Elliptic Quantum Group U_{q,p}(hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

Elliptic Quan tum Group U q ,p ( b sl 2 ) , Hopf Algebroid Stru c tu r e and Elliptic Hyp ergeometric S eries Hitoshi Konno Dep art ment of Mathematics, Gr aduate Scho ol of Scienc e, Hir oshima University, Higashi-Hir oshima 739-8521 , Jap an konno@mis.hir oshima-u.ac.jp Abstract W e prop ose a new realiza tion of the elliptic quantum group equipped with the H -Hopf algebroid structure on the ba sis of the elliptic algebra U q,p ( b sl 2 ). The algebra U q,p ( b sl 2 ) ha s a constructive deﬁnition in terms of the Drinfeld genera tors of the quantum aﬃne algebra U q ( b sl 2 ) and a Heisenberg algebra. This yields a systematic co nstruction of b oth ﬁnite and inﬁnite-dimensional dynamical repr esent atio ns and their par allel structur es to U q ( b sl 2 ). In particular we give a c la ssiﬁcation theor em of the ﬁnite-dimensional irreducible pseudo-highest weigh t r epresentations stated in terms of an elliptic analog ue of the Drinfeld p olynomials. W e also inv estigate a structure of the tensor pro duct of t wo ev a luation r epresentations and derive an elliptic ana logue of the Clebsch-Gordan co eﬃcients. W e show that it is expr essed by using the very-w ell-p oised balanced elliptic hypergeo metric series 12 V 11 . 1 In tro duction In this pap er w e revisit the elliptic algebra U q ,p ( b sl 2 ) and study its coalge br a structur e. The algebra U q ,p ( b sl 2 ) was introd uced in [34] as an elliptic analog ue of the quantum aﬃne al gebra U q ( b sl 2 ) in th e Drinfeld r ealiz ation [9]. It wa s realized in [25] that U q ,p ( b sl 2 ) is constructiv ely deﬁned by using the Drinfeld generators of U q ( b sl 2 ) and the Heisen b erg alg ebra { P, e Q } . Th is construction w as generalized to the elliptic algebra U q ,p ( g ) of all t yp es of unt wisted aﬃne Lie algebras g [25] and of the t wisted type A (2) 2 [32]. It w as also realized that U q ,p ( g ) has an in teresting relation to the deformed cose t Virasoro/ W algebras [2, 15, 19, 46 ]. Namely , the lev el one ( c = 1) elliptic curr en ts of U q ,p ( g ) are identiﬁed with the screening cu r ren ts of the deformed W algebras for g = b sl N [25, 31, 34] and f or g = A (2) 2 [32]. This observ ation led us to a conjecture that the elliptic currents E i ( u ) and F i ( u ) of U q ,p ( g ) ( i = 1 , 2 , · · · , r ank ¯ g ) deﬁne the deformation of the Virasoro/ W a lgebra asso ciated with the coset g ⊕ g / g [25, 34]. 1 A study of coalgebra structur e on U q ,p ( g ) wa s far fr om straigh tforward. W e constru cted the L op erators in terms of the elliptic currents an d derive d the R LL relatio n for the cases g = b sl N [25, 31] and g = A (2) 2 [32]. Ho w ev er it turned out that a naiv e F adeev-Reshetikhin- Skly anin-T akhta jan (FRST) construction [14, 4 7] do es not w ork due to the dynamical shift app earing in the R matrices. Instead w e obtained a connection to the quasi-Hopf algebra B q ,λ ( g ) [25]. That is the isomorphism U q ,p ( g ) ∼ = B q ,λ ( g ) ⊗ { P i , e Q i } ( i = 1 , 2 , · · · , rank ¯ g ) as an asso ciativ e algebra, w here { P i , e Q i } denotes a Heisenb er g algebra. The quasi-Hopf algebra B q ,λ ( g ) (the face t yp e) was int ro du ced b y Jimb o, K onno, Odak e and Shiraishi [24] motiv ated by the w orks of Drinfeld [10], Bab elon, Bernard and Billey [3] and F rønsdal [20]. A t the same time, w e introd uced the v ertex t yp e quasi-Hopf algebra A q ,p ( b sl N ). Both A q ,p ( b sl N ) and B q ,λ ( g ) are isomorph ic to the corresp ondin g quan tum aﬃne algebras U q ( g ) as asso ciativ e algebras, b ut their coalgebra stru ctures are deformed from U q ( g ) by the t wistors E ( r ) and F ( λ ), resp ectiv ely . By t wisting the ob jects in U q ( g ), suc h as the com ultiplication, the unive rsal R matrices and t he v ertex op erators, we ca n derive their quasi-Hopf alge br a coun- terparts [24]. Then the relatio n U q ,p ( g ) ∼ = B q ,λ ( g ) ⊗ { P i , e Q i } al lo ws us to deriv e the U q ,p ( g ) coun terparts fr om those of th e qu asi-Hopf algebra B q ,λ ( g ). Such a strategy led u s to an exten- sion of the algebraic a nalysis sc heme of trigonometric solv able latti ce mo dels ´ a la Jim b o and Miw a [23] to the face t yp e elliptic mo dels [25, 31–34]. Ho w ev er the te ns or pro duct with the Heisen b erg alg ebras breaks down the qu asi-Hopf algebra structure, so that U q ,p ( g ) is not a quasi-Hopf algebra. Mo reov er, the quasi-Hopf algebra itself has a d isad v an tage that its coalgebra structure is not s uitable for a pr actic al calculation d ue to a complication arising from th e same t wist pro cedure. This is a serious defect, for examp le, to dev elop the represen tation theory of th e elliptic quantum group s and th eir harmonic analysis. The aim of this pap er is to sho w that a relev ant coalgebra structure of U q ,p ( g ) is an H -Hopf algebroid and to formulate a new elliptic quan tum group, whic h complemen ts the disadv an tage of th e quasi-Hopf algebra. In this pap er w e consider the case g = b sl 2 . Th e cases of other aﬃne Lie algebra t yp es will b e discussed in fu ture p u blications. The H -Hopf algebroid w as introdu ced by Etingof and V arc henko [12, 13] motiv ated by the w orks of F elder and V arc henko [16, 17]. S ome additional structures were giv en by Ko elink and Rosengren [29, 43]. A goo d review of this su b ject can b e found in [53]. In [12, 30, 51], it w as applied to a formulatio n of F elder’s elliptic quan tum group E τ ,η ( sl 2 ) by u s ing the generalized FRST constru ction on the basis of the RLL relation associated with the elliptic dynamical R matrix. Another form ulation of E τ ,η ( sl 2 ) as a quasi-Hopf algebra w as studied by En riquez and F elder [11]. A similar Hopf algebroid s tr ucture w as in tro duced by Lu [39] and Xu [55]. In [55], 2 Xu also studied the algebra D ⊗ U q ( ¯ g ), wh ere D d enotes the algebra of meromorphic diﬀerenti al op erators on ¯ h ∗ . His algebra is similar to U q ,p ( g ), bu t his ¯ g is a ﬁn ite-dimensional simple Lie algebra. Our form ulation is based on the fact that the RLL relation for U q ,p ( b sl 2 ) obtained in [25] is iden tical to a cen tral extension of the one for F elder’s elliptic quant um group. This enables us to apply the ge neralized FRST construction to our case with a mo diﬁcation d ue to a cent ral extension. The main m od iﬁcation is th at we use b oth the comm uting subalgebra H of U q ,p ( b sl 2 ) and the additiv e Ab elian group ¯ H ∗ ⊂ H ∗ appropriately in the form ulation. Here H conta ins the cen tral element c , whereas ¯ H do es not. W e consid er the ﬁeld of meromorphic fu nctions on H ∗ , but w e u se ¯ H ∗ to deﬁne the bigrading structure of U q ,p ( b sl 2 ). As a result we obtain a n ew face t yp e elliptic quantum group U q ,p ( b sl 2 ) as an H -Hopf algebroid, wh ich is realized in terms of the Drinfeld generators and h as the central extension. W e also sh o w that th e coalgebra stru cture is enough simple for a p r actic al calculatio ns . In comparison with the p revious formulat ions [12, 17, 30, 51], U q ,p ( b sl 2 ) h as the adv anta ge that it has a co nstru ctiv e deﬁnition in terms of the Drinfeld generators of U q ( b sl 2 ). This allo ws a systematic deriv ation of b oth ﬁnite and inﬁnite-dimensional representa tions of U q ,p ( b sl 2 ) from those of U q ( b sl 2 ) and their paralle l stru ctures to U q ( b sl 2 ). As an example, we s tu dy a cla ssiﬁ- cation theorem of the ﬁ nite-dimensional irreducible p s eudo-highest w eigh t represen tations and mak e a statemen t in terms of an elliptic analogue of the Drinfeld p olynomia ls. Th is give s an elliptic analogue of the works by Drinfeld [9] an d b y C hari and Pressley [6]. In addition, we in ve stigate a submo dule stru ctur e of the tensor pr o d uct of t w o ev aluation mo du les. W e obtain the s ingular vect ors explicitly and deriv e an elliptic analogue of the C lebs c h -Gordan co eﬃcien ts. W e sho w that the coeﬃcien ts are giv en by the terminating v ery-w ell-p oised balanced elliptic h yp ergeometric series 12 V 11 , whic h w as in tro duced b y F renk el and T uraev [18] on the basis of the wo rk b y Date, Jim b o, Kuniba, Miw a and Ok ado [8], and extensively studied by Spiridonov and Zhedanov [48, 49]. This provides the alternativ e to the representati on theoretica l deriv ation of 12 V 11 b y Ko elink, v an Norden and Rosengren [30]. In [30], 12 V 11 w as obtained as matrix elemen ts of a co-represen tation of F elder ’s elliptic quantum group. In the separate p ap er [36], w e discuss a f ree ﬁ eld r ep resen tation of the inﬁ nite-dimensional highest w eigh t representat ions of U q ,p ( b sl 2 ) and d er ive th e vertex op erators as in tert wining op- erators of such U q ,p ( b sl 2 )-mo dules. The resultant vertex op erators coincide with those obtained indirectly in [25] on the basis of the qu asi-Hopf algebra stru ctur e of B q ,λ ( b sl 2 ). Th is indicates a consistency of our H -Hopf algebroid structure on U q ,p ( b sl 2 ) even in the case with non -zero cen tral elemen t. W e hence establish the extension of the algebraic analysis scheme to the fusion RSOS 3 mo del on the basis of U q ,p ( b sl 2 ). This pap er is organized as follo ws. In Sect.2, w e giv e a deﬁn ition of the elliptic algebra U q ,p ( b sl 2 ) and review some results on th e RLL relation. In S ect.3 , we recall the deﬁ nition of H - Hopf algebroid from [12, 13, 29]. Then we deﬁne an H -Hopf algebroid structure on U q ,p ( b sl 2 ) and form ulate it as an e lliptic qu an tum group. In Sect.4, after a summary of basic fac ts on dynamical represent ations, w e consider ﬁ nite-dimensional represent ations of U q ,p ( b sl 2 ). In particular, w e in tro duce an elliptic analogue of the Drinfeld p olynomial and s tate a criterion for the ﬁ niteness of irreducible pseudo-highest w eigh t repr esen tation of U q ( b sl 2 ). W e also in vesti gate a su b mo dule structure of the tensor p ro duct of t wo ev aluation representa tions and derive an elliptic analogue of the Clebsc h-Gordan co eﬃcien ts. Sect.5 is devot ed to a discussion on the trigonometric, non- aﬃne and non-dynamical limits of the r esults. In App endix A, w e give a list of the comm utation relations of the L op erator elements. App endix B is devot ed to a pro of of T heorem 4.18. 