Infinitely many commuting operators for the elliptic quantum group U q,p ( s l N ^ )
aa r X i v : . [ n li n . S I] J a n Infinitely many commuting operators forthe elliptic quantum group U q,p ( c sl N ) May 1, 2018
Takeo KOJIMA
Department of Mathematics and Physics, Graduate School of Science and Engineering,Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan
Abstract
We construct two classes of infinitely many commuting operators associated with theelliptic quantum group U q,p ( d sl N ). We call one of them the integral of motion G m , ( m ∈ N )and the other the boundary transfer matrix T B ( z ), ( z ∈ C ). The integral of motion G m is related to elliptic deformation of the N -th KdV theory. The boundary transfer matrix T B ( z ) is related to the boundary U q,p ( d sl N ) face model. We diagonalize the boundary transfermatrix T B ( z ) by using the free field realization of the elliptic quantum group, howeverdiagonalization of the integral of motion G m is open problem even for the simplest case U q,p ( c sl ). The free field approach provides a powerful method to study exactly solvable model [1]. Thebasic idea in this approach is to realize the commutation relations for the symmetry algebra andthe vertex operators in terms of free fields acting on the Fock space. We introduce the elliptic1uantum group U q,p ( d sl N ) [2, 3], and give its free field realization. Using the free field realizations,we introduce two extended currents F N ( z ) [4] and U ( z ) [5] associated with the elliptic quantumgroup U q,p ( d sl N ). We construct two classes of infinitely many commuting operators for the ellipticquantum group U q,p ( d sl N ). We call one of them the integral of motion G m , ( m ∈ N ) [4] and theother the boundary transfer matrix T B ( z ), ( z ∈ C ) [6]. Our constructions are based on thefree field realizations of the elliptic quantum group U q,p ( d sl N ), the extended currents and thevertex operator Φ ( a,b ) ( z ). Commutativity of the integral of motion is ensured by Feigin-Odesskiialgebra [7], and those of the boundary transfer matrix is ensured by Yang-Baxter equation andboundary Yang-Baxter equation [8]. Two classes of infinitely many commuting operators havephysical meanings. The integral of motion G m is two parameter deformation of the monodromyof the N -th KdV theory [9, 10]. The boundary transfer matrix T B ( z ) is related to the boundary U q,p ( d sl N ) face model that is lattice deformation of the conformal field theory. We diagonalize theboundary transfer matrix T B ( z ) by using the free field realization of the elliptic quantum groupand the vertex operators. Diagonalization of the boundary transfer matrix allows us calculatecorrelation functions of the boundary U q,p ( d sl N ) face model [11, 12, 6].The organization of this paper is as follows. In section 2 we introduce the elliptic quantumgroup U q,p ( d sl N ) [2, 3], and give its free field realization. In section 3 we introduce two extendedcurrents F N ( z ) , E N ( z ) [4] and U ( z ) , V ( z ) [5, 13] associated with the elliptic quantum group U q,p ( d sl N ). We give the free field realization of the vertex operators Φ ( a,b ) ( z ), using the extendedcurrent U ( z ). We construct two classes of infinitely many commuting operators associated withthe elliptic quantum group U q,p ( d sl N ). The one is the integral of motion G m [4] and the otheris the boundary transfer matrix T B ( z ) [6]. In section 4 we diagonalize the boundary transfermatrix T B ( z ) by using the free field realization of the vertex operators [5, 13, 6]. U q,p ( c sl N ) In this section we introduce the elliptic quantum group U q,p ( d sl N ) and its free field realization. In this section we recall Drinfeld realization of the quantum group [14]. We fix a complex number q such that 0 < | q | <
