arXiv subjects

Ramón González Rodríguez

Publications and source records attributed to Ramón González Rodríguez.

17 recordsLinked to original sources

arXiv2026

Twist deformations for Hopf coquasigroups

In this paper, we develop a general theory of twist deformations for Hopf coquasigroups in a symmetric monoidal category. To this end, we first introduce and study non-coassociative bimonoids endowed with left and right codivisions, and establish their connection with left and right Hopf coquasigroups. Next, motivated by the classical theory of Drinfeld twists for Hopf algebras, we define twists for non-coassociative bimonoids and prove that they induce deformations of Hopf coquasigroup structures through suitable modifications of the coproduct. In particular, we obtain explicit deformation procedures for right and left Hopf coquasigroups and analyze the corresponding antipodes. Finally, we apply the general theory to construct nontrivial examples arising from Hopf coquasigroups associated with the sphere ${\sf S}^7$, obtaining new examples of twisted Hopf coquasigroups that are neither commutative nor cocommutative.

math.QA↗

arXiv2026

Opposite brace triples, Hopf braces and matched pairs of Hopf algebras

In this paper the category of opposite brace triples is introduced in a general braided monoidal setting. Under cocommutativity, it is proved to be isomorphic to the category of Hopf braces. Furthermore, if one considers the subcategories arising from fixing one of the underlying Hopf algebras, then these two categories are also isomorphic to the category of matched pairs over that Hopf algebra.

math.RA↗

arXiv2026

On modules over a Hopf brace

Let $\mathbb{H}=(H_{1},H_{2})$ be a Hopf brace in a symmetric monoidal category ${\sf C}$. In this article it is proved that the category of modules over $\mathbb{H}$ is isomorphic to the category of modules over the smash product algebra $H_{1}\sharp H_{2}$. Furthermore, the category of modules over $\mathbb{H}$ in the sense of Zhu is characterized by the condition that a certain action lies in the cocommutativity class of $H_{2}$.

math.RA↗

arXiv2025

Twisted Post-Hopf Algebras, Twisted Relative Rota-Baxter Operators and Hopf Trusses

The present article is devoted to studying the categorical relationships between the categories of Hopf trusses, weak twisted post-Hopf algebras, introduced by Wang (2023), and weak twisted relative Rota-Baxter operators. The latter objects are a generalisation of the relative Rota-Baxter operators defined by Li-Sheng-Tang (2024), where the Rota-Baxter condition is modified through a cocycle. Under certain conditions, this work shows that the three aforementioned categories are equivalent.

math.RA↗

arXiv2025

Categorical equivalences for Hopf trusses and their modules

In this paper we introduce the notion of generalized invertible 1-cocycle in a strict braided monoidal category C, and we prove that the category of Hopf trusses is equivalent to the category of generalized invertible 1-cocycles. On the other hand, we also introduce the notions of module for a Hopf truss and for a generalized invertible 1-cocycle. We prove some functorial results involving these categories of modules and we show that the category of modules associated to a generalized invertible 1-cocycle is equivalent to a category of modules associated to a suitable Hopf truss. Finally, assuming that in C we have equalizers, we introduce the notion of Hopf-module in the Hopf truss setting and we obtain the Fundamental Theorem of Hopf modules associated to a Hopf truss.

math.RA↗

arXiv2025

About Hopf braces and crossed products

The present article represents a step forward in the study of the following problem: If $\mathbb{A}=(A_{1},A_{2})$ and $\mathbb{H}=(H_{1},H_{2})$ are Hopf braces in a symmetric monoidal category C such that $(A_{1},H_{1})$ and $(A_{2},H_{2})$ are matched pairs of Hopf algebras, then we want to know under what conditions the pair $(A_{1}\bowtie H_{1},A_{2}\bowtie H_{2})$ constitutes a new Hopf brace. We find such conditions for the pairs $(A_{1}\otimes H_{1},A_{2}\bowtie H_{2})$ and $(A_{1}\bowtie H_{1},A_{2}\sharp H_{2})$ to be Hopf braces, which are particular situations of the general problem described above, and we apply these results to study when the Drinfeld's Double gives rise to a Hopf brace.

math.RA↗

arXiv2024

Relative Rota-Baxter operators, modules and projections

The present article is devoted to introduce, in a braided monoidal setting, the notion of module over a relative Rota-Baxter operator. It is proved that there exists an adjunction between the category of modules associated to an invertible relative Rota-Baxter operator and the category of modules associated to a Hopf brace, which induces an equivalence by assuming certain additional hypothesis. Moreover, the notion of projection between relative Rota-Baxter operators is defined, and it is proved that those which are called ``strong'' give rise to a module according to the previous definition in the cocommutative setting.

math.RA↗

arXiv2024

Projections of Hopf braces

This paper is devoted to the study of Hopf braces projections in a monoidal setting. Given a cocommutative Hopf brace ${\mathbb H}$ in a strict symmetric monoidal category ${\sf C}$, we define the braided monoidal category of left Yetter-Drinfeld modules over ${\mathbb H}$. For a Hopf brace ${\mathbb A}$ in this category, we introduce the concept of bosonizable Hopf brace and we prove that its bosonization ${\mathbb A}\blacktriangleright\hspace{-0.15cm}\blacktriangleleft {\mathbb H}$ is a new Hopf brace in ${\sf C}$ that gives rise to a projection of Hopf braces satisfying certain properties. Finally, taking these properties into account, we introduce the notions of v$_{i}$-strong projection over ${\mathbb H}$, $i=1,2,3,4$, and we prove that there is a categorical equivalence between the categories of bosonizable Hopf braces in the category of left Yetter-Drinfeld modules over ${\mathbb H}$ and the category of v$_{4}$-strong projections over ${\mathbb H}$.

