Table Of Content

Algebras, Rings and Modules Mathematics and Its Applications ManagingEditor: M.HAZEWINKEL CentreforMathematicsandComputerScience,Amsterdam,TheNetherlands Volume586 Algebras, Rings and Modules Volume 2 by MichielHazewinkel CWI, Amsterdam,TheNetherlands NadiyaGubareni TechnicalUniversityofCzestochowa, Poland and V.V. Kirichenko KievTarasShevchenkoUniversity, Kiev,Ukraine AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress. ISBN978-1-4020-5141-8(HB) ISBN978-1-4020-5140-1(e-book) PublishedbySpringer, P.O.Box17,3300AADordrecht,TheNetherlands. www.springer.com Printedonacid-freepaper AllRightsReserved (cid:2)c 2007Springer Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmitted inanyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recording orotherwise,withoutwrittenpermissionfromthePublisher,withtheexception ofanymaterialsuppliedspecificallyforthepurposeofbeingentered andexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Table of Contents Preface.....................................................................ix Chapter 1. Groups and group representations..........................1 1.1 Groups and subgroups. Definitions and examples.......................2 1.2 Symmetry. Symmetry groups...........................................7 1.3 Quotient groups, homomorphisms and normal subgroups..............10 1.4 Sylow theorems.......................................................14 1.5 Solvable and nilpotent groups.........................................21 1.6 Group rings and group representations. Maschke theorem.............26 1.7 Properties of irreducible representations...............................35 1.8 Characters of groups. Orthogonality relations and their applications...38 1.9 Modular group representations........................................47 1.10 Notes and references..................................................49 Chapter 2. Quivers and their representations.........................53 2.1 Certain important algebras............................................53 2.2 Tensor algebra of a bimodule..........................................60 2.3 Quivers and path algebras.............................................67 2.4 Representations of quivers.............................................74 2.5 Dynkin and Euclidean diagrams. Quadratic forms and roots...........79 2.6 Gabriel theorem.......................................................93 2.7 K-species..............................................................99 2.8 Notes and references.................................................100 Appendix to section 2.5. More about Dynkin and extended Dynkin (= Eyclidean) diagrams .......................................105 Chapter 3. Representations of posets and of finite dimensional algebras ..................................................................113 3.1 Representations of posets............................................114 3.2 Differentiation algorithms for posets..................................130 3.3 Representations and modules. The regular representations............135 3.4 Algebras of finite representation type.................................140 v vi TABLE OF CONTENTS 3.5 Roiter theorem.......................................................147 3.6 Notes and references.................................................153 Chapter 4. Frobenius algebras and quasi-Frobenius rings...........161 4.1 Duality properties....................................................161 4.2 Frobenius and symmetric algebras....................................164 4.3 Monomial ideals and Nakayama permutations of semiperfect rings....166 4.4 Quasi-Frobenius algebras.............................................169 4.5 Quasi-Frobenius rings................................................174 4.6 The socle of a module and a ring.....................................177 4.7 Osofsky theorem for perfect rings....................................181 4.8 Socles of perfect rings ...............................................183 4.9 Semiperfect piecewise domains .......................................184 4.10 Duality in Noetherian rings..........................................187 4.11 Semiperfect rings with duality for simple modules ...................190 4.12 Self-injective rings ...................................................193 4.13 Quivers of quasi-Frobenius rings .....................................204 4.14 Symmetric algebras with given quivers ...............................205 4.15 Rejection lemma ....................................................208 4.16 Notes and references.................................................212 Chapter 5. Right serial rings...........................................219 5.1 Homological dimensions of right Noetherian rings....................219 5.2 Structure of right Artinian right serial rings..........................224 5.3 Quasi-Frobenius right serial rings.....................................230 5.4 Right hereditary right serial rings....................................231 5.5 Semiprime right serial rings..........................................233 5.6 Right serial quivers and trees.........................................236 5.7 Cartan matrix for a right Artinian right serial ring...................244 5.8 Notes and references.................................................252 Chapter 6. Tiled orders over discrete valuation rings ...............255 6.1 Tiled orders over discrete valuation rings and exponent matrices .....255 6.2 Duality in tiled orders ...............................................270 6.3 Tiled orders and Frobenius rings.....................................276 6.4 Q-equivalent partially ordered sets...................................279 6.5 Indices of tiled orders................................................287 6.6 Finite Markov chains and reduced exponent matrices ................292 6.7 Finite partially ordered sets, (0,1)-orders and finite Markov chains ...296 6.8 Adjacency matrices of admissible quivers without loops...............301 6.9 Tiled orders and weakly prime rings .................................305 6.10 Global dimension of tiled orders......................................313 6.11 Notes and references.................................................323 TABLE OF CONTENTS vii Chapter 7. Gorenstein matrices........................................327 7.1 Gorenstein tiled orders. Examples....................................327 7.2 Cyclic Gorenstein matrices...........................................338 7.3 Gorenstein (0,1)-matrices ............................................346 7.4 Indices of Gorenstein matrices........................................356 7.5 d-matrices ...........................................................364 7.6 Cayley tables of elementary Abelian 2-groups........................369 7.7 Quasi-Frobenius rings and Gorenstein tiled orders ...................379 7.8 Notes and references.................................................381 Suggestions for further reading ........................................385 Subject index......................................................................389 Name index..............................................................397 Preface This book is the natural continuation of “Algebras, rings and modules. vol.I”. The main part of it consists of the study of special classes of algebras and rings. Topics covered include groups, algebras, quivers, partially ordered sets and their representations, as well as such special rings as quasi-Frobenius and right serial rings, tiled orders and Gorenstein matrices. Representation theory is a fundamental tool for studying groups, algebras and rings by means of linear algebra. Its origins are mostly in the work of F.G.Frobenius, H.Weil, I.Schur, A.Young, T.Molien about century ago. The results of the representation theory of finite groups and finite dimensional algebras playafundamentalroleinmanyrecentdevelopmentsofmathematicsandtheoret- ical physics. The physicalaspects of this theory concernaccounting for and using the concepts of symmetry which appear in various physical processes. We start this book with the main results of the theory of groups. For the convenience of a reader in the beginning of this chapter we recall some basic conceptsandresultsofgrouptheorywhichwillbe necessaryfor the nextchapters of the book. Groups are a central object of algebra. The concept of a group is histori- cally one of the first examples of an abstract algebraic system. Finite groups, in particular permutation groups, are an increasingly important tool in many areas of applied mathematics. Examples include coding theory, cryptography, design theory, combinatorial optimization, quantum computing, and statistics. InchapterIwegiveashortintroductiontothetheoryofgroupsandtheirrep- resentations. We consider the representation theory of groups from the module- theoretical point of view using the main results about rings and modules as recorded in volume I of this book. This theoretical approach was first used by E.Noether who established a close connection between the theory of algebras and the theory of representations. From that point of view the study of the representation theory of groups becomes a special case of the study of modules over rings. In the theory of representations of group a special role is played by the famous Maschke theorem. Taking into accountits greatimportance we give three different proofs of this theorem following J.-P.Serre, I.N.Herstein and M.Hall. As a consequence of the Maschke theorem, the representation theory of groups splits into two different cases depending on the characteristic of a field k: classical and modular(followingL.E.Dickson). In“classical”representationtheoryoneassumes that the characteristic of k does not divide the group order |G| (e.g. k can be the field of complex numbers). In “modular” representation theory one assumes that the characteristic of k is a prime, dividing |G|. In this case the theory is almost completely different from the classical case. ix x PREFACE In this book we consider the results belonging to the classical representation theory of finite groups, such as the characters of groups. We give the basic properties of irreducible charactersand their connection with the ring structure of the corresponding group algebras. A central role in the theory of representations of finite dimensional algebras and rings is played by quivers, which were introduced by P.Gabriel in connection with problemsof representationsoffinite dimensionalalgebrasin1972. The main notions and result concerning the theory of quivers and their representations are given in chapter 2. A most remarkable result in the theory of representations of quivers is the theoremclassifyingthequiversoffiniterepresentationtype,whichwasobtainedby P.Gabrielin 1972. This theorem saysthat a quiver is of finite representationtype over an algebraically closed field if and only if the underlying diagram obtained from the quiver by forgetting the orientations of all arrows is a disjoint union of simple Dynkin diagrams. P.Gabriel also proved that there is a bijection between the isomorphism classes of indecomposable representations of a quiver Q and the set of positive roots of the Tits form correspondingto this quiver. A proof of this theorem is given in section 2.6. Another proof of this theorem in the general case, for an arbitrary field, using reflection functors and Coxeter functors has been obtained by I.N.Berstein, I.M.Gel’fand, and V.A.Ponomarev in 1973. In their work the connection between indecomposablerepresentationsofaquiveroffinite type andpropertiesofits Tits quadratic form is elucidated. Representations of finite partially ordered sets (posets, in short) play an im- portantroleinrepresentationtheory. TheywerefirstintroducedbyL.A.Nazarova andA.V.Roiter. Thefirsttwosectionsofchapter3aredevotedtopartiallyordered sets and their representations. Here are giventhe main results of M.M.Kleiner on representationsofposets offinite type and results of L.A.Nazarovaon representations ofposets ofinfinite type. The mostimportantresultin this theorywasbeen obtained by Yu.A.Drozd who showed that there is a trichotomy between finite, tame and wild representation types for finite posets over an algebraically closed field. One of the main problems of representation theory is to obtain information about the possible structure of indecomposable modules and to describe the isomorphism classes of all indecomposable modules. By the famous theorem on tri- chotomyforfinitedimensionalalgebrasoveranalgebraicallyclosedfield,obtained by Yu.A.Drozd, all such algebras divide into three disjoint classes. The main results on representations of finitely dimensional algebras are given in section 3.4. Here we give structure theorems for some special classes of finite dimensional algebras of finite type, such as hereditary algebras and algebras with zero square radical, obtained by P.Gabriel in terms of Dynkin diagrams. Section 3.5 is devoted to the first Brauer-Thrall conjecture, of which a proof has