1. Let us fix the integer N = 3 , , , · · · . We use q-integer [ n ] q = q a − q − a q − q − .We use the abbreviation,( z ; p , p , · · · , p M ) ∞ = ∞ Y k ,k , ··· ,k M =0 (1 − p k p k · · · p k M M z ) . U q ( d sl N ) is generated by h j , a j,m , x j,n , (1 ≦ j ≦ N − m ∈ Z =0 , n ∈ Z ), c, d . Let us set the generating functions x ± j ( z ) , ψ j ( z ) , ϕ j ( z ), (1 ≦ j ≦ N −
1) by x ± j ( z ) = X n ∈ Z x ± j,n z − n ,ψ j ( q c z ) = q h j exp ( q − q − ) X m> a j,m z − m ! ,ϕ j ( q − c z ) = q − h j exp − ( q − q − ) X m> a j, − m z m ! . The defining relations are given by[ d, x ± j,n ] = nx ± j,n , [ h j , d ] = [ h j , a k,m ] = [ d, a k,m ] = 0 , c : central , [ a j,m , a k,n ] = [ A j,k m ] q [ cm ] q m q − c | m | δ m + n, , [ h j , x ± k ( z )] = ± A j,k x ± k ( z ) , [ a j,m , x + k ( z )] = [ A j,k m ] q m q − c | m | z m x + k ( z ) , [ a j,m , x − k ( z )] = − [ A j,k m ] q m z m x − k ( z ) , ( z − q ± A j,k z ) x ± j ( z ) x ± k ( z ) = ( q ± A j,k z − z ) x ± k ( z ) x ± j ( z ) , [ x + j ( z ) , x − k ( z )] = δ j,k q − q − ( δ ( q − c z /z ) ψ j ( q c z ) − δ ( q c z /z ) ϕ j ( q − c z )) , and Serre relation for | j − k | = 1,( x ± j ( z ) x ± j ( z ) x ± k ( z ) − [2] q x ± j ( z ) x ± k ( z ) x ± j ( z ) + x ± k ( z ) x ± j ( z ) x ± j ( z ))+( x ± j ( z ) x ± j ( z ) x ± k ( z ) − [2] q x ± j ( z ) x ± k ( z ) x ± j ( z ) + x ± k ( z ) x ± j ( z ) x ± j ( z )) = 0 . Here ( A j,k ) ≦ j,k ≦ N − is Cartan matrix of sl N type. Here we used the delta function δ ( z ) = P m ∈ Z z m . In this section we introduce the elliptic quantum group U q,p ( d sl N ) [2, 3], which is elliptic defor-mation of the quantum group U q ( d sl N ). We fix complex numbers r, s such that Re( r ) > s ) >
0. When we change the polynomial ( z − q − z ) in the defining relation of the quantumgroup U q ( d sl N ), ( z − q − z ) x − j ( z ) x − j ( z ) = ( q − z − z ) x − j ( z ) x − j ( z ) , to the elliptic theta function [ u ], we have[ u − u + 1] F j ( z ) F j ( z ) = [ u − u − F j ( z ) F j ( z ) . U q.p ( d sl N ). We set the elliptictheta function [ u ] , [ u ] ∗ by [ u ] = q u r − u Θ q r ( q u ) , [ u ] ∗ = q u r ∗ − u Θ q r ∗ ( q u ) , Θ p ( z ) = ( p ; p ) ∞ ( z ; p ) ∞ ( pz − ; p ) ∞ , where we set z = x u and r ∗ = r − c . The elliptic quantum group U q,p ( d sl N ) is generated by thecurrents E j ( z ) , F j ( z ), H + j ( q c − r z ) = H − j ( q − c + r z ), (1 ≦ j ≦ N − E j ( z ) E j +1 ( z ) = (cid:2) u − u + sN (cid:3) ∗ (cid:2) u − u + 1 − sN (cid:3) ∗ E j +1 ( z ) E j ( z ) , (2.1) E j ( z ) E j ( z ) = [ u − u + 1] ∗ [ u − u − ∗ E j ( z ) E j ( z ) , (2.2) E j ( z ) E k ( z ) = E k ( z ) E j ( z ) , otherwise, (2.3) F j ( z ) F j +1 ( z ) = (cid:2) u − u + sN − (cid:3)(cid:2) u − u − sN (cid:3) F j +1 ( z ) F j ( z ) , (2.4) F j ( z ) F j ( z ) = [ u − u − u − u + 1] F j ( z ) F j ( z ) , (2.5) F j ( z ) F k ( z ) = F k ( z ) F j ( z ) , otherwise, (2.6) H + j ( z ) H + j ( z ) = [ u − u − u − u + 1] ∗ [ u − u + 1][ u − u − ∗ H + j ( z ) H + j ( z ) , (2.7) H + j ( z ) H + j +1 ( z ) = [ u − u + 1 − sN ][ u − u − sN ] ∗ [ u − u − sN ][ u − u + 1 − sN ] ∗ H + j +1 ( z ) H + j ( z ) , (2.8) H + j ( z ) H + k ( z ) = H + k ( z ) H + j ( z ) , otherwise, (2.9) H + j ( z ) E j ( z ) = [ u − u + 1 + c ] ∗ [ u − u − − c ] ∗ E j ( z ) H + j ( z ) , (2.10) H + j ( z ) E j +1 ( z ) = [ u − u + sN + c ] ∗ [ u − u + 1 − sN − c ] ∗ E j +1 ( z ) H + j ( z ) , (2.11) H + j +1 ( z ) E j ( z ) = [ u − u + 1 − sN + c ] ∗ [ u − u − sN − c ] ∗ E j ( z ) H + j +1 ( z ) , (2.12) H + j ( z ) E k ( z ) = E k ( z ) H + j ( z ) , otherwise, (2.13) H + j ( z ) F j ( z ) = [ u − u − − c ][ u − u + 1 + c ] F j ( z ) H + j ( z ) , (2.14) H + j ( z ) F j +1 ( z ) = [ u − u + sN − − c ][ u − u − sN + c ] F j +1 ( z ) H + j ( z ) , (2.15) H + j +1 ( z ) F j ( z ) = [ u − u − sN − c ][ u − u + sN − c ] F j ( z ) H + j +1 ( z ) , (2.16) H + j ( z ) F k ( z ) = F k ( z ) H + j ( z ) , otherwise, (2.17)4 E i ( z ) , F j ( z )] = δ i,j q − q − (cid:16) δ ( q − c z /z ) H + j (cid:16) q c z (cid:17) − δ ( q c z /z ) H − j (cid:16) q − c z (cid:17)(cid:17) , (2.18)and the Serre relations for | j − k | = 1, (cid:26) ( z /z ) r ∗ ( q r ∗ − z/z ; q r ∗ ) ∞ ( q r ∗ − z/z ; q r ∗ ) ∞ ( q r ∗ +1 z/z ; q r ∗ ) ∞ ( q r ∗ +1 z/z ; q r ∗ ) ∞ E j ( q − sN z ) E j ( q − sN z ) E k ( q − sN z ) − [2] q ( q r ∗ − z/z ; q r ∗ ) ∞ ( q r ∗ − z /z ; q r ∗ ) ∞ ( q r ∗ +1 z/z ; q r ∗ ) ∞ ( q r ∗ +1 z /z ; q r ∗ ) ∞ E j ( q − sN z ) E k ( q − sN z ) E j ( q − sN z )+( z/z ) r ∗ ( q r ∗ − z /z ; q r ∗ ) ∞ ( q r ∗ − z /z ; q r ∗ ) ∞ ( q r ∗ +1 z /z ; q r ∗ ) ∞ ( q r ∗ +1 z /z ; q r ∗ ) ∞ E k ( q − sN z ) E j ( q − sN z ) E j ( q − sN z ) (cid:27) × z − r ∗ ( q r ∗ +2 z /z ; q r ∗ ) ∞ ( q r ∗ − z /z ; q r ∗ ) ∞ + ( z ↔ z ) = 0 , (2.19) (cid:26) ( z /z ) − r ( q r +1 z/z ; q r ) ∞ ( q r +1 z/z ; q r ) ∞ ( q r − z/z ; q r ) ∞ ( q r − z/z ; q r ) ∞ F j ( q − sN z ) F j ( q − sN z ) F k ( q − sN z ) − [2] q ( q r +1 z/z ; q r ) ∞ ( q r +1 z /z ; q r ) ∞ ( q r − z/z ; q r ) ∞ ( q r − z /z ; q r ) ∞ F j ( q − sN z ) F k ( q − sN z ) F j ( q − sN z )+( z/z ) − r ( q r +1 z /z ; q r ) ∞ ( q r +1 z /z ; q r ) ∞ ( q r − z /z ; q r ) ∞ ( q r − z /z ; q r ) ∞ F k ( q − sN z ) F j ( q − sN z ) F j ( q − sN z ) (cid:27) × z r ( q r − z /z ; q r ) ∞ ( q r +2 z /z ; q r ) ∞ + ( z ↔ z ) = 0 . (2.20) In this section we give the free field realization of the elliptic quantum group U q.p ( d sl N ) [2, 3, 5].In what follows we restrict our interest to level c = 1. Let us introduce the bosons β jm , (1 ≦ j ≦ N ; m ∈ Z ) by[ β jm , β kn ] = m [( r − m ] q [ rm ] q [( s − m ] q [ sm ] q δ m + n, (1 ≦ j = k ≦ N ) − mq sm sgn( j − k ) [( r − m ] q [ rm ] q [ m ] q [ sm ] q δ m + n, (1 ≦ j = k ≦ N ) . (2.21)We set the bosons B jm , (1 ≦ j ≦ N ; m ∈ Z =0 ) by B jm = ( β jm − β j +1 m ) q − jm , (1 ≦ j ≦ N − . (2.22)They satisfy [ B jm , B kn ] = m [( r − m ] q [ rm ] q [ A j,k m ] q [ m ] q δ m + n, , (1 ≦ j, k ≦ N − , (2.23)where ( A j,k ) ≦ j,k ≦ N − is Cartan matrix of sl N type. Let ǫ µ (1 ≦ µ ≦ N ) be the orthonormalbasis of R N with the inner product ( ǫ µ | ǫ ν ) = δ µ,ν . Let us set ¯ ǫ µ = ǫ µ − ǫ where ǫ = N P Nν =1 ǫ ν .5et α µ (1 ≦ µ ≦ N −
