Twisted Partial Hopf Actions

Updated 2 August 2025

Twisted partial Hopf actions are defined by a partial action map and a twisting cocycle, unifying local symmetries and cohomological deformations in Hopf algebras.
The associated partial crossed product construction guarantees associativity and unitality, generalizing classical twisted crossed and smash product frameworks.
Globalization theorems and cohomological techniques enable the embedding of these actions into global structures, linking them to cleft extensions and Hopf algebroids.

Twisted partial Hopf actions generalize both partial actions and twisted (global) actions of Hopf algebras, unifying two central themes in noncommutative symmetry: local (partial or restricted) symmetry and cohomological twisting by 2-cocycles. Such actions arise naturally when Hopf algebra symmetries are only defined on subspaces or ideals of algebras and are further modified by an additional twisting datum. The modern theory, initiated in (Alves et al., 2011), integrates the paper of partial cleft extensions, cohomological invariants, duality theorems, and nonunital contexts, and supports a wide range of algebraic, categorical, and geometric applications.

1. Definitions and Structural Axioms

A twisted partial Hopf action of a Hopf algebra $H$ on a (typically unital) algebra $A$ is specified by a pair of maps:

a partial action $\alpha: H \otimes A \to A$ , with $h \cdot a := \alpha(h \otimes a)$ ,
a twisting map (partial cocycle) $\omega: H \otimes H \to A$ .

These must satisfy the following foundational axioms (Alves et al., 2011, Guo et al., 2014):

(TP1) Unitality: $1_H \cdot a = a$ for all $a \in A$ .
(TP2) Partial multiplicativity: $h \cdot (ab) = (h_{(1)} \cdot a)(h_{(2)} \cdot b)$ , with $h \cdot 1_A$ acting as a local unit.
(TP3) Twisted associativity: $h \cdot (k \cdot a) = \omega(h_{(1)}, k_{(1)})\, ((h_{(2)}k_{(2)}) \cdot a)\, \omega'(h_{(3)}, k_{(3)})$ , with $\omega'$ a suitable convolution inverse (for symmetric cases).
(TP4) Twisted cocycle normalization: $\omega(h, 1_H) = \omega(1_H, h) = h \cdot 1_A$ for all $h \in H$ .
(TP5) Twisted cocycle equation: $\omega(h_{(1)}, k_{(1)})\; \omega(h_{(2)}k_{(2)}, m) = (h_{(1)} \cdot \omega(k_{(1)}, m_{(1)}))\, \omega(h_{(2)}, k_{(2)}m_{(2)})$ for all $h, k, m \in H$ .

These axioms ensure the resulting partial crossed product algebra (see Section 2) is associative and captures both the restriction to local domains and the cohomological deformation.

2. Partial Crossed Product Construction

Given $(A, \cdot, \omega)$ as above, the associated partial crossed product $A~\#_{\alpha, \omega}~H$ is defined as a subspace of $A \otimes H$ with multiplication (Alves et al., 2011, Guo et al., 2014):

$(a \# h)(b \# k) = a\,(h_{(1)} \cdot b)\, \omega(h_{(2)}, k_{(1)}) \# h_{(3)}k_{(2)}.$

Key properties:

The associativity and unitality of $A~\#_{\alpha, \omega}~H$ are ensured if the above axioms hold and if $1_A \# 1_H$ acts as a unit.
For symmetric twisted partial actions (i.e., when certain idempotents in $\operatorname{Hom}(H \otimes H, A)$ are central and $\omega$ is convolution-invertible in the relevant ideal), the crossed product construction encompasses all partial crossed products up to isomorphism (Alves et al., 2011, Guo et al., 2014).
The partial crossed product generalizes both the classical twisted crossed product and the partial smash product, providing a single algebraic framework for "twisted, not necessarily global" symmetries.

3. Globalization and Enveloping Actions

A central result is that, under appropriate conditions, every (symmetric) twisted partial Hopf action can be "globalized" to a (possibly non-unital) larger $H$ -module algebra with a genuine (global) twisted action (Guo et al., 2014, Alves et al., 2015):

Globalization theorem: There exists a pair $(B, \varphi)$ , with $B$ a unital $H$ -module algebra (with a twisting $u: H \otimes H \rightarrow B$ ), and an injective algebra morphism $\varphi: A \hookrightarrow B$ with $\varphi(A)$ an ideal in $B$ , such that the restriction of the global twisted action $(\cdot, u)$ induces the original twisted partial action $(\cdot, \omega)$ on $A$ .
The inclusion $A~\#_{\alpha, \omega}~H \rightarrow B~\#_{u}~H$ induces Morita equivalence between the partial and global crossed products under suitable regularity assumptions (Guo et al., 2014, Castro et al., 2014).
The conditions for globalizability can be fully characterized by the existence of a convolution invertible map $\tilde{\omega}: H \otimes H \to A$ satisfying $\tilde{\omega}$ -twisted cocycle relations, which reconstruct the original partial cocycle and guarantee that the partial data can be embedded into a global structure (Alves et al., 2015).

