Free hypermodules: a categorical approach

Abstract

In category theory, the existence of free objects is very important, especially free modules that play an important role in homological algebra. Although algebraic hyperstructures are a natural extension of algebraic structures, due to the major difference between them, study-free objects in algebraic hyperstructures become very difficult. In this article, we provide a categorical approach for the construction of free hypermodules. In fact, by considering appropriate morphisms between hypermodules, we characterize free hypermodules from three different perspectives.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.     

Polynomial Filtrations and Lannes'' T Functor

This paper defines and studies the polynomial filtration [p_k ] of the shift functor : F , where F is the category of functors between F-vector spaces over a finite field F. The functors [p_k ] correspond to a system of functors (p_k T):U U, related to Lannes'T-functor on the category U of unstable modules over the Steenrod algebra. The main results concern the behaviour of the quotients ~ _s:=~/[p_s-1~ filtrations by ~_s-nilpotent functors are introduced and it is shown that the full subcategory of _s-nilpotent functors is thick.     

Some pairs of adjoint functors involving track homotopy categories

K. A. Hardie and K. H. Kamps investigated the track homotopy categoryH _B over a fixed spaceB ([1]). They have introduced two pairs of adjoint functors:P _B N _B andm _* m ^*, whereP _B :H _B →H ^B , andm _*:H _A →H _B for a fixed mapm:A→B. We have introduced a split fibration of categoriesL:H _b →H _B and provedL J, J L in [2]. This paper first extendsP _B N _B to for any fixed mapb: . Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categoriesH _b andH ^b whereH ^b is the dual ofH _b . One of our results isN _b P _b . This differs fromP _B N _B . Supported by National Natural Science Foundation of China     

On Full and Faithful Kan Extensions

The paper introduces some mild requirements under which a Kan extension is a full embedding that preserves factorization systems and colimits of -directed diagrams. The existence of such Kan extensions is then used to establish conditions equivalent to algebraic universality of concrete categories such as varieties of algebras of any given similarity type.     

On Lie Algebras in the Category of Yetter—Drinfeld Modules

The category of Yetter—Drinfeld modules over a Hopf algebra K (with bijective antipode over a field k) is a braided monoidal category. If H is a Hopf algebra in this category then the primitive elements of H do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in such that the set of primitive elements P(H) is a Lie algebra in this sense. Also the Yetter—Drinfeld module of derivations of an algebra A in is a Lie algebra. Furthermore for each Lie algebra in there is a universal enveloping algebra which turns out to be a Hopf algebra in .     

Quantum Double for Quasi-Hopf Algebras

We introduce a quantum double quasi-triangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross-product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle .     

Stable equivalences of graded algebras

We extend the notion of stable equivalence to the class of locally finite graded algebras. For such an algebra Λ, we focus on the Krull–Schmidt category gr_Λ of finitely generated -graded Λ-modules with degree 0 maps, and the stable category obtained by factoring out those maps that factor through a graded projective module. We say that Λ and Γ are graded stably equivalent if there is an equivalence that commutes with the grading shift. Adapting arguments of Auslander and Reiten involving functor categories, we show that a graded stable equivalence α commutes with the syzygy operator (where defined) and preserves finitely presented modules. As a result, we see that if Λ is right noetherian (resp. right graded coherent), then so is any graded stably equivalent algebra. Furthermore, if Λ is right noetherian or k is artinian, we use almost split sequences to show that a graded stable equivalence preserves finite length modules. Of particular interest in the nonartinian case, we prove that any graded stable equivalence involving an algebra Λ with socΛ=0 must be a graded Morita equivalence.