08 Fakultät Mathematik und Physik

Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9

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Open Access
Algebraic analogues of resolution of singularities, quasi-hereditary covers and Schur algebras
(2021) Santos Cruz, Tiago Miguel dos; Koenig, Steffen (Prof. Dr.)
In algebraic geometry, a resolution of singularities is, roughly speaking, a replacement of a local commutative Noetherian ring of infinite global dimension by a local commutative Noetherian ring of finite global dimension. In representation theory, an analogous problem is asking to resolve algebras of infinite global dimension by algebras of finite global dimension. In addition, such resolutions should have nicer properties to help us study the representation theory of algebras of infinite global dimension. This motivates us to take split quasi-hereditary covers as these algebraic analogues of resolutions of singularities and measure their quality using generalisations of dominant dimension and deformation results based on change of rings techniques. The Schur algebra, SR(n,d) with n ≥ d, together with the Schur functor is a classical example of a split quasi-hereditary cover of the group algebra of the symmetric group, RSd , for every commutative Noetherian ring R. The block algebras of the classical category O, together with their projective-injective module, are split quasi-hereditary covers of subalgebras of coinvariant algebras. In this thesis, we study split quasi-hereditary covers, and their quality, of some cellular algebras over commutative Noetherian rings. The quality of a split quasi-hereditary cover can be measured by the fully faithfulness of the Schur functor on standard modules and on m-fold extensions of standard modules. Over fields, the dominant dimension controls the quality of the split quasi-hereditary cover of KSd formed by the Schur algebra SK(n,d) and the Schur functor. In particular, this quality improves by increasing the characteristic of the ground field. To understand the integral cases, the classical concept of dominant dimension is not useful since in most cases there are no projective-injective modules. Using relative homological algebra, we develop and study a new concept of dominant dimension, which we call relative dominant dimension, for Noetherian algebras which are projective over the ground ring making this concept suitable for the integral setup. For simplicity, we call Noetherian algebras which are projective over the ground ring just projective Noetherian algebras. While developing the theory of relative dominant dimension, we generalize the Morita-Tachikawa correspondence for projective Noetherian algebras and we prove that computations of relative dominant dimension over projective Noetherian algebras can be reduced to computations of dominant dimension over finite-dimensional algebras over algebraically closed fields. Using relative dominant dimension, concepts like Morita algebras and gendo-symmetric algebras can be defined for Noetherian algebras. We compute the relative dominant dimension of Schur algebras SR(n,d) for every commutative Noetherian ring R. Using such computations together with deformation results that involve the spectrum of the ground ring R we determine the quality of the split quasi-hereditary covers of RSd , (SR(n,d), V⊗d) formed by the Schur algebra SR(n,d) and the Schur functor HomSR(n,d)(V⊗d,−): SR(n,d)-mod → RSd-mod for all regular Noetherian rings. Over local commutative regular rings R, the quality of (SR(n,d), V⊗d) depends only on the relative dominant dimension and on R containing a field or not. For this cover, the quality improves compared with the finite-dimensional case whenever the local commutative Noetherian ring does not contain a field. This theory is also applied to q-Schur algebras and Iwahori-Hecke algebras of the symmetric group. In full generality, we prove that the quality of a split quasi-hereditary cover of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple B-modules. Other split quasi-hereditary algebras that we study in this thesis are deformations of block algebras of the Bernstein-Gelfand-Gelfand category O of a semi-simple Lie algebra. These deformations provide split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras. We compute the relative dominant dimensions of these block algebras and we determine the quality of these covers. In these deformations, the quality dramatically improves compared with the finite-dimensional case. Using approximation theory to generalize once more the concept of dominant dimension to relative dominant dimension with respect to direct summands of the characteristic tilting module, we find new split quasi-hereditary covers. In particular, the relative dominant dimension of a characteristic tilting module of SR(n,d) with respect to V⊗d is a lower bound of the quality of a split quasi-hereditary cover of the cellular algebra EndSR(n,d)(V⊗d)op, independent of the natural numbers n and d. This split quasi-hereditary cover involves the Ringel dual of the Schur algebra SR(n,d). Using this technology for deformations of block algebras of the classical BGG category O of a semi-simple Lie algebra, we obtain a new proof for Ringel self-duality of the blocks of the classical BGG category O of a complex semi-simple Lie algebra. Here, the uniqueness of split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras with higher quality is the crucial factor to deduce Ringel self-duality.
