Quantum Groups and Noncommutative Geometry

120 Pages·2018·2.115 MB·English

by  Yuri I. Manin

CRM Short Courses
Yuri I. Manin
Quantum
Groups and
Noncommutative
Geometry
Second Edition
With a Contribution by Theo Raedschelders
and Michel Van den Bergh
CRM Short Courses
Series Editors
Galia Dafni, Concordia University, Montreal, QC, Canada
Véronique Hussin, University of Montreal, Montreal, QC, Canada
Editorial Board
Mireille Bousquet-Mélou (CNRS, LaBRI, Université de Bordeaux)
Antonio Córdoba Barba (ICMAT, Universidad Autónoma de Madrid)
Svetlana Jitomirskaya (UC Irvine)
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Leonid Polterovich (Tel-Aviv University)
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More information about this series at http://www.springer.com/series/15360
Yuri I. Manin
Quantum Groups
and Noncommutative
Geometry
Second Edition
With a Contribution by Theo Raedschelders
and Michel Van den Bergh
123
YuriI. Manin
MaxPlanckInstitute for Mathematics
Bonn,Germany
ISSN 2522-5200 ISSN 2522-5219 (electronic)
CRM Short Courses
ISBN978-3-319-97986-1 ISBN978-3-319-97987-8 (eBook)
https://doi.org/10.1007/978-3-319-97987-8
LibraryofCongressControlNumber:2018950779
Mathematics Subject Classification (2010): 16S10, 16S37, 16S38, 16T05, 16T15, 16W30, 18D50,
18D10,20C30,20G42
1stedition:©CentredeRecherchesMathématiques1988
2ndedition:©SpringerNatureSwitzerlandAG2018
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Preface to the Second Edition
ThissecondeditionwaspreparedwithgreathelpofTh.Raedschelders,M.Vanden
Bergh, and D. Leites.
Th. Raedschelders and M. Van den Bergh carefully read the whole text and
suggested many editorial corrections. D. Leites, who started working on this book
whenitsRussiantranslationwasbeingpreparedabouttwodecadesago,returnedto
this project now and did a very meticulous job.
I am deeply grateful to them.
For some mathematical updates, see Chapter 1 “Introduction”.
Bonn, Germany Yuri I. Manin
v
Contents
Preface to the Second Edition ..... .... .... .... .... .... ..... .... v
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 The Quantum Group GLq(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Bialgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Quadratic Algebras as Quantum Linear Spaces . . . . . . . . . . . . . . . 19
5 Quantum Matrix Spaces. I. Categorical Viewpoint. . . . . . . . . . . . . 25
6 Quantum Matrix Spaces. II. Coordinate Approach . . . . . . . . . . . . 29
7 Adding Missing Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 From Semigroups to Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
9 Frobenius Algebras and the Quantum Determinant . . . . . . . . . . . . 47
10 Koszul Complexes and the Growth Rate of Quadratic
Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11 Hopf *-Algebras and Compact Matrix Pseudogroups. . . . . . . . . . . 63
12 Yang–Baxter Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
13 Algebras in Tensor Categories and Yang–Baxter Functors. . . . . . . 73
14 Some Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
15 The Tannaka–Krein Formalism and (Re)Presentations
of Universal Quantum Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 117
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 123
vii
Chapter 1
Introduction
We begin with some terminology and background, in particular we define the no-
tionsofHopfalgebrasandquantumgroups.
LetH bethealgebraoffunctionsofaLiegroupG.Thenthemultiplicationmap
G (cid:2)G ! G (resp. the inversion map G ! GW g ! g(cid:2)1, resp. the inclusion of
theidentitypoint)givesrisetoa“comultiplication”(cid:2)W H ! H ˝H (resp.“an-
tipode” iW H ! H, resp. “counit” "W H ! R), which satisfies a set of identities
definingthegeneralalgebraicnotionofaHopfalgebra(cf.Chapter3below).This
setissymmetricwithrespecttoreversingallarrows.However,anasymmetrymight
arise: while H, as algebra, always has a commutative multiplication, it may well
haveanoncocommutativecomultiplication(thisisthecaseifG isnonabelian).
