Sandipan De, Vijay Kodiyalam
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.
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We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a Yetter-Drinfeld module over a crossed group coalgebra H and we prove that both the category of Yetter-Drinfeld modules over H and the center of the category of representations of H as well as the category of representations of the quantum double of H are isomorphic as braided crossed categories.
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.
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.
We introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld centre $\mathcal{Z}(\mathcal{A})$, the notion of $\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\mathcal{Z}(\mathcal{A})$. In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group $G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is shown to be equivalent to the 2-category of $G…
We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed Hopf group coalgebra is both modular and unimodular.
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 $(A\otimes V\otimes W, μ_{A\otimes V\otimes W})$
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.
Generalized Yetter-Drinfeld modules, the center of bi-actegories and groupoid-crossed braided bicategories
本文研究双作用范畴的 $E$-中心，证明其等价于广义 Yetter-Drinfeld 模范畴，并引入双群胚交叉辫状双范畴概念，推广 Turaev 的群交叉辫状幺半范畴。
It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.
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 ribbon category of representations of a crossed module
研究有限群交叉模表示，扩展Drinfeld量子对偶构造得到霍普夫代数D(G,H)，研究其单子代数、有限维模范畴，给出同构及相关公式，发展交叉模表示特征理论，得到带边不变量。
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。
Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, here meaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \to H_{n}$ of Hopf algebra morphisms, such that $\partial \, i = \mathrm{id}$. Given a Hopf algebra projection $(\partial \colon I \to H,i)$ in a braided monoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structure living in the category of Yetter-Drinfeld modules over $H$, due to Radford's theorem. The underlying set of this Hopf algebra is obtained by an…
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.
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.
Let k be a field. Let also (F, G) be a matched pair of groups. We give necessary and sufficient conditions on a pair (σ, τ) of 2-cocycles in order that the crossed product algebra and the crossed coproduct coalgebra k^G{}^τ#_σ kF combine into a braided Hopf algebra. We also discuss diagonal realizations of such braided Hopf algebras in the category of Yetter-Drinfeld modules over a finite group.
We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces…
Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles
本文通过扭曲Drinfeld双构造纽结的量子不变量，推广了缝合流形的扭曲Kuperberg不变量至非对合情形，并将SL(n,C)扭曲Reidemeister挠率表示为Reshetikhin-Turaev不变量。
Let $G$ be a finite group. Starting from the field algebra ${\mathcal{F}}$ of $G$-spin models, one can construct the crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$ such that it coincides with the $C^*$-basic construction for the field algebra ${\mathcal{F}}$ and the $D(G)$-invariant subalgebra of ${\mathcal{F}}$, where $D(G)$ is the quantum double of $G$. Under the natural $\widehat{D(G)}$-module action on ${\mathcal{F}}\rtimes D(G)$,the iterated crossed product $C^*$-algebra can be obtained, which is $C^*$-isomorphic to the $C^*$-basic construction for ${\mathcal{F}}\rtimes D(G)$ a…
Hopf Galois extensions of Hopf algebroids
该研究将Hopf代数的Galois扩张理论推广至Hopf代数胚，引入相关概念并证明了裂扩张等三类结构的等价关系，丰富了Hopf代数胚的相关理论。
We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, etc. I give a construction of \textit{doubles} for \textit{graded} $QS$- and \textit{graded $QP$-manifolds} (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded $QS…
Centers of Algebras in Monoidal 2-Categories
介绍幺半 2- 范畴中代数的左、右和完全中心并证明其莫里塔不变性，对戴维多夫理论进行范畴化，统一阐述融合范畴等的中心理论，还发展了 2- 伴随和模 2- 范畴的基变换构造。
We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double D(H). Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov-Tian- Shansky, both algebras are isomorphic to the algb…
About Hopf braces and crossed products
本文研究对称幺半范畴中Hopf braces的交叉积，给出(A1⊗H1,A2⋈H2)和(A1⋈H1,A2♯H2)成为Hopf braces的条件，并应用于Drinfeld's Double。
Quantum symmetries of the twisted tensor products of C*-algebras
本文研究带正交滤过的C*-代数扭张量积的量子对称群，证明满足扭形式假设时该群为原滤过量子对称群的广义Drinfeld double，给出离散群交叉积的应用及主定理不成立的反例。
Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitria…
We study finite set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over $\C$ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra $\Acal(\C,X,r)$ having a $q$-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of finite abelian group $\Gcal$ of left actions on $X$. We study the structure of $\Acal(\C,X,r)$ and show that they have …
The values at positive integers of the polyzeta functions are solutions of the polynomial equations arising from Drinfeld's associators, which have numerous applications in quantum algebra. Considered as iterated integrals they become periods of the motivic fundamental groupoid of $\mathbb{P}^1\setminus\{0,1,\infty\}$. From there comes a fundamental, yet no more explicit, system of algebraic relations; it implies the system of associators. We focus here on the combinatorics properties of another system of relations, the ``double shuffles'', which comes from elementary series and integrals mani…
We study generalised differential structures $Ω^1,d$ on an algebra $A$, where $A\tens A\to Ω^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs $(Λ^1,ω)$ where $Λ^1$ is a right module and $ω$ a right module map, and the Hopf algebra bicovariant case corresponds to morphisms $ω:A^+\to Λ^1$ in the category of right crossed (or Drinfeld-Radford-Yetter) modules over $A$. When $A=U(g)$ the generalised left-covariant differential structures are classified by cocycles $ω\in Z^1(g,Λ^1)…
Braided tensor product of von Neumann algebras
本文定义了配备三角量子群作用的 von Neumann 代数的辫子张量积，证明了交叉积可实现为辫子张量积，并指出等变正规完全有界映射可进行辫子张量积运算。
We study super-braided Hopf algebras $Λ$ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules $Λ^1$ over a Hopf algebra $A$ which are quotients of the augmentation ideal $A^+$ under right multiplication and the adjoint coaction. Here super-bosonisation $Ω=A\ltimesΛ$ provides a bicovariant differential graded algebra on $A$. We introduce $Λ_{max}$ providing the maximal prolongation, while the canonical braided-exterior algebra $Λ_{min}=B_-(Λ^1)$ provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator $\sharp$ by sup…
Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and H^-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H=\CC G is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their sy…
In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ associated to the Lie algebra $\mathfrak sl_{\infty}$ of infinite rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak sl_{\infty})$ admits both a Hopf algebra structure and a triangular decomposition. In particular, it can be realized as the Drinfeld double of it's certain Hopf subalgebras. We will also study a two-parameter twisted Ringel-Hall algebra $H_{r,s}(A_{\infty})$ associated to the category of all finite dimensional representations of the infinite linear quiver $A_{\infty…
$q$-Analogue of the degree zero part of a rational Cherednik algebra and generalised Van Diejen's Hamiltonians
本文定义了GL_n型双仿射Hecke代数的子代数$\boldsymbol{\textsf{H}}^{\boldsymbol{\textsf{gl}}_n}$，证明其性质并将Van Diejen哈密顿量推广到含两类粒子的带外部场系统。
Twisted unipotent groups
本文研究仿射代数幺幂群上带Hopf 2-上循环形变的余三角Hopf代数的结构与表示，证明其为迭代导子型Hopf Ore扩张，给出交叉积分解、单模分层及有限维单模的群结构。
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