Yinhuo Zhang
Universiteit Hasselt, Wiskunde & Statistics, Faculty Member
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We give the full structure of the Ext algebra of a Nichols algebra of type $A_2$ by using the Hochschild-Serre spectral sequence. As an application, we show that the pointed Hopf algebras $u(\mathcal{D}, \lmd, \mu)$ with Dynkin diagrams... more
We give the full structure of the Ext algebra of a Nichols algebra of type $A_2$ by using the Hochschild-Serre spectral sequence. As an application, we show that the pointed Hopf algebras $u(\mathcal{D}, \lmd, \mu)$ with Dynkin diagrams of type $A$, $D$, or $E$, except for $A_1$ and $A_1\times A_1$ with the order $N_{J}>2$ for at least one component $J$, are wild.
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ABSTRACT Let (H, R) be a finite coquasitriangular Hopf algebra over a commutative ring k with unity. In this article, we show that the exact sequence [image omitted] for the equivariant Brauer group obtained in the article Zhang (2004),... more
ABSTRACT Let (H, R) be a finite coquasitriangular Hopf algebra over a commutative ring k with unity. In this article, we show that the exact sequence [image omitted] for the equivariant Brauer group obtained in the article Zhang (2004), is stable under cocycle twisting for any cocycle :H⊗ H→ k on H.
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In this paper, we investigate the Green ring $r(H_{n,d})$ of the generalized Taft algebra $H_{n,d}$, extending the results of Chen, Van Oystaeyen and Zhang in \cite{Coz}. We shall determine all nilpotent elements of the Green ring... more
In this paper, we investigate the Green ring $r(H_{n,d})$ of the generalized Taft algebra $H_{n,d}$, extending the results of Chen, Van Oystaeyen and Zhang in \cite{Coz}. We shall determine all nilpotent elements of the Green ring $r(H_{n,d})$. It turns out that each nilpotent element in $r(H_{n,d})$ can be written as a sum of indecomposable projective representations. The Jacobson radical $J(r(H_{n,d}))$ of $r(H_{n,d})$ is generated by one element, and its rank is $n-n/d$. Moreover, we will present all the finite dimensional indecomposable representations over the complexified Green ring $R(H_{n,d})$ of $H_{n,d}.$ Our analysis is based on the decomposition of the tensor product of indecomposable representations and the observation of the solutions for the system of equations associated to the generating relations of the Green ring $r(H_{n,d})$.
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A Survey on Multiplier Hopf Algebras Alfons Van Daele Yinhuo Zhang KU Leuven University of Antwerp, UIA Heverlee, Belgium Antwerp, Belgium Abstract For any group G, the group algebra kG over a field k has a natural Hopf alge-bra... more
A Survey on Multiplier Hopf Algebras Alfons Van Daele Yinhuo Zhang KU Leuven University of Antwerp, UIA Heverlee, Belgium Antwerp, Belgium Abstract For any group G, the group algebra kG over a field k has a natural Hopf alge-bra structure. If the group is finite. this Hopf ...
We calculate the Brauer group of the four dimensional Hopf algebra H4 introduced by M. E. Sweedler. This Brauer group BM (k, H4, R0) is defined with respect to a (quasi-) triangular structure on H4, given by an element R0 ∈ H4 ⊗ H4. In... more
We calculate the Brauer group of the four dimensional Hopf algebra H4 introduced by M. E. Sweedler. This Brauer group BM (k, H4, R0) is defined with respect to a (quasi-) triangular structure on H4, given by an element R0 ∈ H4 ⊗ H4. In this paper k is a field. The additive group (k, +) of k is embedded in the Brauer group and it fits in the exact and split sequence of groups: 1 → (k, +) → BM (k, H4, R0) → BW (k) → 1 where BW(k) is the well-known Brauer-Wall group of k. The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.