Hopfalgebrasthatarenotnecessarilycommutativeorcocommutativehavebeen
studied by algebraists for several decades (cf. Abe [1]). Recently, however, some
veryspecificHopfalgebrasemergedinmathematicalphysics;V.G.Drinfeldnamed
them “quantum groups.” Initially, they appeared in the quantum inverse scattering
transformmethod(QIST)developedbyL.D.Faddeevandhisschool(cf.[26,27,56,
58, 59]) and reviewed in V.G. Drinfeld’s Berkeley talk [21]. Closely related work
(also motivated by QIST) was done by M. Jimbo [36, 37] and, from a somewhat
different viewpoint, by S.L. Woronowicz [68, 69]. One of the main ideas behind
these works is that such rigid objects as classical simple groups (or Lie algebras)
admitinfactcontinuousdeformationsatthelevelofHopfalgebrascorresponding
to them, and the deformed objects close to the initial one can be described very
preciselytogetherwiththeirrepresentationtheory.
Inthiswork,wesystematicallydevelopadifferentapproachtoquantumgroups,
based upon the following observation. Suppose that you “quantize” the simplest
phaseplane,imposingonitscoordinatesthecommutationrelation
xy D e(cid:2)yx I
©SpringerNatureSwitzerlandAG2018 1
Y.I.Manin,QuantumGroupsandNoncommutativeGeometry,
CRMShortCourses,https://doi.org/10.1007/978-3-319-97987-8_1
2 1 Introduction
i.e., the integrated version of the Heisenberg commutation relation. Then the or-
dinary symmetry group GL.2/ of the plane breaks down. However the “broken
symmetry”iscompletelyrestoredifoneimposessomenontrivialcommutationre-
lationsupontheentriesofthe.2 (cid:2) 2/-matrices,theelementsofGL.2/.Inthisway,
one arrives at the notion of the quantum group GLq.2/, where q D e(cid:2), which is
describedincompletedetailinChapter2.
Oneremarkablepropertyofthisapproachisitsgenerality.Namely,inChapters4
and5,insteadof“quantumplane,”westartwitha“quantumlinearspace”defined
byarbitraryquadraticrelationsbetweenitsnoncommutativecoordinates.Inthis
wayweobtaina“generallinearquantumgroup,”orratherapair:
“quantumsemigroupofendomorphisms” ! “quantumgroup.”
The first object is a noncommutative space of the same kind (i.e., it is defined by
quadraticrelations)whilethesecondoneisobtainedfromthefirstonebyaprocess
ofnoncommutativelocalizationasadvocatedinalgebrabyP.M.Cohn,andarising
hereforthefirsttimeinanaturalway.Thepointisthatoneinvertsmatricesandnot
justelementsofaring,andtoobtainaHopfalgebra,onemust,generallyspeaking,
invertinfinitelymanymatrices.
We use the word “noncommutative space” in the spirit of Alain Connes. The
maindifferenceisthatwedevelopafragmentofnoncommutativealgebraicgeom-
etrywhileConnesdealswithnoncommutativedifferentialgeometryandtopology.
InChapter11,wediscussawaytointroducea(cid:3)-structureinourgroupsthusmak-
ingitpossibletodefinetheircompactforms.
As in [56, 68, 69], but unlike [21], we work with the noncommutative ring of
functions over a quantum group rather than with the “universal enveloping alge-
bra,”whichisadualobject.Ofcourse,bothobjectsdeservecloseattention,butin
our approach the former appears more naturally. The key technical notion in this
connectionisthatofamultiplicativematrix:cf.Sections3.6–3.10.Itisalsoworth
mentioning that we have no need to consider only small deformations of classi-
calobjects:theparameterspacesofourobjectsaredefinedglobally.Theyarejust
Grassmanniansofquadraticrelations:cf.Section4.2.Thepricepaidforthisisthe
lossofthenotionofa“semisimple”quantumgroup(whichanywayhasneverbeen
formalizedinpreviousworks).
Moreover,wedonotneedtoimposeanyrelationontheYang–Baxterequations
from the start. However, we can and must explain such a relation at a later stage.