2 The Elliptic Algebra U q ,p ( b sl 2 ) In this section, we giv e a deﬁnition of the elliptic algebra U q ,p ( b sl 2 ) in terms of th e Drinfeld generators of the quantum aﬃne algebra U q ( b sl 2 ) and the Heisen b erg algebra { P, e Q } . W e then recall some basic facts on U q ,p ( b sl 2 ) from [25, 34]. 2.1 Quan tum Aﬃne Algebra K [ U q ( b sl 2 )] Throughout this p ap er we ﬁ x a complex n umb er q suc h that q 6 = 0 , | q | < 1. Deﬁnition 2.1. [9] F or a ﬁeld K , the quantum aﬃne algebr a K [ U q ( b sl 2 )] in the Drinfeld r e aliza- tion is an asso ci ative algebr a over K gener ate d by the standar d Drinfeld gener ators a n ( n ∈ Z 6 =0 ) , x ± n ( n ∈ Z ) , h , c and d . The deﬁning r elations ar e given as fol lows . c : c entr al , [ h, d ] = 0 , [ d, a n ] = na n , [ d, x ± n ] = nx ± n , [ h, a n ] = 0 , [ h, x ± ( z )] = ± 2 x ± ( z ) , [ a n , a m ] = [2 n ] q [ cn ] q n q − c | n | δ n + m, 0 , [ a n , x + ( z )] = [2 n ] q n q − c | n | z n x + ( z ) , [ a n , x − ( z )] = − [2 n ] q n z n x − ( z ) , 4 ( z − q ± 2 w ) x ± ( z ) x ± ( w ) = ( q ± 2 z − w ) x ± ( w ) x ± ( z ) , [ x + ( z ) , x − ( w )] = 1 q − q − 1  δ  q − c z w  ψ ( q c/ 2 w ) − δ  q c z w  ϕ ( q − c/ 2 w )  . wher e we use [ n ] q = q n − q − n q − q − 1 , δ ( z ) = P n ∈ Z z n and the Drinfeld curr ents deﬁne d by x ± ( z ) = X n ∈ Z x ± n z − n , ψ ( q c/ 2 z ) = q h exp ( q − q − 1 ) X n> 0 a n z − n ! , ϕ ( q − c/ 2 z ) = q − h exp − ( q − q − 1 ) X n> 0 a − n z n ! . We also denote by K [ U ′ q ( b sl 2 )] the sub algebr a of K [ U q ( b sl 2 )] ge ne r ate d by the same gener ators as K [ U q ( b sl 2 )] exc ept for d exclude d. In the later s ections, we use the symb ols ψ n and φ − n ( n ∈ Z ≥ 0 ) deﬁned by ψ ( q c/ 2 z ) = X n ≥ 0 ψ n z − n , ϕ ( q c/ 2 z ) = X n ≥ 0 φ − n z n . Let α 1 and ¯ Λ 1 = 1 2 α 1 b e the simple ro ot and the fundamenta l w eigh t of sl (2 , C ), resp ectiv ely . W e set ¯ h = C h , Q = Z α 1 and ¯ h ∗ = C ¯ Λ 1 . W e denote by < , > the paring of ¯ h and ¯ h ∗ giv en by < ¯ Λ 1 , h > = 1. 2.2 Deﬁnition of the Elliptic Algebra U q ,p ( b sl 2 ) Let r b e a generic complex num b er. W e set r ∗ = r − c , p = q 2 r and p ∗ = q 2 r ∗ . W e deﬁne the Jacobi theta functions [ u ] and [ u ] ∗ b y [ u ] = q u 2 r − u ( p ; p ) 3 ∞ Θ p ( q 2 u ) , [ u ] ∗ = [ u ] | r → r ∗ , Θ p ( z ) = ( z ; p ) ∞ ( p/z ; p ) ∞ ( p ; p ) ∞ , where ( z ; p 1 , p 2 , · · · , p m ) ∞ = ∞ Y n 1 ,n 2 , ··· ,n m =0 (1 − z p n 1 1 p n 2 2 · · · p n m m ) . Setting p = e − 2 πi τ , [ u ] satisﬁes the quasi-p erio dicit y [ u + r ] = − [ u ], [ u + r τ ] = e − π i (2 u/r + τ ) [ u ]. Let { P , Q } b e a Heisen b erg algebra comm uting with C [ U q ( b sl 2 )] and satisfying [ P , Q ] = − 1 . (2.1) W e set H = C P ⊕ C r ∗ and H ∗ = C Q ⊕ C ∂ ∂ r ∗ . W e denote b y the same sym b ol < , > as the ab o v e the pairing of H and H ∗ deﬁned by < Q, P > = 1 = < ∂ ∂ r ∗ , r ∗ >, 5 the others are zero. W e regard H ∗ ⊕ H as a Heisen b erg algebra b y [ x, y ] = < x, y > . W e also consider th e Ab elian group ¯ H ∗ = Z Q . W e hav e the isomorphism φ : Q → ¯ H ∗ b y nα 1 7→ nQ . W e den ote b y C [ ¯ H ∗ ] the group a lgebra o ver C of ¯ H ∗ . W e denote by e α the elemen t of C [ ¯ H ∗ ] corresp onding to α ∈ ¯ H ∗ . These e α satisfy e α e β = e α + β and ( e α ) − 1 = e − α . In particular, e 0 = 1 is th e iden tit y elemen t. No w we tak e the p o wer series ﬁeld F = C (( P , r ∗ )) as K and consider the semi-dir ect pro duct C -algebra U q ,p ( b sl 2 ) = F [ U q ( b sl 2 )] ⊗ C C [ ¯ H ∗ ] of F [ U q ( b sl 2 )] and C [ ¯ H ∗ ]. W e imp ose the follo win g relation. F or α ∈ ¯ H ∗ , e α f ( P , r ∗ ) e − α = f ( P + < α, P >, r ∗ ) . (2.2) Then the multiplica tion of U q ,p ( b sl 2 ) is deﬁned by ( f ( P , r ∗ ) a ⊗ e α ) · ( g ( P , r ∗ ) b ⊗ e β ) = f ( P , r ∗ ) g ( P + < α, P > , r ∗ ) ab ⊗ e α + β , a, b ∈ C [ U q ( b sl 2 )] , f ( P , r ∗ ) , g ( P , r ∗ ) ∈ F , α, β ∈ ¯ H ∗ . Moreo ver, for f ( P , r ∗ ) ∈ F we r egard the ob ject f ( P + h, r ∗ + c ) as the elemen t of U q ,p ( b sl 2 ) in the sense of completion. W e then hav e the follo wing relations. x ± ( z ) f ( P + h, r ∗ + c ) = f ( P + h ∓ 2 , r ∗ + c ) x ± ( z ) , (2.3) [ f ( P + h, r ∗ + c ) , a n ] = 0 , [ f ( P + h, r ∗ + c ) , d ] = 0 . (2.4) R emark. The relation (2.2) is automatica lly satisﬁed, if one tak es the r ealiz ation Q = ∂ ∂ P . The follo wing automorphism φ r of F [ U q ( b sl 2 )] is the key to our “elliptic d eform atio n” [25]. c 7→ c, h 7→ h, d 7→ d, x + ( z ) 7→ u + ( z , p ) x + ( z ) , x − ( z ) 7→ x − ( z ) u − ( z , p ) , ψ ( z ) 7→ u + ( q c/ 2 z , p ) ψ ( z ) u − ( q − c/ 2 z , p ) , ϕ ( z ) 7→ u + ( q − c/ 2 z , p ) ϕ ( z ) u − ( q c/ 2 z , p ) . Here we set u + ( z , p ) = exp X n> 0 1 [ r ∗ n ] q a − n ( q r z ) n ! , u − ( z , p ) = exp − X n> 0 1 [ r n ] q a n ( q − r z ) − n ! . (2.5) W e d eﬁne the elliptic currents E ( u ) , F ( u ) and K ( u ) in U q ,p ( b sl 2 )[[ u ]] as follo ws. 6 Deﬁnition 2.2 (Elliptic cur ren ts) . E ( u ) = φ r ( x + ( z )) e 2 Q z − P − 1 r ∗ , F ( u ) = φ r ( x − ( z )) z P + h − 1 r , K ( z ) = exp X n> 0 [ n ] q [2 n ] q [ r ∗ n ] q a − n ( q c z ) n ! exp − X n> 0 [ n ] q [2 n ] q [ r n ] q a n z − n ! × e Q z − c 4 rr ∗ (2 P − 1)+ 1 2 r h ˆ d = d − 1 4 r ∗ ( P − 1)( P + 1) + 1 4 r ( P + h − 1)( P + h + 1) , wher e we set z = q 2 u , p = q 2 r . F rom (2.1) and Deﬁnition 2.1, we can d eriv e the follo wing relations. Prop osition 2.3. c : cen tral , [ h, a n ] = 0 , [ h, E ( u )] = 2 E ( u ) , [ h, F ( u )] = − 2 F ( u ) , [ ˆ d, h ] = 0 , [ ˆ d, a n ] = na n , [ ˆ d, E ( u )] =  − z ∂ ∂ z − 1 r ∗  E ( u ) , [ ˆ d, F ( u )] =  − z ∂ ∂ z − 1 r  F ( u ) , [ a n , a m ] = [2 n ] q [ cn ] q n q − c | n | δ n + m, 0 , [ a n , E ( u )] = [2 n ] q n q − c | n | z n E ( u ) , [ a n , F ( u )] = − [2 n ] q n z n F ( u ) , E ( u ) E ( v ) = [ u − v + 1] ∗ [ u − v − 1] ∗ E ( v ) E ( u ) , F ( u ) F ( v ) = [ u − v − 1] [ u − v + 1] F ( v ) F ( u ) , [ E ( u ) , F ( v )] = 1 q − q − 1  δ  q − c z w  H + ( q c/ 2 w ) − δ  q c z w  H − ( q − c/ 2 w )  , wher e z = q 2 u , w = q 2 v , and we set H ± ( z ) = κK  u ± 1 2 ( r − c 2 ) + 1 2  K  u ± 1 2 ( r − c 2 ) − 1 2  , (2.6) κ = lim z → q − 2 ξ ( z ; p ∗ , q ) ξ ( z ; p, q ) , ξ ( z ; p, q ) = ( q 2 z ; p, q 4 ) ∞ ( pq 2 z ; p, q 4 ) ∞ ( q 4 z ; p, q 4 ) ∞ ( pz ; p, q 4 ) ∞ . Moreo ver from (2.2) and (2.3), we ob tain the follo w ing relations. Prop osition 2.4. F or f ( P ) ∈ C (( P )) , K ( u ) f ( P ) = f ( P + 1) K ( u ) , E ( u ) f ( P ) = f ( P + 2) E ( u ) , [ F ( u ) , f ( P )] = 0 , K ( u ) f ( P + h ) = f ( P + h + 1) K ( u ) , [ E ( u ) , f ( P + h )] = 0 , F ( u ) f ( P + h ) = f ( P + h + 2) F ( u ) . 7 Deﬁnition 2.5. We c al l a set ( F [ U q ( b sl 2 )] ⊗ C C [ ¯ H ∗ ] , φ r ) the el liptic algebr a U q ,p ( b sl 2 ) . W e also denote by U ′ q ,p ( b sl 2 ) the sub algebr a F [ U ′ q ( b sl 2 )] ⊗ C C [ ¯ H ∗ ] of U q ,p ( b sl 2 ) . The follo wing relations are also useful. Prop osition 2.6. K ( u ) K ( v ) = ρ ( u − v ) K ( v ) K ( u ) , K ( u ) E ( v ) = [ u − v + 1 − r ∗ 2 ] ∗ [ u − v − 1+ r ∗ 2 ] ∗ E ( v ) K ( u ) , K ( u ) F ( v ) = [ u − v − 1+ r 2 ] [ u − v + 1 − r 2 ] F ( v ) K ( u ) , H + ( u ) H − ( v ) = [ u − v − 1 − c 2 ] [ u − v + 1 − c 2 ] [ u − v + 1 + c 2 ] ∗ [ u − v − 1 + c 2 ] ∗ H − ( v ) H + ( u ) , H ± ( u ) H ± ( v ) = [ u − v − 1] [ u − v + 1] [ u − v + 1] ∗ [ u − v − 1] ∗ H ± ( v ) H ± ( u ) , wher e ρ ( u ) = ρ + ∗ ( u ) ρ + ( u ) , ρ + ∗ ( u ) = ρ + ( u ) | r → r ∗ , ρ + ( u ) = z 1 2 r { pq 2 z } 2 { pz }{ pq 4 z } { z − 1 }{ q 4 z − 1 } { q 2 z − 1 } 2 , { z } = ( z ; p, q 4 ) ∞ . Note that the fun ction ρ ( u ) satisﬁes ρ (0) = 1 , ρ (1) = [1] ∗ [1] , ρ ( u ) ρ ( − u ) = 1 , ρ ( u ) ρ ( u + 1) = [ u + 1] ∗ [ u ] ∗ [ u ] [ u + 1] . In addition, the follo wing formulae indicate a direct construction of H ± ( u ) fr om the Drinfeld currents ψ ( z ) and ϕ ( z ). Prop osition 2.7. H + ( u ) = φ r ( ψ ( z )) e 2 Q  q r − c/ 2 z  − c r r ∗ ( P − 1)+ h r , H − ( u ) = φ r ( ϕ ( z )) e 2 Q  q − r + c/ 2 z  − c r r ∗ ( P − 1)+ h r . 2.3 The R LL -relation for U q ,p ( b sl 2 ) F ollo w ing [25], we sum m arize the resu lts on the L op erator and the RL L -relatio n for U q ,p ( b sl 2 ). In Sect. 3.2, we use the L op erator to deﬁn e th e H -Hopf algebroid s tr ucture of U q ,p ( b sl 2 ). W e ﬁ rst deﬁn e th e half currents E + ( u ) , F + ( u ) and K + ( u ) as follo ws. 8 Deﬁnition 2.8 (Half cu r ren ts) . K + ( u ) = K ( u + r +1 2 ) , (2.7) E + ( u ) = a ∗ I C ∗ E ( u ′ ) [ u − u ′ + c/ 2 − P + 1] ∗ [1] ∗ [ u − u ′ + c/ 2] ∗ [ P − 1] ∗ dz ′ 2 π iz ′ , (2.8) F + ( u ) = a I C F ( u ′ ) [ u − u ′ + P + h − 1] [1] [ u − u ′ ][ P + h − 1] dz ′ 2 π iz ′ . (2.9) Her e the c ontours ar e chosen such that C ∗ : | p ∗ q c z | < | z ′ | < | q c z | , C : | pz | < | z ′ | < | z | , and the c onstants a, a ∗ ar e chosen to satisfy a ∗ a [1] ∗ κ q − q − 1 = 1 . The commutati on relations for the elliptic cur r en ts in P r op ositions 2.3 − 2.6 yield the follo w in g relations for the h alf cur ren ts. Prop osition 2.9. K + ( u 1 ) K + ( u 2 ) = ρ ( u ) K + ( u 2 ) K + ( u 1 ) , (2.10) K + ( u 1 ) E + ( u 2 ) K + ( u 1 ) − 1 = E + ( u 2 ) [1 + u ] ∗ [ u ] ∗ − E + ( u 1 ) [1] ∗ [ P ] ∗ [ P + u ] ∗ [ u ] ∗ , (2.11) K + ( u 1 ) − 1 F + ( u 2 ) K + ( u 1 ) = [1 + u ] [ u ] F + ( u 2 ) − [1] [ P + h ] [ P + h − u ] [ u ] F + ( u 1 ) , (2.12 ) [1 − u ] ∗ [ u ] ∗ E + ( u 1 ) E + ( u 2 ) + [1 + u ] ∗ [ u ] ∗ E + ( u 2 ) E + ( u 1 ) (2.13) = E + ( u 1 ) 2 [1] ∗ [ P − 2] ∗ [ P − 2 + u ] ∗ [ u ] ∗ + E + ( u 2 ) 2 [1] ∗ [ P − 2] ∗ [ P − 2 − u ] ∗ [ u ] ∗ , [1 + u ] [ u ] F + ( u 1 ) F + ( u 2 ) + [1 − u ] [ u ] F + ( u 2 ) F + ( u 1 ) (2.14) = F + ( u 1 ) 2 [1] [ P + h − 2] [ P + h − 2 − u ] [ u ] + F + ( u 2 ) 2 [1] [ P + h − 2] [ P + h − 2 + u ] [ u ] , [ E + ( u 1 ) , F + ( u 2 )] = K + ( u 2 − 1) K + ( u 2 ) [ P − 1 − u ] ∗ [ u ] ∗ [1] ∗ [ P − 1] ∗ − K + ( u 1 ) K + ( u 1 − 1) [ P + h − 1 − u ] [ u ] [1] [ P + h − 1] , (2.15) wher e we set u = u 1 − u 2 . W e n ext deﬁ n e th e L -op erator b L + ( u ) ∈ End( V ) ⊗ U q ,p  b sl 2  with V ∼ = C 2 as follo ws. 9 Deﬁnition 2.10 ( L -op erator) . b L + ( u ) =   1 F + ( u ) 0 1     K + ( u − 1) 0 0 K + ( u ) − 1     1 0 E + ( u ) 1   . (2.16) Then the relatio ns in Prop osition 2.9 c an b e combined in to the follo win g single RLL relation. Prop osition 2.11. The b L + ( u ) op er ator satisﬁes the fol lowing R LL r elation. R +(12) ( u 1 − u 2 , P + h ) b L +(1) ( u 1 ) b L +(2) ( u 2 ) = b L +(2) ( u 2 ) b L +(1) ( u 1 ) R + ∗ (12) ( u 1 − u 2 , P ) , (2.17) wher e R + ( u, P + h ) and R + ∗ ( u, P ) = R + ( u, P ) | r → r ∗ denote the el liptic dynamic al R matric es given by R + ( u, s ) = ρ + ( u )        1 b ( u, s ) c ( u, s ) ¯ c ( u, s ) ¯ b ( u, s ) 1        (2.18) with ρ + ( u ) in P r op osition 2.6, and b ( u, s ) = [ s + 1][ s − 1 ] [ s ] 2 [ u ] [1 + u ] , c ( u, s ) = [1] [ s ] [ s + u ] [1 + u ] , ¯ c ( u, s ) = [1] [ s ] [ s − u ] [1 + u ] , ¯ b ( u, s ) = [ u ] [1 + u ] . One should note that the c = 0 case of the R LL relation (2.17) is iden tical, up to a gauge transformation, with the one stud ied in th e form ulation of F elder’s elliptic quan tum group in [12, 30, 51]. 