Algebraic Geometric Topology paper no Preprint version available at http: www fourier ujf grenoble fr eiserm

Algebraic&GeometricTopology5(2005),paperno.23,537-562.
Preprintversionavailableathttp://www-fourier.ujf-grenoble.fr/
∼
eiserm

Yang-Baxterdeformations
ofquandlesandracks
M
ICHAEL
E
ISERMANN

Givenarack
Q
andaring
A
,onecanconstructaYang-Baxteroperator
c
Q
:
V
⊗
V
→
V
⊗
V
onthefree
A
-module
V
=
A
Q
bysetting
c
Q
(
x
⊗
y
)
=
y
⊗
x
y
forall
x
,
y
∈
Q
.InanswertoaquestioninitiatedbyD.N.YetterandP.J.Freyd,thisarticle
classiesformaldeformationsof
c
Q
inthespaceofYang-Baxteroperators.For
thetrivialrack,where
x
y
=
x
forall
x
,
y
,onehas,ofcourse,theclassicalsetting
ofr-matricesandquantumgroups.Inthegeneralcaseweintroduceandcalculate
thecohomologytheorythatclassiesinnitesimaldeformtaionsof
c
Q
.Inmany
casesthisallowsustoconcludethat
c
Q
isrigid.Intheremainingcases,where
innitesimaldeformationsarepossible,weshowthathigher-orderobstructions
arethesameasinthequantumcase.

Introduction

FollowingM.Gerstenhaber[
16
],analgebraicdeformationtheoryshould
•
denetheclassofobjectswithinwhichdeformationtakesplace,
•
identifyinnitesimaldeformationsaselementsofasuitablecohomology,
•
identifytheobstructionstointegrationofaninnitesimaldeformation,
•
givecriteriaforrigidity,andpossiblydeterminetherigidobjects.
InanswertoaquestioninitiatedbyP.J.FreydandD.N.Yetter[
15
],wecarryoutthis
programmeforracks(linearizedoversomering
A
)andtheirformaldeformationsin
thespaceof
A
-linearYang-Baxteroperators.

2
MichaelEisermann
Arackisaset
Q
withabinaryoperation,denoted(
x
,
y
)
7→
x
y
,suchthat
c
Q
:
x
⊗
y
7→
y
⊗
x
y
denesaYang-Baxteroperatoronthefree
A
-module
A
Q
(see
§
1
fordenitions).
Foratrivialrack,where
x
y
=
x
forall
x
,
y
∈
Q
,weseethat
c
Q
issimplythe
transpositionoperator.Inthiscasethetheoryofquantumgroups[
7
,
27
,
22
,
23
]
producesaplethoraofinterestingdeformations,whichhavereceivedmuchattention
overthelast20years.Itthusseemsnaturaltostudydeformationsof
c
Q
inthegeneral
case,where
Q
isanon-trivialrack.

Outlineofresults
Werstintroduceandcalculatethecohomologytheorythatclassiesinnitesimal
deformationsofracksinthespaceofYang-Baxteroperators.Inmanycasesthis
sufcestodeducerigidity.Intheremainingcases,whereinnitesimaldeformations
arepossible,weshowthathigher-orderobstructionsdonotdependon
Q
:theyarethe
sameasintheclassicalcaseofquantuminvariants.(See
§
1.4
foraprecisestatement.)
FormalYang-Baxterdeformationsofracksthushaveanunexpectedlysimpledescrip-
tion:uptoequivalencetheyarejustr-matriceswithaspecialsymmetryimposedby
theinnerautomorphismgroupoftherack.Althoughthisisintuitivelyplausible,it
requiresacarefulanalysistoarriveatanaccurateformulation.Theprecisenotionof
entropic
r-matriceswillbedenedin
§
1.3
.
Withregardstotopologicalapplications,thisresultmaycomeasadisappointment
inthequestfornewknotinvariants.Toourconsolation,weobtainacompleteand
concisesolutiontothedeformationproblemforracks,whichisquitesatisfactoryfrom
analgebraicpointofview.
Throughoutourcalculationsweconsiderthegenericcasewheretheorder
|
Inn(
Q
)
|
of
theinnerautomorphismgroupoftherack
Q
isinvertibleinthegroundring
A
.We
shouldpointout,however,thatcertainknotinvariantsariseonlyinthemodularcase,
where
|
Inn(
Q
)
|
vanishesin
A
;seetheclosingremarksinSection
6
.

Howthispaperisorganized
Inordertostatetheresultsprecisely,andtomakethisarticleself-contained,Section
1
rstrecallsthenotionsofYang-Baxteroperators(
§
1.1
)andracks(
§
1.2
).Wecan
thenintroduceentropicmaps(
§
1.3
)andstateourresults(
§
1.4
).Wealsodiscusssome
examples(
§
1.5
)andputourresultsintoperspectivebybrieyreviewingrelatedwork
(
§
1.6
).

