Shouchuan Zhang
We obtain that the global dimensions of $R$ and the crossed product $R # _σH$ are the same; meantime, their weak dimensions are also the same, when $H$ is finite-dimensional semisimple and cosemisimple Hopf algebra.
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We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in \cite{BB} with the notion of crossed product with a weak Hopf algebra introduced in~\cite{AG}
In a previous paper we proved a result of the type "invariance under twisting" for Brzezinski's crossed products. In this paper we prove a converse of this result, obtaining thus a characterization of what we call equivalent crossed products. As an application, we characterize cross product bialgebras (in the sense of Bespalov and Drabant) that are equivalent (in a certain sense) to a given cross product bialgebra in which one of the factors is a bialgebra and whose coalgebra structure is a tensor product coalgebra.
If $A\otimes _{R, σ}V$ and $A\otimes _{P, ν}W$ are two Brzeziński crossed products and $Q:W\otimes V\rightarrow V\otimes W$ is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A\otimes _{S, θ}(V\otimes W)$. This construction contains as a particular case the iterated twisted tensor product of algebras.
The relations between the radical of crossed product $R #_σH$ and algebra $R$ are obtained. Using this theory, the author shows that if $H$ is a finite-dimensional semisimple, cosemisimle, and either commutative or cocommutative Hopf algebra, then $R$ is $H$-semiprime iff $R$ is semiprime iff $R#_σH$ is semiprime.
We introduce the Hom-analogue of the L-R-smash product and use it to define the Hom-analogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.
In this note we extend duality theorems for crossed products obtained by M. Koppinen and C. Chen from the case of a base field or a Dedekind domain to the case of an arbitrary noetherian commutative ground ring under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case of arbitrary noetherian ground rings.
We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.
In [J.M. Fernández Vilaboa, R. González Rodríguez and A.B. Rodríguez Raposo: Preunits and weak crossed products. J. of Pure Appl. Algebra 213, 2244-2261 (2009)] the notion of a weak crossed product of an algebra by an object, both living in a monoidal category was presented. Unified crossed products defined in [A. Agore, G. Militaru: Extending structures II: The quantum version. arXiv:1011.2174v3 (2011)] and partial crossed products defined in [M. Muniz S. Alves, E. Batista, M. Dokuchaev, A. Paques: Twisted partial actions of Hopf algebras. preprint (2011)] are crossed product structures defin…
For a finite dimensional Hopf algebra we show that an associated natural inclusion of infinite crossed products is the crossed product by the Drinfeld double, and that this is a characterisation of the double.
We prove a result of the type ''invariance under twisting'' for Brzezinski's crossed products, as a common generalization of the invariance under twisting for twisted tensor products of algebras and the invariance under twisting for quasi-Hopf smash products. It turns out that this result contains also as a particular case the equivalence of crossed products by a coalgebra (due to Brzezinski).
In this paper we give a criterion that characterizes equivalent weak crossed products. By duality, we obtain a similar result for weak crossed coproducts and, as a consequence, we find the conditions that assures the equivalence between two weak crossed biproducts. As an application, we show that the main results proved by Panaite, for Brzeziński's crossed products, admits a substantial reduction in the imposed conditions.
In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may be applied to H^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the "generating matrix" formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.
We define a "mirror version" of Brzezinski's crossed product and we prove that, under certain circumstances, a Brzezinski crossed product D\otimes_{R, σ}V and a mirror version W\bar{\otimes}_{P, ν}D may be iterated, obtaining an algebra structure on W\otimes D\otimes V. Particular cases of this construction are the iterated twisted tensor product of algebras and the quasi-Hopf two-sided smash product.
The notion of crossed product by a coquasi-bialgebra H is introduced and studied. The resulting crossed product is an algebra in the monoidal category of right H-comodules. We give an interpretation of the crossed product as an action of a monoidal category. In particular, necessary and sufficient conditions for two crossed products to be equivalent are provided. Then, two structure theorems for coquasi Hopf modules are given. First, these are relative Hopf modules over the crossed product. Second, the category of coquasi-Hopf modules is trivial, namely equivalent to the category of modules ov…
We introduce the notion of a crossed product of an algebra 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.
