Infinitely many commuting operators for the elliptic quantum group $U_{q,p}(hat{sl_N})$

Infinitely man y comm uting op erators for the elliptic quan tum group U q ,p ( c sl N ) No v ember 16, 2021 T ak eo K OJIMA Dep artment of Mathematics and Physics, Gr aduate Scho ol of Scienc e and Engine ering, Y amagata Univ ersity, Jonan 4-3-16, Y onezawa 992-8510, Jap an Abstract W e co nstruct tw o classes o f infinitely many commu ting op er a tors asso ciated with the elliptic quantum group U q,p ( d sl N ). W e call o ne of them the int egra l of motion G m , ( m ∈ N ) and the other the b o undary transfer matrix T B ( z ), ( z ∈ C ). The in teg ral of motion G m is r elated to elliptic defo rmation of the N -th KdV theory . The b oundar y transfer matrix T B ( z ) is related to the bo undary U q,p ( d sl N ) face mo del. W e diag onalize the b oundar y transfer matrix T B ( z ) by us ing the free field realization of the elliptic q uantum g roup, how ever diagonaliza tion o f the in teg ral of motion G m is op en pro ble m even for the simplest cas e U q,p ( c sl 2 ). 1 In tro duction The free field approac h pr o vid es a p o w erful metho d to study exactly solv able mo del [1]. T h e basic idea i n this a ppr oac h is to rea lize the c omm utation relations for the symmetry algebra and the v ertex op erators in terms of free fields acting on the F o c k space. W e in tro du ce the elliptic 1 quan tum group U q ,p ( d sl N ) [2, 3], and giv e its free field r ealizat ion. Using th e free field realizations, w e int ro du ce tw o extended currents F N ( z ) [4 ] and U ( z ) [5] asso ciated with th e elliptic quantum group U q ,p ( d sl N ). W e constru ct t wo classes of infi nitely many commuting op erators for the elliptic quan tum group U q ,p ( d sl N ). W e call one of th em the integral of motion G m , ( m ∈ N ) [4 ] an d the other the b oundary transfer matrix T B ( z ), ( z ∈ C ) [6 ]. Our constructions are based on th e free field realizations of the elliptic quan tum group U q ,p ( d sl N ), the extend ed curren ts and the v er tex op erator Φ ( a,b ) ( z ). Comm utativit y of the inte gral of motion is ensur ed b y F eigin-Odesskii algebra [7], and those of the b oundary tran s fer matrix is ensured by Y ang-B axter equation and b ound ary Y ang-Baxter equation [8]. Tw o classes of infinitely man y comm u ting op erators ha v e physic al m eanin gs. The integ ral of motion G m is tw o parameter deformation of the mono dromy of th e N -th KdV theory [9, 10]. The b oundary transfer matrix T B ( z ) is r elated to the b oundary U q ,p ( d sl N ) face mo d el that is lattice d eformation of the conformal field theory . W e diagonalize the b ound ary transf er matrix T B ( z ) b y us in g the