Leavitt Path Algebras

296 Pages·2017·5.838 MB·English

by  Gene Abrams,Pere Ara,Mercedes Siles Molina (auth.)

Lecture Notes in Mathematics 2191
Gene Abrams
Pere Ara
Mercedes Siles Molina
Leavitt Path
Algebras
Lecture Notes in Mathematics 2191
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Zurich
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Gene Abrams (cid:129) Pere Ara (cid:129) Mercedes Siles Molina
Leavitt Path Algebras
123
GeneAbrams PereAra
DepartmentofMathematics DepartamentdeMatemàtiques
UniversityofColorado UniversitatAutònomadeBarcelona
ColoradoSprings,USA Barcelona,Spain
MercedesSilesMolina
DepartamentodeÁlgebra,Geometríay
Topología
UniversidaddeMálaga
Málaga,Spain
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-1-4471-7343-4 ISBN978-1-4471-7344-1 (eBook)
DOI10.1007/978-1-4471-7344-1
LibraryofCongressControlNumber:2017946554
Mathematics Subject Classification (2010): 16Sxx, 16D25, 16D40, 16D50, 16E20, 16E40, 16E50,
16N60,16P20,16P40,46L05
©Springer-VerlagLondonLtd.2017
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Preface
The great challenge in writing a book about a topic of ongoing mathematical
researchinterestliesin determiningwhoandwhat.Whoarethereadersforwhom
thebookisintended?Whatpiecesoftheresearchshouldbeincluded?
The topic of Leavitt path algebras presents both of these challenges, in the
extreme.Indeed,muchofthebeautyinherentinthistopicstemsfromthefactthat
itmaybeapproachedfrommanydifferentdirections,andonmanydifferentlevels.
The topic encompasses classical ring theory at its finest. While at first glance
theseLeavittpathalgebrasmayseemsomewhatexotic,infactmanystandard,well-
understoodalgebrasariseinthiscontext:matrixringsandLaurentpolynomialrings,
to namejusttwo. Manyofthe fundamental,classical ring-theoreticconceptshave
beenandcontinuetobeexploredhere,includingtheidealstructure,Z-grading,and
structureoffinitelygeneratedprojectivemodules,tonamejustafew.
Thetopiccontinuesalongtraditionofassociatinganalgebrawithanappropriate
combinatorial structure (here, a directed graph), the subsequent goal being to
establish relationships between the algebra and the associated structures. In this
particular setting, the topic allows for (and is enhanced by) visual, pictorial
representation via directed graphs. Many readers are no doubt familiar with the
by-nowclassical way of associating an algebra overa field with a directed graph,
the standard path algebra. The construction of the Leavitt path algebra provides
another such connection.The path algebra and Leavitt path algebra constructions
areindeedrelated,viaalgebrasofquotients.However,onemayunderstandLeavitt
pathalgebraswithoutanypriorknowledgeofthepathalgebraconstruction.
Thetopichassignificant,deepconnectionswithotherbranchesofmathematics.
For instance, many of the initial results in Leavitt path algebras were guided and
motivatedbyresultspreviouslyknownabouttheiranalyticcousins,thegraphC(cid:2)-
algebras.ThestudyofLeavittpathalgebrasquicklymaturedtoadolescence(when
it became clear that the algebraic results are not implied by the C(cid:2) results), and
almostimmediatelythereaftertoadulthood(wheninfactsomeC(cid:2)results,including
somenewC(cid:2) results,wereshowntofollowfromthealgebraicresults).A number
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viii Preface
oflong-standingquestionsinalgebrahaverecentlybeenresolvedusingLeavittpath
algebrasasatool,thusfurtherestablishingthematurityofthesubject.
The topic continues a deep tradition evident in many branchesof mathematics
in which K-theory plays an important role. Indeed, in retrospect, one can view
Leavitt path algebras as precisely those algebras constructed to produce specified
K-theoretic data in a universal way, data arising naturally from directed graphs.
Much of the currentwork in the field is focusedon better understandingjust how
largearoletheK-theoreticdataplaysindeterminingthestructureofthesealgebras.
Ourgoalinwritingthisbook,theWhy?ofthisbook,simultaneouslyaddresses
boththeWho?andWhat?questions.Weprovidehereaself-containedpresentation
ofthetopicofLeavittpathalgebras,apresentationwhichwillallowreadershaving
different backgroundsand different topical interests to understand and appreciate
these structures. In particular, graduate students having only a first year course in
ringtheoryshouldfindmostofthematerialinthisbookquiteaccessible.Similarly,