2.4 Connection t o the Quasi-Hopf Algebra B q ,λ ( b sl 2 ) It is wo rth to remark a connection of U q ,p ( b sl 2 ) to the qu asi-Hopf algebra B q ,λ ( b sl 2 ). Let us deﬁn e a n ew L op erator L + ( u, P ) b y L + ( u, P ) = b L + ( u )   e − Q 0 0 e Q   . (2.19) Then f rom Deﬁnitions 2.2 and 2.10, one ﬁ nds that L + ( u, P ) is indep endent of Q . W e hence regard L + ( u, P ) as the op erator in F [ U q  b sl 2  ] ha ving P as a parameter. Sub stituting (2.19) in to (2.17), we obtain the f ollo wing statemen t. 10 Prop osition 2.12. The op er ator L + ( u, P ) satisﬁes the fol lowing dynamic al RLL r elation. R +(12) ( u 1 − u 2 , P + h ) L +(1) ( u 1 , P ) L +(2) ( u 2 , P + h (1) ) (2.20) = L +(2) ( u 2 , P ) L +(1) ( u 1 , P + h (2) ) R + ∗ (12) ( u 1 − u 2 , P ) . This RL L relation is identiﬁed with the one for the qu asi-Hopf algebra B q ,λ ( b sl 2 ) und er the parametrization λ = ( r ∗ + 2)Λ 0 + ( P + 1) ¯ Λ 1 [24, 25], wh er e Λ 0 and Λ 0 + ¯ Λ 1 denote the fundamental we ights of b sl (2 , C ). This is due to the fact that u nder this λ the v ector represen tation of the unive rs al d ynamical R matrix R + ( λ ) of B q ,λ ( b sl 2 ) yields the elliptic d y n amical R matrix R + ∗ ( u, P ) [25, 35]. F urthermore w e ha ve the isomorph ism B q ,λ ( b sl 2 ) ∼ = F [ U q  b sl 2  ] as an asso ciativ e algebra. Combining these facts, w e obtain the isomorp hism U q ,p ( b sl 2 ) ∼ = B q ,λ ( b sl 2 ) ⊗ C C [ ¯ H ∗ ] with λ = ( r ∗ + 2)Λ 0 + ( P + 1) ¯ Λ 1 as a semi-direct p ro duct algebra. Note also th at the c = 0 case of (2.20) is iden tical to the one used in [16, 17] to d eﬁne F elder’s elliptic quantum group in its original form. 3 H -Hopf Algebroid In this sectio n, we in tro du ce an H -Hopf a lgebroid structure into the elliptic algebra U q ,p ( b sl 2 ) and formulat e U q ,p ( b sl 2 ) as an elliptic quan tum group. 3.1 Deﬁnition of the H -Hopf Algebroid Let us recall some b asic facts on the H -Hopf algebroid follo w ing the works of Etingof and V arc henko [12, 13] and of Ko elink and R osengren [29]. Let A b e a complex asso ciativ e algebra, H b e a ﬁnite d imensional comm utativ e su balgebra of A , and M H ∗ b e th e ﬁeld of meromorph ic fu nctions on H ∗ the dual sp ace of H . Deﬁnition 3.1 ( H -algebra) . An H -algebr a is a c omplex asso c i ative algebr a A with 1, which is bigr ade d over H ∗ , A = M α,β ∈ H ∗ A αβ , a nd e quipp e d with two al gebr a emb e ddings µ l , µ r : M H ∗ → A 00 (the left and right moment maps), such that µ l ( b f ) a = aµ l ( T α b f ) , µ r ( b f ) a = aµ r ( T β b f ) , a ∈ A αβ , b f ∈ M H ∗ , wher e T α denotes the automorphism ( T α b f )( λ ) = b f ( λ + α ) of M H ∗ . Deﬁnition 3.2 ( H -algebra homomorphism) . An H - algebr a homomorp hism is an algebr a homo- morphism π : A → B b etwe e n two H -algebr as A and B pr e se rvi ng the bigr ading and the moment maps, i.e. π ( A αβ ) ⊆ B αβ for al l α, β ∈ H ∗ and π ( µ A l ( b f )) = µ B l ( b f ) , π ( µ A r ( b f )) = µ B r ( b f ) . 11 Let A and B b e t wo H -algebras. Th e tensor pro du ct A e ⊗ B is the H ∗ -bigraded vecto r space with ( A e ⊗ B ) αβ = M γ ∈ H ∗ ( A αγ ⊗ M H ∗ B γ β ) , where ⊗ M H ∗ denotes the u sual tensor pro duct mo dulo the follo wing relation. µ A r ( b f ) a ⊗ b = a ⊗ µ B l ( b f ) b, a ∈ A, b ∈ B , b f ∈ M H ∗ . (3.1) The tensor p ro duct A e ⊗ B is again an H -algebra with the multiplicati on ( a ⊗ b )( c ⊗ d ) = ac ⊗ bd and the momen t maps µ A e ⊗ B l = µ A l ⊗ 1 , µ A e ⊗ B r = 1 ⊗ µ B r . Let D b e the algebra of automorph isms M H ∗ → M H ∗ D = { X i b f i T β i | b f i ∈ M H ∗ , β i ∈ H ∗ } . Equipp ed with the b igrading D αα = { b f T − α | b f ∈ M H ∗ , α ∈ H ∗ } , D αβ = 0 ( α 6 = β ) and th e momen t maps µ D l , µ D r : M H ∗ → D 00 deﬁned b y µ D l ( b f ) = µ D r ( b f ) = b f T 0 , D is an H -algebra. F or an y H -algebra A , we h a v e th e canonical isomorphism as an H -algebra A ∼ = A e ⊗D ∼ = D e ⊗ A (3.2) b y a ∼ = a e ⊗ T − β ∼ = T − α e ⊗ a for all a ∈ A αβ . Deﬁnition 3.3 ( H -bialgebroid) . An H -bi algebr oid is an H - algebr a A e qui pp e d with two H - algebr a homo morphisms ∆ : A → A e ⊗ A (the c omultiplic ation) and ε : A → D (the c ounit) such that (∆ e ⊗ id) ◦ ∆ = (id e ⊗ ∆) ◦ ∆ , ( ε e ⊗ id) ◦ ∆ = id = (id e ⊗ ε ) ◦ ∆ , under the identiﬁc ation (3.2) . Deﬁnition 3.4 ( H -Hopf algebroid) . An H -Hopf algebr oid is an H -bialgebr oid A e quipp e d with a C -line ar map S : A → A (the antip o de), su c h that S ( µ r ( b f ) a ) = S ( a ) µ l ( b f ) , S ( aµ l ( b f )) = µ r ( b f ) S ( a ) , ∀ a ∈ A, b f ∈ M H ∗ , m ◦ (id e ⊗ S ) ◦ ∆( a ) = µ l ( ε ( a )1) , ∀ a ∈ A, m ◦ ( S e ⊗ id) ◦ ∆( a ) = µ r ( T α ( ε ( a )1)) , ∀ a ∈ A αβ , wher e m : A e ⊗ A → A denotes the multiplic ation and ε ( a )1 is the r esu lt of applying the diﬀer enc e op er ator ε ( a ) to the c onstant function 1 ∈ M H ∗ . 12 R emark. [29] Deﬁnition 3.4 yields that th e an tip o de of an H -Hopf algebroid u niquely exists and giv es th e algebra an tihomomorphism. The H -algebra D is an H -Hopf algebroid with ∆ D : D → D e ⊗D , ε D : D → D , S D : D → D deﬁned by ∆ D ( b f T − α ) = b f T − α e ⊗ T − α , ε D = id , S D ( b f T − α ) = T α b f = ( T α b f ) T α . 3.2 H -Hopf Algebroid Structure on U q ,p ( b sl 2 ) No w let us consider the elliptic algebra U q ,p ( b sl 2 ). Using the isomorp h ism φ : Q → ¯ H ∗ , we deﬁne the ¯ H ∗ -bigrading structur e of U q ,p = U q ,p ( b sl 2 ) as follo ws. U q ,p = M α,β ∈ ¯ H ∗ ( U q ,p ) αβ , ( U q ,p ) αβ =    x ∈ U q ,p       q h xq − h = q <φ − 1 ( α − β ) ,h> x q P xq − P = q <β ,P > x    . (3.3) Noting < φ − 1 ( α ) , h > = < α, P > , we ha v e q P + h xq − ( P + h ) = q <α,P > x (3.4) for x ∈ ( U q ,p ) αβ . R emark. The quan tum aﬃne alg ebr a U q = F [ U q ( b sl 2 )] has the follo wing n atural grading o v er ¯ H ∗ . U q = M α ∈ ¯ H ∗ ( U q ) α , ( U q ) α = { x ∈ U q | q h xq − h = q <φ − 1 ( α ) ,h> x } . W e then ha v e ( U q ,p ) αβ = ( U q ) α − β ⊗ C C e − β . Next let us regard the elemen ts b f = f ( P , r ∗ ) ∈ F as meromorphic fun ctions on H ∗ b y b f ( µ ) = f ( < µ, P >, < µ, r ∗ > ) µ ∈ H ∗ and consider the ﬁ eld of meromorphic functions M H ∗ on H ∗ M H ∗ = n b f : H ∗ → C    b f = f ( P , r ∗ ) ∈ F o . W e d eﬁne tw o em b edd in gs (the left an d righ t moment maps) µ l , µ r : M H ∗ → ( U q ,p ) 00 b y µ l ( b f ) = f ( P + h, r ∗ + c ) , µ r ( b f ) = f ( P , r ∗ ) . (3.5) F rom (2.2) and (2.3), one can verify the follo w in g. 13 Prop osition 3.5. F or x ∈ ( U q ,p ) αβ , we have µ l ( b f ) x = f ( P + h, r ∗ + c ) x = xf ( P + h + < α, P >, r ∗ + c ) = xµ l ( T α b f ) , µ r ( b f ) x = f ( P , r ∗ ) x = xf ( P + < β , P >, r ∗ ) = xµ r ( T β b f ) , wher e we r e gar d T α = e α ∈ C [ ¯ H ∗ ] as the shift op e r ator M H ∗ → M H ∗ ( T α b f ) = e α f ( P , r ∗ ) e − α = f ( P + < α, P >, r ∗ ) . Hereafter we abbreviate f ( P + h, r ∗ + c ) and f ( P , r ∗ ) as f ( P + h ) and f ∗ ( P ), resp ectiv ely . An imp ortan t example of the elemen ts in M H ∗ is th e elliptic dyn amical R matrix elemen ts ( b R + u ) ε ′ 1 ε ′ 2 ε 1 ε 2 ≡ R + ∗ ( u, P ) ε ′ 1 ε ′ 2 ε 1 ε 2 in (2.18), wher e ε i , ε ′ i = + , − ( i = 1 , 2). W e then ha ve µ l (( b R + u ) ε ′ 1 ε ′ 2 ε 1 ε 2 ) = R + ( u, P + h ) ε ′ 1 ε ′ 2 ε 1 ε 2 , µ r (( b R + u ) ε ′ 1 ε ′ 2 ε 1 ε 2 ) = R + ∗ ( u, P ) ε ′ 1 ε ′ 2 ε 1 ε 2 (3.6) in the abbr eviate n otation. Equipp ed with the bigrading structure (3.3 ) and t wo momen t maps (3.5), the elliptic algebra U q ,p ( b sl 2 ) is an H -algebra. W e also consider the H -algebra of th e shift op erators D = { X i b f i T α i | b f i ∈ M H ∗ , α i ∈ ¯ H ∗ } , D αα = { b f T − α } , D αβ = 0 ( α 6 = β ) , µ D l ( b f ) = µ D r ( b f ) = b f T 0 b f ∈ M H ∗ . Then w e h a ve the H -algebra isomorphism U q ,p ∼ = U q ,p e ⊗D ∼ = D e ⊗ U q ,p . No w let us consider the H -Hopf algebroid structure on U q ,p . It is conv eniently giv en b y the L op erator b L + ( u ). W e s h all wr ite the en tries of b L + ( u ) as b L + ( u ) =   b L + ++ ( u ) b L + + − ( u ) b L + − + ( u ) b L + −− ( u )   . (3.7) According to the Gauß decomp osition (2.16), we hav e b L + ++ ( u ) = K + ( u − 1) + F + ( u ) K + ( u ) − 1 E + ( u ) , b L + + − ( u ) = F + ( u ) K + ( u ) − 1 , (3.8) b L + − + ( u ) = K + ( u ) − 1 E + ( u ) , b L + −− ( u ) = K + ( u ) − 1 . One ﬁnd s b L + ε 1 ε 2 ( u ) ∈ ( U q ,p ) − ε 1 Q, − ε 2 Q . (3.9) 14 It is also easy to chec k f ( P + h ) b L + ε 1 ε 2 ( u ) = b L + ε 1 ε 2 ( u ) f ( P + h − ε 1 ) , f ∗ ( P ) b L + ε 1 ε 2 ( u ) = b L + ε 1 ε 2 ( u ) f ∗ ( P − ε 2 ) . (3.10) W e deﬁne t wo H -alge br a homomorphisms, the co-unit ε : U q ,p → D and the co-m ultiplication ∆ : U q ,p → U q ,p e ⊗ U q ,p b y ε ( b L + ε 1 ε 2 ( u )) = δ ε 1 ,ε 2 T ε 2 , ε ( e Q ) = e Q , (3.11) ε ( µ l ( b f )) = ε ( µ r ( b f )) = b f T 0 , (3.12) ∆( b L + ε 1 ε 2 ( u )) = X ε ′ b L + ε 1 ε ′ ( u ) e ⊗ b L + ε ′ ε 2 ( u ) , (3.13) ∆( e Q ) = e Q e ⊗ e Q , (3.14) ∆( µ l ( b f )) = µ l ( b f ) e ⊗ 1 , ∆( µ r ( b f )) = 1 e ⊗ µ r ( b f ) . (3.15) In f act, one can chec k that ∆ preserves the relat ion (2.17). Noting (3.6) and the form ula obta ined from (3.1) f ∗ ( u, P ) a e ⊗ b = a e ⊗ f ( u, P + h ) b a, b ∈ U q ,p , (3.16) w e h a ve ∆( LH S ) = X ε ′ 1 ,ε ′ 2 ∆( R + ( u, P + h ) ε ′ 1 ε ′ 2 ε ′′ 1 ε ′′ 2 )∆( b L + ε ′ 1 ε 1 ( u 1 ))∆( b L + ε ′ 2 ε 2 ( u 2 )) = X ε ′ 1 ,ε ′ 2 ε,ε ′ R + ( u, P + h ) ε ′ 1 ε ′ 2 ε ′′ 1 ε ′′ 2 b L + ε ′ 1 ε ( u 1 ) b L + ε ′ 2 ε ′ ( u 2 ) e ⊗ b L + εε 1 ( u 1 ) b L + ε ′ ε 2 ( u 2 ) = X ε ′ 1 ,ε ′ 2 ε,ε ′ b L + ε ′′ 2 ε ′ 2 ( u 2 ) b L + ε ′′ 1 ε ′ 1 ( u 1 ) R + ∗ ( u, P ) εε ′ ε ′ 1 ε ′ 2 e ⊗ b L + εε 1 ( u 1 ) b L + ε ′ ε 2 ( u 2 ) = X ε ′ 1 ,ε ′ 2 ε,ε ′ R + ∗ ( u, P ) εε ′ ε ′ 1 ε ′ 2 b L + ε ′′ 2 ε ′ 2 ( u 2 ) b L + ε ′′ 1 ε ′ 1 ( u 1 ) e ⊗ b L + εε 1 ( u 1 ) b L + ε ′ ε 2 ( u 2 ) = X ε ′ 1 ,ε ′ 2 ε,ε ′ b L + ε ′′ 2 ε ′ 2 ( u 2 ) b L + ε ′′ 1 ε ′ 1 ( u 1 ) e ⊗ R + ( u, P + h ) εε ′ ε ′ 1 ε ′ 2 b L + εε 1 ( u 1 ) b L + ε ′ ε 2 ( u 2 ) = X ε ′ 1 ,ε ′ 2 ε,ε ′ b L + ε ′′ 2 ε ′ 2 ( u 2 ) b L + ε ′′ 1 ε ′ 1 ( u 1 ) e ⊗ b L + ε ′ 2 ε ′ ( u 2 ) b L + ε ′ 1 ε ( u 1 ) R + ∗ ( u, P ) ε 1 ε 2 εε ′ = ∆( RH S ) . In the fourth line, w e used the prop ert y R + ∗ ( u, P + ε ′ 1 + ε ′ 2 ) ε 1 ε 2 ε ′ 1 ε ′ 2 = R + ∗ ( u, P ) ε 1 ε 2 ε ′ 1 ε ′ 2 . 