arXiv subjects
Publications and source records attributed to Yinhuo Zhang.
Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.
Let $A$ be a Koszul Artin-Schelter regular algebra and $σ$ an algebra homomorphism from $A$ to $M_{2\times 2}(A)$. We compute the Nakayama automorphisms of a trimmed double Ore extension $A_P[y_1, y_2; σ]$ (introduced in \cite{ZZ08}). Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension $A[t; θ]$, where $θ$ is a graded algebra automorphism of $A$. These lead to a characterization of the Calabi-Yau property of $A_P[y_1, y_2; σ]$, the skew Laurent extension $A[t^{\pm 1}; θ]$ and $A[y_1^{\pm 1}, y_2^{\pm 1}; σ]$ with $σ$ a diagonal type.
Let $H$ be a twisted Calabi-Yau (CY) algebra and $σ$ a 2-cocycle on $H$. Let $A$ be an $N$-Koszul twisted CY algebra such that $A$ is a graded $H^σ$-module algebra. We show that the cleft extension $A#_σH$ is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an $N$-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of $U(\mathcal{D},λ)$, a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi-Yau algebras.
Let $(H, R)$ be a finite dimensional quasitriangular Hopf algebra over a field $k$, and $_H\mathcal{M}$ the representation category of $H$. In this paper, we study the braided autoequivalences of the Drinfeld center $^H_H\mathcal{YD}$ trivializable on $_H\mathcal{M}$. We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra $_RH$. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group $\mathrm{BM}(k, H,R)$ in [18]. To this aim, we have to develop the braided bi-Galois theory initiated by Schauenburg in [14,15], which generalizes the Hopf bi-Galois theory over usual Hopf algebras to the one over braided Hopf algebras in a braided monoidal category.
Let $B$ be a generalized Koszul algebra over a finite dimensional algebra $S$. We construct a bimodule Koszul resolution of $B$ when the projective dimension of $S_B$ equals 2. Using this we prove a Poincaré-Birkhoff-Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of $S_B$ is greater than 2, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori.
In this paper, we continue our study of the Green rings of finite dimensional pointed Hopf algebras of rank one initiated in \cite{WLZ}, but focus on those Hopf algebras of non-nilpotent type. Let $H$ be a finite dimensional pointed rank one Hopf algebra of non-nilpotent type. We first determine all non-isomorphic indecomposable $H$-modules and describe the Clebsch-Gordan formulas for them. We then study the structures of both the Green ring $r(H)$ and the Grothendieck ring $G_0(H)$ of $H$ and establish the precise relation between the two rings. We use the Cartan map of $H$ to study the Jacobson radical and the idempotents of $r(H)$. It turns out that the Jacobson radical of $r(H)$ is exactly the kernel of the Cartan map, a principal ideal of $r(H)$, and $r(H)$ has no non-trivial idempotents. Besides, we show that the stable Green ring of $H$ is a transitive fusion ring. This enables us to calculate Frobenius-Perron dimensions of objects of the stable category of $H$. Finally, as an example, we present both the Green ring and the Grothendieck ring of the Radford Hopf algebra.
In this paper, we study the Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between the Poisson homology and the Poisson cohomology, similar to the duality between the Hochschild homology and the Hochschild cohomology of a Frobenius algebra. Using the non-degenerated bilinear form on a Frobenius algebra we construct a Batalin-Vilkovisky structure on the Poisson cohomology ring of a class of Frobenius Poisson algebras.
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 $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some braided Hopf algebra ${}_RH$ in the category $_H\mathscr{M}$. Based on this equivalence, we prove that every braided bi-Galois object $A$ over the braided Hopf algebra ${}_RH$ defines a braided autoequivalence of the category $^H_H\mathscr{YD}$ if and only if $A$ is quantum commutative. In case $H$ is semisimple over an algebraically closed field, i.e. the fusion case, then every braided autoequivalence of $^H_H\mathscr{YD}$ trivializable on $_H\mathscr{M}$ is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in $_H\mathscr{M}$ form a group measuring the Brauer group of $(H,R)$ as studied in [20] in the Hopf algebra case.
Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B^{co(H)}# H which is an H-bicomodule algebra, isomorphic to B.
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-like algebra and hence a bi-Frobenius algebra.
Let $A$ be a Koszul Artin-Schelter regular algebra with Nakayama automorphism $ξ$. We show that the Yoneda Ext-algebra of the skew polynomial algebra $A[z;ξ]$ is a trivial extension of a Frobenius algebra. Then we prove that $A[z;ξ]$ is Calabi-Yau; and hence each Koszul Artin Schelter regular algebra is a subalgebra of a Koszul Calabi-Yau algebra. A superpotential $\hat{w}$ is also constructed so that the Calabi-Yau algebra $A[z;ξ]$ is isomorphic to the derivation quotient of $\hat{w}$. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin-Schelter regular algebra is also discussed.
In this paper, we construct the quasi-Frobenius-Lusztig kernel for any simple Lie algebra $\mathfrak{g}$.
We give a geometric classification of 4-dimensional superalgebras over an algebraic closed field.
We study a class of graded algebras obtained from Ore extensions of graded Calabi-Yau algebras of dimension 2. It is proved that these algebras are graded Calabi-Yau and graded coherent. The superpotentials associated to these graded Calabi-Yau algebras are also constructed.
We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra. The results obtained in this paper will be applied in a subsequent paper.
In this paper, we compute the Clebsch-Gordan formulae and the Green rings of connected pointed tensor categories of finite type.
Let $R$ be a Nichols algebra of Hecke type. In this paper, we show that if $R$ is Noetherian and of finite global dimension, then $R$ has a rigid dualizing complex. We then give a necessary and sufficient condition for $R$ to be a Calabi-Yau algebra.