
VAN DEN BERGH, Michel
In this paper we develop the theory of noncommutative P1bundles over commutative (smooth) schemes. Such noncommutative P1bundles occur in the theory of Dmodules but our definition is more general. We can show that every noncommutative deformation of a Hirzebruch surface is given by a noncommutative P1bundle over P1 in our sense.

Calaque, Damien; Rossi, Carlo A.; Van den Bergh, Michel
In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over polyvector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasiisomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured by Tsygan.) We also rely on the fact  recently proved by the first two authors  that Shoikhet's quasiisomorphism is compatible with...

Ooms, Alfons
Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(g). These facts are then used to finish our earlier work on this subject, i.e. to give an explicit description for the Poisson center of all indecomposable, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide a polynomial, maximal Poisson commutative subalgebra of S(g), enjoying additional properties. As a byproduct we show...

Liu, G. H.; Zhu, Haixing
Let H be a quasitriangular weak Hopf algebra. It is proved that the centralizer subalgebra of its source subalgebra in H is a braided group (or Hopf algebra in the category of left Hmodules), which is cocommutative and also a left braided Lie algebra in the sense of Majid.

Yu, Xiaolan; Zhang, Yinhuo
Let H be a semisimple Hopf algebra and R a braided Hopf algebra in the category of YetterDrinfeld modules over H. When R is a CalabiYau algebra, a necessary and sufficient condition for R # H to be a CalabiYau Hopf algebra is given. Conversely, when H is the group algebra of a finite group and the smash product R # H is a CalabiYau algebra, we give a necessary and sufficient condition for the algebra R to be a CalabiYau algebra. (C) 2012 Elsevier Inc. All rights reserved.

Delvaux, Lydia; Van Daele, A.; Wang, S.
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a coFrobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finitedimensional Hopf algebra, but also that of any Hopf algebra with integrals (coFrobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from wellknown formulas obtained earlier in the theory of regular multiplier Hopf algebras...

Liu, Guohua; Chen, Quanguo; ZHU, Haixing
Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right Hcomodules), which generalizes Majid's transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.

Liu, Guohua; Chen, Quanguo; ZHU, Haixing
Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right Hcomodules), which generalizes Majid's transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.

VAN DEN BERGH, Michel
In this paper, we describe noncommutative versions of P(1) x P(1). These contain the class of noncommutative deformations of P(1) x P(1).

DELVAUX, Lydia; Van Daele, A.; Wang, S. H.
In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A, B) of two regular multiplier Hopf algebras. We assume that B is a right Amodule algebra and that A is a left Bcomodule coalgebra. The right action of A on B gives rise to the smash product A # B. The left coaction of B on A gives a possible coproduct Delta(#) on A # B. We discuss in detail the necessary compatibility conditions between the action and the coaction for Delta(#) to be a proper coproduct on A # B. The result is again a...

DE MAESSCHALCK, Peter; DUMORTIER, Freddy
In this paper, we prove the presence of limit cycles of given multiplicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Lienard equations. The obtained limit cycles are relaxation oscillations. Both classical Lienard equations and generalized Lienard equations are treated.

Keller, Bernhard; VAN DEN BERGH, Michel
We define and investigate deformed nCalabiYau completions of homologically smooth differential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected nonDynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed CalabiYau completions do have the CalabiYau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the CalabiYau property. We show that deformed 3CalabiYau completions of algebras of global dimension at most 2 are quasiisomorphic to Ginzburg...

de la Fe Rodriguez, Pedro Yoelvys; Coddens, Annelies; DEL FAVA, Emanuele; CORTINAS ABRAHANTES, Jose; SHKEDY, Ziv; Maroto Martin, Luis O.; Cruz Munoz, Eduardo; DUCHATEAU, Luc; Cox, Eric; Goddeeris, Bruno Maria
Little information is available on the prevalence of swine enteropathogens in Cuba where diarrheic diseases are responsible for 31% and 37% of the total mortality during the neonatal and postweaning periods. F4(+) and F18(+) enterotoxigenic Escherichia coli and F18(+) verotoxigenic E. coli induce diarrhea and edematous disease in pigs, but their distribution has never been thoroughly studied in the Cuban swine population. Therefore, the present study estimated the prevalence of F4 and F18specific antibodies in sera of 1,044 6monthold gilts distributed in 34 piggeries spread over the Cuban territory. For the data analysis, which included the optical density of individual...

Keller, Bernhard; Murfet, Daniel; VAN DEN BERGH, Michel
In a recent paper, Iyama and Yoshino considered two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper, we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category.

He, JiWei; Van Oystaeyen, Fred; ZHANG, Yinhuo
Let H be a finite dimensional semisimple Hopf algebra, A a differential graded (dg for short) Hmodule algebra. Then the smash product algebra A#H is a dg algebra. For any dg A#Hmodule M, there is a quasiisomorphism of dg algebras: RHom(A) (M, M)#H > RHom(A#H)(M circle times H, M circle times H). This result is applied to dKoszul algebras, CalabiYau algebras and ASGorenstein dg algebras.

DELVAUX, Lydia; Van Daele, A.
Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct Delta on A making the pair (A, Delta) a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group as introduced and studied in [A. Van Daele, Adv. Math. 140 (1998) 323]. Now let H be a finite subgroup of G and consider the subalgebra A(1) of functions in A that are constant on double cosets of H. The coproduct in general will not leave this algebra...

DELVAUX, Lydia; Van Daele, A.
An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum...

VAN ROOSMALEN, AdamChristiaan
In this paper, we show all klinear abelian 1CalabiYau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all klinear abelian 1CalabiYau categories up to equivalence.

He, JiWei; Van Oystaeyen, Fred; ZHANG, Yinhuo
The CalabiYau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional ASGorenstein Hopf algebras. It is shown that the skewgroup algebra of a universal enveloping algebra of a finite dimensional Lie algebra g with a finite subgroup G of automorphisms of g is CalabiYau if and only if the universal enveloping algebra itself is CalabiYau and G is a subgroup of the special linear group SL(g). The Noetherian cocommutative CalabiYau Hopi algebras of dimension not larger than 3 are described. The CalabiYau property of Sridharan enveloping algebras of finite...

He, JiWei; Van Oystaeyen, Fred; ZHANG, Yinhuo
Let H be a Hopf algebra, A/B be an HGalois extension. Let D(A) and D(B) be the derived categories of right Amodules and of right Bmodules, respectively. An object M. is an element of D(A) may be regarded as an object in D(B) via the restriction functor. We discuss the relations of the derived endomorphism rings EA(M.) =. circle plus i is an element of zHom(D(A))(M. , M. [i]) and EB(M.) = circle plus i is an element of z Hom(D(B))(M., M. [i]). If H is a finitedimensional semisimple Hopf algebra, then EA(M.) is a graded subalgebra of EB(M.). In...