Q)$. Thus, linear duality for action bialgebroids is embedded in a broader categorical and Lie-theoretical context.

Quantization and Drinfeld functors: In the $h$-adic context, quantum groupoids appear as topological left or right bialgebroids. Drinfeld functors (denoted $\vee$ and $'$) convert, for instance, quantum formal series into quantum universal enveloping groupoids and vice versa. These dual constructions commute with action bialgebroid formation:

$$(R_h \# F_h)^\vee \cong R_h \# F_h^\vee, \quad (R_h \# U_h)' \cong R_h \# U_h'$$

clarifying the robustness of duality under quantization and the quantum duality principle (Chemla et al., 11 Nov 2025).

5. Hypotheses, Limitations, and Generalizations

The principal hypothesis throughout is the finite generation and projectivity of the relevant modules (e.g., ${}_2 H$ over $A$) to ensure the existence and correct behavior of dual bialgebroids. In practice, this is sometimes relaxed to (co)inductive or completed settings, such as the $h$-adic regime in quantum theory.

Braided-commutativity of $R$ in the Yetter–Drinfeld sense is essential for ensuring that the smash product $R \# H$ retains a bialgebroid structure. All constructions are algebraic and extend without difficulty to completed, topological, or quantum settings.

There are anticipated generalizations—including situations where $H$ possesses a bijective antipode (Hopf algebroids), dualities of two-sided smash products, and settings where $R$ is itself a Hopf algebroid within the center of $H$-mod (\emph{Editor’s term}: central Hopf bialgebroid).

6. Lie Bialgebroid Crossed Modules and Co-Quadratic Manin Triples

Linear duality principles for action bialgebroids extend naturally to the differential-geometric framework of Lie bialgebroid crossed modules (Lang et al., 2019). Given a pair of crossed modules $(X \to A)$ and its dual $(A^* \to X^*)$, their Whitney sums $A \oplus X$ and $A^* \oplus X^*$ inherit Lie bialgebroid structures precisely when the modules together form a matched pair. There is a bijection with co-quadratic Manin triples $(K, Q; P, Q)$, where $K = A \oplus X^*$ and $Q$ is a symmetric bilinear form on $K^*$. This framework situates the algebraic duality phenomena within a broader topological and differential context, reinforcing the ubiquity of the duality commutation property.

7. Duality Phenomena in Hopf Algebroid Theory

For left and right Hopf algebroids $U$ with suitable module-theoretic finiteness, classical duality features persist. Two distinguished duals, $U^* = \operatorname{Hom}_A(U, A)$ and $U_* = \operatorname{Hom}_{A^{op}}(U, A)$, each carry right $A$-bialgebroid structures with explicit, canonically dualized source, target, and multiplication maps (Chemla et al., 2014). The identification between these duals is mediated by a transformation $S^*: U^* \rightarrow U_*$ resembling the transpose of the antipode in Hopf algebras; $S^*$ is an isomorphism exactly when $U$ is both a left and right Hopf algebroid. This property further exemplifies the self-duality principles underlying the commutation of duality with smash-product constructions, extending the structural symmetries witnessed in simpler algebraic contexts.

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Linear duality on action bialgebroids reveals a categorical and algebraic invariance: dualizing after forming an action bialgebroid is equivalent to forming the action bialgebroid of the dual. This commutation property is preserved through quantization, matched pairs, and the passage to quantum groupoids, underpinning broader concepts in bialgebroid theory, representation theory, and noncommutative geometry.