15 Lemma 3.6. The maps ε and ∆ satisfy (∆ e ⊗ id) ◦ ∆ = (id e ⊗ ∆) ◦ ∆ , (3.17) ( ε e ⊗ id) ◦ ∆ = id = (id e ⊗ ε ) ◦ ∆ . (3.18) Pr o of. S traigh t forw ard. W e also h a ve the follo wing form ulae. Prop osition 3.7. ε ( q h ) = ε ( q c ) = T 0 , (3.19) ∆( q h ) = q h e ⊗ q h , ∆( q c ) = q c e ⊗ q c , (3.20) ∆  f ( P , r ∗ ) f ( P + h, r ∗ + c )  = f ( P , r ∗ ) f ( P + h, r ∗ + c ) e ⊗ f ( P , r ∗ ) f ( P + h, r ∗ + c ) . (3.21) Pr o of. (3.19) follo ws from (3.5) and (3.12), whereas (3.20) follo ws from (3.5), (3.15) and (3.1 ). F or example, ∆( q h ) = ∆( q P + h q − P ) = ∆( q P + h )∆( q − P ) = q P + h e ⊗ q − P = q h e ⊗ q h . T o show (3.21) we u s e (3.15) and (3.1) as LHS = ∆( µ r ( b f ))∆( µ l ( b f ) − 1 ) = µ l ( b f ) − 1 e ⊗ µ r ( b f ) = f ( P , r ∗ ) f ( P + h, r ∗ + c ) 1 f ( P , r ∗ ) e ⊗ f ( P , r ∗ ) = RHS . W e n ext deﬁ n e an algebra an tihomomorphism (the an tip o de) S : U q ,p → U q ,p b y S ( b L + ++ ( u )) = b L + −− ( u − 1) , S ( b L + + − ( u )) = − [ P + h + 1] [ P + h ] b L + + − ( u − 1) , S ( b L + − + ( u )) = − [ P ] ∗ [ P + 1] ∗ b L + − + ( u − 1) , S ( b L + −− ( u )) = [ P + h + 1][ P ] ∗ [ P + h ][ P + 1] ∗ b L + ++ ( u − 1) , S ( e Q ) = e − Q , S ( µ r ( b f )) = µ l ( b f ) , S ( µ l ( b f )) = µ r ( b f ) . Note th at S preserve s the R LL relation (2.17 ). T o s h o w this, we use the relatio ns in Prop osition 2.9. F urther m ore we ha v e the follo wing L emm a. Lemma 3.8. The map S satisﬁes m ◦ (id ⊗ S ) ◦ ∆( x ) = µ l ( ε ( x )1) , ∀ x ∈ U q ,p , m ◦ ( S ⊗ id ) ◦ ∆( x ) = µ r ( T α ( ε ( x )1)) , ∀ x ∈ ( U q ,p ) αβ . 16 Pr o of. W e pr o ve the ﬁrst r elatio n f or x = b L ++ ( u ). The other is similar. Using (3.8) and (3.10), LH S = b L ++ ( u ) b L −− ( u − 1) − b L + − ( u ) b L − + ( u − 1) [ P − 1] ∗ [ P ] ∗ = ( K + ( u − 1) + F + ( u ) K + ( u ) − 1 E + ( u )) K + ( u − 1) − 1 − F + ( u ) K + ( u ) − 1 K + ( u − 1) − 1 E + ( u − 1) [ P − 1] ∗ [ P ] ∗ = 1 = µ l ( ε ( b L ++ ( u )1)) . In the last line, we used the relation (2.11) with the replacemen t u 1 7→ u − 1 , u 2 7→ u and u 7→ − 1. F rom Lemmas 3.6 and 3.8, w e h a ve Theorem 3.9. The H -algebr a U q ,p ( b sl 2 ) e quipp e d with (∆ , ε, S ) is an H -Hopf algebr oid. Deﬁnition 3.10. We c al l the H - Hopf algebr oid ( U q ,p ( b sl 2 ) , H , M H ∗ , µ l , µ r , ∆ , ε, S ) the el liptic q uantum gr oup U q ,p ( b sl 2 ) . W e also u se th e follo w ing com ultiplication formulae for the h alf current s. Prop osition 3.11. ∆( K + ( u )) = K + ( u ) e ⊗ K + ( u ) + ∞ X j =1 ( − ) j E + ( u ) j K + ( u ) e ⊗ K + ( u ) F + ( u ) j , ∆( E + ( u )) = 1 e ⊗ E + ( u ) + E + ( u ) e ⊗ K + ( u ) K + ( u − 1) + ∞ X j =1 ( − ) j E + ( u ) j +1 e ⊗ K + ( u ) F + ( u ) j K + ( u − 1) , ∆( F + ( u )) = F + ( u ) e ⊗ 1 + K + ( u − 1) K + ( u ) e ⊗ F + ( u ) + ∞ X j =1 ( − ) j K + ( u − 1) E + ( u ) j K + ( u ) e ⊗ F + ( u ) j +1 , ∆( H ± ( u )) = H ± ( u ) e ⊗ H ± ( u ) + ∞ X j =1 ( − ) j  κK + ( u + C ± ) E + ( u + C ± − 1) j K + ( u + C ± − 1) e ⊗ H + ( u ) F + ( u + C ± − 1) j + E + ( u + C ± ) j H + ( u ) e ⊗ κK + ( u + C ± ) F + ( u + C ± ) j K + ( u + C ± − 1)  + ∞ X i,j = 1 ( − ) i + j κE + ( u + C ± ) i K + ( u + C ± ) E + ( u + C ± − 1) j K + ( u + C ± − 1) e ⊗ κK + ( u + C ± ) F + ( u + C ± ) i K + ( u + C ± − 1) F + ( u + C ± − 1) j , wher e C ± = − r 2 ± ( r 2 − c 4 ) . Pr o of. Use (3.13), (3.8 ) and (2.6) as well as ∆( κ ) = κ e ⊗ κ obtained f rom (3.21). 17 4 Finite-Dimensional Represen tations In this sectio n, we d iscuss r ep resen tations of the elliptic alg ebr a U ′ q ,p = U ′ q .p ( b sl 2 ). T he main results are the criterion for the ﬁniteness of irredu cible representati ons Th eorem 4.11 and the submo du le stru cture of the te ns or pr od uct of t wo ev aluation represen tations Theorem 4 .17 − 4.19. F or br evity , we denote th e entries of b L + ( u ) by b L + ( u ) =   α ( u ) β ( u ) γ ( u ) δ ( u )   . 4.1 Dynamical Representations W e in tro duce the co ncept of dyn amical represent ation, i.e. representat ion as H -algebras [12, 13, 29]. W e follo w s the deﬁnition giv en in [29]. W e then give a constru ction of dynamical represent ations of U ′ q ,p . Let us consider a v ector space b V o v er F , whic h is ¯ h -diagonaliz able, b V = M µ ∈ ¯ h ∗ b V µ , b V µ = { v ∈ V | q ¯ h v = q µ v ( ¯ h ∈ ¯ h ) } . Let us deﬁn e the H -algebra D H, b V of the C -linear op erators on b V by D H, b V = M α,β ∈ ¯ H ∗ ( D H, b V ) αβ , ( D H, b V ) αβ =    X ∈ En d C b V       X ( f ∗ ( P ) v ) = f ∗ ( P − < β , P > ) X ( v ) , X ( b V µ ) ⊆ b V µ + φ − 1 ( α − β ) , v ∈ b V , f ∗ ( P ) ∈ F    , µ D H, b V l ( b f ) v = f ( P + µ ) v, µ D H, b V r ( b f ) v = f ∗ ( P ) v , b f ∈ M H ∗ for v ∈ b V µ . W e follo w the abb reviation men tioned b elow Prop osition 3.5. Deﬁnition 4.1 (Dynamical rep resen tation) . A dynamic al r epr esentation of U ′ q ,p on b V is an H -algebr a homomo rphism b π : U ′ q ,p → D H, b V . The dimension of the dynamic al r epr esentation ( b π , b V ) is dim F b V . Let ( b π V , b V ) , ( b π W , c W ) be t w o dyn amical repr esen tations of U ′ q ,p . W e deﬁ ne the tensor pro du ct b V e ⊗ c W by b V e ⊗ c W = M α ∈ ¯ h ∗ ( b V e ⊗ c W ) α , ( b V e ⊗ c W ) α = M β ∈ ¯ h ∗ b V β ⊗ M H ∗ c W α − β , where ⊗ M H ∗ denotes the u sual tensor pro duct mo dulo the relation f ∗ ( P ) v ⊗ w = v ⊗ f ( P + ν ) w (4.1) 18 for w ∈ c W ν . The action of the scalar f ∗ ( P ) ∈ F on th e tensor sp ace b V e ⊗ c W is deﬁned as follo ws. f ∗ ( P ) . ( v e ⊗ w ) = ∆( µ r ( b f ))( v e ⊗ w ) = v e ⊗ f ∗ ( P ) w . W e ha v e a n atur al H -algebra em b ed d ing θ V W : D H, b V e ⊗D H, c W → D H, b V e ⊗ c W b y X b V e ⊗ X c W ∈ ( D H, b V ) αγ ⊗ M H ∗ ( D H, c W ) γ β 7→ X b V e ⊗ X c W ∈ ( D H, b V e ⊗ c W ) αβ . Hence θ V W ◦ ( b π V ⊗ b π W ) ◦ ∆ : U ′ q ,p → D H, b V e ⊗ c W giv es a dynamical representat ion of U ′ q ,p on b V e ⊗ c W . No w let us c onsid er a construction of dynamical repr esen tations of U ′ q ,p . Let V b e an ¯ h - diagonaliza ble v ector s p ace ov er F . Let V Q b e a vec tor space o v er C , on wh ic h an action of e Q is deﬁned appropriately . Tw o imp ortan t examples of V Q are V Q = C 1 and V Q = ⊕ n ∈ Z C e nQ , where 1 denotes the v acuum state satisfying e Q . 1 = 1. Let us consider the v ector space b V = V ⊗ C V Q , on whic h the actions of f ∗ ( P ) ∈ F and e Q are deﬁned as follo ws. f ∗ ( P ) . ( v ⊗ ξ ) = f ∗ ( P ) v ⊗ ξ , e Q . ( f ∗ ( P ) v ⊗ ξ ) = f ∗ ( P + 1) v ⊗ e Q ξ for f ∗ ( P ) v ⊗ ξ ∈ V ⊗ V Q . The follo wing th eorem sh o ws a construction of d y n amical repr esen- tations. Theorem 4.2. L et V , V Q and b V b e as in the ab ove. L et ( π V : F [ U ′ q ] → End F V , V ) b e a r ep- r esentation of F [ U ′ q ( b sl 2 )] . Deﬁne a map b π V = π V ⊗ id : U ′ q ,p = F [ U ′ q ] ⊗ C C [ ¯ H ∗ ] → End C b V by b π V ( E ( u )) = π V ( φ r ( x + ( z ))) e 2 Q z − P − 1 r ∗ , b π V ( F ( u )) = π V ( φ r ( x − ( z ))) z P + π V ( h ) − 1 r , b π V ( K ( u )) = exp X n> 0 [ n ] q [2 n ] q [ r ∗ n ] q π V ( a − n )( q c z ) n ! exp − X n> 0 [ n ] q [2 n ] q [ r n ] q π V ( a n ) z − n ! × e Q z − c 4 rr ∗ (2 P − 1)+ 1 2 r π V ( h ) . Then ( b π V , b V ) is a dynamic al r epr esentation of U ′ q ,p on b V . Through this pap er we consider the dynamical representati ons obtained in this w a y . 4.2 Pseudo-highest W eight Represen tations W e deﬁ ne the concept of pseu do-highest w eigh t repr esentati ons and write d own some basic results on them. Most of them are parallel to the trigonometric [6] and the r ational [5] cases. W e b eg in by stating an analogue of the P oincar ´ e-Birkhoﬀ-Wit t theorem for U ′ q . 19 Deﬁnition 4.3. L et H (r esp. N ± ) b e the sub algebr as of F [ U ′ q ,p ( b sl 2 )] gene r ate d by c, h and a k ( k ∈ Z 6 =0 ) (r esp. by x ± n ( n ∈ Z ) ). F rom Proposition 3.1 in [6] and a stand ard normal ord ering pr ocedu re on the He isenb er g algebra, we h a ve the follo wing. Theorem 4.4. U ′ q ,p = ( N − ⊗ H ⊗ N + ) ⊗ C [ ¯ H ∗ ] . Her e the last ⊗ should b e understo o d as the semi-dir e ct pr o duct. The follo wing indicates a charact eristic feature of the ﬁnite-dimensional irreducible dynam- ical representati on of U ′ q .p . Theorem 4.5. Every ﬁnite-dimensional i rr e ducible dynamic al r epr esentation ( b π V , b V = V ⊗ V Q ) of U ′ q ,p c ontains a non-zer o ve ctor of the form b Ω = Ω ⊗ 1 , Ω ∈ V such that 1) x + n . b Ω = 0 ∀ n ∈ Z , 2) b Ω is a simultane ous eigenve ctor for the elements of H , 3) e Q . b Ω = b Ω , 4) b V = U ′ q ,p . b Ω . F urthermor e q c acts as 1 or − 1 on b V . Pr o of. Note that for eac h k ∈ Z , C { x + k , x − − k , q h q k c } ∼ = U q ( sl 2 ) is a subalgebra of U ′ q ,p ( b sl 2 ). Then, concerning the action of the F [ U ′ q ( b sl 2 )] part, the existence of a v ector b Ω ′ = Ω ⊗ ξ ∈ b V = V ⊗ V Q satisfying 1) and 2) f ollo ws from Prop osition 3.2 in [6]. There are t wo t yp es of Ω, the one depen ding on P and th e other not. The latter case is simple. e Q acts on b Ω ′ as e Q . b Ω ′ = Ω ⊗ e Q ξ . Th e ﬁniteness and irreducibilit y of b V imply the existence of a un ique non-zero vecto r ξ such that e Q ξ = C ξ with a complex num b er C 6 = 0. Redeﬁning 1 C e Q as e Q , w e identify ξ with 1. F or Ω d ep ending on P , let us write the P dep endence explicitely as b Ω ′ ( P ) = Ω( P ) ⊗ ξ . e Q acts on b Ω ′ ( P ) as e Q . b Ω ′ ( P ) = Ω( P + 1) ⊗ e Q ξ . The ﬁniteness of b V implies that a ﬁnite n umb er of v ectors in { Ω( P + n ) ( n ∈ Z ) } are F -linearly in dep endent . Setting b Ω = P n ∈ Z b Ω( P + n ) ⊗ ξ , we ha v e e Q . b Ω = P n ∈ Z b Ω( P + n ) ⊗ e Q ξ . Th en the same argument as the ﬁrst case leads to ξ = 1, and we obtain b Ω satisfying 3). In b oth cases, Theorem4.4 yields b V = U ′ q ,p . b Ω. As for the action of q c on b V , the statemen t follo ws from C orollary 3.2 in [6]. 