Open Access
Auslander generators for piecewise hereditary algebras
(2012) Müller-Platz, Severin; Koenig, Steffen (Prof. Dr.)
The notion of representation dimension of a finite dimensional algebra has been introduced by Auslander. He has shown that an algebra is of finite representation type if and only if its representation dimension is at most two. This could lead to the expectation that the representation dimension of a representation infinite algebra should measure how far this algebra is from being representation finite. A more realistic expectation is that the representation dimension measures the homological complexity of a representation infinite algebra. The main goal of this theses is the explicit construction of an Auslander-generator for piecewise hereditary algebras. From this we can follow that the representation dimension of a piecewise hereditary algebra (a homological seen "simple" Algebra) is at most three.
Open Access
Beschränkte Lücken zwischen Werten von Normformen
(2008) Hablizel, Markus; Brüdern, Jörg (Prof. Dr.)
In dieser Arbeit wird die Wertefolge der Normform eines beliebigen algebraischen Zahlkörpers untersucht. Wir weisen nach, dass in dieser stets unendlich oft beschränkte Lücken vorhanden sind. Dieses Resultat beweisen wir unter Zuhilfenahme einer sehr flexiblen Methode von Goldston, Pintz und Yildirim, die im Grunde auf Selbergs Sieb basiert und von Thorne weiter ausgebaut wurde.
Open Access
Brauer graph algebras are closed under derived equivalence
(2022) Antipov, Mikhail; Zvonareva, Alexandra
In this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use the rank of the maximal torus of the identity component Out0(A)of the group of outer automorphisms of a symmetric stably biserial algebra A .
Open Access
Centers of Hecke algebras of complex reflection groups
(2023) Chavli, Eirini; Pfeiffer, Götz
We provide a dual version of the Geck-Rouquier Theorem (Geck and Rouquier in Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser Boston, Boston, pp. 251–272, 1997) on the center of an Iwahori-Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank 2, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra H . These models enable concrete calculations inside H . For each of the eight groups, we compute an explicit integral basis of the center of H .
Open Access
A characterisation of Morita algebras in terms of covers
(2021) Cruz, Tiago
A pair (A, P) is called a cover of EndA(P)op if the Schur functor HomA(P,-) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P. By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P)op is self-injective whenever (A, P) is a cover of EndA(P)op for a faithful projective-injective module P.
Open Access
Contributions to the integral representation theory of Iwahori-Hecke algebras
(2002) Soriano Sola, Marcos; Roggenkamp, Klaus W. (Prof. Dr. Dr. h. c.)
Die vorliegende Dissertation beschäftigt sich mit der ganzzahligen Darstellungstheorie der zu endlichen Coxeter Gruppen assoziierten Iwahori-Hecke Algebren. Der einführende erste Teil fasst einige bekannte Ergebnisse und allgemeine Methoden zusammen, die später benötigt werden. Es wird näher auf Ordnungen über kommutativen Noetherschen regulären Ringen der Krull Dimension zwei eingegangen. Der zweite Teil behandelt die Berechnung von Grundringannihilatoren von Erweiterungsgruppen zwischen Gittern für Iwahori-Hecke Algebren. Es wird gezeigt, wie man das Higman Ideal einer Iwahori-Hecke Algebra anhand der Charaktertafel bestimmen kann. Ein wichtiges Ergebnis gibt eine kohomologische Interpretation der bekannten Schur Elemente. Als Anwendung werden Sätze über die Torsionsfreiheit und endliche Erzeugbarkeit über den ganzen Zahlen von gewissen Erweiterungsgruppen von Gittern für Iwahori-Hecke Algebren hergeleitet. Es wird anhand eines Beispiels demonstriert, wie man die Ringstruktur einer Iwahori-Hecke Algebra aus der Kenntnis einer Filtrierung der regulären Darstellung mit einfachen Gittern und gewissen zur Filtrierung assoziierten Annihilatoren herleiten kann. Der dritte und letzte Teil der Arbeit ist kombinatorischer Natur. Es wird gezeigt, daß nach einer geeigneten Lokalisation des Grundrings die Projektion der zu einer endlichen symmetrischen Gruppe assoziierten Iwahori-Hecke Algebra auf die Wedderburn Komponenten, die zu Hakenpartitionen korrespondieren, die Struktur einer Brauerbaum Ordnung hat. Dieses Ergebnis benutzt die explizite Berechnung von gewissen Erweiterungsgruppen in Form von sogenannten modularen Morphismen.
