Category Of Algebras Research Articles

Extensions of the category of comodules of the Taft algebra

We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.

Comodule theories in Grothendieck categories and relative Hopf objects

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category S. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category SHA of relative (A,H)-Hopf modules in S, where H is a Hopf algebra and A is a right H-comodule algebra. We study the cohomological theory in SHA by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in SHA. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in SHA.

Wells-type exact sequence and crossed extensions of algebras with bracket

Abstract We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight-term exact sequence in the cohomology of algebras with bracket.

On abelian subcategories of triangulated categories

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial t-structures, and in particular, full abelian subcategories need not arise as hearts of a t-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.

The Weil descent functor in the category of algebras with free operators

We prove that there exists a version of Weil descent, or Weil restriction, in the category of D-algebras. The objects of this category are k-algebras R equipped with a homomorphism e:R→R⊗kD for some fixed field k and finite-dimensional k-algebra D. We do this under a mild assumption on the so-called associated endomorphisms. In particular, this yields the existence of the Weil descent functor in the category of difference algebras, which, to our knowledge, does not appear elsewhere.

On the representability of actions of Leibniz algebras and Poisson algebras

AbstractIn a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

Projectivity of some Banach modules over semiprime Banach algebras

Let [Formula: see text] be a semiprime Banach algebra and let [Formula: see text] be a nonzero character on [Formula: see text]. In this paper, we discuss the projectivity of certain left Banach [Formula: see text]-modules in terms of left [Formula: see text]-contractibility of Banach algebras. We also apply our results for the category of quantum group algebras and eventually, we present illustrative examples of the projective left Banach modules over the group algebra [Formula: see text] and the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text].

Frobenius-Perron theory of the bound quiver algebras containing loops

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example.

Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.

Gelfand duality for manifolds, and vector and other bundles

In general terms, Gelfand duality refers to a correspondence between a geometric, topological, or analytical category, and an algebraic category. For example, in smooth differential geometry, Gelfand duality refers to the topological embedding of a smooth manifold in the topological dual of its algebra of smooth functions. This is generalised here in two directions. First, the topological embeddings for manifolds are generalised to the cases of real analytic and Stein manifolds, using a unified cohomological argument. Second, this type of duality is extended to vector bundles, affine bundles, and jet bundles by using suitable classes of functions, the topological duals in which the embeddings take their values.

Equivalence of categories of simple Lie algebras in positive characteristic

In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent.

Nodal quintic del Pezzo threefolds and their derived categories

We construct a Kawamata type semiorthogondal decomposition for the bounded derived category of coherent sheaves of nodal quintic del Pezzo threefolds, decomposing the bounded derived category into bounded derived categories of finite dimensional algebras. This is achieved by constructing birational maps from nodal quintic del Pezzo threefolds to quadric surface fibrations over the projective line.

Determination of some almost split sequences in morphism categories

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams.

Gromov ellipticity and subellipticity

Abstract We establish the equivalence of Gromov ellipticity and subellipticity in the algebraic category.

Recollements and Φ-Cohen–Macaulay Auslander–Yoneda algebras

It is shown that a [Formula: see text]-recollement of derived categories of algebras induces a [Formula: see text]-recollement of the corresponding [Formula: see text]-Cohen–Macaulay Auslander–Yoneda algebras. This result not only generalizes the main theorems of Pan [Derived equivalences for [Formula: see text]-Cohen–Macaulay Auslander–Yoneda algebras, Algebr. Represent. Theory 17 (2014) 885–903] and Pan [Derived equivalences for Cohen–Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012) 355–363], but also provides us a useful method to construct a new recollement of derived module categories from a given one.

Operations on the Hochschild bicomplex of a diagram of algebras

A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an L∞-algebra. The same results are true for the reduced and asimplicial subcomplexes and asimplicial cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation.

Complexes with the derived double centralizer property

In representation theory, the double centralizer property is an important property for a module (bimodule). In this paper, we extend this property to complexes in derived categories of algebras, under the name derived double centralizer property. Let A be a finite dimensional algebra over an algebraically closed field. Characterizations for complexes of finitely generated A-modules with the derived double centralizer property and for tilting complexes in the bounded derived category with the endomorphism algebras hereditary are given. In particular, when A is a upper triangular matrix algebra, all complexes of A-A-bimodules with this property are classified.

English

We compare several recent approaches to studying right Bousfield localization and algebras over monads. We prove these approaches are equivalent, and we apply this equivalence to obtain several new results regarding right Bousfield localizations (some classical, some new) for spectra, spaces, equivariant spaces, chain complexes, simplicial abelian groups, and the stable module category. En route, we provide conditions so that right Bousfield localization lifts to categories of algebras, so that right Bousfield localization preserves algebras over monads, and so that right Bousfield localization forms a compactly generated model category.

Skew category algebras and modules on ringed finite sites

Let C be a small category. We investigate ringed sites (C,R) on C and the resulting module categories Mod-R. When C is finite, based on Grothendieck and Verdier's classification of finite topoi, we prove that each Mod-R is equivalent to Mod-R|D[D], where R|D[D] is the skew category algebra, canonically defined on (C,R), for a uniquely determined full subcategory D⊂C and the restriction R|D of R to D.

Exceptional sequences in the derived category of a gentle algebra

In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category {{,mathrm{mathsf {K^b(A)}},}} of a gentle algebra {{,mathrm{textsf{A}},}}. We show that full exceptional sequences in {{,mathrm{mathsf {K^b(A)}},}} exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterize when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we show that the action of the braid group together with the grading shift on full exceptional sequences in {{,mathrm{mathsf {K^b(A)}},}} is transitive. For the case of surfaces of higher genus, we reduce the problem of transitivity to the problem of the existence of certain sequences of pairs of exceptional objects. Finally, we interpret the duality of a full exceptional sequence induced by the longest element in the associated symmetric group using Koszul duality.