Semisimplicity of the Categories of Yetter–Drinfeld Modules and Long Dimodules

Abstract

Let k be a field, and H a Hopf algebra with bijective antipode. If H is commutative, noetherian, semisimple and cosemisimple, then the category _H 𝒴𝒟 ^H of Yetter–Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that H is commutative.     

K(π 1)'s for Artin Groups of Finite Type

We construct K( 1)'s for Artin groups of type C _n and D _n , using the lattice of elements preceding a Coxeter element in the partial order defined by reflection length.     

Cohen–Macaulayness of Rees Algebras of Modules

Abstract

The notion of quasi-polynomials is very important in the theory of functional identities. For example, results on quasi-polynomials were tools in the solution of long-standing Herstein's Lie map conjectures. In this paper, we show that functional identities involving quasi-polynomial of degree one have only standard solutions on d-free sets.     

Derived Brauer–Thrall Type Theorem I for Artin Algebras

We define the global cohomological range for artin algebras, and define the derived bounded algebras to be the algebras with finite global cohomological range, then we prove the first Brauer–Thrall type theorem for bounded derived categories of artin algebras, i.e., derived bounded algebras are precisely the derived finite algebras. Moreover, the main theorem establishes that the derived bounded artin algebras are just piecewise hereditary algebras of Dynkin type, and can be also characterized as those artin algebras with derived dimension zero, which can be regarded as a generalization of the results of Han–Zhang [11 Han, Y., Zhang, C. Brauer-Thrall type theorems for derived category, arXiv:1310.2777. [Google Scholar], Theorem 1] and Chen–Ye–Zhang [4 Chen, X. W., Ye, Y., Zhang, P. (2008). Algebras of derived dimension zero. Comm. Algebra 36:1–10.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem] in the context of finite-dimensional algebras over algebraically closed fields, respectively.     

On the Koszul Modules of Exterior Algebras

We prove that the Koszul modules over an exterior algebra can be filtered by the cyclic Koszul modules. We also introduce the cyclic dimension vector as invariants for studying the Koszul modules over an exterior algebra.     

The Categories of Yetter—Drinfel’d Modules, Doi—Hopf Modules and Two-sided Two-cosided Hopf Modules

We show that the two-sided two-cosided Hopf modules are in some case generalized Hopf modules in the sense of Doi. Then the equivalence between two-sided two-cosided Hopf modules and Yetter—Drinfeld modules, proved in [8], becomes an equivalence between categories of Doi—Hopf modules. This equivalence induces equivalences between the underlying categories of (co)modules. We study the relation between this equivalence and the one given by the induced functor.     

Normal Forms of Modules over Admissible Algebras with Formal Two-ray Modules

The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.     

Modules over Operator Algebras and Its Module Maps

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Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, a left H-comodule algebra  and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules ^C ?(H) in terms of modules. We will also show that for an H-bicomodule algebra  and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) ^C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.     

Coring Categories and Villamayor–Zelinsky Sequence for Symmetric Finite Tensor Categories

Applied Categorical Structures - In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology...     

Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-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–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.      

Primitive Topological Algebras And Modules

In this paper we define a topology that we call canonical on an irreduciblemodule for a topological algebra. Under some additional conditions we provetheorems of the strict density and the uniqueness of the topology for a primi-tive locally convex F-algebra. We also show that a locally finite primitive local-ly convex F-algebra is finite dimensional.     

Chaos for Subshifts of Finite Type

This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Moreover, for any subshift of finite type determined by a matrix, we point out that the cases including positive topological entropy, distributional chaos, chaos and Devaney chaos are mutually equivalent.     

Yetter–Drinfeld Modules over Weak Braided Hopf Monoids and Center Categories

In this paper,we introduce several centralizer constructions in a monoidal context and establish a monoidal equivalence with the category of Yetter–Drinfeld modules over a weak braided Hopf monoid.We apply the general result to the calculus of the center in module categories.     

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 .     

Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.     

Gelfand–Kirillov Dimension of Algebras with a Locally Finite and Multiparameter Indexed Filtration

Let A have a locally finite and multiparameter indexed filtration ?, and let B be a homomorphic image of A. Thus B has the locally finite and multiparameter indexed filtration induced from ?. Here we study a relation between the associated graded algebra of A and that of B and use this result to calculate the Gelfand–Kirillov dimension of several algebras related to quantized algebras and Poisson enveloping algebras.     

A Finite Structure Theorem Between Rings of Endomorphisms of Completely Reducible Modules

In this note we establish a finite structure theorem between rings endomorphisms of homogen-eous completely reducible modules,which generalizes the finite structure theorem between primitiverings as we established before.     

Λ-Cofibration Categories and the Homotopy Categories of Global Actions and Simplicial Complexes

We investigate the homotopy category of a -cofibration category and compare the homotopy categories of Global Actions, Simplicial Complexes and Topological Spaces.