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Let $H$ be a Hopf algebra over a commutative ring $k$ with unity and $\sigma:H\otimes H\longrightarrow k$ be a cocycle on $H$. In this paper, we show that the Yetter-Drinfeld module category of the cocycle deformation Hopf algebra... more
Let $H$ be a Hopf algebra over a commutative ring $k$ with unity and $\sigma:H\otimes H\longrightarrow k$ be a cocycle on $H$. In this paper, we show that the Yetter-Drinfeld module category of the cocycle deformation Hopf algebra $H^{\sigma}$ is equivalent to the Yetter-Drinfeld module category of $H$. As a result of the equivalence, the "quantum Brauer" group BQ$(k,H)$ is isomorphic to BQ$(k,H^{\sigma})$. Moreover, the group $\Gal(\HR)$ constructed in \cite{Z} is studied under a cocycle deformation.
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Let $H$ be a twisted Calabi-Yau (CY) Hopf algebra and $A$ a Koszul twisted CY algebra such that $A$ is a graded $H$-module algebra. We show that the smash product $A\#H$ is also a twisted CY algebra and present its Nakayama automorphism... more
Let $H$ be a twisted Calabi-Yau (CY) Hopf algebra and $A$ a Koszul twisted CY algebra such that $A$ is a graded $H$-module algebra. We show that the smash product $A\#H$ is also a twisted CY algebra and present its Nakayama automorphism explicitly. As a consequence, a necessary and sufficient condition for $A\#H$ to be a CY algebra is obtained. Furthermore, we show that a PBW deformation of $A\#H$, under one condition, is a twisted CY algebra.
The quasi-Frobenius-Lusztig kernel ${Q}\mathbf{u}_{q}(\mathfrak{sl}_{2})$ associated with $\mathfrak{sl}_{2}$ has been constructed in \cite{Liu}. In this paper we study the representations of this small quasi-quantum group. We give a... more
The quasi-Frobenius-Lusztig kernel ${Q}\mathbf{u}_{q}(\mathfrak{sl}_{2})$ associated with $\mathfrak{sl}_{2}$ has been constructed in \cite{Liu}. In this paper we study the representations of this small quasi-quantum group. We give a complete list of non-isomorphic indecomposables and the tensor product decomposition rules for simples and projectives. A description of the Grothendieck ring is provided.
Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an... more
Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an analogue of Kostant--Lusztig ${Z}[v,v^{-1}]$-form for $D_q$ and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple $D_q$-module is the pullback of a simple ${U}_q(sl_2)$-module through certain surjection from $D_q$ onto ${U}_q(sl_2)$, and the category of finite dimensional weight $D_q$-modules is equivalent to a direct sum of $|k^{\times}|$ copies of the category of finite dimensional weight ${U}_q(sl_2)$-modules. As an application, we recover (in a conceptual way) Chen's results as well as Radford's results on the quantum double of Taft algebra. Our main results allow a direct generalization to the quantum double of the Borel subalgebra of the quantized enveloping algebra associated to arbitrary Cart...
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This is a survey of our joint works on graded Calabi-Yau algebras, Calabi-Yau Hopf algebras and their PBW-deformations.
Let $H$ be a finite dimensional semisimple Hopf algebra and $R$ a braided Hopf algebra in the category of Yetter-Drinfeld modules over $H$. When $R$ is a Calabi-Yau algebra, a necessary and sufficient condition for $R#H$ to be a... more
Let $H$ be a finite dimensional semisimple Hopf algebra and $R$ a braided Hopf algebra in the category of Yetter-Drinfeld modules over $H$. When $R$ is a Calabi-Yau algebra, a necessary and sufficient condition for $R#H$ to be a Calabi-Yau Hopf algebra is given. Conversely, when $H$ is the group algebra of a finite group and the smash product $R#H$ is a Calabi-Yau algebra, we give a necessary and sufficient condition for the algebra $R$ to be a Calabi-Yau algebra.
