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Skew group categories, algebras associated to Cartan matrices and folding of root lattices
Abstract
For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic p and the acting group is a cyclic p -group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations , Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.
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Let C be a symmetrizable generalized Cartan matrix with symmetrizer D and orientation . In Geiß et al. (Invent Math 209(1):61–158, 2017) we constructed for any field an -algebra , defined in terms of a quiver with relations, such that the locally free H-modules behave in many aspects like representations of a hereditary algebra of the corresponding type. We define a Noetherian algebra over a power series ring, which provides a direct link between the representation theory of H and of . We define and study a reduction and a localization functor relating the module categories of , and H. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over the three algebras , and H. We show that the indecomposable rigid locally free H-modules are parametrized, via their rank vectors, by the real Schur roots associated to . Moreover, the left finite bricks of H, in the sense of Asai, are parametrized, via their dimension vectors, by the real Schur roots associated to .
We give conditions on when a triangular matrix ring is Gorenstein of a given
selfinjective dimension. We apply the result to the category algebra of a
finite EI category. In particular, we prove that for a finite EI category, its
category algebra is 1-Gorenstein if and only if the given category is free and
projective.
For we consider the K-algebra associated
to a symmetrizable Cartan matrix C, a symmetrizer D, and an orientation
of C, which was defined in Part 1. We construct and analyse a
reduction functor from rep(H(k)) to rep. As a consequence we show
that the canonical decomposition of rank vectors for H(k) does not depend on
k, and that the rigid locally free H(k)-modules are up to isomorphism in
bijection with the rigid locally free -modules. Finally, we show that
for a rigid locally free H(k)-module of a given rank vector the Euler
characteristic of the variety of flags of locally free submodules with fixed
ranks of the subfactors does not depend on the choice of k.
For an abelian category with a Serre duality and a finite group action, we
compute explicitly the Serre duality on the category of equivariant objects.
Special cases and examples are discussed. In particular, an abelian category
with a periodic Serre duality and its equivariantization are studied.
A tree action (G, X), consisting of a group G acting on a tree X, is encoded by a ‘quotient graph of groups’ A=G⧹⧹X. We introduce here the appropriate notion of morphism A→A′= G′⧹⧹X′, that encodes a morphism (G, X)→(G′, X′) of tree actions. In particular, we characterize ‘coverings’ G⧹⧹X→G′⧹⧹X corresponding to inclusions of subgroups G⩽G′. This is a useful tool for producing subgroups of G with prescribed properties. It also yields a strong Conjugacy Theorem for groups acting freely on X.We also prove the following mild generalizations of theorems of Howie and Greenberg. Suppose that G acts discretely on X, i.e. that each vertex stabilizer is finite. Let H and K be finitely generated subgroups of G. Then H∩K is finitely generated. If H and K are commensurable, then H (and K) have finite index in 〈H, K〉.
In this paper we give an explicit algorithm to construct the ordinary quiver
of a finite EI category for which the endomorphism groups of all objects have
orders invertible in the field k. We classify all finite EI categories with
hereditary category algebras, characterizing them as free EI categories (in a
sense which we define) for which all endomorphism groups of objects have
invertible orders. Some applications on the representation types of finite EI
categories are derived.
Let a finite group G act on a differential graded algebra A. This article presents necessary conditions and sufficient conditions for the skew group algebra A*G to be Calabi-Yau. In particular, when A is the Ginzburg dg algebra of a quiver with an invariant potential, then A*G is Calabi-Yau and Morita equivalent to a Ginzburg dg algebra. Some applications of these results are derived to compare the generalised cluster categories of A and A*G when they are defined and to compare the higher Auslander-Reiten theories of A and A*G when A is a finite dimensional algebra.
In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units in \cite{be}. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery in \cite{cm}, and we provide a unification with the results on coverings of quivers and relations by E. Green in \cite{g}. We obtain a confirmation in a quiver and relations free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.
