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Blas Torrecillas

Publications and source records attributed to Blas Torrecillas.

16 recordsLinked to original sources
arXiv2026

Clifford algebras, meson algebras and higher order generalisations

We analyse the homogeneous parts of Clifford and meson algebras and point out that for the Clifford algebra it is related to fermionic statistics, that is, to fermionic parastatistics of order 1 while for the meson algebra it is related to fermionic parastatistics of order 2. We extend these homogeneous algebras into corresponding algebras related to fermionic parastatistics of all orders. We then define correspondingly higher order generalizations of Clifford and meson algebras.

math.QA↗
arXiv2025

New examples of separable cowreaths over Clifford algebras

This paper continues the research we developed in \cite{MT1} and \cite% {MT2}. The main aim of this paper is to investigate separability conditions for a cowreath $(A\otimes H^{op},H,ψ)$ constructed by using the $8$% -dimensional Clifford algebra $A=Cl(α,β_{1},β_{2},γ_{1},γ_{2},λ)$ considered as an $H$-comodulo algebra where $H$ is the $8$-dimensional unimodular ribbon Hopf algebra $E(2)$ introduced by Radford in \cite{R}.

math.QA↗
arXiv2024

Quadratic algebras associated with exterior 3-forms

This paper is devoted to the study of the quadratic algebras with relations generated by superpotentials which are exterior 3-forms. Such an algebra is regular if and only if it is Koszul and is then a 3-Calabi-Yau domain. After some general results we investigate the case of the algebras generated in low dimensions $n$ with $n\leq 7$. We show that whenever the ground field is algebraically closed all these algebras associated with 3-regular exterior 3-forms are regular and are thus 3-Calabi-Yau domains. This result does not generalize to dimensions $n$ with $n\geq 8$ : we describe a counter example in dimension $n=8$.

math.RA↗
arXiv2023

A class of finite-by-cocommutative Hopf algebras

We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a Tannakian perspective. We determine when the new Hopf algebras are co-Frobenius, or cosemisimple, or Noetherian, or regular, or have finite Gelfand-Kirillov dimension.

math.QA↗
arXiv2018

On sovereign, balanced and ribbon quasi-Hopf algebras

We introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively. Examples of balanced and ribbon quasi-Hopf algebras are obtained from a sort of double construction which associates to a braided category (resp. rigid braided) a balanced (resp. ribbon) one.

math.QA↗
arXiv2016

An active attack on a distributed Group Key Exchange system

In this work, we introduce an active attack on a Group Key Exchange protocol by Burmester and Desmedt. The attacker obtains a copy of the shared key, which is created in a collaborative manner with the legal users in a communication group.

cs.CR↗
arXiv2013

Quiver Bialgebras and Monoidal Categories

We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.

math.QA↗
arXiv2013

On Frobenius and separable algebra extensions in monoidal categories. Applications to wreaths

We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobenius algebra and the restriction at $R$ of its Nakayama automorphism is equal to the Nakayama automorphism of $R$. As applications, we obtain several characterizations for an algebra extension associated to a wreath to be Frobenius, respectively separable.

math.QA↗
arXiv2012

Lattices and cohomological Mackey functors for finite cyclic p-groups

For a finite cyclic p-group G and a discrete valuation domain R of characteristic 0 with maximal ideal pR the R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all sub- groups (cf. Thm. A). As a consequence any R[G]-lattice can be presented by R[G]-permutation modules (cf. Thm. C). The proof of these results is based on a detailed analysis of the category of cohomological G-Mackey functors with values in the category of R-modules. It is shown that this category has global dimension 3 (cf. Thm. E). A crucial step in the proof of Theorem E is the fact that a gentle R-order category (with parameter p) has global dimension less or equal to 2 (cf. Thm. D).

math.CT↗
arXiv2012

From Hopf algebras to tensor categories

This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the category of representations of a suitable Hopf algebra.

math.QA↗
arXiv2012

A note on the construction of finitely injective modules

We develop a technique to construct finitely injective modules which are non trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring $R$ is left noetherian if and only if each finitely injective left $R$-module is trivial, thus answering an open question posed by Salce.

math.RA↗
arXiv2009

Generator coalgebras are not necessarily quasi-coFrobenius

We study the problem of whether a coalgebra that generates its category of left (right) comodules is left (right) quasi-coFrobenius or not. We prove it does not hold in general, by giving a method of constructing counterexamples. This gives a negative answer to a question stated in \cite{kn:coalgen}. We also prove it is true for monomial pointed coalgebras and we characterize the quivers $Q$ such that $\Bbbk Q$ admits a monomial subcoalgebra that is left (right) quasi-coFrobenius.

math.RA↗
arXiv2006

Radford's S^4 formula for co-Frobenius Hopf algebras

This note extends Radford's formula for the fourth power of the antipode of a finite dimensional Hopf algebra to co-Frobenius Hopf algebras and studies equivalent conditions to a Hopf algebra being involutory for finite dimensional and co-Frobenius Hopf algebras.

math.QA↗
arXiv2003

Factorizable quasi-Hopf algebras. Applications

We define the notion of factorizable quasi-Hopf algebra by using a categorical point of view. We show that the Drinfeld double $D(H)$ of any finite dimensional quasi-Hopf algebra $H$ is factorizable, and we characterize $D(H)$ when $H$ itself is factorizable. Finally, we prove that any finite dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the Drinfeld double $D(H)$ is a unimodular quasi-Hopf algebra.

math.QA↗