Open Access
Decomposing André-Neto supercharacters of Sylow p-subgroups of Lie type D
(2013) Jedlitschky, Markus; Dipper, Richard (Prof. Dr.)
It is well known that the classification of irreducible characters for the group of upper unitriangular n x n-matrices over a finite field with q elements, denoted U_n(q), is a "wild" problem. N. Yan gave an approximation to a solution of this problem in terms of so-called supercharacters. These supercharacters are not necessarily irreducible but they satisfy many strong properties. For example they are pairwise orthogonal and contain every irreducible character as constituent. They can be classified in a pleasant combinatorial way. Methodically N. Yan's results are essentially obtained by investigating biorbits of a group operation, which is a coarser version of Kirillov's orbit method. More precisely, Yan's biorbits form disjoint unions of Kirillov orbits. Hence N. Yan developed an easily accessible and elementary, but strong, theory for the investigation of U_n(q). This thesis provides a generalization to the Sylow p-subgroups D_n(q) of the orthogonal groups of Lie type D, defined over finite fields of characteristic p. The main result is the construction of a class of combinatorially described modules, the so-called hook-separated staircase modules. These modules are either orthogonal or isomorphic and contain all irreducible modules as constituents. This thesis provides also a generalization in the sense, that many of N. Yan's original results can be obtained as special cases. The most important step is to generalize N. Yan's construction to abstract groups admitting a 1-cocycle. This generalization does not allow to consider biorbits, but only right orbits (or, if one prefers, left orbits). It is an extension of the original method to a more general class than algebra groups, for which P. Diaconis and I.M. Isaacs established a generalization carrying the full strength of biorbits. This is important, since Sylow p-subgroups of finite orthogonal groups of type D are not algebra groups. The 1-cocycle approach is used to construct the mentioned hook-separated staircase modules. As a by-product the results provide a new and elementary proof of C.A.M. André's and A.M. Neto's supercharacter theory for D_n(q) and a purely combinatorial and strong decomposition of their supercharacters into characters of hook-separated staircase modules.
Open Access
Desarguesian and geometric right ℓ-groups
(2021) Dietzel, Carsten; König, Steffen (Prof. Dr.)
Open Access
Diagonale Formen mit polynomialen Koeffizienten und Summen binärer Formen
(2010) Madlener, Alexander; Brüdern, Jörg (Prof. Dr.)
Das Hasseprinzip verbindet die lokale und die globale Lösbarkeit einer Diophantischen Gleichung. Die Lösbarkeit einer Diophantischen Gleichung in ganzen p-adischen Zahlen ist eine notwendige Bedingung für die Lösbarkeit in ganzen Zahlen, und wenn die p-adische Lösbarkeit zusammen mit der reellen Lösbarkeit eine hinreichende Bedingung wird, dann spricht man von der Gültigkeit des Hasseprinzips. In der vorliegenden Arbeit zeigen wir die Gültigkeit des Hasseprinzips für diagonale Formen mit polynomialen Koeffizienten und Summen binärer Formen für eine nur linear vom Grad der Form abhängende Variablenzahl im Mittel.
Open Access
Dominant and global dimension of blocks of quantised Schur algebras
(2021) Fang, Ming; Hu, Wei; Koenig, Steffen
Group algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and Sq(n,r) with n⩾r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59-85, 2021).
Open Access
Dominant dimensions of algebras
(2017) Marczinzik, René; König, Steffen (Prof. Dr.)
Open Access
Dominant dimensions of finite dimensional algebras
(2012) Abrar, Muhammad; König, Steffen (Prof. Dr. rer. nat.)
We study the dominant dimensions of three classes of finite dimensional algebras, namely hereditary algebras, quotient algebras of trees and serial algebras. We see that a branching vertex plays a key role to establish that the dominant dimension (dom.dim) of hereditary algebras (quivers) is at most one. We define arms of a tree and split trees into two classes: trees without arms and trees with arms. Like hereditary algebras, it turns out that the dominant dimension of the quotient algebras of trees can not exceed one. For serial algebras A associated to linearly oriented quiver with n vertices, we give lower and upper bounds of dom.dimA, and show that the bounds are optimal. It is also shown that some of the algebras A satisfy the conditions in the higher dimensional version of the Auslander's correspondence. Further we consider serial algebras corresponding to one-oriented-cycle quiver Q with n vertices, and give optimal bounds for a special subclass of these algebras. We conjecture that for any non self-injective quotient algebra A of Q dom.dimA is at most 2n-3, where the number of vertices n is bigger than 2.. Finally, we construct few examples of algebras having large (finite) dominant dimensions.