By introducing Frobenius morphisms F F on algebras A A and their modules over the algebraic closure F ¯ q {\overline {\mathbb {F}}}_q of the finite field F q {\mathbb {F}}_q of q q elements, we establish a relation between the representation theory of A A over F ¯ q \overline {\mathbb {F}}_q and that of the F F -fixed point algebra A F A^F over F q {\mathbb {F}}_q . More precisely, we prove that the category \textbf {mod}- A F \text {\textbf {mod}-}A^F of finite-dimensional A F A^F -modules is equivalent to the subcategory of finite-dimensional F F -stable A A -modules, and, when A A is finite dimensional, we establish a bijection between the isoclasses of indecomposable A F A^F -modules and the F F -orbits of the isoclasses of indecomposable A A -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over F q {\mathbb {F}}_q can be interpreted as F F -stable representations of the corresponding quiver over F ¯ q \overline {\mathbb {F}}_q . We further prove that every finite-dimensional hereditary algebra over F q {\mathbb {F}}_q is Morita equivalent to some A F A^F , where A A is the path algebra of a quiver Q Q over F ¯ q \overline {\mathbb {F}}_q and F F is induced from a certain automorphism of Q Q . A close relation between the Auslander-Reiten theories for A A and A F A^F is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of A F A^F is obtained by “folding" the Auslander-Reiten quiver of A A . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over F q {\mathbb {F}}_q with a given dimension vector and to generalize Kac’s theorem for all modulated quivers and their associated Kac–Moody algebras defined by symmetrizable generalized Cartan matrices.
This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems.
In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie’s construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.
To each symmetrizable Cartan matrix, we associate a finite free EI category. We prove that the corresponding category algebra is isomorphic to the algebra defined in Geiss et al. (2017) [4], which is associated to another symmetrizable Cartan matrix. In certain cases, the algebra isomorphism provides an algebraic enrichment of the well-known correspondence between symmetrizable Cartan matrices and graphs with automorphisms.
For a quiver with potential (Q,W) with an action of a finite cyclic group G, we study the skew group algebra ΛG of the Jacobian algebra Λ=P(Q,W). By a result of Reiten and Riedtmann, the quiver Q G of a basic algebra η(ΛG)η Morita equivalent to ΛG is known. Under some assumptions on the action of G, we explicitly construct a potential W G on Q G such that η(ΛG)η≅P(Q G ,W G ). The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of G. If Λ is self-injective, then ΛG is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on (Q,W) behave with respect to our construction.
For k ≥1, we consider the K-algebra H(k):= H(C,kD,Ω) associated to a symmetrizable Cartan matrix C, a symmetrizer D and an orientation Ω of C, which was defined in Geiss et al. [8]. We construct and analyze a reduction functor from rep(H(k)) to rep(H(k-1)). As a consequence, we show that the canonical decomposition of rank vectors for H(k) does not depend on k, and that the rigid locally free H(k)-modules are up to isomorphism in bijection with the rigid locally free H(k-1)-modules. Finally, we show that for a rigid locally free H(k)-module of a given rank vector the Euler characteristic of the variety of flags of locally free submodules with fixed ranks of the subfactors does not depend on the choice of k. © The Author(s) 2017. Published by Oxford University Press. All rights reserved.
We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices.
Gabriel's theorem [5] (cf. below for precise satements) was generalized by Dlab-Ringel [3], [4] where Dynkin graphs of type Bn, Cn, F4, G2 also enter in the classification together with the graphs of type ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=450522&tocid=100040719&page=89-97
Let p be a prime and k an algebraically closed field of characteristic p. We construct a functor on the category of finite categories with the property that if is a finite group, then is the orbit category of p-subgroups of G. This leads to an extension of Alperinʼs weight conjecture to any finite category , stating that the number of isomorphism classes of simple -modules should be equal to that of the weight algebra of . We show that the versions of Alperinʼs weight conjecture for finite groups and for finite categories are in fact equivalent.
LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism : ae ofK a linear vector space map V(): V(a)V(e). We classify thoseK having only finitely many isomorphy classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3].
