AKS scheme for face and Calogero-Moser-Sutherland type models
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CRM-2507PUPT-1731LMU-TPW 97-24AKS scheme for face and Calogero-Moser-Sutherland type models Branislav Jurˇc o ?and Peter Schupp ???CRM,Universit′e de Montr′e al Montr′e al (Qc),H3C 3J7,Canada ??Department of Physics,Princeton University Princeton,NJ 08544-0708,USA Abstract We give the construction of quantum Lax equations for IRF mod-
els and di?erence versions of Calogero-Moser-Sutherland models in-troduced by Ruijsenaars.We solve the equations using factoriza-tion properties of the underlying face Hopf algebras/elliptic quantum groups.This construction is in the spirit of the Adler-Kostant-Symes method and generalizes our previous work to the case of face Hopf algebras/elliptic quantum groups with dynamical R-matrices.
1Introduction
Face Hopf algebras[1]have been found to be the algebraic structure that un-derlies some particularly interesting integrable models of statistical physics. They generalize Hopf algebras and quantum groups.Another closely related generalization of quantum groups,the so-called elliptic quantum groups,were introduced by Felder[2]in the context of IRF(face)models[3].Face Hopf algebras and elliptic quantum groups play the same role for face models as quantum groups do for vertex models.The R-matrices are now replaced by dynamical R-matrices,which?rst appeared in the context of Liouville string ?eld theory[4];they can be understood as a reformulation of the Boltz-mann weights in Baxters solutions of the face-type Yang-Baxter equation.A partial classi?cation of the dynamical R-matrices is given in[5].Recently an-other type of integrable quantum systems,known as Ruijsenaars models[6], and their various limiting cases have been shown to be connected to quan-tum groups and elliptic quantum groups[7,8,9,10,11]through di?erent approaches.
The Calogero-Moser-Sutherland class of integrable models describe the motion of particles on a one-dimensional line or circle interacting via pairwise potentials that are given by Weierstrass elliptic functions and their various degenerations.The simplest case is an inverse r2potential.The Ruijsenaars-Schneider model is a relativistic generalization,whose quantum mechanical version,the Ruijsenaars model,is the model that we are interested in here.
The Hamiltonian of the Ruijsenaars model for two particles with coordi-nates x1and x2has the form
H= θ(cηθ(?λ)t(λ)1+θ(cηθ(λ)t(λ)2 ,
whereλ=x1?x2.Here c∈C is the coupling constant,ηis the relativistic deformation parameter,theθ-function is given in(27);we have set =1.
1
The Hamiltonian acts on a wave function as
H ψ(λ)=θ(cηθ(?λ)ψ(λ?η)+θ(cη
θ(λ)
ψ(λ+η),the t (λ)i that appear in the Hamiltonian are hence shift-operators in the vari-able λ;in the present case of two particles they generate a one-dimensional graph:
r r λt 1t 2r r
r ×
space can be chosen to be elements of a Face Algebra F[1](or weak C?-Hopf algebras[12,13],which is essentially the same with a?-structure).
There are two commuting projection operators e i,e i∈F for each vertex of the graph(projectors onto bra’s and ket’s corresponding to vertex i):
e i e j=δij e i,e i e j=δij e i, e i= e i=1.(1) F shall be equipped with a coalgebra structure such that the combination e i j≡e i e j=e j e i is a corepresentation:
?(e i j)= k e i k?e k j,?(e i j)=δij.(2) It follows that?(1)= k e k?e k=1?1(unless the graph has only a single vertex)–this is a key feature of face algebras(and weak C?-Hopf algebras).
So far we have considered matrices with indices that are vertices,i.e. paths of length zero.In the given setting it is natural to also allow paths of ?xed length on a?nite oriented graph G as matrix indices.To illustrate this, here are some paths of length3on a graph that is a square lattice,e.g.on part of a graph corresponding to a particular Ruijsenaars system:
We shall use capital letters to label paths.A path P has an origin(source)·P,an end(range)P·and a length#P.Two paths Q,P can be concatenated to form a new path Q·P,if the end of the?rst path coincides with the start of the second path,i.e.if Q·=·P(this explains our choice of notation).