20 R emark. An example of the vect or b Ω ′ indep endent of P is v l 0 ⊗ 1 in Theorem 4.13, w hereas the one dep end ing on P is v ( s ) in Theorem 4.17. Deﬁnition 4.6 (Elli ptic loop algebra) . The el liptic lo op algebr a U q ,p ( L ( sl 2 )) is the quotient of U ′ q ,p ( b sl 2 ) by the two side d ide al gener ate d by c . Note U q ,p ( L ( sl 2 )) ∼ = F [ U q ( L ( sl 2 ))] ⊗ C C [ ¯ H ∗ ], where F [ U q ( L ( sl 2 ))] denotes the qu an tum lo op algebra obtained as the q u otien t of F [ U q ( b sl 2 )] b y the tw o sided ideal generated b y c [6]. Note also that U q ,p ( L ( sl 2 )) is an H -Hopf algebroid with th e same µ l , µ r , ∆ , ε, S as U ′ q ,p ( b sl 2 ). F urtherm ore the RLL relation for U q ,p ( L ( sl 2 )) is giv en b y (2.17 ) with replacing R + ∗ ( u, P ) with R + ( u, P ). It is identiﬁed with th e one for F elder’s elliptic quan tum group studied in [12, 30, 51]. Hence the corresp onding b L + ( u ) in (2.16) giv es a realizat ion of F elder’s elliptic quan tum group in terms of U q ,p ( L ( sl 2 )). Hereafter we consider dynamical representa tions of U q ,p ( L ( sl 2 )). Deﬁnition 4.7 (Pseudo-highest weigh t represen tation) . A dynamic al r epr ese ntation ( b π V , b V = V ⊗ V Q ) of U q ,p ( L ( sl 2 )) is said to b e pseudo-highest weight, if ther e exists a ve ctor (pseudo-highest weight ve ctor) b Ω ∈ b V such that e Q . b Ω = b Ω and 1) x + n . b Ω = 0 ( n ∈ Z ) 2) ψ n . b Ω = d + n b Ω , φ − n . b Ω = d − − n b Ω ( n ∈ Z ≥ 0 ) , 3) b V = U q ,p ( L ( sl 2 )) . b Ω , with some c omplex numb ers d ± ± n satisfying d + 0 d − 0 = 1 . W e c al l the set d = { d ± ± n } n ∈ Z ≥ 0 the pseudo-highest weight. W e can state the equiv alent conditions in terms of the matrix elemen ts of b L + ( u ). Theorem 4.8. F or a ve ctor b Ω ∈ b V satisfying e Q . b Ω = b Ω , the c onditions 1) and 2) in Deﬁnition 4.7 ar e e quivalent to the fol lowing. i ) γ ( u ) . b Ω = 0 ∀ u, ii ) q h . b Ω = q λ b Ω ∃ λ ∈ C , α ( u ) . b Ω = A ( u ) b Ω , δ ( u ) . b Ω = D ( u ) b Ω with some mer omorphic functions A ( u ) and D ( u ) satisfying D ( u − 1) − 1 = A ( u ) and A ( u ) = z λ 2 r X m ∈ Z ,n ∈ Z ≥ 0 A m,n z m p n A m,n ∈ C , z = q 2 u , p = q 2 r . (4.2) 21 Pr o of. W e sho w th at i ) and ii ) yield 1) and 2). Let us deﬁn e e n ( n ∈ Z ) by φ r ( x + ( z )) = X n ∈ Z e n z − n . F rom (2.8), we ha v e [25] E + ( u ) = e 2 Q a ∗ [1] X n ∈ Z e n 1 1 − q 2( P − 1) p n z − n − P − 1 r . Here we used the follo win g formula. [ u + s ] [ u ][ s ] = − X n ∈ Z 1 1 − q − 2 s p n . Then it follo ws from (3.8) that i ) is equ iv alen t to e n . b Ω = 0 for all n ∈ Z . F urthermore fr om th e deﬁnition of x + ( z ) and (2.5), w e h a v e e n = X k ∈ Z ≥ 0 p k  a − l q r l [ r l ] q  x + n + k . Here p k ( { α l } ) denotes the Sc h ur p olynomial d eﬁned by exp    X n ∈ Z > 0 α n z n    = X k ∈ Z ≥ 0 p k ( { α l } ) z k . p k ( { α l } ) has the follo wing expression. p k ( { α l } ) = X m 1 +2 m 2 + ··· + k m k = k α m 1 1 · · · α m k k m 1 ! · · · m k ! . Expanding p k n a − l q r l [ r l ] q o as a p o wer series in p = q 2 r , it follo ws that the condition e n . b Ω = 0 for all n ∈ Z is equiv alent to x + n . b Ω = 0 for all n ∈ Z . Similarly applyin g ii ), w e can ev aluate H + ( u ) . b Ω as follo ws. ( q r z ) λ r u + ( z , p ) ψ ( z ) u − ( z , p ) . b Ω = A ( u + 1) A ( u ) b Ω . Here we u sed Prop osition 2.7 in the LHS, and (2.6) and (3.8) in the RHS. Note th at d ue to (4.2) fractional p o we rs of z in the b oth hand sides cancel out eac h other. Expanding the b oth sides as a Laurent series in z and a p o w er series in p , one ﬁnds that a k ( k ∈ Z 6 =0 ) are sim ultaneously diagonalized on b Ω and their eigen v alues are determined by the co eﬃcien ts of the series in the righ t h and sid e. Deﬁnition 4.9 (V erma mo du le) . L et d = { d ± ± n } n ∈ Z ≥ 0 b e any se quenc e of c omplex numb ers. The V erma mo dule M ( d ) is the quotient of U q ,p ( L ( sl 2 )) by the left ide al gener ate d by { x + k ( k ∈ Z ) , ψ n − d + n · 1 , φ − n − d − − n · 1 ( n ∈ Z ≥ 0 ) , e Q − 1 } . 22 Prop osition 4.10. The V erma mo dule M ( d ) is a pseudo-highest weight r epr e sentation of pseudo-highest weight d . Every pseudo-highest weight r epr esentation with pseudo-highest weight d is isomo rphic to a quotient of M ( d ) . Mor e over M ( d ) has a unique maximal pr op er submo d- ule N ( d ) , and up to iso morphism, M ( d ) / N ( d ) is the u nique irr e ducible pseudo-highest weight mo dule of U q ,p ( L ( sl 2 )) . 4.3 Elliptic Analogue of the Drinfeld P olynomials W e no w consider a classiﬁcatio n of ﬁnite-dimensional irreducible dynamical represen tations of U q ,p ( L ( sl 2 )). W e introd uce a natural elliptic an alogue of the Drinfeld p olynomials. Theorem 4.1 1. The irr e ducible pseudo-highest weight dynamic al r e pr esentation ( b π V , b V ) of U q ,p ( L ( sl 2 )) is ﬁnite-dimensional if and only if ther e exists an entir e and quasi-p erio dic function P V ( u ) such that H ± ( u ) . b Ω = c V P V ( u + 1) P V ( u ) b Ω , P V ( u + r ) = ( − ) deg P P V ( u ) , P V ( u + r τ ) = ( − ) deg P e − π i P deg P j =1 ( 2( u − α j ) r + τ ) P V ( u ) . Her e b Ω denotes the pseudo-highest weight ve ctor in b V , and τ = − 2 π i log p . The symb ol c V denotes a c onstant given by c V = q r − 1 r deg P deg P Y j =1 a 1 r j , wher e deg P is a numb er of zer os of P V ( u ) in the fundamental p ar al lelo gr am (1 , τ ) (= the de gr e e of the Drinfeld p olynomial P ( z ) = lim r →∞ P V ( u ) , z = q 2 u ), and a j = q 2 α j with α j b eing a zer o of P V ( u ) in the fundamenta l p ar al lelo gr am. The function P V ( u ) is unique up to a sc alar multiple. Pr o of of the “only if” p art. F rom Theorem 4.5, b V has the pseudo-highest we ight vecto r b Ω. F rom Theorem 3.4 in [6], th er e exists the Drinfeld p olynomial P ( z ) su c h that P (0) = 1 and ϕ ( z ) . b Ω = q deg P P ( q − 2 z − 1 ) P ( z − 1 ) b Ω = ψ ( z ) . b Ω . Here the ﬁrst and second equalitie s are in the sense of the p o wer series in z and z − 1 , resp ectiv ely . Then using Prop osition 2.7 and the form ulae u + ( z , p ) = ∞ Y l =0 q h ϕ ( q c/ 2 q 2 r ∗ ( l +1) z ) , u − ( z , p ) = ∞ Y l =0 q − h ψ ( q c/ 2 q − 2 r ( l +1) z ) , 23 w e obtain H + ( u ) . b Ω = ( q r z ) h r ∞ Y l =0 q h ϕ ( q 2 r ( l +1) z ) · ψ ( z ) · ∞ Y l =0 q − h ψ ( q − 2 r ( l +1) z ) b Ω = ( q r z ) h r ∞ Y l =1 q deg P P ( q − 2 q − 2 rl z − 1 ) P ( q − 2 rl z − 1 ) ∞ Y l =0 q deg P P ( q − 2 q 2 rl z − 1 ) P ( q 2 rl z − 1 ) b Ω . Supp osing that the Drinfeld p olynomial P ( z ) is factorized as P ( z ) = Q deg P j =1 (1 − a j z ), we h a v e H + ( u ) . b Ω = ( q r z ) h r q deg P deg P Y j =1 Θ q 2 r ( a j /q 2 z ) Θ q 2 r ( a j /q 2 z ) b Ω = q r − 1 r deg P deg P Y j =1 a 1 r j [ u + 1 − α j ] [ u − α j ] b Ω . This is the desired result with P V ( u ) = Q deg P j =1 [ u − α j ]. The quasi-p erio dicit y of P V ( u ) follo w s from the one of the th eta function [ u ]. The pro of of the “if” p art is giv en in th e next subsection. R emark. W e ca n tak e c V = 1 b y the gauge transformation giv en from (2.11) in [25]. An example is giv en in Corollary 4.15. Prop osition 4.12. L et b V and c W b e ﬁnite dimensional dyna mic al r epr esentations of U q ,p ( L ( sl 2 )) and assume that the tensor pr o duct b V e ⊗ c W is irr e ducible. L et P V ( u ) , P W ( u ) and P V e ⊗ W ( u ) b e the entir e quasi-p erio dic function asso ci ate d to b V , c W and b V e ⊗ c W in The or e m 4.11. Then P V e ⊗ W ( u ) = P V ( u ) P W ( u ) . Pr o of. The statemen t follo ws from the comultiplic ation formulae for the h alf curren ts in Prop o- sition 3.11. 4.4 Ev aluation Represen tations W e consider an elliptic analogue of the ev aluation represent ation of U q ( L ( sl 2 )) [6, 22]. This is an imp ortan t example of the ﬁnite-dimensional irreducible dynamical represen tation of U q ,p ( L ( sl 2 )). Some form ulae presented here were essentia lly obtained in [25]. Corollary 4.15 and Prop osition 4.16 are new. Let u s consider the l + 1- dimen sional ev aluation representa tion ( π l,w , V ( l ) w ) of F [ U q ( L ( sl 2 ))]. Here V ( l ) = ⊕ l m =0 F v l m , V ( l ) w = V ( l ) ⊗ C [ w , w − 1 ], and w e d eﬁne op erators h, S ± on V ( l ) b y hv l m = ( l − 2 m ) v l m , S ± v l m = v l m ∓ 1 , v l m = 0 for m < 0 , m > l. 24 The action of the Drinfeld generators on V ( l ) w is giv en as f ollo ws. π l,w ( a n ) = w n n 1 q − q − 1 (( q n + q − n ) q nh − ( q ( l +1) n + q − ( l +1) n )) , (4.3) π l,w ( x ± ( z )) = S ±  ± h + l + 2 2  q δ  q h ± 1 w z  . Applying this to Prop osition 4.2 and noting Deﬁnition 2.8, w e obtain the follo wing theorem. Theorem 4.13. L et b V ( l ) ( w ) = V ( l ) ( w ) ⊗ C 1 b e the ve ctor sp ac e, on which e Q acts as e Q . ( f ( P ) v ⊗ 1) = f ( P + 1) v ⊗ 1 . The image of the half c u rr ents by the map b π l,w = π l,w ⊗ id on U q ,p ( L ( sl 2 )) ∼ = F [ U q ( L ( sl 2 ))] ⊗ C C [ ¯ H ∗ ] is given, up to f r actional p owers of z , w and q , by b π l,w ( K + ( u )) = − ϕ l ( u − v ) [ u − v − h − 1 2 ] e Q , b π l,w ( E + ( u )) = − e Q S + [ u − v − h +1 2 − P ][ l + h +2 2 ] [ u − v − h +1 2 ][ P ] e Q , b π l,w ( F + ( u )) = S − [ u − v + h − 1 2 + P ][ l − h +2 2 ] [ u − v − h − 1 2 ][ P + h − 1] , b