1) the simple root : α µ = ¯ ǫ µ − ¯ ǫ µ +1 . The type sl N weight lattice is thelinear span of ¯ ǫ µ , P = P N − µ =1 Z ¯ ǫ µ . Let us set P α , Q α ( α ∈ P ) by[ iP α , Q β ] = ( α | β ) , ( α, β ∈ P ) . (2.24)In what follows we deal with the bosonic Fock space F l,k , generated by β j − m ( m >
0) over thevacuum vector | l, k i , where l, k ∈ P . F l,k = C [ { β j − , β j − , · · ·} ≦ j ≦ N ] | l, k i , | l, k i = e i √ rr − Q l − i q r − r Q k | , i , where β jm | l, k i = 0 , ( m > , P α | l, k i = α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r rr − l − r r − r k ! | l, k i . Free field realizations of E j ( z ) , F j ( z ) , H ± j ( z ) (1 ≦ j ≦ N −
1) are given by E j ( z ) = e − i √ rr − Q αj ( q ( sN − j z ) − √ rr − P αj + rr − × : exp − X m =0 m [ rm ] q [( r − m ] q B jm ( q ( sN − j z ) − m : , (2.25) F j ( z ) = e i q r − r Q αj ( q ( sN − j z ) q r − r P αj + r − r × : exp X m =0 m B jm ( q ( sN − j z ) − m : , (2.26) H + j ( q − r z ) = q (1 − sN )2 j e − i √ r ( r − Q αj ( q ( sN − j z ) − √ r ( r − P αj + r ( r − × : exp − X m =0 m [ m ] q [( r − m ] q B jm ( q ( sN − j z ) − m : . (2.27)The free field realization for general level c [17] is completely different from those for level c = 1. In this section we construct two classes of infinitely many commuting operators G m [4] and T B ( z )[6]. E N ( z ) , F N ( z ) In this section we introduce the extended currents E N ( z ) , F N ( z ) [4]. Let us set the extendedcurrent E N ( z ) , F N ( z ) by the similar commutation relations as the elliptic quantum group. The6xtended currents E N ( z ) , F N ( z ) satisfy the following commutation relations. E j ( z ) E j +1 ( z ) = (cid:2) u − u + sN (cid:3) ∗ (cid:2) u − u + 1 − sN (cid:3) ∗ E j +1 ( z ) E j ( z ) , ( j ∈ Z /N Z ) ,E j ( z ) E j ( z ) = [ u − u + 1] ∗ [ u − u − ∗ E j ( z ) E j ( z ) , ( j ∈ Z /N Z ) ,F j ( z ) F j +1 ( z ) = (cid:2) u − u + sN − (cid:3)(cid:2) u − u − sN (cid:3) F j +1 ( z ) F j ( z ) , ( j ∈ Z /N Z ) ,F j ( z ) F j ( z ) = [ u − u − u − u + 1] F j ( z ) F j ( z ) , ( j ∈ Z /N Z ) , [ E j ( z ) , F k ( z )] = δ j,k q − q − (cid:16) δ ( q − z /z ) H + j (cid:16) q z (cid:17) − δ ( qz /z ) H − j (cid:16) q − z (cid:17)(cid:17) , ( j, k ∈ Z /N Z ) , and other defining relations of the elliptic quantum group, in which the suffix j, k should beunderstood as mod. N . Free field realizations of the extended currents E N ( z ) , F N ( z ) and H + N ( q − r z ) = H − N ( q − r z ) are given by E N ( z ) = e − i √ rr − Q αN ( q s − N z ) − √ rr − P ¯ ǫN + r r − z √ rr − P ¯ ǫ + r r − × : exp − X m =0 m [ rm ] q [( r − m ] q B Nm ( q s − N z ) − m : , (3.1) F N ( z ) = e i q r − r Q αN ( q s − N z ) q r − r P ¯ ǫN + r − r z − q r − r P ¯ ǫ + r − r × : exp − X m =0 m B Nm ( q s − N z ) − m : , (3.2) H + N ( q − r z ) = q N − s ) e − i √ rr ∗ Q αN ( q s − N z ) − √ rr ∗ P ¯ ǫN + rr ∗ z √ rr ∗ P ¯ ǫ + rr ∗ × : exp − X m =0 m [ m ] q [( r − m ] q B Nm ( q s − N z ) − m : . (3.3) V ( z ) , U ( z ) In this section we introduce the extended currents V ( z ) , U ( z ) [5, 13]. In this section we considerthe case s = N . For our purpose it is convenient to introduce E j ( z ) = E j ( q − j z ) , F j ( z ) = F j ( q − j z ) , (1 ≦ j ≦ N − . The extended currents U ( z ) , V ( z ) are given by the following commutation relations. (cid:20) u − u + 12 (cid:21) ∗ V ( z ) E ( z ) = (cid:20) u − u + 12 (cid:21) ∗ E ( z ) V ( z ) , (3.4) E j ( z ) V ( z ) = V ( z ) E j ( z ) (2 ≦ j ≦ N ) , (3.5)7 u − u − (cid:21) U ( z ) F ( z ) = (cid:20) u − u − (cid:21) F ( z ) U ( z ) , (3.6) F j ( z ) U ( z ) = U ( z ) F j ( z ) (2 ≦ j ≦ N ) . (3.7) U ( z ) U ( z ) = ( z /z ) r − r N − N ρ ( z /z ) ρ ( z /z ) U ( z ) U ( z ) , (3.8) V ( z ) V ( z ) = ( z /z ) − rr − N − N ρ ∗ ( z /z ) ρ ∗ ( z /z ) V ( z ) V ( z ) , (3.9) U ( z ) V ( z ) = z − N − N Θ q N ( − qz )Θ q N ( − qz − ) V ( z ) U ( z ) , (3.10)where we set ρ ( z ) = ( q z ; q r , q N ) ∞ ( q N +2 r − z ; q r , q N ) ∞ ( q r z ; q r , q N ) ∞ ( q N z ; q r , q N ) ∞ , (3.11) ρ ∗ ( z ) = ( z ; q r ∗ , q N ) ∞ ( q N +2 r − z ; q r ∗ , q N ) ∞ ( q r z ; q r ∗ , q N ) ∞ ( q N − z ; q r ∗ , q N ) ∞ . (3.12)The free field realizations of U ( z ) , V ( z ) are given by U ( z ) = z r − r N − N e − i q r − r Q ¯ ǫ z − q r − r P ¯ ǫ : exp − X m =0 m β m z − m : , (3.13) V ( z ) = z r r − N − N e i √ rr − Q ¯ ǫ z √ rr − P ¯ ǫ × : exp X m =0 m [ rm ] q [( r − m ] q β m ( − z ) − m : . (3.14) In this section we give a class of infinitely many commuting operators G m , ( m ∈ N ) that we callthe integral of motion [4]. In this section we consider the case 0 < Re( s ) < N . Let us set theintegral of motion G m , ( m ∈ N ) by integral of the currents. G m = Z · · · Z N Y t =1 m Y j =1 dz ( t ) j z ( t ) j F ( z (1)1 ) F ( z (1)2 ) · · · F ( z (1) m ) × F ( z (2)1 ) F ( z (2)2 ) · · · F ( z (2) m ) · · · F N ( z ( N )1 ) F N ( z ( N )2 ) · · · F N ( z ( N ) m ) × N Y t =1 Y ≦ j 1) be the fundamental weights, which satisfy( α µ | ω ν ) = δ µ,ν , (1 ≦ µ, ν ≦ N − . Explicitly we set ω µ = P µν =1 ¯ ǫ ν . For a ∈ P we set a µ and a µ,ν by a µ,ν = a µ − a ν , a µ = ( a + ρ | ¯ ǫ µ ) , ( µ, ν ∈ P ) . Here we set ρ = P N − µ =1 ω µ . Let us set the restricted path P + r − N by P + r − N = { a = N − X µ =1 c µ ω µ ∈ P | c µ ∈ Z , c µ ≧ , N − X µ =1 c µ ≦ r − N } . For a ∈ P + r − N , condition 0 < a µ,ν < r, (1 ≦ µ < ν ≦ N − 1) holds. We recall elliptic solutionsof the Yang-Baxter equation of face type. An ordered pair ( b, a ) ∈ P is called admissible ifand only if there exists µ (1 ≤ µ ≤ N ) such that b − a = ¯ ǫ µ . An ordered set of four weights( a, b, c, d ) ∈ P is called an admissible configuration around a face if and only if the orderedpairs ( b, a ), ( c, b ), ( d, a ) and ( c, d ) are admissible. Let us set the Boltzmann weight functions W c db a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u associated with admissible configuration ( a, b, c, d ) ∈ P [19]. For a ∈ P + r − N and µ = ν , we set W a + 2¯ ǫ µ a + ¯ ǫ µ a + ¯ ǫ µ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = R ( u ) , (3.18) W a + ¯ ǫ µ + ¯ ǫ ν a + ¯ ǫ µ a + ¯ ǫ ν a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = R ( u ) [ u ][ a µ,ν − u − a µ,ν ] , (3.19) W a + ¯ ǫ µ + ¯ ǫ ν a + ¯ ǫ ν a + ¯ ǫ ν a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = R ( u ) [ u − a µ,ν ][1][ u − a µ,ν ] . (3.20)The normalizing function R ( u ) is given by R ( u ) = z r − r N − N ϕ ( z − ) ϕ ( z ) , ϕ ( z ) = ( q z ; q r , q N ) ∞ ( q r +2 N − z ; q r , q N ) ∞ ( q r z ; q r , q N ) ∞ ( q N z ; q r , q N ) ∞ . (3.21)Because 0 < a µ,ν < r (1 ≦ µ < ν ≦ N − 1) holds for a ∈ P + r − N , the Boltzmann weight functionsare well defined. The Boltzmann