Algebras, Quivers and Representations: The Abel Symposium 2011

311 Pages·2013·2.89 MB·English

by  Claire Amiot (auth.)

Abel Symposia 8
Aslak Bakke Buan
Idun Reiten
Editors
Øyvind Solberg
Algebras, Quivers
and Representations
The Abel Symposium 2011
ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
Forfurthervolumes:
www.springer.com/series/7462
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Aslak Bakke Buan (cid:2) Idun Reiten (cid:2) Øyvind Solberg
Editors
Algebras, Quivers
and Representations
The Abel Symposium 2011
Editors
AslakBakkeBuan ØyvindSolberg
Dept.ofMathematicalSciences Dept.ofMathematicalSciences
NTNU NTNU
Trondheim,Norway Trondheim,Norway
IdunReiten
Dept.ofMathematicalSciences
NTNU
Trondheim,Norway
ISSN2193-2808 AbelSymposia
ISBN978-3-642-39484-3 ISBN978-3-642-39485-0(eBook)
DOI10.1007/978-3-642-39485-0
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013947781
Mathematics Subject Classification (2010): 13F60, 14F05, 14J70, 16E05, 16E10, 16G20, 16G70,
16E40,18E30,37K10
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Wededicatethisvolumetothememoryofour
friendsandcolleaguesDieterHappeland
Andrei Zelevinsky.Happelpassedaway on
August15th,2012—and ZelevinskyonApril
10th,2013. Bothofthemparticipatedand
each gaveatalkattheAbelSymposium.In
addition,Happelhasajointcontributionto
theproceedingswithDanZacharia.
Happelwasahighlyrespectedexpertin
representation theoryoffinitedimensional
algebras; hemademanyimportant
contributionstothefield,especiallyhis
ground-breaking resultsonthelinkbetween
tiltingtheoryandequivalencesofderived
categoriesandonhereditary categorieswith
a tiltingobject.Hewasalsoa memberofthe
RoyalNorwegian SocietyofScienceand
Letters(DetKongelige Norske Videnskabers
Selskab).
One of Zelevinsky’s major mathematical con-
tributions was the start and development of
the theory of cluster algebras (with Fomin),
which has had an enormous impact in the
fieldofalgebraanditsapplicationsinthelast
decade.
Foreword
The Niels Henrik Abel Memorial Fund was established by the Norwegian gov-
ernment on January 1, 2002. The main objective is to honor the great Norwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstand-
ing scientific work in the field of mathematics. The prize shall contribute towards
raising the status of mathematics in society and stimulate the interest for science
amongschoolchildrenandstudents.InkeepingwiththisobjectivetheBoardofthe
Abel Fund has decided to finance an annual Abel Symposium. The topic may be
selectedbroadlyintheareaofpureandappliedmathematics.TheSymposiashould
beatthehighestinternationallevel,andservetobuildbridgesbetweenthenational
and international research communities. The Norwegian Mathematical Society is
responsiblefortheevents.Ithasalsobeendecidedthatthecontributionsfromthese
Symposia should be presented in a series of proceedings,and Springer Verlag has
enthusiastically agreed to publish the series. The board of the Niels Henrik Abel
Memorial Fund is confident that the series will be a valuable contribution to the
mathematicalliterature.
HelgeHolden
ChairmanoftheboardoftheNielsHenrikAbelMemorialFund
vii
Preface
Thetitleofthe8thAbelsymposiumwas“Algebras,QuiversandRepresentations”.
Representationtheoryofalgebrasisaclassicalandcentraltopicwithinmathemat-
ics with strong links to Lie theory and mathematical physics. Non-commutative,
andespeciallyfinitedimensionalalgebras,canoftenbedescribedintermsofquiv-
ers. Moreover, the representation theory of such algebras can also be understood
in terms of representations of quivers. Mainly due to its concrete and combinato-
rial nature, the theory has recently had an increasing impact in mathematics, with
new connections to many other areas, like commutative algebra, algebraic geom-
etry, algebraic groups, and combinatorics. The symposium focused on several of
these connections, like cluster algebras, homological and representation theoretic
aspectsincommutativealgebraandquivervarieties.
The recent increase in interaction with various other fields of mathematics has
alsohadapositiveimpactforthestudyofinternalrepresentationtheoreticquestions.
Ononehand,ithas givennewtechniques,bothof combinatorial,geometricaland
categoricalnature.Ontheotherhand,ithasmotivatedthestudyofnewclassesof