Open Access
From simplicial groups to crossed squares
(2022) Asiki, Natalia-Maria
Simplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.
Open Access
Gendo-Frobenius algebras and comultiplication
(2022) Yırtıcı, Çiğdem
Gendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.
Open Access
Gendo-Frobenius algebras and comultiplication
(2020) Yirtici, Cigdem; Koenig, Steffen (Prof. Dr.)
Two large classes of algebras, Frobenius algebras and gendo-symmetric algebras, are characterised by thenexistence of a comultiplication with some special properties. Symmetric algebras are both Frobenius and gendo-symmetric. Kerner and Yamagata investigated two variations of gendo-symmetric algebras and in fact these two variations contain gendo-symmetric and Frobenius algebras. We call one of these variations gendo-Frobenius algebras. In this thesis, we construct a comultiplication for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius algebras and on gendo-symmetric algebras. Moreover, we show that Frobenius algebras are precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication.
Open Access
Gluing silting objects along recollements of well generated triangulated categories
(2020) Bonometti, Fabiano; König, Steffen (Prof. Dr.)
We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a ‘nice’ set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a weighted noncommutative regular projective curve of genus 0.
Open Access
Group rings and twisted group rings for a series of p-groups
(2003) Weber, Harald; Roggenkamp, Klaus W. (Prof. Dr. Dr. hc)
In this work we regard integral group rings ZG of non abelian p-groups with maximal cyclic subgroup. For this we consider the integral group ring as pullback describing the more complicate structure of ZG by two simpler stuctures: an integral group ring of an abelian group and a twisted group ring connected by some congruences. The Pascal triangle provides a matrix, which enables us to describe the twisted group ring p-adically. With the aid of this we can solve some classical problems: We construct over ordes, compute the cohomology ring and determine the representation type.
Open Access
Higher Morita-Tachikawa correspondence
(2024) Cruz, Tiago
Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo‐symmetric algebras. Explicitly, the Morita-Tachikawa correspondence describes that endomorphism algebras of generators-cogenerators over finite‐dimensional algebras are exactly the finite‐dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi‐generators and quasi‐cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita-Tachikawa correspondence that take into account relative dominant dimension with respect to a self‐orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring.
Open Access
Interactions between universal localisations, ring epimorphisms and tilting modules
(2015) Marks, Frederik; König, Steffen (Prof. Dr.)
The aim of this thesis is to study the interaction between universal localisations, ring epimorphisms and (generalised) tilting modules. We show that these concepts are intrinsically connected and that they provide various new applications to representation theory. Universal localisations, as defined by Cohn and Schofield, have recently proven to be useful in tilting theory, a fundamental branch of representation theory. In fact, universal localisations were used to classify tilting modules over some rings and they provide interesting decompositions of the derived module category. However, both the structural properties of universal localisations as well as the nature of the various connections to tilting theory are far from being understood. Satisfying answers are only known in special cases. One way to approach localisations is via ring epimorphisms. These are epimorphisms in the category of all rings that are relevant to study certain abelian subcategories of a given module category. Even though it is well-known that universal localisations yield ring epimorphisms, the question of which epimorphisms arise from universal localisations is still widely open. Chapters 2-4 of this thesis provide some answers to the latter question, on the one hand, by looking at finite localisations over any ring and, on the other hand, by focusing on finite dimensional algebras. In particular, over self-injective algebras a classification of certain ring epimorphisms is accessible. We further focus on correspondences between universal localisations and tilting objects. Explicit bijections are established for certain classes of finite dimensional algebras. Chapters 5-7 of this thesis are dedicated to the new concept of silting modules and its relation to localisations. We begin by developing a general theory of silting modules over any ring. These modules generalise tilting modules as well as support τ-tilting modules over a finite dimensional algebra and they turn out to parametrise diverse structures in the derived module category. Subsequently, we show that minimal silting modules classify all universal localisations over a hereditary ring. Also, in the general setup, we can associate a silting object to every localisation. Thus, silting theory provides an adequate setup to study universal localisations.