Integrable Systems and Quantum Groups
Itzhak Roditi
CentroBrasileirodePesquisas Fsias
Rua Dr.XavierSigaud150, Rio deJaneiro, 22290-180,Brazil
Reeived7January,2000
Wepresent someaspetsofthestudyofquantumintegrablesystemsanditsrelationto quantum
groups.
I Introdution
Themainpurposeofthisletureistopresentthestudy
of quantum integrablesystems and its interplay with
thestudyofquantumgroups. Thehistoryofquantum
integrable systems begins, following a ertain
histori-al path, with the rst attempts of W. Heisenberg to
developamirosopitheory thatould explain
ferro-magnetism [1℄. Later on a method was proposed by
H. Bethe [2℄ in order to obtain the spetrum of the
isotropi Heisenberg model. This approah is alled
the Bethe ansatz and it givesthe parameterizationof
the eigenvetorsthroughaset ofequations, alled the
Bethe ansatz equations establishing the solvability of
the model. Sine then a steady evolution has taken
plae, goingthroughtheworksofL.Onsager[3℄,C.N.
Yang andC.P.Yang [4℄, R.Baxter [5℄andmany
oth-ers, ulminating with the adventof the Quantum
In-verse Sattering Method (QISM)[6℄. The time
inter-val between Bethe's solution and the QISM is of
al-most fty years. The QISM was developed to
inves-tigate integrablesystemsin quantum eld theoryand
quantumstatistialphysisanditwaslargelybasedon
an algebrai viewpoint. It provided a unied
frame-workfortreatingthepreviousapproahesusedtosolve
integrable systems, in theoretial and mathematial
physis. Moreover,itisduetotheQISMthatquantum
groupsmadetheirappearaneinthephysissenario.
The study of quantum groups, or quantum
alge-bras[7,8℄is ruialtotheunderstandingofintegrable
systems as well as the development of new integrable
models. It also grewas anindependent areain
math-ematial physis showing interesting onnetions with
other mathematial subjets suh as knottheory and
non-ommutativegeometry.
Atthispointduetospaeandtimelimits,inherent
to aleture, itwouldbeuseful tomakerefereneto a
ertainnumberofexellentreviewswhereoneannd
a deeper insight into the QISM and quantum groups,
aswellasaomprehensivelistofreferenes[9℄.
In the next setion the QISM is introdued and
quantum groupsare dened. The hoie of
presenta-tion followsanalgebraiapproah,givingaquite
gen-eral andmodelindependentframework.
InsetionIIIweshowhowtoobtain
multiparamet-riquantumspinhainsstillbasedonalgebrairesults.
Themotivation of that setion isrelated to thestudy
oftheeetofboundaryonditionsin models
desrib-ing systems of orrelated eletrons. In partiular we
establish aorrespondene between twistedboundary
onditionsandmultiparametrimodels [10,11,12℄
SetionIVisdevotedtomentioningsomeother
de-velopmentsandtodrawsomeindiationsaboutfuture
work.
Before we proeed some aknowledgments are in
order. The present work has muh proted from the
disussionswith myolleagues from CBPF,Maro
R-Monteiroand Ligia Rodrigues aswell aswith my
ol-laborators and friends, Angela Foerster, Katrina
Hib-berd, Antonio Lima-Santosand Jon Links,whose
le-turenotes[13℄ havebeenextremelyuseful.
