Blas Torrecillas

University of Almería | UAL · Department of Mathematics

In this paper we introduce a group key management protocol for secure group communications in a non-commutative setting. To do so, we consider a group ring over the dihedral group with a twisted multiplication using a cocycle. The protocol is appropriate for the so-called post-quantum era and it is shown that the security of the initial key agreeme...

Let R be a ring and $$N \ge 2$$ N ≥ 2 . First, we prove that any deconstructible class of modules $${\mathcal {F}}$$ F over R induces two coreflective subcategories of the homotopy category $$\textbf{K}_N(\mathrm {Mod-}{R})$$ K N ( Mod - R ) of (unbounded) N -complexes of right R -modules: the one whose objects are all N -complexes with components...

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,...

The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath $$ \left( A\otimes H_{4}^{op}, H_{4}, \p...

In this paper we introduce a Group Key Management protocol following the idea of the classical protocol that extends the well-known Diffie–Hellman key agreement to a group of users. The protocol is defined in a non-commutative setting, more precisely, in a twisted dihedral group ring. The protocol is defined for an arbitrary cocycle, extending prev...

Framework and justification: The content of this paper is located on the intersection of two fields: Finance and Algebra. In effect, the current dynamism shown by most financial instruments makes it necessary to endow the foundations of finance with, as general as possible, algebraic structures. Therefore, the objective of this paper is to provide...

Motivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1]. Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category TA♯ of right transfer morphisms through...

We show that a Galois cowreath (A, X) in a monoidal category C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}$$\end{document} is Frobenius if and only if...

The problem of finding an analytical solution for the circuit formed by a diode with series resistance (D-R circuit) has been investigated for a long time and several authors have proposed approximate solutions. Most solutions use the Lambert function and numerical methods. However, an analytical solution independent of experimental parameters is s...

We prove a uniqueness type theorem for (weak, total) integrals on a Frobenius cowreath in a monoidal category. When the cowreath is, moreover, pre-Galois, we construct a Morita context relating the subalgebra of coinvariants and a certain wreath algebra. Then we see that the strictness of the Morita context is related to the Galois property of the...

Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). Th...

We survey results on Frobenius algebras and illustrate their importance to the structure of some generalizations of the notion of Hopf algebra, as well as their connections to topics like monoidal categories, 2-categories, functors, topological quantum field theories, etc.

We classify the monoidal structures for the category of N-complexes which respect the graded structure.

Key management is a central problem in information security. The development of quantum computation could make the protocols we currently use unsecure. Because of that, new structures and hard problems are being proposed. In this work, we give a proposal for a key exchange in the context of NIST recommendations. Our protocol has a twisted group rin...

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 as...

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 ass...

As generalizations of Yetter–Drinfeld module over a Hopf quasigroup, we introduce the notions of Yetter–Drinfeld–Long bimodule and generalize the Yetter–Drinfeld module over a Hopf quasigroup in this paper, and show that the category of Yetter–Drinfeld–Long bimodules (Formula presented.) over Hopf quasigroups is braided, which generalizes the resul...

In this article the experiment carried out by Takahashi et al. [2009 Takahashi, T., T. Hadzibeganovic, S. A. Cannas, T. Makino, H. Fukui, and S. Kitayama. “Cultural Neuroeconomics of Intertemporal Choice.” Neuroendocrinology Letters, 30, (2009), pp. 185–191.[PubMed], [Web of Science ®] [Google Scholar]] is replicated to analyze the influence of cul...

We prove that, if F is the class of torsion free discrete modules over a profinite group G, that is, the class of discrete G-modules which are torsion free as abelian groups, then (F;F?) is a complete cotorsion pair. Moreover, we find a structure theorem for torsion free and cotorsion discrete G-modules and for finitely generated cotorsion discrete...

Entwined modules over cowreaths in a monoidal category are introduced. Monoidal cowreaths can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra $A$ is Frobenius (re...

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.

We extend the main result of Skryabin in [9] to Yetter-Drinfeld Hopf algebras over an arbitrary Hopf algebra H with bijective antipode.