weight functions satisfy the Yang-Baxter equation of the facetype. X g W d ec g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u W c gb a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u W e fg a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u X g W g fb a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u W d eg f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u W d gc b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u . (3.22)We set the normalization function ϕ ( z ) such that the minimal eigenvalue of the corner transfermatrix becomes 1 [21]. The vertex operator Φ ( b,a ) ( z ) and the dual vertex operator Φ ∗ ( a,b ) ( z ) as-sociated with the elliptic quantum group U q,p ( d sl N ), are the operators which satisfy the followingcommutation relations,Φ ( a,b ) ( z )Φ ( b,c ) ( z ) = X g W a gb c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u Φ ( a,g ) ( z )Φ ( g,c ) ( z ) , (3.23)Φ ( a,b ) ( z )Φ ∗ ( b,c ) ( z ) = X g W g ca b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u Φ ∗ ( a,g ) ( z )Φ ( g,c ) ( z ) , (3.24)Φ ∗ ( a,b ) ( z )Φ ∗ ( b,c ) ( z ) = X g W c bg a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u Φ ∗ ( a,g ) ( z )Φ ∗ ( g,c ) ( z ) . (3.25)and the inversion relation, Φ ( a,g ) ( z )Φ ∗ ( g,b ) ( z ) = δ a,b . (3.26)We give free field realization of the vertex operator. In what follows we set l = b + ρ, k = a + ρ ,( a ∈ P + r − N , b ∈ P + r − N − ) and π µ = p r ( r − P ¯ ǫ µ , π µ,ν = π µ − π ν . We give the free fieldrealization of the vertex operators Φ ( a +¯ ǫ µ ,a ) ( z ), (1 ≦ µ ≦ N − 1) [5] byΦ ( a +¯ ǫ ,a ) ( z − ) = U ( z ) , Φ ( a +¯ ǫ µ ,a ) ( z − ) = I · · · I µ − Y j =1 dz j πiz j U ( z ) F ( z ) F ( z ) · · · F µ − ( z µ − ) × µ − Y j =1 [ u j − u j − + − π j,µ ][ u j − u j − − ] . (3.27)Here we set z j = q u j . We take the integration contour to be simple closed curve that encircles z j = 0 , q rs z j − , ( s ∈ N ) but not z j = q − − rs z j − , ( s ∈ N ) for 1 ≤ j ≤ µ − 1. The Φ ( a +¯ ǫ µ ,a ) ( z )is an operator such that Φ ( a +¯ ǫ µ ,a ) ( z ) : F l,k → F l,k +¯ ǫ µ . The free field realization of the dualvertex operator Φ ∗ ( a,b ) ( z ) is given by similar way [5]. The vertex operator Φ ( a,b ) ( z ) plays animportant role in construction of the correlation functions of the U q,p ( d sl N ) face model [5, 11]. In this section we introduce the boundary transfer matrix T B ( z ) [6], following theory of boundaryYang-Baxter equation [8, 16]. In this section we consider the case r ≧ N + 2 , ( r ∈ N ) and s = N .11n order set of three weights ( a, b, g ) ∈ P is called an admissible configuration at a boundary ifand only if the ordered pairs ( g, a ) and ( g, b ) are admissible. Let us set the boundary Boltzmannweight functions K ag b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u for admissible weights ( a, b, g ) as following [15]. K aa + ¯ ǫ µ b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = z r − r N − N − r a h ( z ) h ( z − ) [ c − u ][ a ,µ + c + u ][ c + u ][ a ,µ + c − u ] δ a,b . (3.28)In this paper, we consider the case of continuous parameter 0 < c < 1. The normalizationfunction h ( z ) is given by following [6]. h ( z ) = ( q r +2 N − /z ; q r , q N ) ∞ ( q N +2 /z ; q r , q N ) ∞ ( q r /z ; q r , q N ) ∞ ( q N /z ; q r , q N ) ∞ × ( q N +2 c /z ; q r , q N ) ∞ ( q r − c /z ; q r , q N ) ∞ ( q N +2 r − c − /z ; q r , q N ) ∞ ( q c +2 /z ; q r , q N ) ∞ (3.29) × N Y j =2 ( q r +2 N − c − a ,j /z ; q r , q N ) ∞ ( q c +2 a ,j /z ; q r , q N ) ∞ ( q r +2 N − c − a ,j − /z ; q r , q N ) ∞ ( q c +2+2 a ,j /z ; q r , q N ) ∞ . The boundary Boltzmann weight functions and the Boltzmann weight functions satisfy theboundary Yang-Baxter equation [8]. X f,g W c fb a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u W c df g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + u K gf a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u K ed g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = X f,g W c df e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u W c fb g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + u K ef g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u K gb a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u . (3.30)We set the normalization function h ( z ) such that the minimal eigenvalue of the boundary transfermatrix T B ( z ) becomes 1. We define the boundary transfer matrix T B ( z ) for the elliptic quantumgroup U q,p ( d sl N ). T B ( z ) = N X µ =1 Φ ∗ ( a,a +¯ ǫ µ ) ( z − ) K aa + ¯ ǫ µ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u Φ ( a +¯ ǫ µ ,a ) ( z ) . (3.31)12he boundary T B ( z ) commute with each other.[ T B ( z ) , T B ( z )] = 0 , for any z , z . (3.32)This commutativity is consequence of the commutation relations of the vertex operators (3.23),(3.24), (3.25), and boundary Yang-Baxter equation (3.30). In this section we diagonalize the boundary transfer matrix T B ( z ), using free field realizationof the vertex operators [6, 5, 13]. In this section we consider the case r ≧ N + 2, ( r ∈ N ) and s = N . We call the eigenvector | B i with the eigenvalue 1 the boundary state. T B ( z ) | B i = | B i . (4.1)We construct the free field realization of the boundary state | B i , analyzing those of the transfermatrix T B ( z ). The free field realization of the boundary state | B i is given as following [6]. | B i = e F | k, k i . (4.2)Here we have set F = − X m> N − X j =1 N − X k =1 m [ rm ] q [( r − m ] q I j,k ( m ) B j − m B k − m + X m> N − X j =1 m D j ( m ) β j − m , (4.3)where D j ( m ) = − θ m [( N − j ) m/ q [ rm/ + q q (3 j − N − m [( r − m/ q + q ( j − m [( − r + 2 π ,j + 2 c − j + 2) m ] q [( r − m ] q + [ m ] q q ( r − c +2 j − m [( r − m ] q N − X k = j +1 q − mπ ,k + q (2 j − N ) m [( r − π ,N − c + N − m ] q [( r − m ] q , (4.4)13 j,k ( m ) = [ jm ] q [( N − k ) m ] q [ m ] q [ N m ] q = I k,j ( m ) (1 ≦ j ≦ k ≦ N − . (4.5)Here we have used [ a ] + q = q a + q − a and θ m ( x ) = x, m : even0 , m : odd . In this section we construct diagonalization of the boundary transfer matrix T B ( z ) by usingthe boundary state | B i and type-II vertex operator Ψ ∗ ( b,a ) ( z ). Let us introduce type-II vertexoperator Ψ ∗ ( b,a ) ( z ) [13] by the following commutation relations,Ψ ∗ ( a,b ) ( z )Ψ ∗ ( b,c ) ( z ) = X g W ∗ a gb c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u − u Ψ ∗ ( a,g ) ( z )Ψ ∗ ( g,c ) ( z ) , (4.6)Φ ( d,c ) ( z )Ψ ∗ ( b,a ) ( z ) = χ ( z /z )Ψ ∗ ( b,a ) ( z )Φ ( d,c ) ( z ) , (4.7)Φ ∗ ( c,d ) ( z )Ψ ∗ ( b,a ) ( z ) = χ ( z /z )Ψ ∗ ( b,a ) ( z )Φ ∗ ( c,d ) ( z ) , (4.8)where we have set χ ( z ) = z − N − N Θ q N ( − qz )Θ q N ( − qz − ) and W ∗ a gb