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The important point is,that the symbols T A B ,where #A =#B ≥0,with
relations
? T A B = A ′
T A A ′?T A ′B
(#A =#A ′=#B )(3)?(T A B )=δAB
(4)T A B T C D =δA ·,·C δB ·,·D T A ·C B ·D (5)span an object that obeys the axioms of a face algebra.Relations (3)and (4)
make T A B a corepresentation;(5)is the rule for combining representations.
The axioms of a face algebra can be found in [1,14].
Pictorial representation:
T A B ～6-
B A T A B T
C
D ～6-B
A -66-
as if it was the canonical element T12of U?F,where U is the dual of F via the pairing , .1
T,f =f, T1?T1,f =?f, T1T2,f?g =fg;f,g∈F
A face Hopf algebra has an anti-algebra and anti-coalgebra endomor-phism,called the antipode and denoted by S or—in the universal tensor formalism—by?T: ?T,f =S(f).The antipode satis?es some compatibility conditions with the coproduct that are given in the appendix.
Remark:In the limit of a graph with a single vertex a face Hopf algebra is the same as a Hopf algebra.Ordinary matrix indices correspond to closed loops in that case.
1.2Boltzmann weights
By dualization we can describe a coquasitriangular structure of F by giving a quasitriangular structure for U.The axioms[1]for a quasitriangular face algebra are similar to those of a quasitriangular Hopf algebra;there is a universal R∈U?U that controls the non-cocommutativity of the coproduct in U and the non-commutativity of the product in F,
RT1T2=T2T1R,?RT2T1=T1T2?R,?R≡(S?id)(R),(6) however the antipode of R is no longer inverse of R but rather
?RR=?(1),R?R=?′(1).(7)
The numerical“R-matrix”obtained by contracting R with two face corep-resentations is given by the face Boltzmann weight W:
R,T A B?T C D =R AC BD≡W C B A D ～?-A B C D
The pictorial representation makes sense since the Boltzmann weight is zero unless C ·A and B ·D are valid paths with common source and range as will
be discussed in more detail below.Also note that T A B → R ,T A B ?T C D is a
representation of the matrix elements of (T A B )while T C D → R ,T A B ?T C D is
an anti-representation .Consistent with our pictorial representation for the T -matrices we see that the orientation of the paths in F -space remain the same for the ?rst case but are reversed for the latter:
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-B A ～?-A B rep .?→?-
A B C D anti ?rep .←?? C D ～6
-D C De?nition:For f ∈F we can de?ne two algebra homomorphisms F →U :
R +(f )= R,f ?id ,R ?(f )= ?R,id ?f (8)Yang-Baxter Equation.As a consequence of the axioms of a quasitriangular face Hopf algebra R satis?es the Yang-Baxter Equation
R 12R 13R 23=R 23R 13R 12∈U ?U ?U .(9)
Contracted with T A B ?T C D ?T E F this expression yields a numerical Yang-Baxter
equation with the following pictorial representation [3]:
S S S S S S /S w ---R 13R 23R 12R 23
R 13R 12=B A E C F D B A
E C F
D The inner edges are paths that are summed over.Moving along the outer edges of the hexagon we will later read of the shifts in the Yang-Baxter equation for dynamical R -matrices.
So far we have argued heuristically that the Ruijsenaars system naturally leads to graphs and face algebras.The formal relation between face Hopf
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algebras and oriented graphs it is established by a lemma of Hayashi(Lemma 3.1of[14])which says that any right or left comodule M of a face Hopf algebra F decomposes as a linear space to a direct sum M=⊕i,j M ij with indices i,j running over all vertices.So we can naturally associate paths from j to i to any pair of indices such that M ij=?.So we may speak(and we really do in the paper)of the vectors of a comodule or a module as of paths.As the dual object U to a face Hopf algebra is again a face Hopf algebra characterized by the same set of vertices[14],the same applies to its comodules.It is convenient to choose the orientation of paths appearing in the decomposition of a comodule of U(and hence in the module of F) opposite to the convention that one uses in the case of F.