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Let Uq(sl2)Uq(sl2) be the quantized enveloping algebra associated to the simple Lie algebra sl2sl2. In this paper, we study the quantum double DqDq of the Borel subalgebra Uq((sl2)⩽0)Uq((sl2)⩽0) of Uq(sl2)Uq(sl2). We construct an analogue... more
Let Uq(sl2)Uq(sl2) be the quantized enveloping algebra associated to the simple Lie algebra sl2sl2. In this paper, we study the quantum double DqDq of the Borel subalgebra Uq((sl2)⩽0)Uq((sl2)⩽0) of Uq(sl2)Uq(sl2). We construct an analogue of Kostant–Lusztig Z[v,v−1]Z[v,v−1]-form for DqDq and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple DqDq-module is the pull-back of a simple Uq(sl2)Uq(sl2)-module through certain surjection from DqDq onto Uq(sl2)Uq(sl2), and the category of finite-dimensional weight DqDq-modules is equivalent to a direct sum of |k×||k×| copies of the category of finite-dimensional weight Uq(sl2)Uq(sl2)-modules. As an application, we recover (in a conceptual way) Chen's results [H.X. Chen, Irreducible representations of a class of quantum doubles, J. Algebra 225 (2000) 391–409] as well as Radford's results [D.E. Radford, On oriented quantum algebras derived from representations of the quantum double of a f...
Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions... more
Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable $H$-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable $H$-modules by virtue of almost split sequences. The Green ring $r(H)$ of $H$ will be presented in terms of generators and relations. It turns out that the Green ring $r(H)$ is commutative and is generated by one variable over the Grothendieck ring $G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius and symmetric with dual bases associated to almost split sequences, and its Jacobson radical is a principal ideal. Finally, we show that the stable Green ring, the Green ring of the stable module category, is isomorphic to the quotient ring of $r(H)$ modulo the principal ideal generated by the projective cover of the trivial module. It turns out that the complexified stable Green algebra is a group...
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We give the full structure of the Ext algebra of a Nichols algebra of type $A_2$ by using the Hochschild-Serre spectral sequence. As an application, we show that the pointed Hopf algebras $u(\mathcal{D}, \lmd, \mu)$ with Dynkin diagrams... more
We give the full structure of the Ext algebra of a Nichols algebra of type $A_2$ by using the Hochschild-Serre spectral sequence. As an application, we show that the pointed Hopf algebras $u(\mathcal{D}, \lmd, \mu)$ with Dynkin diagrams of type $A$, $D$, or $E$, except for $A_1$ and $A_1\times A_1$ with the order $N_{J}>2$ for at least one component $J$,
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ABSTRACT Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C = A°be its dual coalgebra. If A is Artin–Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted... more
ABSTRACT Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C = A°be its dual coalgebra. If A is Artin–Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule 1C σ* with σ a coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twisted bimodule σ*A 1. If σ is an inner automorphism, then A is Calabi–Yau. An appendix is included to prove a duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra.
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Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A\#H$ is a dg algebra. For any dg $A\#H$-module $M$, there is a quasi-isomorphism of dg... more
Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A\#H$ is a dg algebra. For any dg $A\#H$-module $M$, there is a quasi-isomorphism of dg algebras: $\mathrm{RHom}_A(M,M)\#H\longrightarrow \mathrm{RHom}_{A\#H}(M\ot H,M\ot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras
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We consider the Brauer group ${\rm BM}'(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence $$1\longrightarrow {\rm Br}'(k)\longrightarrow {\rm... more
We consider the Brauer group ${\rm BM}'(k,G)$ of a group $G$ (finite or infinite) over a commutative ring $k$ with identity. A split exact sequence $$1\longrightarrow {\rm Br}'(k)\longrightarrow {\rm BM}'(k,G)\longrightarrow {\rm Gal}(k,G) \longrightarrow 1$$ is obtained. This generalizes the Fr\"ohlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of $G$ to the infinite case of $G$. Here ${\rm Br}'(k)$ is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.