math.RA↗

arXiv2024

Weak Hopf quasigroups and matched pairs of quasigroupoids

In this paper we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that if $({\sf A}, {\sf H})$ is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid ${\sf A}\bowtie {\sf H}$ called the double cross product of ${\sf A}$ and ${\sf H}$. Also, we show that, if a quasigroupoid ${\sf B}$ admits an exact factorization, there exists a matched pair of quasigroupoids $({\sf A}, {\sf H})$ and an isomorphism of quasigroupoids between ${\sf A}\bowtie {\sf H}$ and ${\sf B}$. Finally, if ${\mathbb K}$ is a field, we show that every matched pair of quasigroupoids $({\sf A}, {\sf H})$ induce, thanks to the quasigroupoid magma construction, a pair $({\mathbb K}[{\sf A}], {\mathbb K}[{\sf H}])$ of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup ${\mathbb K}[{\sf A}]\bowtie{\mathbb K}[{\sf H}]$ isomorphic to ${\mathbb K}[{\sf A}\bowtie {\sf H}]$ as weak Hopf quasigroups.

math.RA↗

arXiv2024

Distributive laws and Hopf quasigroups

In this paper we introduce the notion of $a$-monoidal distributive law between two Hopf quasigroups $A$ and $H$. We prove that every $a$-monoidal distributive law induce a product on $A\otimes H$, called the wreath product, thanks to which $A\otimes H$ becomes in a Hopf quasigroup. Finally, using this construction, we show that double cross products of Hopf quasigroups, cross products of Hopf quasigroups with a skew pairing between them, Hopf quasigroups defined by the twisted double method, smash products of Hopf quasigroups and twisted smash products of Hopf quasigroups are examples of wreath products associated to $a$-monoidal distributive laws.

math.RA↗

arXiv2024

Hopf bracoids

The present article is devoted to introduce the notion of Hopf bracoid in the braided monoidal framework as the quantum version of skew bracoids, which have been presented by Martin-Lyons and Paul J. Truman. Taking into account that Hopf braces are particular examples of Hopf bracoids, in this paper we generalize many properties of Hopf braces to the Hopf bracoid setting and we obtain a categorical isomorphism between certain full subcategories of the Hopf bracoids category and the 1-cocycles category.

math.RA↗

arXiv2023

Categorical isomorphisms for Hopf braces

In this paper, we introduce the category of brace triples in a braided monoidal setting and prove that it is isomorphic to the category of s-Hopf braces, which are a generalization of cocommutative Hopf braces. After that, we obtain a categorical isomorphism between finite cocommutative Hopf braces and a certain subcategory of cocommutative post-Hopf algebras, which suppose an expansion to the braided monoidal setting of the equivalence obtained by Y. Li, Y. Sheng and R. Tang for the category of vector spaces over a field $\mathbb{K}$.

math.RA↗

arXiv2023

Modules over invertible 1-cocycles

In this paper we introduce in a braided setting the notion of left module for an invertible 1-cocycle and we prove some categorical equivalences between categories of modules associated to an invertible 1-cocycle and categories of modules associated to Hopf braces.

math.RA↗

arXiv2022

Factorizations of Hopf quasigroups

In this paper we introduce the notion of factorization in the Hopf quasigroup setting and we prove that, if $A$ and $H$ are Hopf quasigroups such that their antipodes are isomorphisms, a Hopf quasigroup $X$ admits a factorization as $X=AH$ iff $X$ is isomorphic to a double cross product $A\bowtie H$ as Hopf quasigroups.

math.RA↗

arXiv2021

Cohomological Obstructions and Weak Crossed Products over Weak Hopf Algebras

Let $H$ be a cocommutative weak Hopf algebra and let $(B, φ_{B})$ a weak left $H$-module algebra. In this paper, for a twisted convolution invertible morphism $σ:H\otimes H\rightarrow B$ we define its obstruction $θ_σ$ as a degree three Sweedler 3-cocycle with values in the center of $B$. We obtain that the class of this obstruction vanish in third Sweedler cohomology group $\mathcal{H}^3_{φ_{Z(B)}}(H, Z(B))$ if, and only if, there exists a twisted convolution invertible 2-cocycle $α:H\otimes H\rightarrow B$ such that $H\otimes B$ can be endowed with a weak crossed product structure with $α$ keeping a cohomological-like relation with $σ$. Then, as a consequence, the class of the obstruction of $σ$ vanish if, and only if, there exists a cleft extension of $B$ by $H$.

math.RA↗

arXiv2015

Equivalences for weak crossed products

In this paper we give a criterion that characterizes equivalent weak crossed products. By duality, we obtain a similar result for weak crossed coproducts and, as a consequence, we find the conditions that assures the equivalence between two weak crossed biproducts. As an application, we show that the main results proved by Panaite, for Brzeziński's crossed products, admits a substantial reduction in the imposed conditions.

math.QA↗

arXiv2015

Strong Hopf modules for weak Hopf quasigroups

In this paper we introduce the category of strong Hopf modules for a weak Hopf quasigroup H in a braided monoidal category. We also prove that this category is equivalent to the category of right modules over the image of the target morphism of H.

math.QA↗