arXiv subjects

Fabio Calderón

Publications and source records attributed to Fabio Calderón.

4 recordsLinked to original sources

arXiv2026

Quotients of graded quasi-Hopf algebras

We classify the quasi-Hopf ideals of quasi-Hopf algebras faithfully graded by arbitrary groups. Each ideal determines a normal subgroup, recording which degrees are identified in the quotient, together with an invariant ideal in the corresponding neutral component. We refine this description using an ideal of the original neutral component and a retraction of a naturally associated intermediate quotient. We prove that this intermediate quotient splits as the tensor product of its neutral component with the group algebra of the normal subgroup. For fixed subgroup and neutral-component data, the possible ideals, whenever they exist, form a torsor described by equivariant homomorphisms into central group-like elements. Our results require neither finite dimensionality nor bijectivity of the antipode. We apply them to cocentral abelian extensions with possibly infinite grading, finite-dimensional neutral component, and nontrivial reassociator.

math.RA↗

arXiv2024

Symmetries of algebras captured by actions of weak Hopf algebras

In this paper, we present a generalization of well-established results regarding symmetries of $\Bbbk$-algebras, where $\Bbbk$ is a field. Traditionally, for a $\Bbbk$-algebra $A$, the group $\Bbbk$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\Bbbk$-linear monoidal categories of modules, we introduce an object $\operatorname{Sym}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\Bbbk$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework.

math.QA↗

arXiv2022

Some interactions between Hopf Galois extensions and noncommutative rings

In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.). We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors (also known as quantum heaps) and Hopf Galois systems, while for some families of noncommutative rings we present advances in the characterization of ring-theoretic and homological properties. Every developed topic is exemplified with abundant references to classic and current works, so this paper serves as a survey for those interested in either of the two theories. Throughout, interactions between both are presented.

math.RA↗

arXiv2021

Algebraic properties of face algebras

Prompted an inquiry of Manin on whether a coacting Hopf-type structure $H$ and an algebra $A$ that is coacted upon share algebraic properties, we study the particular case of $A$ being a path algebra $\Bbbk Q$ of a finite quiver $Q$ and $H$ being Hayashi's face algebra $\mathfrak{H}(Q)$ attached to $Q$. This is motivated by the work of Huang, Wicks, Won, and the second author, where it was established that the weak bialgebra coacting universally on $\Bbbk Q$ (either from the left, right, or both sides compatibly) is $\mathfrak{H}(Q)$. For our study, we define the Kronecker square $\widehat{Q}$ of $Q$, and show that $\mathfrak{H}(Q) \cong \Bbbk \widehat{Q}$ as unital algebras. Then we obtain ring-theoretic and homological properties of $\mathfrak{H}(Q)$ in terms of graph-theoretic properties of $Q$ by way of $\widehat{Q}$.

math.RA↗