
He, Jiwei; Van Oystaeyen, Fred; Zhang, Yinhuo
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 SB equals two. Using this we prove a Poincaré–Birkhoff–Witt (PBW) type theorem for a deformation
of a generalized Koszul algebra. When the projective dimension of SB is greater than two, 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...

Huang, Hualin; Van Oystaeyen, Fred; ZHANG, Yinhuo
We study division algebras in an arbitrary linear Grcategory, i.e., a category of finitedimensional vector spaces graded by a groupwith associativity constraint given by a 3cocycle. When the 3cocycle is noncoboundary, this provides some interesting classes of nonassociative division algebras. In particular,
when we work on Grcategories over the field of real numbers, some quasiassociative version of the quaternions and octonions appear.

Presotto, Dennis; Van den Bergh, Michel
In this paper we generalize some classical birational transformations to the noncommutative case. In particular we show that 3dimensional quadratic Sklyanin algebras (noncommutative projective planes) and 3dimensional cubic Sklyanin algebras (noncommutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for noncommutative projective planes.

WANG, Zhihua; Li, Libin; ZHANG, Yinhuo
Let H be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable Hmodules up to isomorphism, and then establish the ClebschGordan formulas for the decompositions of the tensor products of indecomposable Hmodules 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 G0(H) of H modulo one relation. Moreover, r(H) is Frobenius and symmetric with dual bases associated to almost...

Lowen, Wendy; Van den Bergh, Michel
We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's result on compact generation of quasicompact separated schemes. We prove an application of our theorem to noncommutative deformations of such schemes, based upon a change from Koszul complexes to ChevalleyEilenberg complexes. (C) 2013 Elsevier Inc. All rights reserved.

Armour, Aaron; Zhang, Yinhuo
In this paper, we give a geometric classification of 4dimensional superalgebras over an algebraic closed field. It turns out that the number of irreducible components of the variety of 4dimensional superalgebras Salg$_4$ under the Zariski topology is between 20 and 22. One of the significant differences between the variety Alg$_n$ and the variety Salg$_n$ is that Salg$_n$ is disconnected while Alg$_n$ is connected. Under certain conditions on $n$, one can show that the variety Salg$_n$ is the disjoint union of $n$ connected subvrieties. We shall present the degeneration diagrams of the 4 disjoint connected subvarieties Salg$^i_4$ of Salg$_4$.

Zhu, Can; Van Oystaeyen, Fred; ZHANG, Yinhuo
In this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the nondegenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a BatalinVilkovisky structure on the Poisson cohomology ring making it into a BatalinVilkovisky algebra.

Chen, Huixiang; Van Oystaeyen, Fred; ZHANG, Yinhuo
We compute the Green ring of the Taft algebra $H_n(q)$,
where $n$ is a positive integer greater than 1, and $q$ is an $n$th root of unity. It turns out that the Green ring $r(H_n(q))$ of the Taft algebra $H_n(q)$ is a commutative ring generated by two elements subject to certain relations defined recursively. Concrete examples for $n=2,3, ... , 8$ are given.

Smith, S. Paul; VAN DEN BERGH, Michel
The 4dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3space. The degreetwo component of the algebra contains a 2dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of noncommutative quadric surfaces. We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3space. There are exactly four singular quadrics in the pencil. The singular and nonsingular quadrics are characterized by whether they have one or two rulings by noncommutative lines. The Picard groups of the...

YU, Xiaolan; ZHANG, Yinhuo
We study the Calabi???Yau property of pointed Hopf algebra U(D,??) of finite Cartan type. It turns out that this class of pointed Hopf algebras constructed by N. Andruskiewitsch and H.J. Schneider contains many Calabi???Yau Hopf algebras. To give concrete examples of new Calab???Yau Hopf algebras, we classify the Calabi???Yau pointed Hopf algebras U(D,??) of dimension less than 5.

He, J. W.; Torrecillas, B.; Van Oystaeyen, F.; ZHANG, Yinhuo
Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C=A degrees be its dual coalgebra. If A is ArtinSchelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule C1(sigma*) with sigma a coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twisted bimodule (sigma*)A(1). If sigma is an inner automorphism, then A is CalabiYau. An appendix is included to prove a duality theorem of the bounded derived category of quasifinite comodules over an artinian coalgebra.

de Volcsey, Louis de Thanhoffer; van den Bergh, Michel
We prove a technical result which allows us to establish the nondegeneracy of potentials on quivers in some previously unknown or nonobvious cases. Our result applies to certain McKay quivers and also to potentials derived from geometric helices on Del Pezzo surfaces.

Huang, Hualin; Van Oystaeyen, Fred; Yang, Yuping; ZHANG, Yinhuo
In this paper, we compute the Clebsch???Gordan formulae and the Green rings of connected pointed tensor categories of finite type.

He, JiWei; Van Oystaeyen, Fred; ZHANG, Yinhuo
This is a survey of our joint works on graded CalabiYau algebras, CalabiYau Hopf algebras and their PBWdeformations.

DELLO, Jeroen; ZHANG, Yinhuo
We investigate the Brauer group BRM k H of a cocommutative Hopf algebra H (H can be nonfinitely generated). BRM k H is the bigger Brauer group of Hmodule Taylor???Azumaya algebras (which do not necessarily contain a unit). An exact sequence
1>Br(k)>BM(k,H)> Gal(k,H)
is obtained. Furthermore, we prove the surjectivity of ?? under certain circumstances. This generalizes Beattie???s exact sequence to the ???infinite??? case.

VAN DEN BERGH, Michel
Mathematics

He, Jiwei; Van Oystaeyen, Fred; Zhang, Yinhuo
Let A be a Koszul Artin–Schelter regular algebra with Nakayama automorphism ξ . We show that the Yoneda Extalgebra 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 ˆw is also constructed so that the Calabi–Yau algebra A[z; ξ ] is isomorphic to the derivation quotient of ˆw . The Calabi–Yau property of a skew polynomial algebra with coefficients in a PBWdeformation of a Koszul Artin–Schelter regular...

He, Jiwei; Van Oystaeyen, Fred; ZHANG, Yinhuo
We compute the Nakayama automorphism of a PoincareBirkhoffWitt (PBW)deformation of a Koszul ArtinSchelter (AS) Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBWdeformation of a Koszul CalabiYau algebra to be CalabiYau. The relations between the CalabiYau property of augmented PBWdeformations and that of nonaugmented cases are discussed. The Nakayama automorphisms of PBWdeformations of Koszul ASGorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBWdeformation of a graded CalabiYau algebra is still CalabiYau, then it is defined by a potential under some mild conditions. Some classical results are also...

Li, Libin; Zhang, Yinhuo
In this paper, we investigate the Green ring r(Hn,d) of the generalized
Taft algebra Hn,d, extending the results of Chen, Van Oystaeyen and
Zhang (to appear in Proc. of AMS). We shall determine all nilpotent elements
of the Green ring r(Hn,d). It turns out that each nilpotent element in r(Hn,d)
can be written as a sum of indecomposable projective representations. The
Jacobson radical J(r(Hn,d)) of r(Hn,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(Hn,d) of Hn,d.
Our analysis is based on the decomposition of the tensor product of indecomposable
representations...

Yu, Xiaolan; Zhang, Yinhuo
We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2
by using the Hochschild???Serre spectral sequence. As an application, we show that the
pointed Hopf algebras u(D, ??, ??) with Dynkin diagrams of type A, D, or E, except for
A1 and A1
?? A1
with the order NJ
> 2 for at least one component J, are wild.