algebras.Intheconference,severalclassicalrepresentationtheoreticquestionswere
alsoaddressed.
This Abel symposium proceedings contains a selection of papers dealing with
theabovementionedsubjects.ItwasheldinBalestrandattheSognefjordfromthe
20th to 23rd of June 2011 in the Kviknes Hotel. There were 43 participants, and
participation was by invitation only. The scientific committee for the symposium
consistedof:
AslakBakkeBuan(Trondheim)
BernhardKeller(Paris)
IdunReiten(Trondheim)
ClausM.Ringel(Bielefeld)
SverreO.Smalø(Trondheim)
ØyvindSolberg(Trondheim).
Theeditorsofthisproceedingsvolumegratefullyacknowledgethehelp,theassis-
tanceandthesupportoftheothermembersinthescientificcommitteeinorganiz-
ix
x Preface
ingthesymposiumwithspecialthankstotheforeignmembersandthelocalPhD-
students and post doctoral fellows at the Department of Mathematical Sciences,
NTNU.
Thesymposiumhad18talks:
LuchezarAvramov Reversehomologicalalgebraoverlocalrings
Ragnar-OlafBuchweitz Thefundamentalclassofanalgebra
KarinErdmann CohomologyforHeckealgebrasandrelatedalgebras
SergeyFomin Clusterstructuresinclassicalringsofinvariants
DieterHappel Algebrasoffiniteglobaldimension
BernhardKeller Ongenericbasesinclusteralgebras(afterPlamondon)
MaximKontsevich Clusteralgebrasfromthemirrorsymmetryperspective
HenningKrause AnencounterbetweenGabriel-RoiterandZiegler
BernardLeclerc Clusterstructuresonquantumcoordinaterings
HelmutLenzing Trianglesingularities,weightedprojectivelines,andnilpo-
tentoperators
RobertMarsh From triangulated categories to module categories via lo-
calisation
HirakuNakajima HandsawquivervarietiesandfiniteW-algebras
SteffenOppermann n-representationinfinitealgebras
ClausM.Ringel Tensorproductsofpathalgebras
JanSchröer Clustervariablesareirreducible
AndrzejSkowron´ski Modulecategorieswithheart
LaurenWilliams Positivebasesforclusteralgebrasfromsurfaces
AndreiZelevinsky Tropicaldualitiesinclusteralgebras.
In addition to the talks, the participants had the chance to attend a concert in St.
Olaf’s church on Monday night with traditional Norwegian folk music and a con-
ference dinner celebrating Sverre O. Smalø’s 60th birthday on Tuesday. In addi-
tion,theyhadtheoptiontotakepartinanexcursiononWednesday,departingfrom
BalestrandbytheboatM/SSognefjellandenjoyinglunchonthewaytotheNorwe-
gian Glacier Museum. After visiting the museum, the boat stopped at Hopperstad
stavechurchandthenFindenGard,whereparticipantshaddinner,beforereturning
toBalestrand.
The43participantsatthesymposiumwere:
ClaireAmiot(Bonn) SergeyFomin(Michigan)
JavadAsadollahi(IPM&Isfahan) ToreForbregd(Trondheim)
LuchezarAvramov(Lincoln) ChristofGeiss(MexicoCity)
FedraBabaei(Trondheim) EdwardL.Green(Blacksburg)
PetterAndreasBergh(Trondheim) BenedikteGrimeland(Trondheim)
AslakBakkeBuan(Trondheim) DieterHappel(Chemnitz)
Ragnar-OlafBuchweitz(Toronto) MartinHerschend(Trondheim)
KarinErdmann(Oxford) DaveJorgensen(Arlington)
Preface xi
KatharinaKahrs(Trondheim) JanSchröer(Bonn)
BernhardKeller(Paris) AndrzejSkowron´ksi(Torun)
MaximKontsevich(Bures-sur-Yvette) SverreO.Smalø(Trondheim)
HenningKrause(Bielefeld) ØyvindSolberg(Trondheim)
BernardLeclerc(Caen)
JohanSteen(Trondheim)
HelmutLenzing(Paderborn)
HughThomas(NewBrunswick)
DagOskarMadsen(Fredrikstad)
HermundA.Torkildsen(Trondheim)
RobertMarsh(Leeds)
GordanaTodorov(Boston)
HirakuNakajima(Kyoto)
LaurenK.Williams(Berkeley)
NilsNornes(Trondheim)
SteffenOppermann(Trondheim) DanZacharia(Syracuse)
IdunReiten(Trondheim) AndreiZelevinsky(Boston)
ClausM.Ringel(Bielefeld) GrzegorzZwara(Torun)
ShumbanaMohamedSaid(Trondheim) AnetteWrålsen(Trondheim)
We thank the Norwegian Mathematical Society for nominating our application
andtheBoardfortheNielsHenrikAbelMemorialFundforchoosingtosupportit.
Specialthanksalsogoestothecontributorstothisvolumeandtotheparticipantsand
thespeakersfortheirparttowardsthesuccessofthemeeting.Wewouldalsoliketo
acknowledgetheexcellentservicewereceivedfromKjetilNesseandthestaffatthe
KviknesHotelduringourstaythereaswellasthecrewoftheM/SSognefjelland
thestaffattheFindenGardrestaurant,whoallcontributedtomakingtheexcursion
onWednesdayamemorableexperience.
Finally,wethankSpringerandRuthAllewelt,forgreatpatienceandservicedur-
ingthepreparationofthisvolume.
Trondheim, AslakBakkeBuan
May2013 IdunReiten
ØyvindSolberg