arXiv subjects
Publications and source records attributed to Ana Belén Rodríguez Raposo.
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 $H_{2}$.
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.
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 ${\mathbb H}$.
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.
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 $H$.
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.