We relate the homological behavior of an associative ring R and those of the rings R/xR and R-x when x is a regular central element in R. For left weak global dimensions we prove wgldim(R) <= max{1 + wgldim(R/xR),wgldim(R-x)} with equality if wgldim(R/xR) is finite. The key point is a formula for flat dimensions of R-modules: fd(R) M = max{fd(R/xR)...

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.

The purpose of this paper is to introduce the concept of a twisting element based on a Hom- bialgebra and to use it to provide twists or deformations of Hom-associative algebras. Moreover we review the module theory in Hom-setting and show that a twisting element based on a bialgebra gives rise to a twisting element based on a Hom-bialgebra.

We introduce the concept of homological Frobenius functors as the natural generalization of Frobenius functors in the setting of triangulated categories, and study their structure in the particular case of the derived categories of those of complexes and modules over a unital associative ring. Tilting complexes (modules) are examples of homological...

A ring is left Gorenstein regular if the classes of left modules with finite projective dimension and finite injective dimension coincide and the injective and projective finitistic left dimensions are finite. Let A and B be rings and U a (B,A)(B,A)-bimodule such that UB has finite projective dimension and UAUA has finite flat dimension. In this pa...

For a large class of groups necessary and sufficient conditions are given for their group algebras to have the bounded splitting property.

We consider two new algebras from an H-biquasimodule algebra A and a Hopf quasigroup H: twisted smash product A ⊛ H and L-R smash product A⋇H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, if H is a cocommutative Hopf quasigr...

We study entwining structures on a monoidal category C and their corresponding categories of entwined modules. Examples can be constructed from lax Doi-Koppinen and lax Yetter-Drinfeld structures in C. If C is symmetric then lax Yetter-Drinfeld structures appear as special cases of lax Doi-Koppinen structures, at least if we work over a so-called l...

A classical question of Yoneda asks when the tensor product of two injective modules is injective. A complete answer to this question was given by Enochs and Jenda in 1991. In this paper the analogue question for pure-injective modules is studied.

Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C = A°be its dual coalgebra. If A is Artin–Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule 1C σ* with σ a coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twist...

The aim of this paper is to introduce the concept of right and left semidualizing adjoint pair of functors and study its main properties. This concept generalizes the concept of semidualizing module and allows one to consider semidualizing comodules, graded modules, etc. We also study tilting adjoint pair of functors as a particular case. We show g...

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 ar...

Let (A;E) be an exact category and F⊆Ext a subfunctor. A morphism φ in A is an F-phantom if the pullback of an E-conflation along φ is a conflation in F. If the exact category (A;E) has enough injective objects and projective morphisms, it is proved that an ideal I of A is special precovering if and only if there is a subfunctor F⊆Ext with enough i...

For a finite cyclic pp-group GG and a discrete valuation domain RR of characteristic 0 with maximal ideal pRpR the R[G]R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups (cf. Theorem A). As a consequence any R[G]R[G]-lattice can be presented by R[G]R[G]-permutation modules (cf. Theorem C...

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate sinc...

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 separab...

In this paper we introduce the notion of rooted ring with several objects (rooted small preadditive category). Then, we obtain characterizations of projective and flat functors in the corresponding functor category, and use them to give new results concerning right perfect rooted rings with several objects and pure semisimple finitely presented add...

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.

We characterize strict Mittag-Leffler modules in terms of free realizations of positive primitive formulas, and rings over which (pure-) projectives are trivial in terms of various notions of separability of (flat) strict Mittag-Leffler modules. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We present a method for constructing the minimal injective resolution of a simple comodule of a path coalgebra of quivers with relations. Dual to the Calabi–Yau condition of algebras, we introduce the concept of a Calabi–Yau coalgebra, and then describe the Calabi–Yau coalgebras of low global dimensions.

We prove that a profinite algebra whose left (right) cyclic modules are torsionless is finite dimensional and QF. We give a relative version of the notion of left (right) PF ring for pseudocompact algebras and prove it is left-right symmetric and dual to the notion of quasi-co-Frobenius coalgebras. We also prove two ring theoretic conjectures of Fa...

Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^*$ where $C$ is a coalgebra. We show that the if the semiartinian part (the "Dickson" part) of every $A$-module $M$ splits off in $M$, then $A$ is semiartinian, also giving a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conj...

