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Fred Van Oystaeyen

Publications and source records attributed to Fred Van Oystaeyen.

At least 19 recordsLinked to original sources
arXiv2019

Glider representations of group algebra filtrations of nilpotent groups

We continue the study of glider representations of finite groups $G$ with given structure chain of subgroups $e \subset G_1 \subset \ldots \subset G_d = G$. We give a characterization of irreducible gliders of essential length $e \leq d$ which in the case of $p$-groups allows to prove some results about classical representation theory. The paper also contains an introduction to generalized character theory for glider representations and an extension of the decomposition groups in the Clifford theory. Furthermore, we study irreducible glider representations for finite nilpotent groups.

math.RT↗
arXiv2018

Generalized characters for glider representations of groups

Glider representations can be defined for a finite algebra filtration FKG determined by a chain of subgroups 1 < G_1 < ... < G_d = G. In this paper we develop the generalized character theory for such glider representations. We give the generalization of Artin's theorem and define a generalized inproduct. For finite abelian groups G with chain 1 < G, we explicitly calculate the generalized character ring and compute its semisimple quotient. The papers ends with a discussion of the quaternion group as a first non-abelian example.

math.GR↗
arXiv2016

Clifford theory for glider representations

Classical Clifford theory studies the decomposition of simple $G$-modules into simple $H$-modules for some normal subgroup $H \triangleleft G$. In this paper we deal with chains of normal subgroups $1 \triangleleft G_1 \triangleleft \cdots \triangleleft G_d =G$, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field $K$ and relate different representations of the groups appearing in the chain. Picking some normal subgroup $H \triangleleft G$ one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.

math.RT↗
arXiv2016

Localization and sheaves of glider representations

The notion of a glider representation of a chain of normal subgroups of a group is defined by a new structure, i.e. a fragment for a suitable filtration on the group ring. This is a special case of general glider representations defined for a positively filtered ring $R$ with filtration $FR$ and subring $S = F_0R$. Nice examples appear for chains of groups, chains of Lie algebras, rings of differential operators on some variety or $V$-gliders for $W$ for algebraic varieties $V$ and $W$. This paper aims to develop a scheme theory for glider representations via the localizations of filtered modules. With an eye to noncommutative geometry we allow schemes over noncommutative rings with particular attention to so-called almost commutative rings. We consider particular cases of $\mathrm{Proj}~ R$ (e.g. for some P.I. ring $R$) in terms of prime ideals, $R$-tors in terms of torsion theories and $\underline{\mathcal{W}}(R)$ in terms of a noncommutative Grothendieck topology based on words of Ore set localizations.

math.RA↗
arXiv2016

Nakayama automorphisms of double Ore extensions of Koszul regular algebras

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.

math.RA↗
arXiv2014

Cleft extensions of Koszul twisted Calabi-Yau algebras

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.

math.RA↗
arXiv2014

PBW deformations of Koszul algebras over a nonsemisimple ring

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.

math.RA↗
arXiv2014

On the (co)homology of Frobenius Poisson algebras

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.

math.RA↗
arXiv2014

Representations of the small quasi-quantum group ${Q}\mathbf{u}_{q}(\mathfrak{sl}_{2})$

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.

math.QA↗
arXiv2013

Skew polynomial algebras with coefficients in Koszul Artin-Schelter regular algebras

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.

math.RA↗
arXiv2012

The Green Rings of Taft algebras

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.

math.RT↗
arXiv2010

Derived $H$-module endomorphism rings

Let $H$ be a Hopf algebra, $A/B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^\cdot\in D(A)$ may be regarded as an object in $D(B)$ via the restriction functor. We discuss the relations of the derived endomorphism rings $E_A(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(A)}(M^\cdot,M^\cdot[i])$ and $E_B(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(B)}(M^\cdot,M^\cdot[i])$. If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^\cdot)$ is a graded subalgebra of $E_B(M^\cdot)$. In particular, if $M$ is a usual $A$-module, a necessary and sufficient condition for $E_B(M)$ to be an $H^*$-Galois graded extension of $E_A(M)$ is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.

math.RA↗
arXiv2010

Hopf algebra actions on differential graded algebras and applications

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

math.RA↗
arXiv2009

New Families of Finite Coherent Orthoalgebras without Bivaluations

In the present paper we study the following problem: how to construct a coherent orthoalgebra which has only a finite number of elements, but at the same time does not admit a bivaluation (i.e. a morphism with a codomain being an orthoalgebra with just two elements). This problem is important in the perspective of Bell-Kochen-Specker theory, since one can associate such an orthoalgebra to every saturated non-colorable finite configuration of projective lines. The first result obtained in this paper provides a general method for constructing finite orthoalgebras. This method is then applied to obtain a new infinite family of finite coherent orthoalgebras that do not admit bivaluations. The corresponding proof is combinatorial and yields a description of the groups of symmetries for these orthoalgebras.

quant-ph↗
arXiv2009

The Center of Crystalline Graded Rings

In the first section of the paper, we will give some basic definitions and properties about Crystalline Graded Rings. In the following section we will provide a general description of the center. Afterwards, the case where the grading group is Abelian finite will be handled. The center will have some properties of a crystalline graded ring, but not all. We will call this Arithmetically Crystalline Graded. The center is crystalline graded if the part of degree zero is a principal ideal domain. The last section deals with the case where the grading group is non-Abelian finite. Since this situation is much more complicated than the Abelian case, we primarily focus on the conditions to have a trivial center. The fact that the center is Arithmetically Crystalline Graded also holds in this case.

math.RA↗