This book is an introduction to the contemporary representation theory of Artin algebras, by three very distinguished practitioners in the field. Beyond assuming some first-year graduate algebra and basic homological algebra, the presentation is entirely self-contained, so the book is a suitable introduction for any mathematician (especially graduate students) to this field. The main aim of the book is to illustrate how the theory of almost split sequences is used in the representation theory of Artin algebras. However, other foundational aspects of the subject are developed. These results give concrete illustrations of some of the more abstract concepts and theorems. The book includes complete proofs of all theorems, and numerous exercises.
Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. Transporter categories are general-izations of subgroups of G, and we shall demonstrate the finite generation of their cohomology. We record a generalized Frobenius reciprocity and use it to examine some important quotient categories of transporter categories, customarily called local categories.
By using the Ringel-Hall algebra approach we first present a new proof of Kac's theorem for species over a finite field. The method used here is quite different from earlier techniques. In [SV] Sevenhant and Van den Bergh have discovered an interesting relation between a conjecture of Kac on representations of quivers and the structure of the Ringel-Hall algebras in some special cases. We second show that this relation still holds for species.
We characterize the finite EI categories whose representations are standardly stratified with respect to the natural preorder on the simple representations. The orbit category of a finite group with respect to any set of subgroups is always such a category. Taking the subgroups to be the p-subgroups of the group, we reformulate Alperin's weight conjecture in terms of the standard and proper costandard representations of the orbit category. We do this using the properties of the Ringel dual construction and a theorem of Dlab, which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules. We indicate that these results hold in the generality of an algebra whose simple modules are preordered, rather than partially ordered.
Let G be a finite group. Over any finite G-poset P we may define a
transporter category as the corresponding Grothendieck construction. The
classifying space of the transporter category is the Borel construction on the
G-space BP, while the k-category algebra of the transporter category is the
(Gorenstein) skew group algebra on the G-incidence algebra kP.
We introduce a support variety theory for the category algebras of
transporter categories. It extends Carlson's support variety theory on group
cohomology rings to equivariant cohomology rings. In the mean time it provides
a class of (usually non selfinjective) algebras to which Snashall-Solberg's
(Hochschild) support variety theory applies. Various properties will be
developed. Particularly we establish a Quillen stratification for modules.
By introducing Frobenius morphisms F on algebras A and their modules over the algebraic closure {{\bar \BF}}_q of the finite field \BF_q of q elements, we establish a relation between the representation theory of A over {{\bar \BF}}_q and that of the F-fixed point algebra over \BF_q. More precisely, we prove that the category \modh A^F of finite dimensional -modules is equivalent to the subcategory of finite dimensional F-stable A-modules, and, when A is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the F-orbits of the isoclasses of indecomposable A-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over \BF_q can be interpreted as F-stable representations of a corresponding quiver over {{\bar \BF}}_q. We further prove that every finite dimensional hereditary algebra over \BF_q is Morita equivalent to some , where A is the path algebra of a quiver Q over {{\bar \BF}}_q and F is induced from a certain automorphism of Q. A close relation between the Auslander-Reiten theories for A and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by "folding" the Auslander-Reiten quiver of A. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver with a given dimension vector and to establish part of Kac's theorem for all finite dimensional hereditary algebras over a finite field.
A theorem of Kac on quiver representations states that the dimension vectors of indecomposable representations are precisely
the positive roots of the associated symmetric Kac–Moody Lie algebra. Here this result is generalised to representations respecting
an admissible quiver automorphism, and the positive roots of an associated symmetrisable Kac–Moody Lie algebra are obtained.
Also the relationship with species of valued quivers over finite fields is discussed. It is known that the number of isomorphism
classes of indecomposable representations of a given dimension vector for a species is a polynomial in the size of the base
field. It is shown that these polynomials are non-zero if and only if the dimension vector is a positive root of the corresponding
symmetrisable Kac–Moody Lie algebra.
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