free fi eld realizatio n of the elliptic quantum group and the vertex op erators. Diagonali zation of th e b oun dary transfer matrix allo ws us calculate correlation fun ctions of the b ound ary U q ,p ( d sl N ) face mo del [11, 12, 6]. The organization of this pap er is as follo w s. In section 2 we int ro d u ce the elliptic qu an tum group U q ,p ( d sl N ) [2, 3], and giv e its free field realization. In section 3 we introd uce t wo extended currents F N ( z ) , E N ( z ) [4] and U ( z ) , V ( z ) [5, 13] asso ciated with the elliptic quan tum group U q ,p ( d sl N ). W e giv e th e fr ee field realization of the verte x op erators Φ ( a,b ) ( z ), using the extended current U ( z ). W e construct t wo classes of infinitely m an y comm uting op erators asso ciated with the elliptic quant um group U q ,p ( d sl N ). The one is the integ ral of motion G m [4] and the other is the b oundary transfer matrix T B ( z ) [6]. In section 4 we diagonaliz e the b ound ary trans f er matrix T B ( z ) by using the fr ee field realization of th e v ertex op erators [5, 13, 6]. 2 Elliptic quan tum group U q ,p ( c sl N ) In this section we intro d uce the elliptic qu an tum group U q ,p ( d sl N ) and its free field realizatio n. 2.1 Quan tum group In this section we recall Drinf eld r ealization of th e quantum group [14]. W e fix a complex n umber q suc h th at 0 < | q | < 1. Let us fix the in teger N = 3 , 4 , 5 , · · · . W e use q -intege r [ n ] q = q a − q − a q − q − 1 . W e use the abbr eviation, ( z ; p 1 , p 2 , · · · , p M ) ∞ = ∞ Y k 1 ,k 2 , ··· ,k M =0 (1 − p k 1 1 p k 2 2 · · · p k M M z ) . 2 The quantum group U q ( d sl N ) is generated by h j , a j,m , x j,n , (1 ≦ j ≦ N − 1 : m ∈ Z 6 =0 , n ∈ Z ), c, d . L et us set the generating fu nctions 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 2 z ) = q h j exp ( q − q − 1 ) X m> 0 a j,m z − m ! , ϕ j ( q − c 2 z ) = q − h j exp − ( q − q − 1 ) X m> 0 a j, − m z m ! . The definin g relations are giv en by [ d, x ± j,n ] = nx ± j,n , [ h j , d ] = [ h j , a k ,m ] = [ d, a k ,m ] = 0 , c : cent ral , [ a j,m , a k ,n ] = [ A j,k m ] q [ cm ] q m q − c | m | δ m + n, 0 , [ 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 1 − q ± A j,k z 2 ) x ± j ( z 1 ) x ± k ( z 2 ) = ( q ± A j,k z 1 − z 2 ) x ± k ( z 2 ) x ± j ( z 1 ) , [ x + j ( z 1 ) , x − k ( z 2 )] = δ j,k q − q − 1 ( δ ( q − c z 1 /z 2 ) ψ j ( q c 2 z 2 ) − δ ( q c z 1 /z 2 ) ϕ j ( q − c 2 z 2 )) , and Serre relation for | j − k | = 1, ( x ± j ( z 1 ) x ± j ( z 2 ) x ± k ( z ) − [2] q x ± j ( z 1 ) x ± k ( z ) x ± j ( z 2 ) + x ± k ( z ) x ± j ( z 1 ) x ± j ( z 2 )) +( x ± j ( z 2 ) x ± j ( z 1 ) x ± k ( z ) − [2] q x ± j ( z 2 ) x ± k ( z ) x ± j ( z 1 ) + x ± k ( z ) x ± j ( z 2 ) x ± j ( z 1 )) = 0 . Here ( A j,k ) 1 ≦ j,k ≦ N − 1 is Cartan matrix of sl N t yp e. Here w e used the delta function δ ( z ) = P m ∈ Z z m . 