II Quantum Inverse Sattering
Method: a telegraphi
ver-sion
Letus introdue herein a verybrief way someof the
algebraistruturedening theQISM.LetT i
j
(); 1
i;j n , be n 2
generators of an abstrat algebra A
and 2C is alled the spetral parameter. Consider
aninvertibleoperator
R ()2End(V V);
where V is an n-dimensionalvetorspae. R maybe
writteninomponentformas
R ()= X
ijk l R
jl
ik ()e
i
j e
k
l :
Onean set
T()= X
ij e
i
j T
j
i ();
andimposegeneralizedommutationrelationsinAvia
R
12
( )T
13 ()T
23 ()
where the subsripts refer to the embeddings in End
(V V)A. R k l
ij
()maybeseentogivethestruture
onstantsofthealgebraA,ifonewrites (1)as
R jl
ik
( )T
p
j ()T
q
l
()=T j
k ()T
l
i ()R
pq
lj
( ):
From (1)andtheylipropertyof thetraeone an
showthattheelements
T
13 ()T
23
()=R 1
12
( )T
23 ()T
13 ()R
12
( ) (2)
()=(trI)T()= n
X
i=1 T
i
i ();
ofthealgebraAommute
[();()℄=0; 8;2C:
The fat that () provides a one-parameter family
of mutually ommuting elements of the algebra A is
deeply related to the denition of integrability as we
shall disuss later. A suÆient ondition for A to be
assoiativeandthenbeabletoobtainmatrix
represen-tationsisthat
R
12
( )R
13
( w)R
23
(v w) (3)
=R
23
(v w)R
13
( w)R
12
( );
onthespaeEnd(V V V),whihistheelebrated
Yang-Baxterequation. We all the algebraA dened
insuhawayaYang-BaxteralgebraoraYangian
alge-bra. The fatthat R ()must satisfytheYang-Baxter
equationprovidesuswiththeadjointrepresentation,
T j
i ()
= X
k ;l R
jl
ik ()e
k
l :
Furthermore,theYangianalgebraAhasastruture
ofabi-algebrawhihanbeseenfromtheexisteneof
thefollowingalgebrahomomorphisms,theo-produt
:A!AA
denedby
T j
i ()
= X
k T
j
k
()T k
i ()
andtheo-unit
:A!C
T j
i ()
= Æ j
i :
The signiane of the o-produt is that it allows
ustobuild tensorprodutrepresentationsofA.
If
1 ;
2
aretworepresentationsofAwhihaton
thespaesV
1 ;V
2
respetivelythen
(
1
2 )
T j
i ()
=
1 T
k
i ()
2
T j
k ()
givesarepresentationonV
1 V
2
. Wemayalsousethe
oppositeo-produtgivenby
T j
()
=T k
()T j
()
whihisalsoanalgebrahomomorphism.
II.1 Quasi-triangularHopf algebras
Inthe above we havedened the Yangian algebra
Aandseenthatithasabi-algebrastruture. The
dis-ussedpropertiesarepartoftheonstrutionof
quan-tumgroups,orquasi-triangularHopfalgebras. AHopf
algebraanbedenedasabi-algebrathatpossessesan
antihomomorphism
S:A!A;
suhthat
m(SI)(a)=m(IS)(a)=i(a);
where m andi arerespetively themultipliationand
unit elementmapsonA,denedby
m(ab)=ab
and
i()=I;
8a2A and 2C:
The key ingredient in the denition of aquantum
groupisthe existeneof anelement thatestablishes a
relation between the o-produt and the opposite
o-produtdevelopinganisomorphismbetweenthetensor
produt
1
2 !
2
1
oftworepresentations. More
preisely, aHopfalgebraAis quasi-triangularifthere
existsaninvertibleelementR2AA,alledthe
uni-versalR-matrix,that satisesthefollowingrelations,
(a)R = R(a);8a2A (4)
(I)R = R
13 R
12
(I)R = R
13 R
23
and
(SI)R=(IS 1
)R=R 1
:
AsaresultoftheaboverelationstheuniversalR-matrix
isasolutionoftheYang-Baxterequation,
R
12 R
13 R
23 =R
23 R
13 R
12
(5)
One of best known examples of a quantum group is
U
q (sl
2
), whih is assoiatedto the anisotropi
Heisen-berg model. Muh of theabove disussionan be
ex-tended to superalgebrasand to aÆne (super)algebras.
TheuniversalR-matrixRpresentedabovedoesnot
de-pendonaspetralparameter. AYang-Baxterequation
thatdependsonaspetralparametermaybeobtained
forquantumaÆne(super)algebrasanditssolution
pro-videsaspetralparameterdependentR-matrix,byuse
Baxter-that throughBaxterization itis possible to obtain
R-matriesrelatedtonewintegrablesystems.
II.2 Integrable systems: anoverview
Using the representations of a Yangian algebra A
one is in position to onstrut a one-dimensional
ab-stratintegrablequantumsystem. Forthispurposelet
usdenetheLaxoperatorastheadjointrepresentation
L j
i ()=
X
k ;l R
jl
ik ()e
k
l :
WheretheR-matrixsatisestheYang-Baxter
equa-tiononV V foragivenvetorspaeV. Those
Lax-operatorsareloalmatrixoperators,meaningthatthey
atonsomeinternalspaeatapartiularsite. Onthe
(N+1)-foldtensorprodutspaewemayonstrutthe
monodromymatrix
T()=L
0N ()L
0(N 1) ()::::L
01 ();
whihgivesarepresentationofAonthespaeV N
as
R
12
( )L
13 ()L
23
()=L
23 ()L
13 ()R
12
( );
where thesubsript3refersto thespaeV N
andthe
spae to whih thesubsript 0refers is usually alled
the auxiliaryspae. Onealso denes,in this
represen-tation, thetransfermatrix
t()=tr
0
T()=(())= X
i T
i
i ()
!
and itiseasytoverify thatthetransfermatries
om-mute.