Particularly we have for any matrix corepresentation T A B of F with sym-bols A,B used to label some linear basis in M,the linear span T A B of all T A B decomposes as linear space(bicomodule of the face Hopf algebra)as a direct sum
i,j,k,l e i e j T A B e k e l = T A B
i.e.a sum over paths with?xed starting and ending vertices.The upper indices i and k?x the beginning and the end of the path A and the lower indices j and l?x the beginning and the end of the path B.
Let us assume that the matrix elements T A B of a corepresentation of F act in a module of paths that we shall label by greek charactersα,β,etc.The de?nition of the dual face Hopf algebra implies that the matrix element(T A B)αβis nonzero only if·α=·B,·A=α·,B·=·βand A·=β·,i.e.if pathsα·A and B·βhave common starting and endpoints.This justi?es the pictorial representation used in the paper.It also follows immediately that in the case of a coquasitriangular Face Hopf algebra R,T A B?T C D =R AC BD≡W C B A D is zero unless·C=·B,B·=·D,C·=·A and A·=D·.
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To make a contact with the Ruijsenaars type of models,we have to assume that the(coquasitriangular)face Hopf algebra F is generated by the matrix elements of some fundamental corepresentation of it.We shall postulate the paths of the corresponding corepresentation to be of length1.The paths belonging to the n-fold tensor product of the fundamental corepresentation are then by de?nition of length n.Taking an in?nite tensor product of the fundamental corepresentation we get a graph that corresponds to the one generated by the shift operators of the related integrable model.
In the next section we are going to formulate a quantum version of the so-called Main Theorem which gives the solution by factorization of the Heisenberg equations of motion.For this construction F needs to have a coquasitriangular structure–this will also?x its algebra structure.
2Quantum factorization
The cocommutative functions in F are of particular interest to us since they form a set of mutually commutative operators.We shall pick a Hamiltonian from this set.Cocommutative means that the result of an application of comultiplication?is invariant under exchange of the two resulting factors. The typical example is a trace of the T-matrix.The following theorem gives for the case of face Hopf algebras what has become known as the“Main theorem”for the solution by factorization of the equations of motion[15,16, 17,18,19].The following theorem is a direct generalization of our previous results for Hopf algebras/Quantum Groups[20,21]:
Theorem2.1(Main theorem for face algebras)
(i)The set of cocommutative functions,denoted I,is a commutative sub-
algebra of F.
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(ii)The Heisenberg equations of motion de?ned by a Hamiltonian H∈I are of Lax form
dT
i
g±(t)=M±(t)g±(t),g±(0)=1.(14)
dt
Proof:The proof is similar to the one given in[21]for factorizable qua-sitriangular Hopf algebras.Here we shall only emphasize the points that are di?erent because we are now dealing with face algebras.An important relation that we shall use several times in the proof is
?(1)T1T2=T1T2=T1T2?(1).(15) (Note that the generalization is not trivial since now?(1)=1?1in general.)
(i)Let f,g∈I?F,then fg∈I.Let us show that f and g commute:
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fg= T1T2,f?g = ?(1)T1T2,f?g = ?RRT1T2,f?g =
?RT2T1R,f?g = T2T1R?R,f?g = T2T1?′(1),f?g =
T2T1,f?g =gf.
(?R≡(S?id)(R)can be commuted with T2T1R in the?fth step because
f and
g are bot
h cocommutative.)
(ii)We need to show that R±(H(2))?H(1),T =0.This follows from the cocommutativity of H and(6):
T1R±21T2,H?id = R±21T1T2,H?id = T2T1R±21,H?id . (iii)We need to show that[m+,m?]=0;then the proof of[21]applies.
m+m?= T1R13T2?R32,H?H?id ≡ T1T2R13?R32,H?H?id
= ?R12T2T1R12R13?R32,H?H?id = ?R12T2T1?R32R13R12,H?H?id = R12?R12T2T1?R32R13,H?H?id = ?′(1)T2T1?R32R13,H?H?id = T2?R32T1R13,H?H?id =m?m+.