2.2 Elliptic quan tum group In this section we in tro duce the elliptic quan tum group U q ,p ( d sl N ) [2 , 3], which is elliptic d efor- mation of the quantum group U q ( d sl N ). W e fix complex num b ers r , s suc h that Re( r ) > 1 and Re( s ) > 0. When w e c hange the p olynomial ( z 1 − q − 2 z 2 ) in the d efining relation of the quantum group U q ( d sl N ), ( z 1 − q − 2 z 2 ) x − j ( z 1 ) x − j ( z 2 ) = ( q − 2 z 1 − z 2 ) x − j ( z 2 ) x − j ( z 1 ) , to the elliptic th eta function [ u ], we h a ve [ u 1 − u 2 + 1] F j ( z 1 ) F j ( z 2 ) = [ u 1 − u 2 − 1] F j ( z 2 ) F j ( z 1 ) . 3 This is one of the defin ing relations of the elliptic quantum group U q .p ( d sl N ). W e set the elliptic theta function [ u ] , [ u ] ∗ b y [ u ] = q u 2 r − u Θ q 2 r ( q 2 u ) , [ u ] ∗ = q u 2 r ∗ − u Θ q 2 r ∗ ( q 2 u ) , Θ p ( z ) = ( p ; p ) ∞ ( z ; p ) ∞ ( pz − 1 ; p ) ∞ , where w e set z = x 2 u and r ∗ = r − c . The elliptic qu an tum group U q ,p ( d sl N ) is generated b y the currents E j ( z ) , F j ( z ), H + j ( q c 2 − r z ) = H − j ( q − c 2 + r z ), (1 ≦ j ≦ N − 1). The defin ing relations are giv en b y E j ( z 1 ) E j +1 ( z 2 ) =  u 2 − u 1 + s N  ∗  u 1 − u 2 + 1 − s N  ∗ E j +1 ( z 2 ) E j ( z 1 ) , (2.1) E j ( z 1 ) E j ( z 2 ) = [ u 1 − u 2 + 1] ∗ [ u 1 − u 2 − 1] ∗ E j ( z 2 ) E j ( z 1 ) , (2.2) E j ( z 1 ) E k ( z 2 ) = E k ( z 2 ) E j ( z 1 ) , other w ise, (2.3) F j ( z 1 ) F j +1 ( z 2 ) =  u 2 − u 1 + s N − 1   u 1 − u 2 − s N  F j +1 ( z 2 ) F j ( z 1 ) , (2.4) F j ( z 1 ) F j ( z 2 ) = [ u 1 − u 2 − 1] [ u 1 − u 2 + 1] F j ( z 2 ) F j ( z 1 ) , (2.5) F j ( z 1 ) F k ( z 2 ) = F k ( z 2 ) F j ( z 1 ) , other w ise, ( 2.6) H + j ( z 1 ) H + j ( z 2 ) = [ u 1 − u 2 − 1][ u 1 − u 2 + 1] ∗ [ u 1 − u 2 + 1][ u 1 − u 2 − 1] ∗ H + j ( z 2 ) H + j ( z 1 ) , (2.7) H + j ( z 1 ) H + j +1 ( z 2 ) = [ u 1 − u 2 + 1 − s N ][ u 1 − u 2 − s N ] ∗ [ u 1 − u 2 − s N ][ u 1 − u 2 + 1 − s N ] ∗ H + j +1 ( z 2 ) H + j ( z 1 ) , (2.8) H + j ( z 1 ) H + k ( z 2 ) = H + k ( z 2 ) H + j ( z 1 ) , other w ise, (2.9) H + j ( z 1 ) E j ( z 2 ) = [ u 1 − u 2 + 1 + c 4 ] ∗ [ u 1 − u 2 − 1 − c 4 ] ∗ E j ( z 2 ) H + j ( z 1 ) , (2.10) H + j ( z 1 ) E j +1 ( z 2 ) = [ u 2 − u 1 + s N + c 4 ] ∗ [ u 1 − u 2 + 1 − s N − c 4 ] ∗ E j +1 ( z 2 ) H + j ( z 1 ) , (2.11) H + j +1 ( z 1 ) E j ( z 2 ) = [ u 2 − u 1 + 1 − s N + c 4 ] ∗ [ u 1 − u 2 − s N − c 4 ] ∗ E j ( z 2 ) H + j +1 ( z 1 ) , (2.12) H + j ( z 1 ) E k ( z 2 ) = E k ( z 2 ) H + j ( z 1 ) , other w ise, (2.13) H + j ( z 1 ) F j ( z 2 ) = [ u 1 − u 2 − 1 − c 4 ] [ u 1 − u 2 + 1 + c 4 ] F j ( z 