π l,w ( H ± ( u )) = [ u − v − l +1 2 ][ u − v + l +1 2 ] [ u − v − h − 1 2 ][ u − v − h +1 2 ] e 2 Q . wher e z = q 2 u , w = q 2 v , and ϕ l ( u ) = − z − l 2 r ρ + 1 l ( z , p ) − 1 [ u + l + 1 2 ] , ρ + k l ( z , p ) = q kl 2 { pq k − l +2 z }{ pq − k + l +2 z } { pq k + l +2 z }{ pq − k − l +2 z } { q k + l +2 /z }{ q − k − l +2 /z } { q k − l +2 /z }{ q − k + l +2 /z } . F urthermor e ( b π l,w , b V ( l ) ( w )) is the l + 1 -dimensional irr e ducible dynamic al r epr esentation of U q ,p ( L ( sl 2 )) with the pseudo-highest weight ve ctor v l 0 ⊗ 1 . Pr o of. O n e can directly c hec k that b π l,w ( K + ( u )) , b π l,w ( E + ( u )) and b π l,w ( F + ( u )) satisfy the r ela- tions in Th eorem 2.9. In the pro cess, we u se the form ula ϕ l ( u ) ϕ l ( u − 1) = [ u − l + 1 2 ][ u + l + 1 2 ] . (4.4) F rom Deﬁnition 2.10, w e obtain the im age of the matrix elements of b L + ( u ) as follo ws. 25 Theorem 4.14. b π l,w ( α ( u )) = − [ u − v + h +1 2 ][ P − l − h 2 ][ P + l + h +2 2 ] ϕ l ( u − v )[ P ][ P + h + 1] e Q , b π l,w ( β ( u )) = − S − [ u − v + h − 1 2 + P ][ l − h +2 2 ] ϕ l ( u − v )[ P + h − 1] e − Q , b π l,w ( γ ( u )) = S + [ u − v − h +1 2 − P ][ l + h +2 2 ] ϕ l ( u − v )[ P ] e Q , b π l,w ( δ ( u )) = − [ u − v − h − 1 2 ] ϕ l ( u − v ) e − Q . Corollary 4.15. The el liptic analo gue of the Drinfeld p olynomial asso ciate d to b V ( l ) ( q 2 v ) is given by P l,v ( u ) = [ u − v − l − 1 2 ][ u − v − l − 1 2 + 1] · · · [ u − v + l − 1 2 ] . Pr o of. Noting δ ( u ) = K + ( u ) − 1 , from (2.6), (2.7) an d Theorem 4.14, we obtain b π l,w ( H ± ( u ))( v l 0 ⊗ 1) = [ u − v + l +1 2 ] [ u − v − l − 1 2 ] v l 0 ⊗ 1 . (4.5) Then the enti reness and the quasi-p erio dicit y of P l,v ( u ) yield the desired result. Note that the zeros of P l,v ( u ) coincides with those of the Drinfeld p olynomial corresp ond ing to the ev aluation represen tation V ( l ) ( q 2 v ) of U ′ q ( L ( sl 2 )) mo du lo Z r + Z r τ . Note also that w e ha v e no c V factor in (4.5) d ue to the remark b elow Theorem 4.11. Pr o of of the “i f ” p art of The or em 4.11. Let P V ( u ) b e an y ent ire quasi-p erio dic fu n ction s atis- fying the conditions in the Theorem 4.11 , and let its zeros in the fundamen tal p arallelo gram (1 , τ ) b e α 1 , · · · , α r . F rom Th eorem 4.8, P V ( u ) d etermin es the set of eigen v alues d of ψ k and φ − k ( k ∈ Z ≥ 0 ) uniquely . Consider the represen tation b V = b V (1) ( q 2 α 1 ) e ⊗ · · · e ⊗ b V (1) ( q 2 α 1 ). Let v 1 0 ( i ) = v 1 0 ⊗ 1 denote the p seudo-highest weigh t v ector in b V (1) ( q 2 α i ) and set b Ω = v 1 0 (1) e ⊗ · · · e ⊗ v 1 0 ( r ). Then up to a scalar m ultiple, b Ω is a unique pseudo-highest w eigh t v ector such that q h . b Ω = q r b Ω. Let us consider the sub mo dule b V ′ = U q ,p ( L ( sl 2 )) . b Ω of b V . b V ′ has a uniqu e maximal submo dule b V ′′ . Then the qu otien t mo dule b V ′ / b V ′′ is irreducible, and from Corollary 4.15 and Th eorem 4.12, b V ′ / b V ′′ has the entire qu asi-p eriod ic fu nction give n b y f P V ( u ) = Q r j =1 [ u − α j ] . f P V ( u ) has th e same quasi-p erio dicit y and zeros as P V ( u ). Hence f P V ( u ) coincides w ith P V ( u ) up to a scalar m ultiple. The follo wing Prop osition indicates a consistency of our construction of b π l,w and the fu sion construction of the d ynamical R matrices (=face t yp e Boltzmann weig hts). 26 Prop osition 4.16. L et us deﬁne the matrix e lements of b π l,w ( b L + ε 1 ε 2 ( u )) by b π l,w ( b L + ε 1 ε 2 ( u )) v l m = l X m ′ =0 ( b L + ε 1 ε 2 ( u )) µ m ′ µ m v l m ′ , wher e µ m = l − 2 m . Then we have ( b L + ε 1 ε 2 ( u )) µ m ′ µ m = R + 1 l ( u − v , P ) ε 2 µ m ε 1 µ m ′ . Her e R + 1 l ( u − v , P ) is the R matrix fr om (C.17) in [25]. The c ase l = 1 , R + 11 ( u − v , P ) c oincides with the image ( π 1 ,z ⊗ π 1 ,w ) of th e univ e rsal R matrix R + ( λ ) [24] given in (2.18) . The c ase l > 1 , R + 1 l ( u − v , P ) c oincides with the R matrix obtaine d by fusing R + 11 ( u − v , P ) l - times. In p articular the matrix element R + 1 l ( u − v , P ) ε ′ µ ′ εµ is gauge e quivalent to the fusion fac e weight W l 1 ( P + ε ′ , P + ε ′ + µ ′ , P + µ, P | u − v ) fr om (4) in [7]. 4.5 T ensor Pro duct R epresen t ations In this sub section, we inv estigat e a sub m od ule stru ctur e of th e tensor pro duct s p ace b V ( l 1 ) ( q 2 a ) e ⊗ b V ( l 2 ) ( q 2 b ) and derive an elliptic analogue of the Clebsc h-Gordan co eﬃcien ts. W e abbreviate the p seudo- highest we ight v ectors v l 1 0 ⊗ 1 and v l 2 0 ⊗ 1 of b V ( l 1 ) ( q 2 a ) and b V ( l 2 ) ( q 2 b ) as v l 1 0 and v l 2 0 , resp ectiv ely . Theorem 4.17. Ther e exists a ve ctor v ( s ) ∈ b V ( l 1 ) ( q 2 a ) e ⊗ b V ( l 2 ) ( q 2 b ) satisfying the c onditions 1) ∼ 3) in the b elow, if and only if b − a = l 1 + l 2 − 2 s 2 + 1 ( s = 0 , 1 , · · · , min { l 1 , l 2 } ) . 1) q h .v ( s ) = q l 1 + l 2 − 2 s v ( s ) ( h ∈ H ) , 2) ∆( γ ( u )) .v ( s ) = 0 ∀ u, 3) ∆( α ( u )) .v ( s ) = A ( u ) v ( s ) , ∆( δ ( u )) .v ( s ) = D ( u ) v ( s ) ∀ u, wher e A ( u ) = [ u − a − l 1 +1 2 ][ u − a + l 1 +1 2 ] ϕ l 1 ( u − a ) ϕ l 2 ( u − b ) , D ( u ) = [ u − a − l 1 − 1 2 + s ][ u − a − l 1 − 1 2 − l 2 + s − 1] ϕ l 1 ( u − a ) ϕ l 2 ( u − b ) . Explicitely, the ve ctor v ( s ) is given b y v ( s ) = s X m 1 =0 C s m 1 ( P ) v l 1 m 1 e ⊗ v l 2 s − m 1 , C s m 1 ( P ) = C s 0 [ P − l 2 + s − m 1 ] s − m 1 [ l 2 − s + 1] m 1 [ P + 1] s − m 1 [ − l 1 ] m 1 . (4.6) 27 Pr o of. W e solv e the conditions 1) ∼ 3). The ﬁrst condition yields v ( s ) = s X m 1 =0 C s m 1 ( u, P ) v l 1 m 1 e ⊗ v l 2 s − m 1 (4.7) with unkn o w n co eﬃcien ts C s m 1 ( u, P ). By using 2) and (3.10), we obtain ∆( γ ( u )) v ( s ) = X m 1 n C s m 1 ( u, P + 1) γ ( u ) v l 1 m 1 e ⊗ α ( u ) v l 2 s − m 1 + C s m 1 ( u, P − 1) δ ( u ) v l 1 m 1 e ⊗ γ ( u ) v l 2 s − m 1 o = 0 . Apply Theorem 4.14 and mov e all the co eﬃcien ts in the second tensor sp ace to the ﬁrst one b y using the follo wing form ula obtained from (4.1) v e ⊗ f ( u, P ) v l 2 m 2 = v e ⊗ f ( u, P + h − ( l 2 − 2 m 2 )) v l 2 m 2 = f ( u, P − ( l 2 − 2 m 2 )) v e ⊗ v l 2 m 2 . (4.8) Here one shou ld note f ∗ ( u, P ) = f ( u, P ), i.e. c = 0, in the ev aluation r ep resen tations. W e th us obtain the follo wing recursion relation. C s m 1 ( u, P ) = − C s m 1 − 1 ( u, P − 2) [ u − a − l 1 +1 2 + m 1 ][ u − b + l 2 − 2 s +1 2 + m 1 + 1 − P ] [ u − a − l 1 +1 2 + m 1 + 1 − P ][ u − b + l 2 − 2 s +1 2 + m 1 ] × [ l 2 − s + m 1 ][ P ][ P − 1] [ l 1 + 1 − m 1 ][ P − l 2 + s − 1 − m 1 ][ P + s − m 1 ] . (4.9) Then one ﬁnd s that all the u d ep enden t facto rs cance l out eac h other, if and only if b − a = l 1 + l 2 − 2 s 2 + 1 ( s = 0 , 1 , · · · , min { l 1 , l 2 } ). W e h ence obtain C s m 1 ( P ) = − C s m 1 − 1 ( P − 2) [ l 2 − s + m 1 ][ P ][ P − 1] [ l 1 + 1 − m 1 ][ P − l 2 + s − 1 − m 1 ][ P + s − m 1 ] . (4.10) Here we rewr ote C s m 1 ( u, P ) as C s m 1 ( P ). Solving this, we obtain C s m 1 ( P ) = C s 0 ( P − 2 m 1 ) [ l 2 − s + 1] m 1 [ P − 2 m 1 + 1] 2 m 1 [ − l 1 ] m 1 [ P − l 2 + s − 2 m 1 ] m 1 [ P + s − 2 m 1 + 1] m 1 . (4.11) Here [ u ] m denotes the elliptic shifted factorial [ u ] m = [ u ][ u + 1] · · · [ u + m − 1] . 28 Finally the third condition yields, for δ ( u ), ∆( δ ( u )) v ( s ) = s X m 1 =0 C s m 1 ( P + 1) γ ( u ) v l 1 m 1 e ⊗ β ( u ) v l 2 s − m 1 + s X m 1 =0 C s m 1 ( P − 1) δ ( u ) v l 1 m 1 e ⊗ δ ( u ) v l 2 s − m 1 = s X m 1 =0 C s m 1 ( P − 1) ϕ l 1 ( u − a ) ϕ l 2 ( u − b )[ P − l 2 + s − m 1 − 1][ P + s − m 1 ] ×  [ P − l 2 + s − m 1 − 1][ P + s − m 1 ][ u − a − l 1 − 2 m 1 − 1 2 ][ u − b − l 2 − 2( s − m 1 ) − 1 2 ] +[ s − m 1 ][ l 2 − s + m 1 + 1][ u − a − l 1 − 2 m 1 − 1 2 − P ][ u − b − l 2 − 2( s − m 1 ) − 1 2 + P ]  v l 1 m 1 e ⊗ v l 2 s − m 1 . In the second equ alit y , we u sed (4.10). By using the theta f unction iden tit y [ u + x ][ u − x ][ v + y ][ v − y ] − [ u + y ][ u − y ][ v + x ][ v − x ] = [ x − y ][ x + y ][ u + v ][ u − v ] , w e obtain ∆( δ ( u )) v ( s ) = D ( u ) s X m 1 =0 C s m 1 ( P − 1) [ P ][ P − l 2 + 2 s − 2 m 1 − 1] [ P − l 2 + s − m 1 − 1][ P + s − m 1 ] v l 1 m 1 e ⊗ v l 2 s − m 1 with D ( u ) app ea rin g in the statemen t of th e theorem. Th en the necessary and su ﬃ cien t cond i- tion that ∆( δ ( u )) is diagonalized on v ( s ) with the eigen v alue D ( u ) is C s m 1 ( P ) = C s m 1 ( P − 1) [ P ][ P − l 2 + 2 s − 2 m 1 − 1] [ P − l 2 + s − m 1 − 1][ P + s − m 1 ] . Solving this, we obtain C s m 1 ( P ) = C s m 1 [ P − l 2 + s − m 1 ] s − m 1 [ P + 1] s − m 1 . (4.12) Here C s m 1 is a co eﬃcient indep endent of P , which can b e determined by substituting (4.12) in to (4.10). W e hence obtain C s m 1 ( P ) in th e form in (4.6). W e can c heck that for ∆( α ( u )), the similar argumen t leads to the same result (4.6). R emark. A similar statemen t w as obtained in [17]. By applying β ( u ) on v ( s ) rep eatedly , we can compute the other weig ht v ectors as follo ws . 