AnimportantthingtorememberisthataYang–Baxteroperatorgeneratesquantum
groups by two very different procedures. One is to consider the quantum auto-
morphism group of a quadratic algebra which is just a “Yang–Baxter symmetric
algebra.”ThegroupsappearinginQISTareofthistype.
1 Introduction 3
Anotherwaystartswitharelativizationofthenotionsofquadraticalgebra,quan-
tum group, etc. by replacing everywhere the transposition of factors in a tensor
product by a Yang–Baxter operator. The simplest example is the “super version”
of our construction, which is fairly obvious. Other examples were not considered
before[46].WeexplaintheseideasrathersuccinctlyinChapters12and13.
An earlier version of this work is [46]. Many details and some new results are
addedhere.IwouldliketomentioninparticularthegeneralconstructionofaHopf
algebrastartingfromabialgebrawithageneratingmultiplicativematrix(seeChap-
ter8).Thesenotesarenotmeanttobeasurveyofthisquicklygrowingsubject;the
bibliography is very incomplete.1 The ideas of this paper were first developed in
mylecturesinMoscowUniversityduringthewinterof1986–1987andthesubse-
quentseminarwhereYu.KobyzevsuggestedthedescriptionofGLq.2/whichwas
seminalforallthatfollows.
These notes were written for a series of lectures given at the Centre de recher-
chesmathématiquesattheUniversitédeMontréalinJune1988,whileIwasinvited
asAisenstadtProfessor.IamgratefultomanypeoplewhomademystayinMont-
réalveryagreeableandproductive,inparticulartoAndréAisenstadt,whomitwas
my pleasure to meet. I would like to thank Luis Alvárez-Gaumé and John Harnad
for their assistance in the preparation of these notes. Finally I would like to thank
NathalieBrunet,LouiseLetendre,andAngèlePatenaudefortheircarefultypingof
themanuscript.
1Severaltextbooksandreviewsonquantumgroupswerepublished:[14,19,39,40,45].Theycontaina
morecompletebibliographyandanswersomeofthequestionsposedinChapter14ofthisbook.
Chapter 2
.2/
The Quantum Group GL
q
2.1 NoncommutativeSpaces:PointsandRingsofFunctions
In this paper, we fix once and for all a field K. A ring (or an algebra) means an
associative K-algebra with unit, not necessarily commutative. It is suggestive to
imagine the ring A as a ring of (polynomial) functions on a space which is an
object of noncommutative, or “quantum,” geometry. Morphisms of spaces corre-
spondtoringhomomorphismsintheoppositedirection.ForAandB fixed,theset
HomK(cid:2)alg.A;B/isalsocalledthesetofB-pointsofthespacedefinedbyA.
There is no harm in giving a formal definition of the category of noncommuta-
tivespacesas.K(cid:2)Alg/opaslongasoneremembersthatsomeofthemostcommon
categoricalprejudicescouldbemisleadingwhenworkingwith.K(cid:2)Alg/op.Toquote
justone,thetensorproductinK(cid:2)Algdoesnotdefineadirectproductin.K(cid:2)Alg/op,
butmorallydoescorrespondtoa“directproduct”ofquantumspaces.
2.2 TwoQuantumPlanesandQuantumMatrices
Fixq 2 K,whereq ¤ 0.ThequantumplaneA2qj0 isdefinedbythering
A2j0 WD Khx;yi=.xy (cid:2)q(cid:2)1yx/ ; (2.1)
q
where Khx1;:::;xni is the associative K-algebra freely generated by x1;:::;xn.
This plane A2qj0 is a deformation of the usual plane corresponding to q D 1. We
alsoneedadeformationofthe0j2-dimensional“plane”ofsupergeometry:
A0j2 WD Kh(cid:2);(cid:3)i=.(cid:2)2;(cid:3)2;(cid:2)(cid:3)Cq(cid:3)(cid:2)/ : (2.2)
q
©SpringerNatureSwitzerlandAG2018 5
Y.I.Manin,QuantumGroupsandNoncommutativeGeometry,
CRMShortCourses,https://doi.org/10.1007/978-3-319-97987-8_2