Equivalence of crossed product of linear categories and generalized Maschke theorem

Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#＇_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.

CROSSED PRODUCTS BY FINITE GROUP ACTIONS WITH CERTAIN TRACIAL ROKHLIN PROPERTY

We introduce a special tracial Rokhlin property for unital C~＊-algebras. Let A be a unital tracial rank zero C~＊-algebra（or tracial rank no more than one C~＊-algebra）. Suppose that α ： G → Aut（A） is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~＊-algebra. Then, the crossed product C~＊-algebra C~＊（G, A, α） has tracia rank zero（or has tracial rank no more than one）. In fact,we get a more general results.

Crossed products for Hopf group-algebras

First,the group crossed product over the Hopf group-algebras is defined,and the necessary and sufficient conditions for the group crossed product to be a group algebra are given.The cleft extension theory of the Hopf group algebra is introduced,and it is proved that the crossed product of the Hopf group algebra is equivalent to the cleft extension.The necessary and sufficient conditions for the crossed product equivalence of two Hopf groups are then given.Finally,combined with the equivalence theory of the Hopf group crossed product and cleft extension,the group crossed product constructed by the general 2-cocycle as algebra is determined to be isomorphic to the group crossed product of the 2-cocycle with a convolutional invertible map of the 2-cocycle.The unit property of a general 2-cocycle is equivalent to the convolutional invertible map of the 2-cocycle,and the combination condition of the weak action is equivalent to the convolutional invertible map of the 2-cocycle and the combination condition of the weak action.Similarly,crossed product algebra constructed by the general 2-cocycle is isomorphic to the Hopfπ-crossed product algebra constructed by the 2-cocycle with a convolutional invertible map.

Diagonal crossed product of multiplier Hopf algebras

Let A and B be two regular multiplier Hopf algebras.First,the notion of diagonal crossed product B#A of multiplier Hopf algebras is constructed using the bimodule algebra,which is a generalization of the diagonal crossed product in the sense of Hopf algebras.The result that the product in B#A is non-degenerate is given.Next,the definition of the comultiplicationΔ#on B#A is introduced,which is composed of the multiplierΔB(b)on B⊗B and the multiplierΔA(a)on A⊗A,and the elementΔ#(b⊗a)is a two-side multiplier of B#A⊗B#A,for any b∈B and a∈A.Then,a sufficient condition for B#A to be a regular multiplier Hopf algebra is described.In particular,Delvaux's main theorem in the case of smash products is generalized.Finally,these integrals on a diagonal crossed product of multiplier Hopf algebras are considered.

Cotorsion Dimension of Weak Crossed Products

Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.

Reversible Properties of Monoid Crossed Products

We study the reversible properties of monoid crossed products. The new class of strongly CM-reversible rings is introduced and characterized. This class of rings is a generalization of those of strongly reversible rings, skew strongly reversible rings and strongly M-reversible rings. Some well-known results on this subject are generalized and extended.

Homological Dimension of Crossed Products of Hopf Algebras

Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra. In this paper, we characterize the projectivity (injectivity) of M as a left A#σH-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σH, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.

Reduced Crossed Products of Pro-Banach Algebras

This paper gives the concept of the reduced pro-Banach algebra crossed product associated with inversely pro-Banach algebra dynamical system,and shows that the reduced crossed product is an inverse limit of an inverse system of Banach algebra crossed products.Also,the authors show that if the locally compact group is amenable,then the crossed product and the reduced crossed product are isometrically isomorphic.

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.

Similarity Degrees for the Crossed Product of von Neumann Algebras

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.

C-Crossed Products by R, III： Traces

When a flow on a C＊-algebra has KMS states we study traces on the associated crossed product to show that they are determined by KMS states under a mild assumption on the flow.

The Global Dimensions of Crossed Products and Crossed Coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.

AF-Embedding of Crossed Products of Certain Graph C＊-algebras by Quasi-free Actions （II）

Let E be a row-finite directed graph, let G be a locally compact abelian group with dual group G = F, let w be a labeling map from E＊ to F, and let （C＊（E）, G,a^w） be the C＊-dynamical system defined by w. Some mappings concerning the AF-embedding construction of C＊ （E） X（aw） G are studied in more detail. Several necessary conditions of AF-embedding and some properties of almost proper labeling map are obtained. Moreover it is proved that if E is constructed by attaching some l-loops to a directed graph T consisting of some rooted directed trees and G is compact, then oJ is k almost proper, that is a sufficient condition for AF-embedding, if and only if ∑j^Kk=1^wγ j ≠fi 1r for any loop γi, γ2 …γk attached to one path in T

Crossed Products on the Flow of Groupoid with Quasi-Invariant Measures

Let G be a second countable groupoid with Haar system {λ~u}, A be an abelian group which left invariant acts on G. Then we have a C*-dynamic system (C*(G), A, β). In this paper we have studied the existence of quasi-invariant measure with certain properties; using these measures some important results about crossed products and groupoid C*-algebras have been obtained.

Generalized Crossed Pro ducts and L-R Smash Pro ducts of Multiplier Hopf Algebras

In this paper we generalize the notions of crossed products and L-R smash products in the context of multiplier Hopf algebras. We use comodule algebras to define generalized diagonal crossed products, L-R smash products, two-sided smash products and two-sided crossed products and prove that they are all associative algebras. Then we show the isomorphic relations of them.

Magnetic Field Curves and Magnetic Equipotential Surfaces around Crossing Electrical Wires Replacing Classical Magnetic Field Lines

This article is based on a recent model specifically defining magnetic field values around electrical wires. With this model, calculations of field around parallel wires were obtained. Now, this model is extended with the new concept of magnetic equipotential surface to magnetic field curves around crossing wires. Cases of single, double, and triple wires are described. Subsequent article will be conducted for more general scenarios where wires are neither infinite nor parallel.

Production of p-rich nuclei with Z = 20-25 based on radioactive ion beams

Within the framework of isospin-dependent Boltzmann-Langevin model,the production cross sections of proton-rich nuclei with Z=20-25 are investigated.According to the reaction results for different isospin of projectiles^(48)Ni,^(49)Ni,and^(50)Ni,proton-rich fragments tend to be more easily produced in reactions with the protonrich projectile^(48)Ni.The production cross sections of the unknown nuclei in the vicinity of the projectile are sensitive to incident energy.It is observed that incident energy of 345 MeV/u is appropriate for producing proton-rich nuclei with Z=20-25.In projectile fragmentation reactions based on the radioactive ion beam of48Ni at 345MeV/u,several unknown proton-rich nuclei near the proton drip line are generated in the simulations.All these new nuclei are near-projectile elements near Z=28.The production cross sections of the new nuclei^(34)Ca,37,38Sc,^(38)Ti,^(40,41,42)V,^(40,41)Cr,and^(42,43,44,45)Mn are in the range of 10-2-102mb.Hence,projectile fragmentation of radioactive ion beams of Ni is a potential method for generating new proton-rich nuclei with Z=20-25.

M_(αβ) X-ray production cross sections of Pb and B iby 9–40 keV electron impact

The experimental data of Mαβ X-ray production cross sections for Pb and Bi by 9–40 keV electron impact have been given. Thin films with thick carbon substrates are used in the experiment. The effects of target structure on the Mαβ X-ray production cross sections are corrected by using the Monte Carlo method. The corrected experimental data are compared with calculated cross sections in terms of the distorted-wave Born approximation(DWBA) theory. The measured Mαβ X-ray production cross sections for Pb and Bi are lower than the DWBA calculations. The atomic relaxation parameters used in comparing the DWBA values with experimental results affect the degree of difference.

A Unified Definition of Electrostatic and Magnetic Fields

This article originates from the observation that field lines are drawn using distinctive rules in magnetic field and electrostatic fields. It aims at reconciliating the definitions of these fields and thus reaching a consensus on the interpretation of field lines. Our unified field definition combines three orthogonal vectors and a unique scalar value. Field lines are then defined as isovalue lines of the scalar value, rendering it simpler to interpret in both field types. Specific to our field definition is the use of square root of vector’s cross product so that all vectors have the same physical unit. This enhanced field definition also enables a more efficient calculation of Biot-Savart law. This article is the first of a series allowing the drawing of isovalue contour lines.