Yang-Baxterdeformationsofquandlesandracks
3
Theproofsaregiveninthenextfoursections:Section
2
introducesYang-Baxtercoho-
mologyandexplainshowitclassiesinnitesimaldeformaitons.Section
3
calculates
thiscohomologyforracks.Section
4
generalizesourclassicationfrominnitesimal
tocompletedeformations.Section
5
examineshigher-orderobstructionsandshows
thattheyarethesameasintheclassicalcaseofquantuminvariants.Section
6
,nally,
discussessomeopenquestions.

1Reviewofbasicnotionsandstatementofresults
1.1Yang-Baxteroperators
Let
A
beacommutativeringwithunit.Inthesequelallmoduleswillbe
A
-modules,
andalltensorproductswillbeformedover
A
.Forevery
A
-module
V
wedenoteby
V
⊗
n
the
n
-foldtensorproductof
V
withitself.Theidentitymapof
V
isdenotedby
I:
V
→
V
,and
I
=
I
⊗
Istandsfortheidentitymapof
V
⊗
V
.
Denition1
A
Yang-Baxteroperator
on
V
isanautomorphism
c
:
V
⊗
V
→
V
⊗
V
thatsatisesthe
Yang-Baxterequation
,alsocalled
braidrelation
,
(
c
⊗
I)(I
⊗
c
)(
c
⊗
I)
=
(I
⊗
c
)(
c
⊗
I)(I
⊗
c
)inAut
A
(
V
⊗
3
)
.
Thisequationrstappearedintheoreticalphysics,inapaperbyC.N.Yang[
28
]onthe
many-bodyprobleminonedimension,inworkofR.J.Baxter[
2
,
3
]onexactlysolvable
modelsinstatisticalmechanics,andlaterinquantumeldtheory[
13
]inconnection
withthequantuminversescatteringmethod.Italsohasaverynaturalinterpretationin
termsofArtin'sbraidgroups[
1
,
4
]andtheirtensorrepresentations:
Remark2
Recallthatthebraidgroupon
n
strandscanbepresentedas
σ
i
σ
j
=
σ
j
σ
i
for
|
i
−
j
|≥
2
B
n
=
σ
1
,...,σ
n
−
1

σ
i
σ
j
σ
i
=
σ
j
σ
i
σ
j
for
|
i
−
j
|
=
1
,
wherethebraid
σ
i
performsapositivehalf-twistofthestrands
i
and
i
+
1.Ingraphical
notation,braidscanconvenientlyberepresentedasinFigure
1
.
GivenaYang-Baxteroperator
c
,wecandeneautomorphisms
c
i
:
V
⊗
n
→
V
⊗
n
for
i
=
1
,...,
n
−
1bysetting
c
i
=
I
⊗
(
i
−
1)
⊗
c
⊗
I
⊗
(
n
−
i
−
1)
.TheArtinpresentationensures
thatthereexistsauniquebraidgrouprepresentation
ρ
cn
:B
n
→
Aut
A
(
V
⊗
n
)denedby
ρ
cn
(
σ
i
)
=
c
i
.

4
MichaelEisermann
Hereweadoptthefollowingconvention:braidgroupswillactontheleft,sothat
compositionofbraidscorrespondstocompositionofmaps.ThebraidinFigure
1
,forexample,reads
β
=
σ
1
−
2
σ
22
σ
1
−
1
σ
21
σ
1
−
1
σ
21
;itisrepresentedbytheoperator
ρ
c
3
(
β
)
=
c
1
−
2
c
22
c
1
−
1
c
21
c
1
−
1
c
21
actingon
V
⊗
3
.
11iii+1i+1
nnFigure1:Elementarybraids
σ
i
+
1
,
σ
i
−
1
;amorecomplexexample
β
NoticethatArtin,afterhavingintroducedhisbraidgroups,couldhavewrittendown
theYang-Baxterequationinthe1920s,butwithoutanynon-trivialexamplesthetheory
wouldhaveremainedvoid.ItisaremarkablefactthattheYang-Baxterequationadmits
anyinterestingsolutionsatall.Manyofthemhaveonlybeendiscoveredsincethe
1980s,andourrstexamplerecallsthemostprominentone:
Example3
Forevery
A
-module
V
thetransposition
τ
:
V
⊗
V
→
V
⊗
V
givenby
τ
(
a
⊗
b
)
=
b
⊗
a
isaYang-Baxteroperator.Thisinitselfisnotverysurprising,but
deformationsof
τ
canbeveryinteresting:Supposethat
V
isfreeofrank2andchoose
abasis(
v
,
w
).Ifweequip
V