In this paper we show how iterate weak crossed products with common monoid. More concretely, if $(A\otimes V, μ_{A\otimes V})$ and $(A\otimes W, μ_{A\otimes W})$ are weak crossed products, we find sufficient conditions to obtain a new weak crossed product $(A\otimes V\otimes W, μ_{A\otimes V\otimes W})$
L-R-crossed products
本文通过给定结合代数H、线性空间U及满足特定公理的线性映射，在U\otimes H上定义了L-R-交叉积这一新的结合代数结构，并证明其涵盖了Brzezinski交叉积和准双代数上的L-R-smash积等先前构造作为特例。
The main properties of the crossed product in the category of Hopf algebras are investigated. Let $A$ and $H$ be two Hopf algebras connected by two morphism of coalgebras $\triangleright : H\ot A \to A$, $f:H\ot H\to A$. The crossed product $A #_{f}^{\triangleright} H$ is a new Hopf algebra containing $A$ as a normal Hopf subalgebra. Furthermore, a Hopf algebra $E$ is isomorphic as a Hopf algebra to a crossed product of Hopf algebras $A #_{f}^{\triangleright} H$ if and only if $E$ factorizes through a normal Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1_{E} \in H$. The universality o…
Two-sided crossed products
本文在结合代数 A、C 与线性空间 V 上，通过满足特定条件的线性映射构造了双侧交叉积代数结构，并统一了迭代扭曲张量积与拟双代数上的双侧交叉积。
We show that when a co-involutive Hopf C*-algebra $S$ coacts via $δ$ on a C*-algebra $A$, there exists a full crossed product $A\times_δS$, with universal properties analogous to those of full crossed products by locally compact groups. The dual Hopf C*-algebra is then defined by $\hat S:={\Bbb C}\times_{\id} S$.
If H is a finite dimensional Hopf algebra, C. Cibils and M. Rosso found an algebra X having the property that Hopf bimodules over H^* coincide with left X-modules. We find two other algebras, Y and Z, having the same property; namely, Y is the "two-sided crossed product" H^*#(H\otimes H^{op})# H^{* op} and Z is the "diagonal crossed product" (H^*\otimes H^{*op})\bowtie (H\otimes H^{op}) (both concepts are due to F. Hausser and F. Nill). We also find explicit isomorphisms between the algebras X, Y, Z.
For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld's quantum double construction. We study C*-algebraic properties of these double crossed products and several links between double crossed products and bicrossed products. In an appendix, we study the Radon-Nikodym derivative of a weight under a quantum group action, following Yamanouchi and obtain, as a corollary, a new characterization of closed quantum subgroups.
A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor categories, graded monoidal functors, monoidal natural transformations, and braiding in terms of coherent outer $G$-actions over tensor categories.
The concept of a crossed tensor product of algebras is studied from a few points of views. Some related constructions are considered. Crossed enveloping algebras and their representations are discussed. Applications to the noncommutative geometry and particle systems with generalized statistics are indicated.
We show that for a C*-algebra A and a discrete group G with an action of G on A, the reduced crossed product C*-algebra possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.
In this paper, we will prove that if $A$ is a $C^*$-algebra with an effective coaction $ε$ by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac $C^*$-algebras.
It is shown that quantum Euclidean groups $E_q(2)$, $E_κ(2)$ and $E_κ(3)$ have the structure of generalised crossed products.
We show that for a closed embedding $\mathbb{H}\le \mathbb{G}$ of locally compact quantum groups (LCQGs) with $\mathbb{G}/\mathbb{H}$ admitting an invariant probability measure, a unitary $\mathbb{G}$-representation is type-I if its restriction to $\mathbb{H}$ is. On a related note, we also prove that if an action $\mathbb{G}\circlearrowright A$ of an LCQG on a unital $C^*$-algebra admits an invariant state then the full group algebra of $\mathbb{G}$ embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). W…
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