Duality theorem for L-R crossed coproducts

In this paper, the notion of L-R crossed coproduct is introduced as a unified approach for smash coproducts, crossed coproducts and L-R smash coproducts of Hopf algebras. A duality theorem for L-R crossed coproduct is proved.     

作用在n维欧氏空间R~n上的离散交叉积

主要研究作用在n维欧氏空间R~n上的离散交叉积■_n■_αG_A,b_1,…,b_n的代数性质与因子分类.首先证明了该交叉积是内射的;其次给出了该交叉积为因子的一些充要条件;最后讨论该交叉积为因子时究竟是几型因子.     

2- 上循环交叉积

利用 von Neumann 代数 M 和带有正规的2- 上循环 μ 的离散群 G, 定义了2- 上循环交叉积, 推广了经典的离散交叉积, 并证明了2- 上循环交叉积具有结合律.     

Witt equivalence of global fields

A Simplicial derivation procedure for crossed module valued functors is established. As a consequence, a theory of higher Baer invariants and relative homology is settle on crossed modules. Then we illustrative how this method is efficient, as it extends classical formulations and results, like various known five term sequences on crossed modules by Conduché and Ellis (1989), the author (1993) and Ladra and R-Grandjeán (1994).     

Drawing complete multipartite graphs on the plane with restrictions on crossings

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near independent crossings （say NIC-planar graph） is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex. The full characterization of NIC-planar complete and complete multipartite graphs is given in this paper.     

A Note on Regular Crossed Products and Galois Representations

A crossed product representing an associative finite dimensional central simple algebra over a field is called regular if all values of the corresponding cocycle are roots of unity. Under a certain assumption such a crossed product is shown to allow the construction of Galois representations. The case of number fields is investigated more closely and several examples are discussed.     

Crossed Products for Weak Hopf Algebras

Weak crossed products by a weak bialgebra are defined. The resulting structure is not that of a unital algebra but an associative algebra with preunit. A general formula of such a product is given in terms of a weak 2-cocycle and a weak measuring. The relation with weak cleft extensions is studied. Equivalences of weak crossed products are defined and are related to gauge transformations. A relation between cleaving maps is described in terms of gauge transformations.     

Preunits and weak crossed products

Using the notion of a preunit and the properties of idempotent morphisms, we give a general notion of a crossed product of an algebra A and an object V both living in a monoidal category C. We endow A⊗V with a multiplication and an idempotent morphism, whose image inherits the multiplication. Sufficient conditions for these multiplications to be associative are given. If the product on A⊗V has a preunit, the related idempotent is given in terms of the preunit, and its image has an algebra structure. A characterization of crossed products with preunit is given, and it is used to recover classical examples of crossed products and to study crossed products in weak contexts. Finally crossed products of an algebra by a weak bialgebra are recovered using this theory.     

Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

A family of Hecke C^*-algebras can be realised as crossed productsby semigroups of endomorphisms. It is shown by dilating representationsof the semigroup crossed product that the category of representationsof the Hecke algebra is equivalent to the category of continuousunitary representations of a totally disconnected locally compactgroup.     

Pontryagin Duality for a Class of Locally Compact Quantum Groups

A class of locally compact quantum semigroups in the sense of WORONOWICZ , containing the quantized Heisenberg group, is studied. The underlying quantum space is described in terms of a crossed product. We show the existence of a universal corepresentation and a Pontryagin dual, which turns out to be a twisted crossed product. The Pontryagin duality for these quantum semigroups is investigated.     

An elementary Green imprimitivity theorem for inverse semigroups

Semigroup Forum - A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let S be an...     

Crossed products by a coalgebra

We introduce the notion of a crossed product of an al¬gebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.     

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.     

Capability of Crossed Modules of Lie Algebras

In this paper, we introduce the concept of capability for crossed modules of Lie algebras, which is a generalization of capability in Lie algebras and groups. By using a special central ideal of a crossed module, we give a sufficient condition for the capability of a crossed module of Lie algebras. Also, we will extend the five-term exact sequence on homology of crossed modules of Lie algebras one term further and study the connection between the capability of crossed modules and this sequence. Finally, we study the relation between the capability and the center of a cover of a crossed module.     

Hopf代数的双交叉积

本文定义并详细讨论了交叉余积，考虑交叉积与交叉余积合起未成为双代数的问题，讨论了由内作用，内余作用构造的双交叉积．我们还用正合序列刻划了双交叉积．     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

The Spectrum of a Harmonic Oscillator Operator Perturbed by {\delta}-Interactions

We define reduced crossed products associated with a Banach algebra dynamical system. If the group is amenable, we prove that the reduced crossed product and the crossed product are isometrically isomorphic.     

余拟三角弱Hopf群代数与辫子Monoidal范畴

In this paper, we first give the definitions of a crossed left π-H-comodules over a crossed weak Hopf π-algebra H, and show that the category of crossed left π-H-comodules is a monoidal category. Finally, we show that a family σ = {σα,β: Hα Hβ→ k}α,β∈πof k-linear maps is a coquasitriangular structure of a crossed weak Hopf π-algebra H if and only if the category of crossed left π-H-comodules over H is a braided monoidal category with braiding defined by σ.     

1-planar Graphs without 4-cycles or 5-cycles are 5-colorable

A graph is 1-planar if it can be drawn on the Euclidean plane so that each edge is crossed by at most one other edge. A proper vertex k-coloring of a graph G is defined as a vertex coloring from a set of k colors such that no two adjacent vertices have the same color. A graph that can be assigned a proper k-coloring is k-colorable. A cycle is a path of edges and vertices wherein a vertex is reachable from itself. A cycle contains k vertices and k edges is a k-cycle. In this paper, it is proved t...     

Crossed products of operator systems

In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under G-equivariant maps, tensor products, and the canonical $C^{?}$-covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects $C^{?}$-nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey.