
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.

DE MAESSCHALCK, Peter; RebolloPerdomo, S.; Torregrosa, J.
This paper concerns the study of smallamplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as an impassable grain. The canonical form of the unperturbed vector field is like a degenerate flow box. Near the singularity, the phase portrait consists of parallel fibers, all but one of which have no singular points, and at the singular fiber, there is one node. We demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic. (C) 2014 Elsevier...

Huzak, Renato; De Maesschalck, Peter
Using techniques from singular perturbations we show that for any n???6 and m???2 there are Li??nard equations {x??=y???F(x), y??=G(x)}, with F a polynomial of degree n and G a polynomial of degree m, having at least 2[n???2/2]+[m/2] hyperbolic limit cycles, where [???] denotes "the greatest integer equal or below".

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.

HUZAK, Renato; DE MAESSCHALCK, Peter; DUMORTIER, Freddy
In this paper we continue the study of ``large" smallamplitude limit cycles in slowfast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blowup and good normal forms.

DE MAESSCHALCK, Peter; HUZAK, Renato
We significantly improve lower bounds for the number of limit cycles for polynomial classical Li??nard equations, aiming at stating lower bounds that are reasonable enough to be optimal. The techniques used are the notion of slow divergence integral from the geometric theory of planar slowfast systems.

DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Roussarie, Robert
We introduce transitory canard cycles for slow???fast vector fields in the plane. Such cycles separate ???canards without head??? and ???canardswithhead???,like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle underwhatever condition on the related slow divergence integralI,including the challenging caseI=0.

Bonckaert, Patrick
Given a 1:1 resonant saddle singularity of a planar analytic vector field, we provide a linearization procedure using a series expansion in compensators of Mourtadatype, and show that this series has Gevrey1 asymptotics. In case of an analytic Poincar\'eDulac normal form we show that this transformation is analytic as a function of the compensators

DE MAESSCHALCK, Peter; Popovic, Nikola; KUTAFINA, Ekaterina
We consider an extended threedimensional Bonhoeffervan der Pol oscillator which generalises the planar FitzHughNagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. and by Freire and Gallas. Focussing on a parameter regime
which has hitherto been neglected, and in which the governing equations evolve on three distinct timescales, we propose a reduction to a model problem that was formulated by Krupa et al. as a canonical form for such systems. Based on previously obtained results in, we characterise completely the mixedmode dynamics of the resulting three timescale extended Bonhoeffervan der Pol oscillator from the point of...

KUTAFINA, Ekaterina
Following the paper of Shimizu et al. (Phys Lett A 375:1566, 2011), we consider the Bonhoeffervan der Pol oscillator with nonautonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometric singular perturbation theory. The considered model can be rewritten as a fourdimensional (locally threedimensional) autonomous system, which under certain conditions has a folded saddlenode singularity and additionally can be treated as a three time scale one.

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.