Frobenius Algebras I: Basic Representation Theory

662 Pages·2011·4.18 MB·English

by  Andrzej Skowronski

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Andrzej Skowron´ ski
Kunio Yamagata
Frobenius Algebras I
Basic Representation Theory
Authors:
Andrzej Skowron´ski Kunio Yamagata
Faculty of Mathematics Department of Mathematics
and Computer Science Tokyo University of Agriculture
Nicolaus Copernicus University and Technology
Chopina 12/18 Nakacho 2-24-16, Koganei
87-100 Torun´ Tokyo 184-8588
Poland Japan
E-mail: [email protected] E-mail: [email protected]
2010 Mathematical Subject Classification (primary; secondary): 16-01; 13E10, 15A63, 15A69, 16Dxx,
16E30, 16G10, 16G20, 16G70, 16K20, 16W30, 51F15
Key words: Algebra, module, representation, quiver, ideal, radical, simple module, semisimple module,
uniserial module, projective module, injective module, simple algebra, semisimple algebra, separable
algebra, hereditary algebra, Nakayama algebra, Frobenius algebra, symmetric algebra, selfinjective
algebra, Brauer tree algebra, enveloping algebra, Coxeter group, Coxeter graph, Hecke algebra,
coalgebra, comodule, Hopf algebra, Hopf module, syzygy module, periodic module, periodic algebra,
irreducible homomorphism, almost split sequence, Auslander–Reiten translation, Auslander–Reiten
quiver, extension spaces, projective dimension, injective dimension, category, functor, Nakayama
functor, Nakayama automorphism, Morita equivalence, Morita–Azumaya duality
ISBN 978-3-03719-102-6
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
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Contents
Introduction ix
I Algebrasandmodules 1
1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Representationsofalgebrasandmodules . . . . . . . . . . . . . . 12
3 TheJacobsonradical . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 TheKrull–Schmidttheorem . . . . . . . . . . . . . . . . . . . . . 41
5 Semisimplemodules . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Semisimplealgebras . . . . . . . . . . . . . . . . . . . . . . . . . 56
7 TheJordan–Höldertheorem . . . . . . . . . . . . . . . . . . . . . 67
8 Projectiveandinjectivemodules . . . . . . . . . . . . . . . . . . . 73
9 Hereditaryalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10 Nakayamaalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11 TheGrothendieckgroupandtheCartanmatrix . . . . . . . . . . . 105
12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
II Moritatheory 123
1 Categoriesandfunctors . . . . . . . . . . . . . . . . . . . . . . . . 123
2 Bimodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3 Tensorproductsofmodules . . . . . . . . . . . . . . . . . . . . . 133
4 Adjunctionsandnaturalisomorphisms . . . . . . . . . . . . . . . . 140
5 Progenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6 Moritaequivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7 Morita–Azumayaduality . . . . . . . . . . . . . . . . . . . . . . . 175
8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
III Auslander–Reitentheory 203
1 Theradicalofamodulecategory . . . . . . . . . . . . . . . . . . . 203
2 TheHarada–Sailemma . . . . . . . . . . . . . . . . . . . . . . . . 207
3 Thespaceofextensions . . . . . . . . . . . . . . . . . . . . . . . . 209
4 TheAuslander–Reitentranslations . . . . . . . . . . . . . . . . . . 232
5 TheNakayamafunctors. . . . . . . . . . . . . . . . . . . . . . . . 247
6 TheAuslander–Reitenformulas . . . . . . . . . . . . . . . . . . . 252
7 Irreducibleandalmostsplithomomorphisms . . . . . . . . . . . . 257
8 Almostsplitsequences . . . . . . . . . . . . . . . . . . . . . . . . 269