4. Cleft Extensions and Partially Cleft Extensions

There is a deep relationship between twisted partial Hopf actions, partial crossed products, and cleft (or partially cleft) extensions (Alves et al., 2011, Batista et al., 2017):

Partial cleft extensions: Given an $H$ -comodule algebra $B$ with coinvariants $A$ , the extension $A \subset B$ is called partially cleft if there exist linear maps $y, y': H \to B$ satisfying weakened convolution invertibility and comodule compatibility.
The crossed product $A~\#_{\alpha, \omega}~H$ is isomorphic to $B$ if and only if $A \subset B$ is partially cleft, thus generalizing the classical correspondence between crossed products and cleft extensions (Alves et al., 2011).
In the partial and twisted setting, these results extend: $A \subset A~\#_{\omega}~H$ is a partially cleft extension, and—when $H$ is suitably cocommutative and $A$ commutative—such extensions correspond to cleft extensions by Hopf algebroids (Batista et al., 2017).

5. Cohomological Aspects and Obstruction Theory

A full cohomology theory for partial Hopf actions, extending Sweedler's and group cohomology, is established for cocommutative Hopf algebras over commutative algebras (Batista et al., 2017):

Partial cohomology group $H_{\text{par}}^2(H, A)$ classifies twisted partial cocycles modulo coboundaries. Two partial 2-cocycles $\omega, \omega'$ define isomorphic partial crossed products precisely when they are cohomologous.
Explicit computation of the cohomology is facilitated by reduction to idempotent-supported complexes and the construction of new Hopf algebras over the subalgebra $E(A)$ generated by the partial "local units" $h \cdot 1_A$ .
The cohomological approach controls obstruction theory for the existence of (twisted) partial crossed product and their cleft/Hopf algebroid extensions.

6. Examples, Constructions, and Applications

Twisted partial Hopf actions have been constructed for a wide spectrum of examples:

Algebraic groups and linear algebraic groups: Partial and twisted partial actions arise via the restriction of global coactions (e.g., on coordinate rings of maximal tori in $GL_n(k)$ ) and by incorporating nontrivial 2-cocycles (e.g., Schur multipliers of finite abelian groups) [(Alves et al., 2011), Example Section].
Sweedler Hopf algebra and low-dimensional Hopf algebras: Explicit classification of all partial actions and computations of twisted cocycles are available, including those acting on split (semi-)quaternion algebras, matrix algebras, and the base field (Martini et al., 2020, Quanguo, 26 Jul 2025).
Generalized matrix algebras and Ore extensions: Criteria for partial (twisted) Hopf actions on non-simple and non-unital matrix algebras, and extensions to rank-one Hopf algebras and Nichols algebras via Hopf–Ore extensions, have been completely characterized (Bagio et al., 12 Dec 2024, Giraldi et al., 25 Oct 2024).
Weak Hopf algebras and C*-algebraic versions: The partial crossed product/Morita theory works analogously for weak Hopf algebras and in the $C^*$ -algebra setting for quantum symmetries (Castro et al., 2014, Kraken et al., 2017).

7. Connections to Duality, Partial Representations, and Hopf Algebroids

The structural theory of twisted partial Hopf actions interacts with several other foundational areas:

Duality: There is a rich duality between partial actions and partial coactions, extended in the twisted case to cover relationships with partial module/comodule (co)algebras and dual cohomological theories (Batista et al., 2014).
Partial representations and module categories: Partial representations factor via a universal partial Hopf algebroid $H_{par}$ , and algebra objects in the monoidal category of $H_{par}$ -modules correspond precisely to (twisted) partial Hopf actions (Alves et al., 2013). Twisted phenomena can be encoded in the monoidal structure and in higher-cocycle conditions.
Hopf algebroids: Twisted partial crossed products in commutative/cocommutative settings yield natural Hopf algebroids, and cotwist techniques allow passage between twisted and untwisted settings via Drinfeld cotwists (Han et al., 2022).

In summary, the theory of twisted partial Hopf actions integrates local symmetry and cohomological deformation, provides a robust construction for associative crossed products, admits globalization into global twisted Hopf actions, relates deeply to partial and cleft extensions, and is governed by a well-developed cohomology. Its reach covers algebraic, categorical, and operator-algebraic settings, situates classical group-theoretic constructions in a broader context, and has strong algorithmic and computational underpinnings for determining explicit partial/twisted actions in concrete algebraic families.