2 ) H + j ( z 1 ) , (2.14) H + j ( z 1 ) F j +1 ( z 2 ) = [ u 2 − u 1 + s N − 1 − c 4 ] [ u 1 − u 2 − s N + c 4 ] F j +1 ( z 2 ) H + j ( z 1 ) , (2.1 5) H + j +1 ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 − s N − c 4 ] [ u 1 − u 2 + s N − 1 + c 4 ] F j ( z 2 ) H + j +1 ( z 1 ) , (2.16) H + j ( z 1 ) F k ( z 2 ) = F k ( z 2 ) H + j ( z 1 ) , other w ise, (2.17) 4 [ E i ( z 1 ) , F j ( z 2 )] = δ i,j q − q − 1  δ ( q − c z 1 /z 2 ) H + j  q c 2 z 2  − δ ( q c z 1 /z 2 ) H − j  q − c 2 z 2  , (2.18) and the Serre relations for | j − k | = 1,  ( z 2 /z ) 1 r ∗ ( q 2 r ∗ − 1 z /z 1 ; q 2 r ∗ ) ∞ ( q 2 r ∗ − 1 z /z 2 ; q 2 r ∗ ) ∞ ( q 2 r ∗ +1 z /z 1 ; q 2 r ∗ ) ∞ ( q 2 r ∗ +1 z /z 2 ; q 2 r ∗ ) ∞ E j ( q 1 − 2 s N z 1 ) E j ( q 1 − 2 s N z 2 ) E k ( q 1 − 2 s N z ) − [2] q ( q 2 r ∗ − 1 z /z 1 ; q 2 r ∗ ) ∞ ( q 2 r ∗ − 1 z 2 /z ; q 2 r ∗ ) ∞ ( q 2 r ∗ +1 z /z 1 ; q 2 r ∗ ) ∞ ( q 2 r ∗ +1 z 2 /z ; q 2 r ∗ ) ∞ E j ( q 1 − 2 s N z 1 ) E k ( q 1 − 2 s N z ) E j ( q 1 − 2 s N z 2 ) +( z /z 1 ) 1 r ∗ ( q 2 r ∗ − 1 z 1 /z ; q 2 r ∗ ) ∞ ( q 2 r ∗ − 1 z 2 /z ; q 2 r ∗ ) ∞ ( q 2 r ∗ +1 z 1 /z ; q 2 r ∗ ) ∞ ( q 2 r ∗ +1 z 2 /z ; q 2 r ∗ ) ∞ E k ( q 1 − 2 s N z ) E j ( q 1 − 2 s N z 1 ) E j ( q 1 − 2 s N z 2 )  × z − 1 r ∗ 1 ( q 2 r ∗ +2 z 2 /z 1 ; q 2 r ∗ ) ∞ ( q 2 r ∗ − 2 z 2 /z 1 ; q 2 r ∗ ) ∞ + ( z 1 ↔ z 2 ) = 0 , (2.19)  ( z 2 /z ) − 1 r ( q 2 r +1 z /z 1 ; q 2 r ) ∞ ( q 2 r +1 z /z 2 ; q 2 r ) ∞ ( q 2 r − 1 z /z 1 ; q 2 r ) ∞ ( q 2 r − 1 z /z 2 ; q 2 r ) ∞ F j ( q 1 − 2 s N z 1 ) F j ( q 1 − 2 s N z 2 ) F k ( q 1 − 2 s N z ) − [2] q ( q 2 r +1 z /z 1 ; q 2 r ) ∞ ( q 2 r +1 z 2 /z ; q 2 r ) ∞ ( q 2 r − 1 z /z 1 ; q 2 r ) ∞ ( q 2 r − 1 z 2 /z ; q 2 r ) ∞ F j ( q 1 − 2 s N z 1 ) F k ( q 1 − 2 s N z ) F j ( q 1 − 2 s N z 2 ) +( z /z 1 ) − 1 r ( q 2 r +1 z 1 /z ; q 2 r ) ∞ ( q 2 r +1 z 2 /z ; q 2 r ) ∞ ( q 2 r − 1 z 1 /z ; q 2 r ) ∞ ( q 2 r − 1 z 2 /z ; q 2 r ) ∞ F k ( q 1 − 2 s N z ) F j ( q 1 − 2 s N z 1 ) F j ( q 1 − 2 s N z 2 )  × z 1 r 1 ( q 2 r − 2 z 2 /z 1 ; q 2 r ) ∞ ( q 2 r +2 z 2 /z 1 ; q 2 r ) ∞ + ( z 1 ↔ z 2 ) = 0 . (2.20) 2.3 F ree field r ealization In this section w e give the free field realization of the elliptic quan tum group U q .p ( d sl N ) [2, 3, 5]. In wh at follo ws w e restrict our in terest to lev el c = 1. Let us introd uce th e b osons β j m , (1 ≦ j ≦ N ; m ∈ Z ) b y [ β j m , β k n ] =        m [( r − 1) m ] q [ r m ] q [( s − 1) m ] q [ sm ] q δ m + n, 0 (1 ≦ j = k ≦ N ) − mq sm sgn ( j − k ) [( r − 1) m ] q [ r m ] q [ m ] q [ sm ] q δ m + n, 0 (1 ≦ j 6 = k ≦ N ) . (2.21) W e set the b osons B j m , (1 ≦ j ≦ N ; m ∈ Z 6 =0 ) b y B j m = ( β j m − β j +1 m ) q − j m , (1 ≦ j ≦ N − 1) . (2.22) They satisfy [ B j m , B k n ] = m [( r − 1) m ] q [ r m ] q [ A j,k m ] q [ m ] q δ m + n, 0 , (1 ≦ j, k ≦ N − 1) , (2.23) where ( A j,k ) 1 ≦ j,k ≦ N − 1 is Cartan matrix of sl N t yp e. Let ǫ µ (1 ≦ µ ≦ N ) b e th e orth onormal basis of R N with the inn er pro duct ( ǫ µ | ǫ ν ) = δ µ,ν . Let us set ¯ ǫ µ = ǫ µ − ǫ where ǫ = 1 N P N ν =1 ǫ ν . 