[t();t()℄=0;8;2C:
Letusnowexpandthetransfermatrixinapowerseries
t()= X
k
k
k
;
from theommutationrelationsitfollowsthat
[
k ;
j
℄=0 8k;j;
and one may interpret the operators
k
as
onserva-tion lawsfor some abstrat one-dimensionalquantum
system ating on the spae V N
. We onsider that
a model is integrable when the number of onserved
quantitiesandthenumberofdegreesoffreedomofthe
systemareequal.
From theintertwining relation forthe monodromy
matrixwhen=itisreasonabletoexpetthatR (0)
anbemadeproportionaltothepermutationoperator
P,withthatinminditmaybepossibletoextratfrom
the transfer matrixsome operators for whih one an
giveaphysialinterpretation.
In partiular, assuming that the R-matrix an be
normalizedsothatR (0)=P,onehas
t(0)=tr L(0)=P P ::::P :
Thisoperatorwillatasashiftoperatoronaperiodi
lattieofN sites.
AlsousuallyonedenestheHamiltonianas
H =t 1
(0)t 0
(0)=t 0
(0)t 1
(0);
wheretheprimedenotesdierentiationwithrespetto
thevariable. Furthermore itispossibletoseethat
t 1
(0)t 0
(0)= N
X
i=1 h
i(i+1) ;
where
h= d
d
( PR ()) j
=0 ;
andN+11.
Thus we have dened a Hamiltonian that is the
sum of loal two-site Hamiltonians ating on a
one-dimensionallattiewithperiodiboundaryonditions.
Most integrable models on one dimensional latties,
suh as the XXZ or anisotropi Heisenberg model
[4℄, the supersymmetri t-J model [15-19℄ and the U
model[20, 21℄, fall into the abovedesription. Within
the framework of the QISM in order to obtain
solu-tionsof partiularmodelsandgetthespetrumofthe
ommuting familyof operatorst()one usesthe
alge-braiBetheansatz(ABA) [6, 9℄. The ABA providesa
rigorousmethod for obtainingtheBethe ansatz
equa-tions. As the main interest here is to reveal some of
the interplay between quantum groups and quantum
integrablesystems we refer to the exellent literature
available[6,9℄forthedetailsontheABA.
III Multiparametri Integrable
Systems
Hereweshowhowtoobtainmultiparametriquantum
spin hains using Reshetikhin's onstrution [22℄ for
multiparametri quantum algebras. We also
demon-strate that under appropriate onstraints these
mod-els may be transformed to quantum spin hains with
twistedboundaryonditions[12℄.
Let (A; ; R ) denote a quasitriangular Hopf
al-gebra where and R denote the o-produt and
R-matrixrespetively. Suppose that there exists an
ele-mentF 2AAsuh that
(I)(F)= F
13 F
23
; (I)(F) =F
13 F
12 ;
F
12 F
13 F
23
= F
23 F
13 F
12
; F
12 F
21
=I (6)
Theorem 1 of [22℄ states that (A; F
; R F
) is also a
quasitriangular Hopf algebra with o-produt and
R-matrixrespetivelygivenby
F
=F
12 F
21
; R
F
=F
21 R F
21
: (7)
Intheasethat(A; ; R )isanaÆnequantumalgebra
wehavefrom[22℄ thatF anbehosentobe
F =exp X
(H
i H
j H
j H
i )
ij
wherefH
i
gisabasisfortheCartansubalgebraofthe
aÆne quantum algebra and the
ij
; i < j are
arbi-trary omplex parameters. For our purposes we will
extendtheCartansubalgebrabyanadditionalentral
extension(not theusualentral harge)H
0
whih will
atasasalarmultiple oftheidentityoperatorin any
representation.
