Quantum Groups in Three-Dimensional Integrability

330 Pages·2022·5.658 MB·English

by  Atsuo Kuniba

Theoretical and Mathematical Physics
Atsuo Kuniba
Quantum Groups in
Three-Dimensional
Integrability
Quantum Groups in Three-Dimensional
Integrability
Theoretical and Mathematical Physics
Thisseries,foundedin1975andformerlyentitled(until2005)TextsandMonographs
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SeriesEditors
PiotrChrusciel,Wien,Austria
Jean-PierreEckmann,Genève,Switzerland
HaraldGrosse,Wien,Austria
AnttiKupiainen,Helsinki,Finland
HartmutLöwen,Düsseldorf,Germany
KasiaRejzner,York,UK
LeonTakhtajan,StonyBrook,NY,USA
JakobYngvason,Wien,Austria
Atsuo Kuniba
Quantum Groups
in Three-Dimensional
Integrability
AtsuoKuniba
InstituteofPhysics,GraduateSchool
ofArtsandSciences
UniversityofTokyo
Komaba,Tokyo,Japan
ISSN 1864-5879 ISSN 1864-5887 (electronic)
TheoreticalandMathematicalPhysics
ISBN 978-981-19-3261-8 ISBN 978-981-19-3262-5 (eBook)
https://doi.org/10.1007/978-981-19-3262-5
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Preface
Inintegrablesystemsinquantumfieldtheoryin(1+1)-dimensionalspacetimeand
statisticalmechanicalmodelsontwo-dimensionallattices,acentralroleisplayedby
theYang–Baxterequation.Itsalgebraicaspectsarenowwellunderstoodintermsof
quantumgroups.
Anobviouschallengeistoexplorethehigherdimensions.Thefirstattemptinsuch
adirectionwasmadebyA.B.Zamolodchikovin1980wholaunchedthetetrahedron
equation as a generalization of the Yang–Baxter equation in three dimensions. In
his seminal work in 1983, Baxter referred to it as “immensely more complicated”
and to Zamolodchikov’s conjectural solution which he proved as “what appears
to be an extraordinary feat of intuition”. Over nearly forty years since the initial
breakthrough, much effort has been made and many results continue to emerge
despitethecomplexity.
This book is the first monograph devoted to the subject. It is a selective but
expositoryintroductiontoaquantumgrouptheoreticalapproachtothetetrahedron
equationsandtheirrelativeswhichhavebeenshapedduringsuchdevelopments.It
explainsthenaturaloriginoftheseequations,prototypicalsolutionsandtheirnotable
aspects.Thelatterhalfofthebookalsoencompassesfeedbackstothetwo-oreven
lowerone-dimensionalsystemsfromtheviewpointofmathematicalphysics.
Thecontentsareelementaryandpresentedinacasualstyleforthesakeofread-
ability.Asaresult,asubstantialpartofthetexthasbecomeacollectionofalgebraic
manipulations, which are straightforward in principle but sometimes too tedious
withoutthehelpofacomputer.Hopefully,suchcalculationsarenotjustlaborious
butwouldberewardingandfunforthosewhoenjoytheprogramming.
Thetitleofthebookmaysoundsomewhatoddortoostrong,sinceadmittedlyit
actuallyachievesonlyaglimpseintothequantumintegrabilityinthreedimensions.
Itismyhope,however,thatitstimulatesthesubject,nowinitsadolescence,tomake
atransitionintothenextphase.
TherearemanypeopletowhomIamindebtedforthedelightfulaswellaschal-
lengingopportunitytowritethisbook.IamgratefultoMasatoOkadoforadviceon
theplanofthebookandcollaborationonmanyrelatedprojects;VladimirMangazeev,
Vincent Pasquier, Sergey Sergeev and Yasuhiko Yamada for sharing their insights
v
vi Preface
andToshiyukiTanisakiforusefulcommunications.SpecialthanksgotoRodneyJ.
Baxter,VladimirV.Bazhanov,EtsuroDate,MichioJimbo,TetsujiMiwaandthelate
MikiWadatiforinspiringmeinthewondersofintegrablesystemsformanyyears.
IhavehadkindsupportontexthandlingfromMasayukiNakamurafromSpringer
Japan.Lastbutnotleast,Ithankmyfamilyforlettingmeworkcomfortablyfrom
homethroughtheturbulentCOVID-19yearsofwriting.
Thefirstmanuscriptofthisbook,whichwasalmostinthefinalform,wassentto
thepublisheronOctober52021.
Komaba,Tokyo,Japan AtsuoKuniba
Contents
1 Introduction .................................................. 1
1.1 QuantumIntegrabilityinTwoDimensions .................. 1
1.2 Quantization:IntroducingtheThirdDimension .............. 3
1.3 QuantizedCoordinateRing ............................... 4
1.4 Compatibility: Tetrahedron, 3D Reflection and F
4
Equations .............................................. 5
1.5 Feedbackto2D ......................................... 6
1.6 LayoutoftheBook ...................................... 7
2 TetrahedronEquation ......................................... 9
2.1 3D R .................................................. 9
2.2 TetrahedronEquationofType RRRR = RRRR ............. 10
2.3 3D L .................................................. 13
2.4 TetrahedronEquationofType RLLL = LLLR ............. 13
2.5 QuantizedYang–BaxterEquation .......................... 15
2.6 TetrahedronEquationofType MMLL = LLMM ........... 16
2.7 BibliographicalNotesandComments ...................... 18
3 3D RFromQuantizedCoordinateRingofTypeA ............... 21
3.1 QuantizedCoordinateRing Aq(An−1) ...................... 21
3.2 RepresentationTheory ................................... 23
3.3 IntertwinerforCubicCoxeterRelation ..................... 26
3.4 ExplicitFormulafor3D R ................................ 30
3.5 SolutiontotheTetrahedronEquations ...................... 36
3.5.1 RRRR = RRRRType ........................... 36
3.5.2 RLLL = LLLRType ........................... 39
3.5.3 MMLL = LLMM Type ......................... 43
3.6 FurtherAspectsof3D R ................................. 45
3.6.1 BoundaryVector ................................ 45
3.6.2 CombinatorialandBirationalCounterparts .......... 47