Let $A$ and $B$ be algebras and coalgebras in a braided monoidal category $\Cc$, and suppose that we have a cross product algebra and a cross coproduct coalgebra structure on $A\ot B$. We present necessary and sufficient conditions for $A\ot B$ to be a bialgebra, and sufficient conditions for $A\ot B$ to be a Hopf algebra. We discuss when such a cr...

A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual coalgebra. If $A$ is Artin-Schelter regular, then the local cohomology of $A$ is isomorphic to a shift of twisted bi...

We recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic ve...

Let λ be an infinite regular cardinal. It is proved, under the assumption of the Generalized Continuum Hypothesis, that any λ-accessible and λ-accessibly embedded subcategory of a category of modules closed under direct sums gives rise to non-trivial κ-separable modules, for arbitrarily large regular cardinals κ ≥ λ, when some modules of are not di...

We show that in a finitely accessible additive category every class of objects closed under direct limits and pure epimorphic images is covering. In particular, the classes of flat objects in a locally finitely presented additive category and of absolutely pure objects in a locally coherent category are covering.

We describe flat covers and cotorsion envelopes over formal triangular matrix rings. To describe the flat covers we use Quillen factorizations of linear maps relative to the classical cotorsion pair. Using flat covers over formal triangular matrix rings we prove the existence and minimality of Quillen factorization.

Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in...

The paper has been withdrawn due to a crucial error in section 3. Comment: This paper has been withdrawn

We establish equivalent conditions under which every object in an Abelian category has a monic X -cover and an epic X -envelope respectively, when X is a class of objects closed under extensions and, more particularly, an n-th Ext-orthogonal class of a given class of objects. We give applications to categories of modules and comodules.

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 Nastasescu et al. (Algebr Represent Theory 11(2):179-190, 2008). We...

Let R be an associative ring with identity and κ, an infinite regular cardinal. A left R-module M is said to be κ ≤-generated (resp. κ <-generated) if there exist a generator set {m i }I of M of cardinality at most κ (resp. strictly smaller than κ And a module M is called κ-separable if any subset X of M of cardinality strictly smaller than κ is co...

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.

In this note we show the interlacing between homological algebra and the category of relative Hopf modules. We show that many well-known concepts related with differential graded algebras can be performed as purely Hopf algebraic phenomenons.

We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple.

We show that a (not necessarily unitary) ring with enough idempotents is left perfect if and only if there exists a cardinal number ℵ such that every flat strict Mittag-Leffler module is a direct sum of ℵ-generated modules. Several applications are given to the decomposition properties of modules into direct summands.

We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra $H$ and of its quantum double $D(H)$, within the rigid braided category of finite dimensional left $D(H)$-modules.

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.

For a quasi-Hopf algebra H, an H-bicomodule algebra and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules is equivalent to the category of right–left generalized Yetter–Drinfeld modules . Using alternative versions of this result we will recover the category isomorphism between the categories of left–lef...

Without Abstract

In [[6]6. Doi , Y. 1985. Algebras With Total Integrals. Comm. Algebra, 13: 2137–2159. [Web of Science ®]View all references, Theorem 2.2] Doi gave a Hopf-algebraic proof of a generalization of Oberst's theorem on affine quotients of affine schemes. He considered a commutative Hopf algebra H over a field, coacting on a commutative H-comodule algebr...

We give the structure of a class of pseudoprojective modules over a semiperfect ring. Moreover, we describe all self-pseudoprojective modules over an Artinian serial ring. As an application, we give the number of (non-necessarily hereditary) torsion theories over such a ring.

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕ σ∈G R σ with finite support if R e has Gabriel dimension then R-grR\hbox{-}{\rm gr} has Gabriel dimension. Moreover, adding some lattice results, we prove that if R-modR\hbox{-}{\rm mod} has Gabriel dimension...

We show the Tychonoff's theorem for a Grothendieck category with a set of small projective generators. Strictly quasi-finite objects for semiartinian Grothendieck categories are characterized. We apply these results to the study of the Morita duality of dual algebra of a coalgebra.