29 Theorem 4.18. Setting l = l 1 + l 2 − 2 s , we have for 0 ≤ m ≤ l ∆( β ( u ) β ( u + 1) · · · β ( u + m − 1) ) v ( s ) = [ P ] Q m i =1 ϕ l 1 ( u − a + i − 1) ϕ l 2 ( u − a − l 2 + i − 1) × min( l 1 ,s + m ) X k =max( 0 ,s + m − l 2 ) ( − ) k C s 0 [ P + m − 2 k − l 2 + s ] s [ P + m − 2 k + 1] s [ u − a + l 1 + 1 2 ] m − k [ u − a − l + l 1 − 1 2 + m − k + P ] m − k × [ − u + a − l 1 − 1 2 − m + k − P ] k [ − u + a + l − l 1 − 1 2 − m + 1] k × [ − m ] k [ P − k ] m − k [ P + l 1 − k + 1] m − k [ s + 1] m − k [ P − k + 1] m − k [ P ] m − k [ P + l 1 − 2 k + 1] m × 12 V 11  P + m − 2 k ; − k , − s, P − k, l 2 − s + 1 , − u + a − l 1 − 1 2 , u − a − l + l 1 − 1 2 + 2 m − 2 k + P , P + m − 2 k + l 1 + 1  v l 1 k e ⊗ v l 2 m + s − k . (4.13) Mor e over, we have for l < m ∆( β ( u ) β ( u + 1) · · · β ( u + m − 1) ) v ( s ) = 0 . Here 12 V 11 denotes the v ery-well-poised balanced e lliptic h yp ergeometric series deﬁned by [18, 48, 49] s +1 V s ( u 0 ; u 1 , · · · , u s − 4 ) = ∞ X j =0 [ u 0 + 2 j ] [ u 0 ] s − 4 Y i =0 [ u i ] j [ u 0 + 1 − u i ] j with the balancing cond ition s − 4 X i =1 u i = s − 7 2 + s − 5 2 u 0 . W e give the pro of in App end ix B. The vect or v ( s ) obtained in Theorem 4.17 dep ends on th e dynamical p arameter P . Let u s write its P dep endence explicitely as v ( s ) ( P ). It satisﬁes e Q .v ( s ) ( P ) = v ( s ) ( P + 1). Setting b v ( s ) = P n ∈ Z v ( s ) ( P + n ), w e ha v e e Q . b v ( s ) = b v ( s ) . Hence b v ( s ) is a pseud o-highest we ight v ector. Note that b v ( s ) is a v ector in the F -linear sp ace spaned b y the v ectors v l 1 m 1 e ⊗ v l 2 s − m (0 ≤ m 1 ≤ s ). Let us consider the p seudo-highest w eight U q ,p ( L ( sl 2 ))-mod ule W ( s ) generated b y b v ( s ) . Theorem 4.19. If b − a = l 1 + l 2 − 2 s 2 + 1 0 < s ≤ m in( l 1 , l 2 ) , the pseudo-highest weight U q ,p ( L ( sl 2 )) -mo dule W ( s ) is a unique pr op er submo dule of b V = b V ( l 1 ) ( q 2 a ) e ⊗ b V ( l 2 ) ( q 2 b ) . Mor e- over we have W ( s ) ∼ = b V ( l 1 − s ) ( q 2( a − s 2 ) ) e ⊗ b V ( l 2 − s ) ( q 2( b + s 2 ) ) , (4.14) b V /W ( s ) ∼ = b V ( s − 1) ( q 2( a + l 1 − s +1 2 ) ) e ⊗ b V ( l 1 + l 2 − s +1) ( q 2( b − l 1 − s +1 2 ) ) . (4.15) 30 Pr o of. F rom Th eorem 4.11, it is enough to sho w that the entire quasi-p erio dic fun ctions asso- ciated with the r epresen tations in the b oth sid es coincide with eac h other under the condition b − a = l 1 + l 2 − 2 s 2 + 1 for 0 < s ≤ min( l 1 , l 2 ). In fact, one can ev aluate the action of the op erators H ± ( u ) on the h ighest weigh t v ectors b v ( s ) and v l 1 − s 0 e ⊗ v l 2 − s 0 of W ( s ) and b V ( l 1 − s ) ( q 2( a − s 2 ) ) e ⊗ b V ( l 2 − s ) ( q 2( b + s 2 ) ), resp ectiv ely , and ﬁnd that the eigen v alues coincide w ith eac h other. T hey are given by [ u − a − l 1 +1 2 ][ u − a + l 1 +1 2 ] [ u − a − l 1 − 1 2 + s ][ u − a − l 1 +1 2 − 1 + s ] . Similarly , the eigen v alues of H ± ( u ) on the h ighest weigh t v ectors v l 1 0 e ⊗ v l 2 0 and v s − 1 0 e ⊗ v l 1 + l 2 − s +1 0 of b V /W ( s ) and b V ( s − 1) ( q 2( a + l 1 − s +1 2 ) ) e ⊗ b V ( l 1 + l 2 − s +1) ( q 2( b − l 1 − s +1 2 ) ), resp ectiv ely , coincide and are giv en by [ u − a + l 1 +1 2 ][ u − b + l 2 +1 2 ] [ u − a − l 1 − 1 2 ][ u − b − l 2 − 1 2 ] . R emark. A similar statemen t w as p resen ted in [17] without p ro of. 5 Discussions In this section, w e consider the limits trigonometric r → ∞ , non-aﬃne u → ∞ and non - dynamical P → ∞ of the resu lts and mak e some remarks on their algebraic structures and relations. In the limits, the elliptic dynamical R m atrix degenerates as follo ws. R + ( u, P ) r →∞ − → R + trig . ( u, P ) u →∞ − → R + ( P ) P → ∞ − → R + , ց P → ∞ ր u →∞ R + ( u ) where setting x = q 2 P w e ha ve R + trig . ( u, P ) = ρ + trig . ( z )        1 0 0 0 0 (1 − q 2 x )(1 − q − 2 x ) (1 − x ) 2 b ( z ) 1 − xz 1 − x c ( z ) 0 0 1 − xz − 1 1 − x z c ( z ) b ( z ) 0 0 0 0 1        , R + ( P ) = q 1 / 2        1 0 0 0 0 q (1 − q 2 x )(1 − q − 2 x ) (1 − x ) 2 1 − q 2 1 − x 0 0 − x (1 − q 2 ) 1 − x q 0 0 0 0 1        , R + ( u ) = ρ + trig . ( z )        1 0 0 0 0 b ( z ) c ( z ) 0 0 z c ( z ) b ( z ) 0 0 0 0 1        , 31 R + = q 1 / 2        1 0 0 0 0 q 1 − q 2 0 0 0 q 0 0 0 0 1        , ρ + trig . ( z ) = q 1 / 2 ( z − 1 ; q 4 ) ∞ ( q 4 z − 1 ; q 4 ) ∞ ( q 2 z − 1 ; q 4 ) 2 ∞ , b ( z ) = q (1 − z ) 1 − q 2 z , c ( z ) = 1 − q 2 1 − q 2 z . Corresp ondin gly n aiv e dege neration limits of the RLL relation (2. 17 ) imply the follo wing diagram of quantum algebras. U q ,p ( b sl 2 ) r →∞ − → U q ,x ( b sl 2 ) u →∞ − → U q ,x ( sl 2 ) P → ∞ − → U q ( sl 2 ) . ց P → ∞ ր u →∞ U q ( b sl 2 ) (5.1) Here U q ,x ( b sl 2 ) is the dynamical quan tum aﬃne a lgebra suggested in [1]. Ho w ev er neither its generators nor the L op erators h as yet b een giv en explicitely . Let us mak e some sp eculat ions on it. The U q ,x ( b sl 2 ) should b e a semi-direct pro d uct C -algebra isomorphic to F [ U q ( b sl 2 )] ⊗ C C [ ¯ H ∗ ] and c haracterized by the RLL relation of the typ e (2.17) asso ciated with R + trig . ( u, P ). In fact, the b L + ( u ) op erator as wel l as the h alf current s in Deﬁnition 2.8 d o surv ive in the trigonometric limit. Th e H -Hopf alg ebroid structure of U q ,p ( b sl 2 ) also survives in th e limit. W e hence exp ec t that U q ,x ( b sl 2 ) is an H -Hopf algebroid. W e w ill discuss this sub ject in elsewhere. U q ,x ( sl 2 ) denotes the dynamical quan tum alge br a in tro duced b y Bab elon [4], and U q ( b sl 2 ), U q ( sl 2 ) are the standard quan tum aﬃne and non-aﬃne alge br as by Drinfeld-Jimb o, resp ectiv ely . The quasi-Hopf algebra structur e of U q ,x ( sl 2 ) wa s stud ied in [3], whereas the generalized FRST form ulation and the H -Hopf alge br oid stru cture were discussed in [12, 13, 29]. The FRS T formu- lation and Hopf alg ebr a structure of U q ( b sl 2 ) and U q ( sl 2 ) w ere giv en in [41] and [14], resp ectiv ely . Concerning th e FRS T formulatio ns, we sh ould remark that in the elliptic alg ebra U q ,p ( b sl 2 ) as w ell as in B q ,λ ( b sl 2 ) the L op erators b L + ( u ) and b L − ( u ) are not indep endent [24]. This is also true for its trigonometric and n on-aﬃne limits. Ho wev er this is not true after the non-dynamical limit P → ∞ , so that w e need tw o L op erators L + and L − for U q ( b sl 2 ) and U q ( sl 2 ). It is also interesting to see the limits of 12 V 11 obtained in (4.13). Corresp onding to the upp er 32 series in (5.1), we ﬁnd the follo wing. 12 V 11  P + m − 2 k ; − s , − k , P − k, l 2 − s + 1 , − u + a − l 1 − 1 2 , u − a − l + l 1 − 1 2 + 2 m − 2 k + P , P + m − 2 k + l 1 + 1  r →∞ − → 10 W 9  q 2( P + m − 2 k ) ; q − 2 s , q − 2 k , q 2( P − k ) , q 2( l 2 − s +1) , q 2( − u + a − l 1 − 1 2 ) , q 2( u − a − l + l 1 − 1 2 +2 m − 2 k + P ) , q 2( P + m − 2 k + l 1 +1) ; q 2 , q 2  u →∞ − → 8 W 7  q 2( P + m − 2 k ) ; q − 2 s , q − 2 k , q 2( P − k ) , q 2( l 2 − s +1) , q 2( P + m − 2 k + l 1 +1) ; q 2 , q − 2( l − m )  = ( q 2( P + m − 2 k +1) , q 2( m +1) , q 2( P + m − k − l 2 + s ) , q 2( k − l 1 ) ; q 2 ) s ( q 2( P + m − k +1) , q 2( m − k +1) , q 2( P + m − 2 k − l 2 + s ) , q − 2 l 1 ; q 2 ) s q − 2 sk × 4 φ 3   q − 2 s , q − 2 k , q − 2( P + m − k + s ) , q 2( l − m +1) q − 2( s + m ) , q − 2( P + m − k + l 1 − l − 1) , q 2( l 1 +1 − s − k ) ; q 2 , q 2   P → ∞ − → ( q 2( m +1) ; q 2 ) s ( q 2( m − k +1) ; q 2 ) s 3 φ 2   q − 2 s , q − 2 k , q − 2( s + l +1) q − 2( s + m ) , q − 2 l 1 ; q 2 , q 2   , (5.2) where ( a 1 , a 2 , · · · , a m ; q 2 ) s = m Y i =1 ( a i ; q 2 ) s , ( a ; q 2 ) s = (1 − a )( 1 − aq 2 ) · · · (1 − aq 2( s − 1) ) . Here w e follo we d the notations in [21]. After the second limit, we used the transf orm ation form ula from (2.17) in [29] wh ereas after th e third limit, th e formula fr om (3.2.2) in [21]. As sho wn b y Rosengren [44], 10 W 9 in the ab ov e giv es a system of biorthogonal functions iden tical to the one obtained b y Wilson [54]. The 4 φ 3 part is identiﬁed with the q -Raca h p olynomia l, and the 3 φ 2 part with the q -Ha hn p olynomial. A representa tion theoretical deriv ation of 3 φ 2 or the q -Clebsc h-Gordan co eﬃcients was done on the basis of U q ( sl 2 ) in [27, 28, 52]. The case of 8 W 7 or the q -Racah p olynomials, or Ask ey- Wilson p olynomial, has an in teresting history . It w as ﬁr st done on the basis of the quan tum group S U q (2) with considering the so-called twisted primitiv e element in [38] and [40]. Later an alternativ e d eriv ation was carried out on the basis of the co-representa tions of U q ,x ( sl 2 ) in [29]. The relation b etw een these tw o deriv ations can b e found in [42] and [50]. As for th e case 12 V 11 , a representat ion theoretical deriv ation w as ﬁr s t d one in [30 ] on the basis of co-representati ons of F elder’s elliptic quantum group. In this pap er we ha v e giv en an alternative deriv ation on the basis o f representat ions of U q ,p ( b sl 2 ). C omp aring (5.1) and (5.2), w e conjecture that Wilson’s biorthogonal fun ctions 10 W 9 can b e deriv ed similarly on the basis of U q ,x ( b sl 2 ). 33 Moreo ver it is instructiv e to note that reading the diagram (5.2) in in ve rse direction th e dy- namical parameter P m od iﬁes th e q -3 j -sym b ols ( 3 φ 2 or the Cle bs c h -Gordan coeﬃcien ts) into the q -6 j -symb ols ( 4 φ 3 or q -Racah p olynomials), the aﬃnization parameter u mo diﬁes the orthogonal p olynomials ( q -Racah, q -Hahn p olynomials ) in to the biorthogonal fu nctions (W ilson’s biorthog- onal function). As a result 12 V 11 is an elliptic an alogue of the q -6 j -sym b ol and is biorthogonal. W e think that this observ ation should b ecome a guid ing pr inciple in choosing a suitable t yp e of quan tum groups, suc h as dynamical or non-dynamical, aﬃne or non-aﬃn e, in a deriv ation of elliptic analogues of the q -sp ecial functions. It is also interesting to note that the ab ov e degeneration diag ram of 12 V 11 coincides with the one of the h yp ergeometric type sp ecial solutions of the discrete P ainlev ´ e equations [26] corresp onding to the follo wing degeneration of aﬃne W eyl group symmetries [45]. E (1) 8 → E (1) 7 → E (1) 6 → D (1) 5 → · · · . It should b e int eresting if one could ﬁnd a direct connection b et w een this diagram or the discrete P ainlev ´ e equations themselv es and the quantum group s in (5.1 ). Ac kno wledgmen ts The author wo uld lik e to thank Mic hio Jim b o, Anatol K irillo v, Atsushi Nak ay ashiki, Masatoshi Noumi, Hjalmar Rosengren and T adashi Sh ima f or stim ulating discussions and v aluable s u gges- tions. He is also grateful to T etsuo Deguc hi, Jonas Hartwig, Masahiko I to, Masaki Kash iwara, Christian Korf , Barry McCo y , T etsu j i Miw a, T omoki Nak anishi, Masato Ok ado, Vitaly T araso v and Y asuhik o Y amada for their in terests and useful conv ers ations. He also th ank Hjalmar Rosen- gren for his kind hospitalit y during a s ta y in Charmers Unive rsity of T ec hnology and G¨ o teb org Univ ersit y . This w ork is sup p orted b y the Grant -in-Aid for Scien tiﬁc Researc h (C)19540 033, JSPS Japan. 