Shouchuan Zhang
Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., $$ (R # H)# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *}) \hbox {as algebras in} {\cal C}.$$ Also, it is proved that the Drinfeld double $(D(H),[b])$ is a quasi-triangular Hopf algebra in ${\cal C}$.
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Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finite dimensional and $α,β\in Aut_{Hopf}(H_1), γ,δ\in Aut_{Hopf}(H_2)$. In this paper, we introduce a category ${}_{H_1}\mathcal{LR}_{H_2}(α, β, γ, δ)$, generalizing Yetter-Drinfeld-Long bimodules and construct a braided $T$-category $\mathcal{LR}(H_1,H_2)$ containing all the categories $_{H_1}\mathcal{LR}_{H_2}(α, β, γ, δ)$ as components. We also prove that if $(α, β, γ, δ)$ admits a quadruple in involution, then ${}_{H_1}\mathcal{LR}_{H_2}(α, β, γ, δ)$ is isomorphic to the usual category ${}_{H_1}\mathcal{LR}_{H_2}$ of Yetter-Dr…
In this paper, we generalizing the main result in Liu[10] to weak Hopf coquasigroups case. We first define and study group-cograded weak Hopf quasigroups, which generalize both group-cograded Hopf quasigroups and weak Hopf group-coalgebras. Then we introduce the notion of p-Yetter-Drinfeld weak quasimodule over group-cograded weak Hopf quasigroups H. If the antipode of H is bijective, we show that the category YDWQ(H) of Yetter-Drinfeld weak quasimodules over H is a crossed category, and the subcategory YD(H) of Yetter-Drinfeld modules is a braided crossed category.
Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce all possible categories of Yetter-Drinfeld modules over $H$, and give explicitly the monoidal and braided structure of them. Then we prove that the category $^H_H\mathcal{YD}^{fd}$ of finite-dimensional left-left Yetter-Drinfeld modules is rigid. Finally we will study the braided cocommunitivity of $H_0$ in $^H_H\mathcal{YD}$.
We study the Yetter--Drinfeld D(B)-module algebra structure on the Heisenberg double H(B^*) endowed with a "heterotic" action of the Drinfeld double D(B). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_q(s…
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…
Let $H$ be a finite dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra $HøH^*$ as monoidal category. Moreover if $H$ is a Hopf algebra with bijective antipode, the isomorphism is braided.
This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by Canepeel and De Lombaerde. Then we define the $p$-Yetter-Drinfeld category over a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided $T$-categories.
Let $π$ be a group. The aim of this paper is to construct the category of Yetter-Drinfeld modules over the quasi-Turaev group coalgebra $H=(\{H_\a\}_{\a\inπ},Δ,\varepsilon,S,Φ)$, and prove that this category is isomorphic to the center of the representation category of $H$. Therefore a new Turaev braided group category is constructed.
Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the subcategory $\mathscr Y\mathscr D(H)$ of Yetter-Drinfeld modules is a braided crossed category.
For a quasi-triangular Hopf algebra $\left( H,R\right) $, there is a notion of transmuted braided group $H_{R}$ of $H$ introduced by Majid. The transmuted braided group $H_{R}$ is a Hopf algebra in the braided category $_{H}\mathcal{M}$. The $R$-adjoint-stable algebra associated with any simple left $H_{R}$-comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in ${}_{H}^{H} \mathcal{YD}$. In this note, we prove for a semisimple factorizable Hopf algebra $ \left( H,R\right) $ that any simple subcoalgebra of $H_R$ is $H$-stable …
The aim of this paper is to define and study Yetter-Drinfeld modules over weak Hom-Hopf algebras. We show that the category ${}_H{\cal WYD}^H$ of Yetter-Drinfeld modules with bijective structure maps over weak Hom-Hopf algebras is a rigid category and a braided monoidal category, and obtain a new solution of quantum Hom-Yang-Baxter equation. It turns out that, If $H$ is quasitriangular (respectively, coquasitriangular)weak Hom-Hopf algebras, the category of modules (respectively, comodules) with bijective structure maps over $H$ is a braided monoidal subcategory of the category ${}_H{\cal WYD}…
Let $(H,α)$ be a monoidal Hom-Hopf algebra and $^{H}_{H}\mathcal{HYD}$ the Hom-Yetter-Drinfeld category over $(H,α)$. Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in $^{H}_{H}\mathcal{HYD}$ gives rise to a braided Hom-Lie algebra. Second, we prove that if $(A,β)$ is a sum of two $H$-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal $[A,A]$ of $A$ is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a constru…
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。
Let $(H,\a_H)$ be a Hom-Hopf algebra, $(A,\a_A)$ a right $H$-comodule algebra and $(C,\a_C)$ a left $H$-module coalgebra. Then we have the category $_A\mathcal{M}(H)^C$ of Hom-type Doi-Hopf modules. The aim of this paper is to make the category $_A\mathcal{M}(H)^C$ into a braided monoidal category. Our construction unifies quasitriangular and coquasitriangular Hom-Hopf algebras and Hom-Yetter-Drinfeld modules. We study tensor identities for monoidal categories of Hom-type Doi-Hopf modules. Finally we show that the category $_A\mathcal{M}(H)^C$ is isomorphic to $A\#C^*$-module category.
In continuation of our recent work about smash product Hom-Hopf algebras in \cite{MLY}, we introduce Hom-Yetter-Drinfeld category $_H^H{\mathbb{YD}}$ via Radford biproduct Hom-Hopf algebra, and prove that the Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_H^H{\mathbb{YD}}$ is a pre-braided tensor category, where $(H, \b, S)$ is a Hom-Hopf algebra. Furthermore, we obtain that $(A^{\natural}_{\diamond} H,\aø\b)$ is a Radford biproduct Hom-Hopf algebra if and only if $(A,\a)$ is a Hopf algebra in the category $_H^H{\mathbb{YD}}$. At last, some examples and…
Let $H$ be an infinite-dimensional braided Hopf algebra and assume that the braiding is symmetric on $H$ and its quasi-dual $H^d$. We prove the Blattner-Montgomery duality theorem, namely we prove $$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$ In particular, we present two duality theorems for infinite braided Hopf algebras in the Yetter-Drinfeld module category.
Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal M…
In this letter, we use quantum quasi-shuffle algebras to construct Rota-Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota-Baxter algebras, which is the relevant object of Rota-Baxter algebras in a braided tensor category. Examples of such new algebras are provided by using quantum multi-brace algebras in a category of Yetter-Drinfeld modules.
Duality for action bialgebroids
研究线性对偶及量子对偶原理下Drinfeld函子对作用双代数胚的影响，证明辫子交换Yetter-Drinfeld代数在对偶双代数胚上的性质及作用双代数胚构造与对偶/量子对偶函子的交换性。
We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Second, we give the maps between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras give rise to functors between the category of Doi Hom-Hopf modules and study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf mo…
We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show …
The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^{d}$-submodule $H^{d rat}$ of $H^{d}$ is found. That is, $H^{d rat}\cong \int ^l_{H^d} \otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category $(^B_B{\cal YD},C)$ are given.
Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups and we exploit computations from our paper with Shroff for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler, we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the brai…
A decomposition Theorem for pointed braided Hopf algebras
本文推广了带点辫子Hopf代数的分解定理，证明在底层普通Hopf代数余半单时，其张量积分解与Yetter-Drinfeld范畴的余模结构相容。
Finite-dimensional Nichols algebras of simple Yetter-Drinfeld modules (over groups) of prime dimension
该研究在特征零域下确定了所有生成有限维Nichols代数的素维数群上绝对不可约Yetter-Drinfeld模，通过引入辫子向量空间的阶并研究其退化与特化实现了这一目标。
Let $\mathcal{C}$ be a finite braided multitensor category. Let $B$ be Majid's automorphism braided group of $\mathcal{C}$, then $B$ is a cocommutative Hopf algebra in $\mathcal{C}$. We show that the center of $\mathcal{C}$ is isomorphic to the category of left $B$-comodules in $\mathcal{C}$, and the decomposition of $B$ into a direct sum of indecomposable $\mathcal{C}$-subcoalgebras leads to a decomposition of $B$-$\operatorname*{Comod}_{\mathcal{C}}$ into a direct sum of indecomposable $\mathcal{C}$-module subcategories. As an application, we present an explicit characterization of the struc…
Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$ with dim $B< \infty $ . In particular, generalized classical braided m-Lie algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of …
For a Hopf algebra B with bijective antipode, we show that the Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra over the Drinfeld double D(B). The braiding structure allows generalizing H(B^*) = B^{*cop}\braid B to "Heisenberg n-tuples" and "chains" ...\braid B^{*cop}\braid B \braid B^{*cop}\braid B\braid..., all of which are Yetter--Drinfeld D(B)-module algebras. For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_qsl(2).
We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter-Drinfeld structure has universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum D…
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Be…
Let $\left( H,R\right) $ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${} {}_{H}^{H}\mathcal{YD}.$ Let $H_{R}$ be the Majid's transmuted braided group of $\left( H,R\right) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}\oplus\cdots\oplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $\left( 1\leq i\leq r\right) $, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define th…
On Braided Differential Calculi and Quantum G-structures
该研究在辫子幺正范畴发展微分演算理论，结合双积构造协变smash积微分演算，引入量子G-结构并以实例说明，为量子几何提供新理论框架。
We study Galois extensions Coinv(M)<M for M an H-comodule algebra and H a Frobenius Hopf algebroid. We obtain generalizations of various theorems in Hopf-Galois theory by Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. An algebra extension is Galois precisely if it is balanced, depth 2, and Frobenius. Then we show that Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by the Brzezinski-Militaru Theorem. Contravariant "fiber functors" are used to prove an analogue of Ulbrich's Theorem and t…
In braided tensor categories we show the Maschke's theorem and give the necessary and sufficient conditions for double cross biproducts and crossbiproducts and biproducts to be bialgebras. We obtain the factorization theorem for braided Hopf algebras; In symmetric tensor categories, we show the duality theorem and construct quantum double; In ordinary vector space category with ordinary twist, we obtain the relation between the global dimension, the weak dimension, the Jacbson radical, Baer radical of algebra $R$ and its crossed product $R #_σH$. We also give the relation between the decomposi…
We consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in works of N. Andruskiewitsch, I. Angiono, I. Heckenberger, and its restricted version in odd characteristic introduced by the same authors. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in their work. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of $\mathfrak{osp}(1|2)$.
Scalar extension Hopf algebroids
本文补全了 Brzeziński 和 Militaru 关于 smash product 代数上 Hopf algebroid 的 antipode 反同态性质的完整证明，并提出不要求 $H$ 的 antipode 可逆的广义对称 Hopf algebroid 构造。
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