Remark:The objects in this theorem(M±,m±,g±(t),T(t))can be inter-preted(a)as elements of U?F,(b)as maps F→F or(c),when a repre-sentation of U is considered,as matrices with F-valued matrix elements.
3Dynamical operators
In the main theorem we dealt with expressions that live in U?F(and should be understood as maps from F into itself).In this section we want to write expressions with respect to one?xed vertex.Like we mentioned in the introduction we are particularly interested in the action of the Hamiltonian with respect to a?xed vertex.Since the Hamiltonian is an element of F,we shall initially?x the vertex in this space;due to the de?nition of the dual face Hopf algebra U this will also?x a corresponding vertex in that space.
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We shall proceed as follows:We will ?x a vertex in the T -matrix with the help of e λ,e λ∈F and will also introduce the corresponding universal T (λ).Next we will contstruct dynamical R -matrices R ±(λ)as R ±(T (λ))–this is an algebra homomorphism –and will give the Yang-Baxter and RT T equations with shifts as an illustration.Finally we shall plug everything into the main theorem.
Convention for corepresentation T with respect to a ?xed vertex:T (λ)A B is zero unless the range (end)of path A is equal to the ?xed vertex λ.Such a T will map the vector space spanned by vectors v A with A ·=λ?xed to itself.With the help of the projection operator e λ∈F we can give the following explicit expression:
T (λ)A B =T A B e λ～
6-×λB A (16)
The universal T (λ)is an abstraction of this an is de?ned analogously.
T 12(λ)=T 12(e λ)2～6-×λ(17)Coproducts of T .Expressions for the coproducts ?1T (λ)and ?2T (λ)follow either directly from the de?nition of T (λ)or can be read of the corresponding pictorial representations.
(i)Coproduct in F -space [2]:
?2T (λ)=T 12(λ)T 13(λ?h 2)～×λ×λ?h 2
6-6-(18)
The shift operator h in F -space that appears here is
h (F )= η,μ(μ?η)e μη∈F,
(19)
11
where we assume some appropriate (local)embedding of the vertices of the underlying graph in C n so that the
di?erence of vertices makes sense.Proof:?2T 12(e λ)2=
ηT 12T 13(e λη)2(e η)3=
η,μT 12(e λe μ)2(e η)2T 13(e η)3=
η,μT 12(λ)(e μη)2T 13(λ+η?μ)=T 12(λ)T 13(λ?h 2).In the second and third step we used e λe μ∝δλ,μ.
(ii)Coproduct in U -space:(gives multiplication in F )
?1T (λ)=T 13(λ?h 2)T 23(λ)～6--
Pictorially:
×λZ Z Z ~Z Z Z ~ > >
>Z Z Z ~666
T 2T 1R 12Shifts h 1,h 2are in U -space,shift h 3is in F -space.Twice contracted with R the dynamical RT T -equation yields the dynamical Yang-Baxter equation:Dynamical Yang-Baxter equation [4]
R 12(λ)R 13(λ?h 2)R 23(λ)=R 23(λ?h 1)R 13(λ)R 12(λ?h 3)
(24)Pictorially:
×λZ Z Z ~Z Z Z ~ > > >Z Z Z ~666R 13R 23R 12=×λZ Z Z ~ > >Z Z Z ~ >Z Z Z ~66R 23R 13
R 12Hamiltonian and Lax operators in dynamical setting
The Hamiltonian H should be a cocomutative element of F .In the case of the Ruijsenaars model it can be chosen to be the trace of a T -matrix,i.e.the U -trace of T in an appropriate representation ρ:H =tr (ρ)1T 1.This can be written as a sum over vertices λof operators that act in the respective subspaces corresponding to paths ending in the vertex λ:
H = Q,#Q ?xed
T Q Q = λH (λ)
(25)with H (λ)=H e λ=
T (λ)Q Q .The pictorial representation of the Hamil-
tonian is two closed dashed paths (F -space)connected by paths Q of ?xed length that are summed over.In H (λ)the end of path Q is ?xed.When we
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look at a representation on Hilbert space the paths Q with endpointλthat appear in the component H(λ)of the Hamiltonian H will shift the argument of a stateψ(λ)corresponding to the vertexλto a new vertex corresponding to the starting point of the path Q.In the next section we will see in detail how this construction is applied to the Ruijsenaars system.For this we will have to take a representation of the face Hopf algebra F.We shall then denote the resulting Lax operator by L.The Hamiltonian will contain a sum (coming from the trace)over shift operators.