researchers who don’t self-identify as algebraists (e.g., people working in C(cid:2)-
algebrasorsymbolicdynamics)willbe able to understandhow these Leavittpath
algebras stem from, or apply to, their own research interests. While most of the
resultscontainedherehaveappearedelsewhereintheliterature,afewofthecentral
results appear here for the first time. The style will be relatively informal. We
willoftenprovidehistoricalmotivationandoverview,bothtoincreasethereader’s
understanding of the subject and to play up the connections with other areas of
mathematics. Although space considerations clearly require us to exclude some
otherwise interesting and important topics, we provide an extensive bibliography
forthosereaderswhoseekadditionalinformationaboutvarioustopicswhicharise
herein.
More candidly, our real Why? for writing this book is to share what we know
aboutLeavittpathalgebrasin sucha waythatothersmightbecomeprepared,and
subsequentlyinspired,tojoininthegame.
ColoradoSprings,USA GeneAbrams
Barcelona,Spain PereAra
Málaga,Spain MercedesSilesMolina
Acknowledgments
Throughout the years during which this book was being written, dozens of
individualsofferedtopicsuggestions,pointedouttypos,andhelpedtoimprovethe
finalversion.Wearedeeplyappreciativeoftheircontributions.
While much of the time and energy which went into writing this book was
expendedwhileeachoftheauthorswasinresidenceather/hisownuniversity,the
threeauthorswereableonanumberofoccasionstospendtimevisitingeachother’s
home institutions to help more effectively move the project toward completion.
We collectivelythankthe UniversitatAutònomade Barcelona,the Universidadde
Málaga,andtheUniversityofColoradoColoradoSpringsfortheirwarmhospitality
duringthesevisits.
The first authorwas partiallysupportedby a SimonsFoundationCollaboration
Grant,#20894.
The second author was partially supported by the Spanish MEC and Fondos
FEDER through projects MTM2008-06201-C02-01, MTM2011-28992-C02-01,
and MTM2014-53644-P and by the Generalitat de Catalunya through project
2009SGR1389.
The third author was partially supported by the Spanish MEC and Fondos
FEDER throughprojects MTM2010-15223,MTM2013-41208-P,and MTM2016-
76327-C3-1-PandbytheJuntadeAndalucíaandFondosFEDER,jointly,through
projectsFQM-336andFQM-02467.
ix
Contents
1 TheBasicsofLeavittPathAlgebras:Motivations,Definitions
andExamples................................................................. 1
1.1 AMotivatingConstruction:TheLeavittAlgebras .................... 2
1.2 LeavittPathAlgebras................................................... 5
1.3 TheThreeFundamentalExamplesofLeavittPathAlgebras......... 10
1.4 ConnectionsandMotivations:TheAlgebrasofBergman,and
GraphC(cid:2)-Algebras ..................................................... 12
1.5 TheCohnPathAlgebrasandConnectionstoLeavittPath
Algebras................................................................. 14
1.6 DirectLimitsintheContextofLeavittPathAlgebras................ 23
1.7 ABriefRetrospectiveontheHistoryofLeavittPathAlgebras....... 30
2 Two-SidedIdeals............................................................. 33
2.1 TheZ-Grading .......................................................... 36
2.2 TheReductionTheoremandtheUniquenessTheorems.............. 46
2.3 AdditionalConsequencesoftheReductionTheorem................. 54
2.4 GradedIdeals:BasicPropertiesandQuotientGraphs ................ 56
2.5 TheStructureTheoremforGradedIdeals,andtheInternal
StructureofGradedIdeals.............................................. 65
2.6 TheSocle ............................................................... 76
2.7 TheIdealGeneratedbytheVerticesinCyclesWithoutExits ........ 83
2.8 TheStructureTheoremforIdeals,andtheInternalStructure
ofIdeals.................................................................. 89
2.9 AdditionalConsequencesoftheStructureTheoremforIdeals.
TheSimplicityTheorem................................................ 97
3 Idempotents,andFinitelyGeneratedProjectiveModules .............. 103
3.1 PurelyInfiniteSimplicity,andtheDichotomyPrinciple.............. 104
3.2 FinitelyGeneratedProjectiveModules:TheV-Monoid.............. 108
3.3 TheExchangeProperty................................................. 116
3.4 VonNeumannRegularity............................................... 121
3.5 PrimitiveNon-MinimalIdempotents................................... 125
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