5 Let α µ (1 ≦ µ ≦ N − 1) the simple r o ot : α µ = ¯ ǫ µ − ¯ ǫ µ +1 . The t yp e sl N w eigh t lattice is the linear span of ¯ ǫ µ , P = P N − 1 µ =1 Z ¯ ǫ µ . Let us set P α , Q α ( α ∈ P ) b y [ iP α , Q β ] = ( α | β ) , ( α, β ∈ P ) . (2.24) In what follo ws w e deal with the b osonic F o c k space F l,k , generate d by β j − m ( m > 0) o ver the v acuum v ector | l , k i , where l, k ∈ P . F l,k = C [ { β j − 1 , β j − 2 , · · · } 1 ≦ j ≦ N ] | l, k i , | l, k i = e i √ r r − 1 Q l − i q r − 1 r Q k | 0 , 0 i , where β j m | l, k i = 0 , ( m > 0) , P α | l, k i = α      r r r − 1 l − r r − 1 r k ! | l, k i . F ree field realizations of E j ( z ) , F j ( z ) , H ± j ( z ) (1 ≦ j ≦ N − 1) are giv en b y E j ( z ) = e − i √ r r − 1 Q α j ( q ( 2 s N − 1) j z ) − √ r r − 1 P α j + r r − 1 × : exp   − X m 6 =0 1 m [ r m ] q [( r − 1) m ] q B j m ( q ( 2 s N − 1) j z ) − m   : , (2.25) F j ( z ) = e i q r − 1 r Q α j ( q ( 2 s N − 1) j z ) q r − 1 r P α j + r − 1 r × : exp   X m 6 =0 1 m B j m ( q ( 2 s N − 1) j z ) − m   : , (2.26) H + j ( q 1 2 − r z ) = q (1 − 2 s N )2 j e − i √ r ( r − 1) Q α j ( q ( 2 s N − 1) j z ) − 1 √ r ( r − 1) P α j + 1 r ( r − 1) × : exp   − X m 6 =0 1 m [ m ] q [( r − 1) m ] q B j m ( q ( 2 s N − 1) j z ) − m   : . (2.27) The fr ee field r ealizati on for general lev el c [17] is completely d ifferen t fr om those for level c = 1. 3 Comm uting op erators In this section w e constru ct t w o classes of infinitely m any commuting op erators G m [4] and T B ( z ) [6]. 3.1 Extended curren t s E N ( z ) , F N ( z ) In this section we in tro d uce the extended curr en ts E N ( z ) , F N ( z ) [4]. Let us set the extended current E N ( z ) , F N ( z ) by the similar commutat ion relations as the elliptic quan tum group. The 6 extended currents E N ( z ) , F N ( z ) s atisfy the follo wing commutati on relations. E j ( z 1 ) E j +1 ( z 2 ) =  u 2 − u 1 + s N  ∗  u 1 − u 2 + 1 − s N  ∗ E j +1 ( z 2 ) E j ( z 1 ) , ( j ∈ Z / N Z ) , E j ( z 1 ) E j ( z 2 ) = [ u 1 − u 2 + 1] ∗ [ u 1 − u 2 − 1] ∗ E j ( z 2 ) E j ( z 1 ) , ( j ∈ Z / N Z ) , F j ( z 1 ) F j +1 ( z 2 ) =  u 2 − u 1 + s N − 1   u 1 − u 2 − s N  F j +1 ( z 2 ) F j ( z 1 ) , ( j ∈ Z / N Z ) , F j ( z 1 ) F j ( z 2 ) = [ u 1 − u 2 − 1] [ u 1 − u 2 + 1] F j ( z 2 ) F j ( z 1 ) , ( j ∈ Z / N Z ) , [ E j ( z 1 ) , F k ( z 2 )] = δ j,k q − q − 1  δ ( q − 1 z 1 /z 2 ) H + j  q 1 2 z 2  − δ ( q z 1 /z 2 ) H − j  q − 1 2 z 2  , ( j, k ∈ Z / N Z ) , and other