For the Lax operators it is convenient to?x two verticesλandμcor-responding to paths fromμtoλ.(We did not need to do this for the Hamiltonian since it acts only on closed paths.)The T that appears in the Lax equation(10)becomes
T12(λ,μ)=T12(eμeλ)2=(Eλ)1T12(Eμ)1;
it should be taken in some representation of F on the appropriate Hilbert space.On space U we are interested in a?nite-dimensional representation that is going to give us matrices.Both is done in the next section and we shall call the resulting operator L(λ,μ).Similarly we proceed with the other two Lax operators M±:
M±(λ,μ)≡(Eλ?1)M±(Eμ?1)=Eλδλ,μ?H?m±(λ,μ).
(Recall that EλEμ=Eλδλ,μ.)If we de?ne M±012=T02(1?R±01)then M±12= M±012,H?id2 and we can write dynamical Lax equations in an obvious notation as
dT(λ,μ)
i
Remark:There is another possible choice of conventions for the?xed vertex. We could worked with T12{ν}=(eν)2T12instead of T12(λ)=T12(eλ)2.This would have?xed the lower left vertex in the pictorial representation of T A B and the vertex that is starting-point for all paths in R{ν}.The dynamical equations would of course look a little di?erent from the ones that we have given.
4The Lax Pair
Here we give as an example the Lax pair for the case of the N-particle quantum Ruijsenaars model.We may think of the face Hopf algebra F as of the elliptic quantum group associated to sl(N)introduced by Felder[2]. We will continue to use the the symbol F for it.In that case there is an additional spectral parameter entering all relations in the same way as it is in the case ordinary quantum groups.
Let h be the Cartan subalgebra of sl(N)and h?its dual.The actual graph related to the elliptic quantum group F is h?～C N?1.This is the huge graph of the remark made in the introduction.However it decomposes into a continuous family of disconnected graphs,each one isomorphic to ?η.Λ,the?η∈C multiple of the weight latticeΛof sl(N),and we can restrict ourselves to this one component for simplicity.Correspondingly the shifts in all formulas are rescaled by a factor?η.The space h?～C N?1 itself will be considered as the orthogonal complement of C N=⊕i=1,...,N Cεi, εi,εj =δij with respect to i=1,...,Nεi.We write the orthogonal projection ?i=εi?1
V ρ,λ～C η.?k and a corresponding path of length 1to any pair λ,ρ∈η.Λ,such that ρ?λ=η?k ,for some k =1,...,N .We let V ρ,λ=?for all other pairs of vertices.The vector space V of the fundamental corepresentation is formed by all paths of length 1
V = λ,ρV ρ,λ～ λ,ρ-××ρλ= λ,i
-××λ+η?i λ
As all spaces V ρ,λare at most one-dimensional,we can characterize the numer-ical R -matrix R AC BD in the fundamental corepresentation by just four indices referring to the vertices of the “square”de?ned by paths A,B,C,D in the case of nonzero matrix element R AC BD .Let us set ·B =·C =ν,·D =B ·=μ,D ·=A ·=λand ·A =C ·=ρand also
R AC BD =W C B
A D ≡W νρμ
λ ～-?νρμ
λ
.Then the non-zero Boltzmann weights as given by [22]are:(i =j )
W λ+2η?i λ+η?i
λ+η?i
λ u =1～--λ×,
W λ+η(?i +?j )λ+η?i λ+η?i
λ u =θ(η)θ(?u +λij )?
-
θ(u +η)θ(λij )～--
λ
×.