9 TheAuslander–Reitenquiver . . . . . . . . . . . . . . . . . . . . . 282
10 TheAuslandertheorem . . . . . . . . . . . . . . . . . . . . . . . . 301
viii Contents
11 TheBautista–Smaløtheorem . . . . . . . . . . . . . . . . . . . . . 312
12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
IV Selfinjectivealgebras 332
1 TheFrobeniustheorem . . . . . . . . . . . . . . . . . . . . . . . . 332
2 TheBrauer–Nesbitt–Nakayamatheorems . . . . . . . . . . . . . . 336
3 Frobeniusalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 345
4 Symmetricalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 355
5 Simplealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
6 TheNakayamatheorems . . . . . . . . . . . . . . . . . . . . . . . 375
7 Non-Frobeniusselfinjectivealgebras . . . . . . . . . . . . . . . . . 386
8 Thesyzygyfunctors . . . . . . . . . . . . . . . . . . . . . . . . . 392
9 Thehigherextensionspaces . . . . . . . . . . . . . . . . . . . . . 402
10 Periodicmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
11 Periodicalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
12 TheGreen–Snashall–Solbergtheorems . . . . . . . . . . . . . . . 442
13 DynkinandEuclideangraphs . . . . . . . . . . . . . . . . . . . . 447
14 CanonicalmeshalgebrasofDynkintype . . . . . . . . . . . . . . . 452
15 TheRiedtmann–Todorovtheorem . . . . . . . . . . . . . . . . . . 455
16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
V Heckealgebras 489
1 Finitereflectiongroups . . . . . . . . . . . . . . . . . . . . . . . . 489
2 Coxetergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
3 TheCoxetertheorems . . . . . . . . . . . . . . . . . . . . . . . . 507
4 TheIwahoritheorem . . . . . . . . . . . . . . . . . . . . . . . . . 515
5 Heckealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
VI Hopfalgebras 539
1 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
2 Hopfalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
3 TheLarson–Sweedlertheorems . . . . . . . . . . . . . . . . . . . 584
4 TheRadfordtheorem . . . . . . . . . . . . . . . . . . . . . . . . . 595
5 TheFischman–Montgomery–Schneiderformula. . . . . . . . . . . 609
6 Themodulecategory . . . . . . . . . . . . . . . . . . . . . . . . . 616
7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
Bibliography 637
Index 645
Introduction
The major concern of this book is the representation theory of finite dimensional
associativealgebraswithanidentityoverafield. Insimplestterms,itisanapproach
totheproblemofdescribinghowafinitenumberoflineartransformationscanact
simultaneously on a finite dimensional vector space over a field. The representa-
tiontheoryoffinitedimensionalalgebrastracesitsorigintothemiddlepartofthe
nineteenthcenturywithHamilton’sdiscoveryofthequaternions,thefirstnoncom-
mutativefield,andinvestigationsoffinitegroupsviatheirrepresentationsinmatrix
algebrasoverthefieldofcomplexnumbers. Themainachievementsoftherepre-
sentation theory of algebras of the latter part of the nineteenth and the beginning
ofthetwentiethcenturyconcernedthestructureofsemisimplefinitedimensional
algebrasoverfieldsandtheirrepresentations. Anewandfundamentalviewonthe
representationtheoryoffinitedimensionalalgebrasoverfieldscameinthe1930s
from the papers by Noether who gave the theory its modern setting by interpret-
ing representations as modules. The module theoretical approach allowed one to
applyintherepresentationtheoryoffinitedimensionalalgebrasthelanguageand
techniquesofcategorytheoryandhomologicalalgebra. Themodernrepresentation
theoryoffinitedimensionalalgebrasoverfieldscanberegardedasthestudyofthe
categoriesoftheirfinitedimensionalmodulesandtheassociatedcombinatorialand
homologicalinvariants.