defining relations of th e elliptic q u an tum group, in w hic h the suffi x j, k should b e understo o d as mod. N . F ree field r ealizat ions of the extended currents E N ( z ) , F N ( z ) and H + N ( q 1 2 − r z ) = H − N ( q 1 2 − r z ) are giv en by E N ( z ) = e − i √ r r − 1 Q α N ( q 2 s − N z ) − √ r r − 1 P ¯ ǫ N + r 2( r − 1) z √ r r − 1 P ¯ ǫ 1 + r 2( r − 1) × : exp   − X m 6 =0 1 m [ r m ] q [( r − 1) m ] q B N m ( q 2 s − N z ) − m   : , (3.1) F N ( z ) = e i q r − 1 r Q α N ( q 2 s − N z ) q r − 1 r P ¯ ǫ N + r − 1 2 r z − q r − 1 r P ¯ ǫ 1 + r − 1 2 r × : exp   − X m 6 =0 1 m B N m ( q 2 s − N z ) − m   : , (3.2) H + N ( q 1 2 − r z ) = q 2( N − 2 s ) e − i √ r r ∗ Q α N ( q 2 s − N z ) − 1 √ r r ∗ P ¯ ǫ N + 1 2 rr ∗ z 1 √ r r ∗ P ¯ ǫ 1 + 1 2 rr ∗ × : exp   − X m 6 =0 1 m [ m ] q [( r − 1) m ] q B N m ( q 2 s − N z ) − m   : . (3.3) 3.2 Extended curren t s V ( z ) , U ( z ) In this s ection we in tro duce the extended curr en ts V ( z ) , U ( z ) [5, 13]. In th is section we consid er the case s = N . F or our purp ose it is con venien t to introdu ce E j ( z ) = E j ( q − j z ) , F j ( z ) = F j ( q − j z ) , (1 ≦ j ≦ N − 1) . The extended currents U ( z ) , V ( z ) are given b y the follo w ing comm utation relations.  u 1 − u 2 + 1 2  ∗ V ( z 1 ) E 1 ( z 2 ) =  u 2 − u 1 + 1 2  ∗ E 1 ( z 2 ) V ( z 1 ) , (3.4) E j ( z 1 ) V ( z 2 ) = V ( z 2 ) E j ( z 1 ) (2 ≦ j ≦ N ) , (3.5) 7  u 1 − u 2 − 1 2  U ( z 1 ) F 1 ( z 2 ) =  u 2 − u 1 − 1 2  F 1 ( z 2 ) U ( z 1 ) , (3.6) F j ( z 1 ) U ( z 2 ) = U ( z 2 ) F j ( z 1 ) (2 ≦ j ≦ N ) . (3.7) U ( z 1 ) U ( z 2 ) = ( z 1 /z 2 ) r − 1 r N − 1 N ρ ( z 2 /z 1 ) ρ ( z 1 /z 2 ) U ( z 2 ) U ( z 1 ) , (3.8) V ( z 1 ) V ( z 2 ) = ( z 1 /z 2 ) − r r − 1 N − 1 N ρ ∗ ( z 2 /z 1 ) ρ ∗ ( z 1 /z 2 ) V ( z 2 ) V ( z 1 ) , (3.9) U ( z 1 ) V ( z 2 ) = z − N − 1 N Θ q 2 N ( − q z ) Θ q 2 N ( − q z − 1 ) V ( z 2 ) U ( z 1 ) , (3.10) where we set ρ ( z ) = ( q 2 z ; q 2 r , q 2 N ) ∞ ( q 2 N +2 r − 2 z ; q 2 r , q 2 N ) ∞ ( q 2 r z ; q 2 r , q 2 N ) ∞ ( q 2 N z ; q 2 r , q 2 N ) ∞ , (3.11) ρ ∗ ( z ) = ( z ; q 2 r ∗ , q 2 N ) ∞ ( q 2 N +2 r − 2 z ; q 2 r ∗ , q 2 N ) ∞ ( q 2 r z ; q 2 r ∗ , q 2 N ) ∞ ( q 2 N − 2 z ; q 2 r ∗ , q 2 N ) ∞ . (3.12) The free field realizations of U ( z ) , V ( z ) are giv en b y U ( z ) = z r − 1 2 r N − 1 N e − i q r − 1 r Q ¯ ǫ 1 z − q r − 1 r P ¯ ǫ 1 : exp   − X m 6 =0 1 m β 1 m z − m   : , (3.13) V ( z ) = z r 2( r − 1) N − 1 N e i √ r r − 1 Q ¯ ǫ 1 z √ r r − 1 P ¯ ǫ 1 × : exp   X m 6 =0 1 m [ r m ] q [( r − 1) m ] q β 1 m ( − z ) − m   : . (3.14) 3.3 In t egral of motion In this section we give a class of infinitely many commuting op erators G m , ( m ∈ N ) th at w e call the integ ral of motion [4]. In this section w e consider