Here λij ≡λi ?λj = λ,?i ??j ,θ(u )is the Jacobi theta function
θ(u )= j ∈Z
e πi (j +12)(u +1
the vertices that ?x the length-1paths A and B .Let us use the following notation for it (i,j =1,...,N )
T A B (λ|u )= μL i j (μ,λ|u )e μ～6-××××μ+η?j μ
λ+η?i λA B where μ=B ·,λ=A ·,μ+η?j =·B ,λ+η?i =·A .
As the lattice η.Λthat we consider is just one connected component of a continuous family of disconnected graphs,the vertices λ,μ,...are allowed to take any values in C (N ?1).This will be assumed implicitly in the rest of this section.Now we need to specify the appropriate representation of F which can be read of from [7].We can characterize it by its path decomposition.To
any pair of vertices λ,μwe associate a one-dimensional vector space ?V
λ,μ～C (path from μto λ).The representation space ?V
is then ?V = λ,μ∈h ?
?V
λ,μ.The matrix element T A B (λ)≡L i j (λ,μ|u )for ?xed A,B and hence also with
?xed i ,j ,λ,μis obviously non-zero only if restricted to act from ?V
λ,μto ?V λ+η?i ,μ+η?j
in which case it acts as multiplication by L i j (λ,μ|u )=θ(cη
θ(u ) k =i θ(cηθ(λk ?λi ).(28)
Here we used notation λi = λ,?i for λ∈h ?.c ∈C will play the role of coupling constant.
The Hamiltonian is chosen in accordance with Section 3as H = P T P P ,i.e.the trace in the “fundamental”corepresentation.In accordance with the discussion of the previous sections it is non-zero only when acting on the diagonal subspace (closed paths)
H =
λ∈h ?H λ≡ λ∈h ?
?V λ,λ17
of ?V
.So this is the actual state space of the integrable system under consid-eration.The Lax operator M ±acts from ?V λ,μto ?V λ+η?i ,μ+η?j
as multiplication by M ±l k (λ,μ|u,v )=(1?H )l k (λ,μ|v )?m ±l k (λ,μ|u,v )with
(1?H )l k (λ,μ|v )=δλ,μδl k θ(
cη
θ(v ) j ′=i θ(cηθ(λj ′
,i ),(29)m +l k (λ,μ|u,v )=
δ?i +?k ,?j +?l δλ,μ+η?j i,j L i j (μ+η?k ,μ|v )W
λ+η?l μ+η?k λμ v ?u
(30)
～??
--×
×××??μ
λ
λ+η?l μ+η?k μλm ?is given by a similar formula with the inverse Boltzmann weight.Dynamical Lax Equation
i dL i
k (λ,μ|u )
N +λj,i )
Acknowledgement
We would like to thank Pavel Winternitz for constant interest and support, Koji Hasegawa and Jan Felipe van Diejen for interesting discussions and Elliott Lieb for hospitality.
Appendix
For many reasons it is very convenient to use a formalism based on the so-called universal T.The expressions formally resemble those in a matrix representation but give nevertheless general face Hopf algebra statements. This greatly simpli?es notation but also the interpretation and application of the resulting expressions.When we are dealing with quantizations of the functions on a group we need to keep track both of the non-commutativity of the quantized functions and the residue of the underlying group structure. Both these structures can be encoded in algebraic relations for the universal T which easily allows to control two non-commutative structures.In fact T can be regarded as a universal group element.Universal tensor expressions can formally be read in two ways:Either as“group”operations(or rather operations in U)or as the corresponding pull-back maps in the dual space. This simpli?es the heuristics of“dualizing and reversing arrows”and allows us to keep track of the classical limit.Example:Multiplication in U,x?y→xy:The corresponding pull-back map in the dual space F is the coproduct ?.Both operations are summarized in the same universal expression T12T13.
Sometimes T can be realized as the canonical element U?F,but we do not need to limit ourselves to these cases an will instead de?ne T as the identity map from F to itself and will use the same symbol for the identity map U→U.
For the application of this identity map to an element of F we shall nevertheless use the same bracket notation as we would for a true canonical
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