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this artic...

Finiteness conditions of reflexive objects of a Morita duality of Grothendieck category is studied. It is observed that the relation between coproduct and product is a fundamental fact. We show that for coalgebras with a duality, the class of reflexive objects coincides with the class of quasi-finite comodules.

This paper is devoted to the study of flat module representations. We characterize such representations for quivers not containing the path ·• → • (the so called rooted quivers). Then, we prove that for every such quiver any representation has a flat cover and a cotorsion envelope. We finally observe that if Q is one of those quivers and if ℱ denot...

Let R be a ring and let t be a torsion preradical, R is said to have the splitting property, provided that for every left R-module M, the torsion submodule t(M) of M is a direct summand of M. The characterization of rings with this property is a classical problem (in particular the Goldie and Dickson torsion theories have been studied) that for non...

For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an $H$-bicomodule algebra $\mb{A}$ and an $H$-bimodule coalgebra $C$ the category of generalized Yetter-Drinfeld modu...

The injective right comodules appearing in the minimal injective resolution of a finite-dimensional comodule need not to be of finite dimension or even quasi-finite. The obstruction here is that factor comodules of quasi-finite comodules are not in general quasi-finite. This note is mainly devoted to the study of coalgebras for which the class of a...

In this paper we obtain a general version of Gabriel–Popescu theorem representing any Grothendieck category A as a quotient category of the category of modules over a ring (not necessarily with unit) with enough idempotents to right using a family of generators (Ui)i∈I of A where Ui are not supposed to be small. Applications to locally finite categ...

We study projective resolutions for quasi-finite comodules over a semiperfect coalgera. We show that the number of the indecomposable projective comodules in the i-th term of the projective resolution for these comodules is finite, which gives an invariant, the so-called Betti numbers. We study the behavior of these invariants under duality and loc...

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 unimodula...

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories.

We study graded left semiartinian rings with finite support. It is shown that the semiartinian property is preserved when we pass to the smash product in the sense of Quinn. We apply these results to investigate left perfect graded rings.

We study Gorenstein injective and projective modules over Zariski filtered rings and obtain relations between the Gorenstein dimensions on the category of filtered modules from the associated category of graded modules over the associated graded ring.

Let script G sign be an abstract class (closed under isomorpic copies) of left R-modules. In the first part of the paper some sufficient conditions under which script G sign is a precover class are given. The next section studies the script G sign-precovers which are script G sign-covers. In the final part the results obtained are applied to the he...

We consider rings admitting a Matlis dualizing module E. We argue that if R admits two such dualizing modules, then a module is reflexive with respect to one if and only if it is reflexive with respect to the other. Using this fact we argue that the number (whether finite or infinite) of distinct dualizing modules equals the number of distinct inv...

. Since S.D. Berman showed in 1967 that cyclic codes and Reed Muller codes can be studied as ideals in a group algebra KG (over a finite field K and G is considered, in each case, a finite cyclic group and a 2-group respectively), several authors have investigated these codes. It has been observed, that the presence of additional algebraic structur...

Let $$\sigma = ({\mathcal{T}},{\mathcal{F}})$$ be a hereditary torsion theory for the category $${\mathcal{R}}$$ -mod of unital left $${\mathcal{R}}$$ -modules over an associative ring $${\mathcal{R}}$$ with an identity element. The purpose of this note is to prove that if the associated Gabriel filter $${\mathcal{L}}$$ consists of finitely pr...

The notion of an almost G-precover as a generalization of a G-precover is introduced. Among other results it is shown that if a is any hereditary torsion theory for the category R-mod, then every module has an almost σ-torsionfree precover and an almost Coprod(J)-precover, where J denotes the class all σ-torsionfree σ-injective modules. Especially,...

The authors prove in this paper that an involutory Hopf algebra with non-zero integral over a field of characteristic zero is cosemisimple.

In this paper we study strongly graded coalgebras and its relation to the Picard group. A classification theorem for this kind of coalgebras is given via the second Doi's cohomology group. The strong Picard group of a coalgebra is introduced in order to characterize those graded coalgebras with strongly graded dual ring. Finally, for a Hopf algebra...