the case 0 < Re( s ) < N . Let us set the in tegral of motion G m , ( m ∈ N ) by in tegral of the cur ren ts. G m = Z · · · Z N Y t =1 m Y j =1 dz ( t ) j z ( t ) j F 1 ( z (1) 1 ) F 1 ( z (1) 2 ) · · · F 1 ( z (1) m ) × F 2 ( z (2) 1 ) F 2 ( z (2) 2 ) · · · F 2 ( z (2) m ) · · · F N ( z ( N ) 1 ) F N ( z ( N ) 2 ) · · · F N ( z ( N ) m ) × N Y t =1 Y 1 ≦ j 0 N − 1 X j =1 N − 1 X k =1 1 m [ r m ] q [( r − 1) m ] q I j,k ( m ) B j − m B k − m + X m> 0 N − 1 X j =1 1 m D j ( m ) β j − m , (4.3) where D j ( m ) = − θ m   [( N − j ) m/ 2] q [ r m/ 2] + q q (3 j − N − 1) m 2 [( r − 1) m/ 2] q   + q ( j − 1) m [( − r + 2 π 1 ,j + 2 c − j + 2) m ] q [( r − 1) m ] q + [ m ] q q ( r − 2 c + 2 j − 2) m [( r − 1) m ] q   N − 1 X k = j +1 q − 2 mπ 1 ,k   + q (2 j − N ) m [( r − 2 π 1 ,N − 2 c + N − 1) m ] q [( r − 1) m ] q , (4.4) 13 I j,k ( m ) = [ j m ] q [( N − k ) m ] q [ m ] q [ N m ] q = I k ,j ( m ) (1 ≦ j ≦ k ≦ N − 1) . (4.5) Here we hav e used [ a ] + q = q a + q − a and θ m ( x ) =    x, m : ev en 0 , m : od d . 4.2 Excited st at es In this section we construct diagonal ization of the b oundary trans fer matrix T B ( z ) b y using the b oun dary state | B i and typ e-I I v ertex op erator Ψ ∗ ( b,a ) ( z ). Let u s introd uce t yp e-I I vertex op erator Ψ ∗ ( b,a ) ( z ) [13] b y the follo wing comm u tation relations, Ψ ∗ ( a,b ) ( z 1 )Ψ ∗ ( b,c ) ( z 2 ) = X g W ∗   a g b c       u 1 − u 2   Ψ ∗ ( a,g ) ( z 2 )Ψ ∗ ( g, c ) ( z 1 ) , (4.6) Φ ( d,c ) ( z 1 )Ψ ∗ ( b,a ) ( z 2 ) = χ ( z 2 /z 1 )Ψ ∗ ( b,a ) ( z 2 )Φ ( d,c ) ( z 1 ) , (4.7) Φ ∗ ( c,d ) ( z 1 )Ψ ∗ ( b,a ) ( z 2 ) = χ ( z 1 /z 2 )Ψ ∗ ( b,a ) ( z 2 )Φ ∗ ( c,d ) ( z 1 ) , (4.8) where w e hav e set χ ( z ) = z − N − 1 N Θ q 2 N ( − q z ) Θ q 2 N ( − q z − 1 ) and W ∗   a g b c       u   is obtained by su b stitution r → r ∗ of the Boltzmann weigh t functions W   a g b c       u   defined in (3.18), (3.19 ), (3.20). Let us set l = b + ρ, k = a + ρ , ( a ∈ P + r − N , b ∈ P + r − N − 1 ). The fr ee field realizat ion of the type-I I v er tex op erators Ψ ∗ ( b,a ) µ ( z ), (1 ≦ µ ≦ N − 1) are giv e b y Ψ ∗ ( b +¯ ǫ 1 ,b ) ( z − 1 0 ) = V ( z 0 ) , Ψ ∗ ( b +¯ ǫ µ ,b ) ( z − 1 0 ) = I · · · I µ − 1 Y j =1 dz j 2 π iz j V ( z 0 ) E 1 ( z 1 ) E 2 ( z 2 ) · · · E µ − 1 ( z µ − 1 ) × µ − 1 Y j =1 [ u j − u j − 1 − 1 2 + π j,µ ] ∗ [ u j − u j − 1 + 1 2 ] ∗ . (4.9) W e tak e the integrat ion con tour to b e sim p le closed curv e that encircles z j = 0 , q − 1+2 r ∗ s z j − 1 , ( s ∈ N ) but not z j = q 1 − 2 r ∗ s z j − 1 , ( s ∈ N ) for 1 ≤ j ≤ µ − 1. The Ψ ∗ ( b +¯ ǫ µ ,b ) ( z ) is an op erator suc h that Ψ ∗ ( b +¯ ǫ µ ,b ) ( z ) : F l,k → F l +¯ ǫ µ ,k . W e in tro duce the v ectors | ξ 1 , ξ 2 , · · · , ξ M i µ 1 ,µ 2 , ··· ,µ M (1 ≦ µ 1 , µ 2 , · · · , µ M ≦ N ). | ξ 1 , ξ 2 , · · · , ξ M i µ 1 ,µ 2 , ··· ,µ M = Ψ ∗ ( b +¯ ǫ µ 1 +¯ ǫ µ 2 + ··· +¯ ǫ µ M ,b +¯ ǫ µ 2 + ··· +¯ ǫ µ M ) ( ξ 1 ) × · · · × · · · Ψ ∗ ( b +¯ ǫ µ M − 1 +¯ ǫ µ M ,b +¯ ǫ µ M ) ( ξ M − 1 )Ψ ∗ ( b +¯ ǫ µ M ,b ) ( ξ M ) | B i . (4.10) 14 W e construct many eigen v ectors of T B ( z ). T B ( z ) | ξ 1 , ξ 2 , · · · , ξ M i µ 1 ,µ 2 , ··· ,µ M = M Y j =1 χ ( ξ j /z ) χ (1 /ξ j z ) | ξ 1 , ξ 2 , · · · , ξ M i µ 1 ,µ 2 , ··· ,µ M . (4.11) The vec tors | ξ 1 , ξ 2 , · · · , ξ M i µ 1 ,µ 2 , ··· ,µ M are the basis of the sp ace of the state of the b oundary U q ,p ( d sl N ) face mo del [11, 12, 6]. It is though t that our m etho d can b e extended to more general elliptic quantum group U q ,p ( g ). Ac kno w ledgemen ts The auth or would lik e to thank to Brank o Drago vic h , Vladimir Dobrev, Sergei V erno v, P aul Sorba, Igor Salom, Drag an Savic and the organizing committee of 6th Mathematical Physic s Meetings : Su mmer School and C onference on Mo d ern Mathematical Physics, h eld in Belgrade, Serbia. Th is w ork is su pp orted b y the Grant -in Aid for Scienti fic Researc h C (21540 228) from Japan So ciet y for Promotion of Science. References [1] M.Jim b o and T.Miw a, Algebr aic Analysis of Solvable L attic e Mo dels CBMS Regional C on - ference Series in Mathematics 85 AMS 1994. [2] M.Jim b o,H.Konno,S.Od ak e and J.Shir aish i, Commun.Math.Phys. 199 (1999) 605. [3] T.Ko jima and H.Konno, Commun.Math.Phys. 239 (2003) 405. [4] T.Ko jima and J.Shiraishi, Commun.Math.Phys. 283 (2008) 795. [5] Y.A sai,M.Jim b o,T.Miw a and Y a.Pugai, J.P hys. A29 (1996) 6595. [6] T.Ko jima, accepte d for pu b lication in J.Math.Phys. (2010) [arXiv.1007.37 95]. [7] B. F eigin and A.Odesskii, Internat.Math.R es.N otic es 11 (1997 ) 531. [8] E.Skly anin, J.Phys. A21 (198 8) 74. [9] V.B azhano v, S.Lu ky ano v, Al.Zamolod c h ik ov, Commun.Math.Phys. 177 (1996) 381. [10] T.K o jima, J.Phys. A41 (2008) 355206 (16pp). 15 [11] S .Luky ano v and Y a.Pugai, No cl.Phys. B473 (1996) 631. [12] T.Miwa and R.W eston, Nucl.Phys. B486 (1997) 517. [13] H.F uru tsu, T.Ko jima and Y.Quano, Int.J.Mo d.Phys. A15 (2000) 1533. [14] V.G.Drinfeld, Soviet.Math.Dokl. 36 (1988) 212. [15] M.T.Batc h elor, V.F ridkin, A.Kun iba and Y.K.Zhou, Phys.L ett. B276 (1996) 266. [16] M.Jimb o,R.Kedem.T.Ko jima,H.Konno and T.Miwa, Nucl.Phys. B441 (1995) 437. [17] T.K o jima, Int.J.Mo d.Phys. A24 (2009) 5561. [18] T.K o jima and J.Shiraishi, J.Ge ometry, Inte gr ability and Quantization X (2009) 183. [19] M.Jimb o, T.Miw a and S .O dak e, Nucl.Phys. B300 (1988) 507. [20] H.F uru tsu and T.Ko jima, J.M ath.Phys. 41 (2000) 4413. [21] R.Baxter, Exactly Solve d Mo dels in Statistic al Me chanics , Academic Press 1982. 16