vii
viii Contents
3.6.3 BilinearizationandGeometricInterpretation ......... 50
3.7 BibliographicalNotesandComments ...................... 52
4 3DReflectionEquationandQuantizedReflectionEquation ....... 53
4.1 Introduction ............................................ 53
4.2 3D K .................................................. 54
4.3 3DReflectionEquation .................................. 55
4.4 QuantizedReflectionEquation ............................ 59
4.5 BibliographicalNotesandComments ...................... 63
5 3D K FromQuantizedCoordinateRingofTypeC ............... 65
5.1 QuantizedCoordinateRing A (C ) ........................ 65
q n
5.2 FundamentalRepresentations ............................. 67
5.3 InterwtinersforQuadraticandCubicCoxeterRelations ....... 69
5.4 IntertwinerforQuarticCoxeterRelation .................... 70
5.5 ExplicitFormulafor3D K ............................... 74
5.6 Solutiontothe3DReflectionEquation ..................... 79
5.7 SolutiontotheQuantizedReflectionEquation ............... 81
5.8 FurtherAspectsof3D K ................................. 84
5.8.1 BoundaryVector ................................ 84
5.8.2 CombinatorialandBirationalCounterparts .......... 86
5.9 BibliographicalNotesandComments ...................... 89
6 3D K FromQuantizedCoordinateRingofTypeB ............... 91
6.1 QuantizedCoordinateRing A (B ) ........................ 91
q n
6.2 FundamentalRepresentations ............................. 92
6.3 Intertwiners ............................................ 95
6.4 3DReflectionEquation .................................. 96
6.5 CombinatorialandBirationalCounterparts .................. 99
6.6 ProofofProposition6.5 .................................. 100
6.6.1 MatrixProductFormulaoftheStructureFunction .... 100
6.6.2 RTT Relation ................................... 102
6.6.3 ρTT Relations .................................. 103
6.7 BibliographicalNotesandComments ...................... 105
7 IntertwinersforQuantizedCoordinateRing A (F ) .............. 107
q 4
7.1 FundamentalRepresentations ............................. 107
7.2 Intertwiners ............................................ 108
7.3 F AnalogueoftheTetrahedron/3DReflectionEquations ..... 109
4
7.4 Reductionto3DReflectionEquations ...................... 112
7.5 BibliographicalNotesandComments ...................... 116
8 IntertwinerforQuantizedCoordinateRing A (G ) .............. 117
q 2
8.1 Introduction ............................................ 117
8.2 QuantizedCoordinateRing A (G ) ........................ 118
q 2
8.3 FundamentalRepresentations ............................. 118
8.4 Intertwiner ............................................. 120
Contents ix
8.5 QuantizedG ReflectionEquation ......................... 123
2
8.5.1 3D L ........................................... 123
8.5.2 QuantizedG ScatteringOperator J ................ 124
2
8.5.3 QuantizedG ReflectionEquation ................. 127
2
8.6 FurtherAspectsof F .................................... 131
8.6.1 BoundaryVector ................................ 131
8.6.2 CombinatorialandBirationalCounterparts .......... 131
8.7 DataonRelevantQuantum RMatrix ....................... 133
8.8 BibliographicalNotesandComments ...................... 136
9 Comments on Tetrahedron-Type Equation
forNon-crystallographicCoxeterGroups ........................ 139
9.1 FiniteCoxeterGroups ................................... 139
9.2 Tetrahedron-TypeEquationfortheCoxeterGroup H ........ 141
3
9.3 DiscussionontheQuinticCoxeterRelation ................. 143
10 ConnectiontoPBWBasesofNilpotentSubalgebraofU .......... 147
q
10.1 QuantizedUniversalEnvelopingAlgebraU (g) ............. 147
q
10.1.1 Definition ...................................... 147
10.1.2 PBWBasis ..................................... 148
10.2 QuantizedCoordinateRing A (g) ......................... 149
q
10.2.1 Definition ...................................... 149
10.2.2 RightQuotientRing A (g) ....................... 153
q S
10.3 MainTheorem .......................................... 154
10.3.1 DefinitionsofγA and(cid:4)A ......................... 154
B B
10.3.2 ProofofTheorem10.6forRank2Cases ............ 155
10.4 ProofofProposition10.7 ................................. 157
10.4.1 ExplicitFormulasfor A .......................... 158
2
10.4.2 ExplicitFormulasforC .......................... 161
2
10.4.3 ExplicitFormulasforG ......................... 165
2
10.5 Tetrahedronand3DReflectionEquationsfromPBW
Bases .................................................. 170
10.6 χ-Invariants ............................................ 172
10.7 BibliographicalNotesandComments ...................... 174
11 TraceReductionsof RLLL = LLLR ........................... 177
11.1 Introduction ............................................ 177
11.2 TraceReductionOvertheThirdComponentof L ............ 179
11.3 TraceReductionOvertheFirstComponentof L ............. 184
11.4 TraceReductionOvertheSecondComponentof L .......... 186
11.5 IdentificationwithQuantum RMatricesof A(1) ............. 188
n−1
11.5.1 Str3(z) .......................................... 189
11.5.2 Str1(z) .......................................... 191
11.5.3 Str2(z) .......................................... 193
11.6 CommutingLayerTransferMatricesandDuality ............ 194
11.7 BibliographicalNotesandComments ...................... 197