SupposethatisalooprepresentationoftheaÆne
quantum algebra. Welet R (), R F
() be the matrix
representativesof R and R F
respetively, whih both
satisfytheYang-Baxterequation
R 12 ( )R 13 ()R 23
()=R
23 ()R 13 ()R 12 ( ):
If R ()j
=0
= P with P the permutation
opera-tor then R F
()
=0
=P as a result of (6). We may
onstrutthetransfermatrix
t F
()=tr
0 (N+1) I F N R F 01 = tr 0 R F 0N ()R F 0(N 1) ()::::R F 01 () ; (9) where F N
isdenedreursivelythrough
F
N
= II:::: F F N 1 = F
I::::I
F
N 1
: (10)
Thesubsripts0and1,2,...,N denotetheauxiliaryand
quantum spaes respetively and tr
0
is the traeover
thezerothspae. FromtheYang-Baxterequationit
fol-lows that the multiparametritransfer matriest F
()
form a ommuting family. The assoiated
multipara-metrispinhain Hamiltonianisgivenby
H F = t F () 1 d du t F () =0 = N 1 X i=1 h F i;i+1 +h F N1 ; (11) with h F = d du PR F () =0 :
Throughuseof(6)wemayalternativelywrite
J N = G N 1 G N 2 ::::G 1 ; G i = F iN F i(N 1) ::::F i(i+1) ; (12)
weandene anewtransfermatrix
t()=J 1 N t F ()J N =tr 0 (N+1) (I N )(F 10 R 01 F 10 ) ;
wherewehaveemployedtheonventiontoletF denote
boththealgebraiobjetanditsmatrixrepresentative.
Throughfurtheruseof(6)wemayshowthat
t()=tr
0 F 10 F 20 ::::F N0 R 0N ()R 0(N 1) () (13)
:::R ()F ::::F );
andtheassoiatedHamiltonianisgivenby
H = (t()) 1 d du t() =0 (14) = N 1 X i=1 h i;i+1 + F N(N 1) ::::F N1 2 h N1 F 1N ::::F (N 1)N 2 where h= d du PR ()
=0 :
The above Hamiltonian desribes a losed system
where instead of the usual periodi boundary
ondi-tions we now have a more general type of boundary
ondition. The boundary termin theabove
Hamilto-nianisaglobaloperator;i.e. itatsnon-triviallyonall
sites. Howeverweaninfatinterpretthistermasa
lo-aloperatorwhihouplesonlythesiteslabeled1and
N. Itanbeshownthattheboundarytermommutes
withtheloalobservablesh
i;i+1
fori6=1; N 1. This
situation is analogous to the losed quantum algebra
invarianthainsdisussedin [23℄.
Fromtheaboveonstrutionwemayalsoyield
mod-elswithtwistedboundaryonditionsbyanappropriate
hoie of F. Reallthat we extendthe Cartan
subal-gebrabythe entralelementH
0
. Letthis elementat
asI in the representation where issomeomplex
number. If we now hoose
ij
= 0 for i 6= 0 in the
expression(8)thematrixF fatorizesasF =M 1
1 M
2
with
M=exp l X i=1 0i H i ! ;
and l is the rank of the underlying quantum algebra
U
q
(g). Usingthefatthat theR-matrixsatises
[R (); I H
i +H
i
I℄=0; i=1;2;:::;l
tellsusthat
[R (); M
1 M
2 ℄=0:
InthisasetheHamiltonian(15)reduesto
H = N 1 X i=1 h i;i+1 +M 2N 1 h N1 M 2N 1 ; (15)
whih is preisely the form for asystem with twisted
boundaryonditions(see[24℄).
IV Final Remarks
Weareawarethatinthisshortandhighlybiasedreview
it is virtuallyimpossibleto onveyallthe rihness
in-volvedinthestudyofquantumintegrablesystemsand
quantumgroups. Neverthelessitis importantto
men-tionsomeofitshighlightsastheintersetionwith
on-formal eld theories (see [25℄ and referenes therein).
In fat from the saling limit of integrable models it
ispossibletoobtaininformationaboutonformaleld
the Bethe ansatz equations of some models it is
pos-sible to alulate suh quantities asthe entral harge
andtheonformaldimensions.
Alsoduetothequestforanappropriatetheory
ex-plaining high T
superondutivity there has been an
inreasinginterestin integrablemodelsdesribing
or-relatedeletrons. Examples ofsuhmodelsarethe
su-persymmetrit Jmodel[15-19℄,theEKS[26℄andthe
U model[20, 21℄ that are generalizations of the
Hub-bardmodel(see[27,28℄),aswellasmodelsrelatedtoa
Temperley-Liebalgebra[29-31℄. Morereentlyan
inte-grable modelof orrelatedeletronsbasedonanso(5)
algebra andshowingo-diagonallong-rangeorder has
beenproposed[32℄.
The future points towards a multitude of
di-retions some of them being the investigation of
reation-diusion models [33℄,the study of q-vertex
operators[34℄, the relation of integrable models with
quantummatrixmodels[35℄andthereentlyproposed
integrableladdermodels[36℄.
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