34 A The RLL relation (2.17) at c = 0 W e write do wn the RLL relatio n (2.17 ) in terms of the matrix elemen ts of b L + ( u ) in th e case c = 0. [ α ( u 1 ) , α ( u 2 )] = 0 , [ δ ( u 1 ) , δ ( u 2 )] = 0 , (A.1) [ β ( u 1 ) , β ( u 2 )] = 0 , [ γ ( u 1 ) , γ ( u 2 )] = 0 , (A.2) α ( u 1 ) β ( u 2 ) = ¯ c ( u, P ) α ( u 2 ) β ( u 1 ) + b ( u, P ) β ( u 2 ) α ( u 1 ) , (A.3) β ( u 1 ) α ( u 2 ) = ¯ b ( u, P ) α ( u 2 ) β ( u 1 ) + c ( u, P ) β ( u 2 ) α ( u 1 ) , (A.4) γ ( u 1 ) δ ( u 2 ) = ¯ c ( u, P ) γ ( u 2 ) δ ( u 1 ) + b ( u, P ) δ ( u 2 ) γ ( u 1 ) , (A.5) δ ( u 1 ) γ ( u 2 ) = ¯ b ( u, P ) γ ( u 2 ) δ ( u 1 ) + c ( u, P ) δ ( u 2 ) γ ( u 1 ) , (A.6) c ( u, P + h ) γ ( u 1 ) α ( u 2 ) + b ( u, P + h ) α ( u 1 ) γ ( u 2 ) = γ ( u 2 ) α ( u 1 ) , (A.7) ¯ b ( u, P + h ) γ ( u 1 ) α ( u 2 ) + ¯ c ( u, P + h ) α ( u 1 ) γ ( u 2 ) = α ( u 2 ) γ ( u 1 ) , (A.8) c ( u, P + h ) δ ( u 1 ) β ( u 2 ) + b ( u, P + h ) β ( u 1 ) δ ( u 2 ) = δ ( u 2 ) β ( u 1 ) , (A.9) ¯ b ( u, P + h ) δ ( u 1 ) β ( u 2 ) + ¯ c ( u, P + h ) β ( u 1 ) δ ( u 2 ) = β ( u 2 ) δ ( u 1 ) , (A.1 0) c ( u, P + h ) γ ( u 1 ) β ( u 2 ) + b ( u, P + h ) α ( u 1 ) δ ( u 2 ) = ¯ c ( u, P ) γ ( u 2 ) β ( u 1 ) + b ( u, P ) δ ( u 2 ) α ( u 1 ) , (A.11) ¯ b ( u, P + h ) γ ( u 1 ) β ( u 2 ) + ¯ c ( u, P + h ) α ( u 1 ) δ ( u 2 ) = b ( u, P ) β ( u 2 ) γ ( u 1 ) + ¯ c ( u, P ) α ( u 2 ) δ ( u 1 ) , (A.12) b ( u, P + h ) β ( u 1 ) γ ( u 2 ) + c ( u, P + h ) δ ( u 1 ) α ( u 2 ) = ¯ b ( u, P ) γ ( u 2 ) β ( u 1 ) + c ( u, P ) δ ( u 2 ) α ( u 1 ) , (A.13) ¯ c ( u, P + h ) β ( u 1 ) γ ( u 2 ) + ¯ b ( u, P + h ) δ ( u 1 ) α ( u 2 ) = c ( u, P ) β ( u 2 ) γ ( u 1 ) + ¯ b ( u, P ) α ( u 2 ) δ ( u 1 ) . (A.14) B Pro of of Theorem 4.18 In order to p ro v e the theorem, we need the follo wing four Lemmas. Lemma B.1. F or c = 0 , α ( u ) β ( v 1 ) · · · β ( v l ) = [ P + 1][ P − l ][ u − v l ] [ P ][ P − l + 1 ][ u − v 1 + 1] β ( v 1 ) · · · β ( v l ) α ( u ) + l X k =1 [ P + 1][ P − k + 1 − u + v k ][1] [ P ][ u − v 1 + 1][ P − k + 2] β ( v 1 ) · · · α ( v k ) · · · β ( v l ) β ( u ) . 35 Pr o of. Use (A.3) r ep eatedly . Lemma B.2. ∆( β ( u ) β ( u + 1) · · · β ( u + m − 1) ) = m X j =0 D m j ( P ) α ( u + m − 1) · · · α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) e ⊗ δ ( u ) · · · δ ( u + m − j − 1) β ( u + m − j ) · · · β ( u + m − 1) , wher e D m j ( P ) = [1] m [1] j [1] m − j [ P ][ P − m + 2 j ] [ P + j ][ P − m + j ] m ∈ Z ≥ 0 . (B.1) Pr o of. W e pro v e the statemen t b y ind u ction on m . The case m = 1 is j ust the com ultiplication form ula for β ( u ). Assu me th at the statemen t is tr ue for m . Then ∆( β ( u ) β ( u + 1) · · · β ( u + m − 1))∆( β ( u + m )) = m X j =0 D m j ( P ) α ( u + m − 1 ) · · · α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) α ( u + m ) e ⊗ δ ( u ) · · · δ ( u + m − j − 1) β ( u + m − j ) · · · β ( u + m − 1) β ( u + m ) + m X j =0 D m j ( P ) α ( u + m − 1) · · · α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) β ( u + m ) e ⊗ δ ( u ) · · · δ ( u + m − j − 1) β ( u + m − j ) · · · β ( u + m − 1) δ ( u + m ) = α ( u + m − 1 ) · · · α ( u ) α ( u + m ) + β ( u + m − 1 ) · · · β ( u ) β ( u + m ) + m X j =1 n D m j − 1 ( P ) α ( u + m − 1) · · · α ( u + m − j + 1) β ( u + m − j ) · · · β ( u ) α ( u + m ) + [ P + j − m ] [ P + 2 j − m ] D m j ( P ) α ( u + m − 1) · · · α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) β ( u + m )  e ⊗ δ ( u ) · · · δ ( u + m − j − 1) δ ( u + m − j ) β ( u + m − j + 1) · · · δ ( u + m ) . T o obtain the second equ alit y w e used the p r op ert y of e ⊗ in (3.16) with pu tting c = 0 and the follo wing r elatio n obtained from (A.10) with putting u 1 = v and u 2 = v + 1 δ ( v ) β ( v + 1) = [ P + h + 1] [ P + h ] β ( v ) δ ( v + 1) . Therefore we need to s h o w D m +1 j ( P − j + 1 ) α ( u + m ) β ( u + m − j ) · · · β ( u ) = D m j − 1 ( P − j + 1 ) β ( u + m − j ) · · · β ( u ) α ( u + m ) + [ P − m + 1] [ P + j − m + 1] D m j ( P − j + 1 ) α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) β ( u + m )(B.2) 36 for j = 1 , 2 , · · · , m . Sp ecializing l → m − j, u → u + m − j, v k → u + k − 1 (1 ≤ k ≤ m − j ) in Lemma B.1, we ha v e α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) = m − j +1 X k =1 [ P + 1][ P − m + j ][1] [ P ][ m − j + 1][ P − k + 2] β ( u ) · · · α ( u + k − 1) · · · β ( u + m − j ) . Substituting this into th e second term in the RHS of (B.2), w e obtain D m +1 j ( P − j + 1 ) α ( u + m ) β ( u + m − j ) · · · β ( u ) = D m j − 1 ( P − j + 1 ) β ( u + m − j ) · · · β ( u ) α ( u + m ) + D m j ( P − j + 1 ) m − j +1 X k =1 [ P − m + 1][ P + 1][ P − m + j ][1] [ P + j − m + 1][ P ][ m − j + 1][ P − k + 2] × β ( u ) · · · α ( u + k − 1) · · · β ( u + m − j ) β ( u + m ) . (B.3) Similarly , sp ecializing l → m − j + 1 , u → u + m, v k → u + k − 1 (1 ≤ k ≤ m − j + 1) in Lemma B.1 , we obtain α ( u + m ) β ( u + m − j ) · · · β ( u ) = [ P + 1][ P − m + j − 1][ j ] [ P ][ P − m + j ][ m + 1] β ( u ) · · · β ( u + m − j ) α ( u + m ) + m − j +1 X k =1 [ P + 1][ P − m ][1] [ P ][ m + 1 ][ P − k + 2] β ( u ) · · · α ( u + k − 1) · · · β ( u + m − j ) β ( u + m ) . (B.4) W e compare this with (B.3). These tw o relations coincide w ith eac h other if and only if D m j − 1 ( P − j + 1 ) D m +1 j ( P − j + 1 ) = [ P + 1][ P − m + j − 1] [ j ] [ P ][ P − m + j ][ m + 1] , D m j ( P − j + 1 ) D m +1 j ( P − j + 1 ) = [ P − m ][ P − m + j + 1][ m − j + 1] [ P − m + j ][ P − m + 1][ m + 1 ] . Therefore we obtain D m j ( P ) D m j − 1 ( P ) = [ P − m + j − 1][ P − m + 2 j ][ m − j + 1][ P + j − 1 ] [ P − m + j ][ P − m + 2( j − 1)][ P + j ][ j ] . Solving this with the initial condition D m 0 ( P ) = 1, we obtain (B.1). Lemma B.3. F or v l 1 m 1 ∈ b V ( l 1 ) ( q 2 a ) , we have α ( u + m − 1 ) · · · α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) v l 1 m 1 = ( − ) k + m 1 + m [ u − a + l 1 +1 2 ] m − k [ P − k ] m − k [ P + l 1 − k + 1] m − k [ − u + a − m − l 1 − 1 2 − P + k ] k [1] k Q m i =1 ϕ l ( u − a + i − 1) [ P ] m − k [ P + l 1 − 2 k + 1] m × [ − u + a − l 1 +1 2 + 1] m 1 [ P − 2 k + m ] m 1 [ P + l 1 + m − 2 k + 1] m 1 [ P + m − k ] m 1 [ u − a + m + l 1 − 1 2 + P − 2 k + 1] m 1 [1] m 1 v l 1 k . 37 Her e we set k = m 1 + m − j . Pr o of. App lying Theorem4.14, w e ev aluate the LHS as LHS = ( − ) m j Y i =1 [ u − a + m − i + h +1 2 ][ P + i − 1 − l 1 − h 2 ][ P + i − 1 + l 1 + h +2 2 ] ϕ l ( u − a + m − i )[ P + i − 1][ P + h + i ] × m − j Y i =1 [ u − a + m + h − 1 2 + P − i + 1][ l 1 − h +2 2 − i ] ϕ l ( u − a + m − j − i )[ P + h + j + i ] v l 1 m 1 + m − j = ( − ) k + m 1 [ − u + a − m − l 1 +1 2 + 1 + k ] m 1 + m − k [ P − k ] m 1 + m − k [ P + l 1 − k + 1] m 1 + m − k Q m i =1 ϕ l ( u − a + i − 1) [ P ] m 1 + m − k × [ − u + a − m − l 1 − 1 2 + k − P ] k − m 1 [ − k ] k − m 1 [ P + l 1 − 2 k + 1] m v l 1 k . Then using the follo wing formulae, we obtain the desired result. F or b, k, m 1 , k − m 1 ∈ Z ≥ 0 , [ a ] m 1 + b = [ a ] b [ a + b ] m 1 , [ a − b ] m 1 + b = ( − ) b [ − a + 1] b [ a ] m 1 , [ a + k ] k − m 1 = ( − ) m 1 [ a + k ] k [ − a − 2 k + 1] m 1 . Lemma B.4. F or v l 2 s − m 1 ∈ b V ( l 2 ) ( q 2 b ) , we have v l 1 k e ⊗ δ ( u ) · · · δ ( u + m − j − 1) β ( u + m − j ) · · · β ( u + m − 1) v l 2 s − m 1 = ( − ) m + k [ − u + b + l 2 − 1 2 − m − s + 1] k [ u − b − l 2 +1 2 + m + s + 1 − k + P ] m − k [ s + 1] m − k Q m i =1 ϕ l 2 ( u − b + i − 1)[ P − k + 1] m − k × [ u − b − l 2 +1 2 + 2 m + s + 1 − 2 k + P ] m 1 [ P − k + 1] m 1 [ − s ] m 1 [ − u + b + l 2 − 1 2 − m − s + 1] m 1 [ P + m − 2 k + 1] 2 m 1 v l 1 k e ⊗ v l 2 m + s − k . Her e we set k = m 1 + m − j . Pr o of. App lying Theorem 4.14, w e ev aluate the LHS as v l 1 k e ⊗ ( − ) m m − j Y i =1 [ u − b + i − 1 − h − 1 2 ] ϕ l 2 ( u − b + i − 1) j Y i =1 [ u − b + h − 1 2 + P + i ][ l 2 − h +2 2 − i ] ϕ l 2 ( u − b + m − j + i − 1)[ P + h − m + j + i ] v l 2 s − m 1 + j = v l 1 k e ⊗ ( − ) m 1 + k [ u − b − l 2 − 1 2 + m + s − k ] k − m 1 [ u − b − l 2 +1 2 + m + s + P + h + 1 − k ] m 1 + m − k Q m i =1 ϕ l 2 ( u − b + i − 1) [ P + h + m 1 − k + 1] m 1 + m − k × [ − m − s + k ] m 1 + m − k v l 2 m + s − k . Then the statemen t follo w s fr om (4.8) and [ a − k ] k − m 1 = ( − ) k + m 1 [ − a + 1] k [ − a + 1] m 1 , [ a + m 1 − k ] m 1 + m − k = [ a − k ] m − k [ a + m − 2 k ] 2 m 1 [ a − k ] m 1 k , m, m 1 ∈ Z ≥ 0 , m ≥ k ≥ m 1 . 38 Pr o of of the ﬁrst statement in The or em 4.18. By using Lemma B.2 and (4.7), we obtain LHS = s X m 1 =0 m X j =0 D m j ( P ) C s m 1 ( P − m + 2 j ) α ( u + m − 1) · · · α ( u + m − j ) β ( u + m − j − 1) · · · β ( u ) v l 1 m 1 e ⊗ δ ( u ) · · · δ ( u + m − j − 1) β ( u + m − j ) · · · β ( u + m − 1) v l 2 s − m 1 . Then usin g (4.6) and Lemma B.3, B.4, and c hange th e summation v ariable fr om j to k by k = m 1 + m − j , we obtain the desired result. In the p ro cess, 12 V 11 is iden tiﬁed with the p art asso ciate d with the summation with resp ect to m 1 o v er max(0 , k − m ) ≤ m 1 ≤ min( k , s ). There w e also man ip ulate (4.6) b y the form ula [ P − l 2 + s − m 1 ] s − m 1 [ P + 1] s − m 1 = [ P − l 2 + s − 2 m 1 ] s [ P − 2 m 1 + 1] 2 m 1 [ P − 2 m 1 + 1] s [ P − l 2 + s − 2 m 1 ] m 1 [ P + s − 2 m 1 + 1] m 1 . Pr o of of the se c ond statement. Let u s set m = l 1 + l 2 − 2 s + 1. Then l 1 − s + 1 ≤ k ≤ l 1 . W e sho w that 12 V 11 part in (4.13) v anish es for k = l 1 − s + n ( n = 1 , 2 , · · · , s ). In fact, substituting m = l 1 + l 2 − 2 s + 1 and k = l 1 − s + n , we ﬁnd that the 12 V 11 part is redu ced to s X m 1 =0 [ P − l 1 + l 2 + 1 − 2 n + 2 m 1 ] [ P − l 1 + l 2 + 1 − 2 n ] × [ P − l 1 + l 2 + 1 − 2 n ] m 1 [ − l 1 + s − n ] m 1 [ − s ] m 1 [ P − l 1 + s − n ] m 1 [ l 2 + 1 − s ] m 1 [ P + l 2 + 2 − 2 n ] m 1 [1] m 1 [ P + l 2 − s + 2 − n ] m 1 [ P − l 1 + l 2 + 2 + s − 2 n ] m 1 [ l 2 − s + 2 − n ] m 1 [ P − l 1 + s + 1 − 2 n ] m 1 [ − l 1 ] m 1 = [1 − n ] s [ − P + l 1 − l 2 − 2 s − 1 + 2 n ] s [ l 1 + l 2 − 2 s + 2] s [ − P − s + n ] s [ l 1 − s + 1] s [ − P − l 2 − 1 + n ] s [ l 2 − s + 2 − n ] s [ − P + l 1 − 2 s + 2 n ] s . (B.5) In the last line we used the Jac kson-F renkel-T uraev s ummation formula [18] 10 V 9 ( β − γ − s ; − s , α − γ , − α − γ + 1 − s, β + δ , β − δ ) = [ γ − β , γ + β , α + δ, α − δ ] s [ α − β , α + β , γ + δ , γ − δ ] s . (B.5) v anish es f or n = 1 , 2 , · · · , s . 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Elliptic Quantum Group U_{q,p}(hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

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