c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u is obtained by substitution r → r ∗ of the Boltzmann weight functions W a gb c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u defined in (3.18), (3.19), (3.20). Letus set l = b + ρ, k = a + ρ , ( a ∈ P + r − N , b ∈ P + r − N − ). The free field realization of the type-IIvertex operators Ψ ∗ ( b,a ) µ ( z ), (1 ≦ µ ≦ N − 1) are give byΨ ∗ ( b +¯ ǫ ,b ) ( z − ) = V ( z ) , Ψ ∗ ( b +¯ ǫ µ ,b ) ( z − ) = I · · · I µ − Y j =1 dz j πiz j V ( z ) E ( z ) E ( z ) · · · E µ − ( z µ − ) × µ − Y j =1 [ u j − u j − − + π j,µ ] ∗ [ u j − u j − + ] ∗ . (4.9)We take the integration contour to be simple closed curve that encircles z j = 0 , q − r ∗ s z j − , ( s ∈ N ) but not z j = q − r ∗ s z j − , ( s ∈ N ) for 1 ≤ j ≤ µ − 1. The Ψ ∗ ( b +¯ ǫ µ ,b ) ( z ) is an operatorsuch that Ψ ∗ ( b +¯ ǫ µ ,b ) ( z ) : F l,k → F l +¯ ǫ µ ,k . We introduce the vectors | ξ , ξ , · · · , ξ M i µ ,µ , ··· ,µ M (1 ≦ µ , µ , · · · , µ M ≦ N ). | ξ , ξ , · · · , ξ M i µ ,µ , ··· ,µ M = Ψ ∗ ( b +¯ ǫ µ +¯ ǫ µ + ··· +¯ ǫ µM ,b +¯ ǫ µ + ··· +¯ ǫ µM ) ( ξ ) × · · ·× · · · Ψ ∗ ( b +¯ ǫ µM − +¯ ǫ µM ,b +¯ ǫ µM ) ( ξ M − )Ψ ∗ ( b +¯ ǫ µM ,b ) ( ξ M ) | B i . (4.10)14e construct many eigenvectors of T B ( z ). T B ( z ) | ξ , ξ , · · · , ξ M i µ ,µ , ··· ,µ M = M Y j =1 χ ( ξ j /z ) χ (1 /ξ j z ) | ξ , ξ , · · · , ξ M i µ ,µ , ··· ,µ M . (4.11)The vectors | ξ , ξ , · · · , ξ M i µ ,µ , ··· ,µ M are the basis of the space of the state of the boundary U q,p ( d sl N ) face model [11, 12, 6]. It is thought that our method can be extended to more generalelliptic quantum group U q,p ( g ). Acknowledgements The author would like to thank to Branko Dragovich, Vladimir Dobrev, Sergei Vernov, PaulSorba, Igor Salom, Dragan Savic and the organizing committee of 6th Mathematical PhysicsMeetings : Summer School and Conference on Modern Mathematical Physics, held in Belgrade,Serbia. This work is supported by the Grant-in Aid for Scientific Research C (21540228) fromJapan Society for Promotion of Science. References [1] M.Jimbo and T.Miwa, Algebraic Analysis of Solvable Lattice Models CBMS Regional Con-ference Series in Mathematics AMS 1994.[2] M.Jimbo,H.Konno,S.Odake and J.Shiraishi, Commun.Math.Phys. (1999) 605.[3] T.Kojima and H.Konno, Commun.Math.Phys. (2003) 405.[4] T.Kojima and J.Shiraishi, Commun.Math.Phys. (2008) 795.[5] Y.Asai,M.Jimbo,T.Miwa and Ya.Pugai, J.Phys. A29 (1996) 6595.[6] T.Kojima, accepted for publication in J.Math.Phys. (2010) [arXiv.1007.3795].[7] B.Feigin and A.Odesskii, Internat.Math.Res.Notices (1997) 531.[8] E.Sklyanin, J.Phys. A21 (1988) 74.[9] V.Bazhanov, S.Lukyanov, Al.Zamolodchikov, Commun.Math.Phys. (1996) 381.[10] T.Kojima, J.Phys. A41 (2008) 355206 (16pp).1511] S.Lukyanov and Ya.Pugai, Nocl.Phys. B473 (1996) 631.[12] T.Miwa and R.Weston, Nucl.Phys. B486 (1997) 517.[13] H.Furutsu, T.Kojima and Y.Quano, Int.J.Mod.Phys. A15 (2000) 1533.[14] V.G.Drinfeld, Soviet.Math.Dokl. (1988) 212.[15] M.T.Batchelor, V.Fridkin, A.Kuniba and Y.K.Zhou, Phys.Lett. B276 (1996) 266.[16] M.Jimbo,R.Kedem.T.Kojima,H.Konno and T.Miwa, Nucl.Phys. B441 (1995) 437.[17] T.Kojima, Int.J.Mod.Phys. A24 (2009) 5561.[18] T.Kojima and J.Shiraishi, J.Geometry, Integrability and Quantization X (2009) 183.[19] M.Jimbo, T.Miwa and S.Odake, Nucl.Phys. B300 (1988) 507.[20] H.Furutsu and T.Kojima,