Aprominentroleintherepresentationtheoryoffinitedimensionalalgebrasover
fieldsisplayedbytheFrobeniusalgebras. Thisisawideclassofalgebrascontain-
ingthesemisimplealgebras, blocksofgroupalgebrasoffinitegroups, theHecke
algebrasoffiniteCoxetergroups,thefinitedimensionalHopfalgebras,andtheorbit
algebrasoftherepetitivecategoriesofalgebras. TheFrobeniusalgebrashavetheir
origininthe1903papersbyFrobeniuswhodiscoveredthattheleftandrightregular
representationsofafinitedimensionalalgebraoverafield,definedintermsofstruc-
tureconstantswithrespecttoafixedlinearbasis,areequivalentifandonlyifthere
isaveryspecialinvertiblematrixintertwiningbothrepresentations. LaterBrauer,
NesbittandNakayamarealizedthatthestudyoffinitedimensionalalgebraswiththe
propertythattheleftandrightregularrepresentationsareequivalentiscrucialfora
betterunderstandingofthestructureofnonsemisimplealgebrasandtheirmodules,
andcalledthemFrobeniusalgebras. Inaseriesofpapersfrom1937–1941,Brauer,
Nesbitt and Nakayama established characterizations of Frobenius algebras which
were independent of the choice of a linear basis of the algebra. In particular, we
maysaythatafinitedimensionalalgebraAoverafieldK isaFrobeniusalgebraif
thereexistsanondegenerateK-bilinearform.(cid:2);(cid:2)/W A(cid:3)A ! K whichisasso-
ciative,inthesensethat.ab;c/D.a;bc/forallelementsa,b,c ofA. Moreover,
if such a nondegenerate, associative, K-bilinear form is symmetric, A is called a
symmetric algebra. We also mention that the Frobenius algebras are selfinjective
x Introduction
algebras (projective and injective modules coincide), and the module category of
everyfinitedimensionalselfinjectivealgebraoverafieldisequivalenttothemodule
categoryofaFrobeniusalgebra.
Themainaimofthebookistoprovideacomprehensiveintroductiontotherep-
resentationtheoryoffinitedimensionalalgebrasoverfields,viatherepresentation
theoryofFrobeniusalgebras. Thebookisprimarilyaddressedtograduatestudents
startingresearchintherepresentationtheoryofalgebrasaswellasmathematicians
working in other fields. It is hoped that the book will provide a friendly access
totherepresentationtheoryoffinitedimensionalalgebrasastheonlyprerequisite
is a basic knowledge of linear algebra. We present complete proofs of all results
exhibitedinthebook. Moreover,arichsupplyofexamplesandexerciseswillhelp
thereadertounderstandthetheorypresentedinthebook.
Wedividethebookintotwovolumes.
Theaimofthefirstvolumeofthebookistwofold. Firstly,itservesasageneral
introductiontobasicresultsandtechniquesofthemodernrepresentationtheoryof
finitedimensionalalgebrasoverfields,withspecialattentiontotherepresentation
theory of Frobenius algebras. The second aim is to exhibit prominent classes of
Frobeniusalgebras,ormoregenerallyselfinjectivealgebras.
The first volume of the book is divided into six chapters, each of which is
subdivided into sections. We start with Chapter I presenting background on the
finite dimensional algebras over a field and their finite dimensional modules. In
ChapterIIwepresenttheMoritaequivalencesandtheMorita–Azumayadualities
forthemodulecategoriesoffinitedimensionalalgebrasoverfields. ChapterIIIis
devoted to presenting background on theAuslander–Reiten theory of irreducible
homomorphismsandalmostsplitsequences,andtheassociatedcombinatorialand
homologicalinvariants. ChapterIVformstheheartofthefirstvolumeofthebook
and contains fundamental classical and recent results concerning the selfinjective
algebrasandtheirmodulecategories. InChapterVwepresenttheclassificationof
finitereflectiongroupsofrealEuclideanspacesviatheassociatedCoxetergraphs
and show that they provide a wide class of symmetric algebras over an arbitrary
field, called the Hecke algebras. In the final Chapter VI we describe the basic
theory of finite dimensional Hopf algebras over fields and show that they form a
distinguishedclassofFrobeniusalgebrasforwhichtheNakayamaautomorphisms
areoffiniteorder.
The main aim of the second volume of the book, “FrobeniusAlgebras II. Or-
bitAlgebras”istostudytheFrobeniusalgebrasastheorbitalgebrasofrepetitive
categoriesoffinitedimensionalalgebraswithrespecttoactionsofadmissibleau-
tomorphism groups. In particular, we will introduce covering techniques which
frequentlyallowustoreducetherepresentationtheoryofFrobeniusalgebrastothe
representation theory of algebras of small homological dimension. A prominent
roleintheseinvestigationswillbeplayedbytiltingtheoryandtheauthorstheory
ofselfinjectivealgebraswithdeformingideals.