A new current algebra and the reflection equation

A NEW CURREN T ALGEBRA AND THE RE FLECTION EQUA TION P . BASEILHAC AND K. SHIGECHI Abstract. W e establish an explicit algebra isomorphism b etw een the quan tum reflection algebra for the U q ( c sl 2 ) R -matrix and a new type of curren t algebra. These t wo algebras are sho wn to be tw o realizations of a special case of tridiagonal algebras ( q − Onsager). MSC: 81R50; 81R10; 81U15. Keywords : Current algebra; R efl ection equation; q − Onsager algebra; Quantum integrable mo dels 1. Introduction Discov ered in the context of the quantum inverse scatter ing metho d for solving quantu m integrable sys- tems, qua ntum groups app ear ed in the literatur e through different wa ys (see [Cha] for references). On one hand, s tarting fro m the fundament al indep endent discov ery o f Drinfeld [Dr1] and Jimbo [Jim] the quantum affine alg e bras U q ( b g ) were initially form ulated using a q − deformed v ersio n o f the commutation relatio ns b e- t ween the elemen ts o f the Chev alley presentation of b g . Later on [Dr2], Drinfeld pr op osed a new rea lization of U q ( b g ) in terms of elements { x ± i,k , ϕ i,m , ψ i,n | i = 1 , ..., l ; k ∈ Z , m ∈ − Z + , n ∈ Z + } with l = r ank ( g ) generated through op erator - v alued functions x ± i ( u ) , ϕ i ( u ) , ψ i ( u ) of the formal v ariable u , the so- called curren ts. In some sense, the Drinfeld’s realizatio n is a quantum analog ue of the lo op realiza tion of affine Lie algebra s. Although Drinfeld stated the isomor phis m b etw een the tw o r e alizations, the pro of only app ear ed later on [Be, Jin]. In pa r ticular, in [Be] (see also [Dam]) Lusztig ’s theory of bra id gro up action [L] on the qua ntum env eloping algebras was used from which an explicit homomorphism from Drinfeld’s new realizatio n [Dr2] to the initial one [Dr1, Jim] w as obtained. On the other hand, an alternative rea lization o f qua ntum affine alg ebras U q ( b g ) by means of solutions of the quantum Y ang-Bax ter equation [KRS, KS, F1] - calle d the R − matr ix - and the “RLL” a lgebraic relations of the quantum in verse scattering metho d was prop osed by Reshetikhin and Semenov-Tian-Shansky in [RS], extending the previous results o f F addeev-Res hetikhin-T akhta jan [FR T1] for finite dimensio nal simple Lie alg ebra g . In view of these realizatio ns , in [DiF] Ding a nd F renkel exhibited an explic it iso morphism b etw een the “RLL” formulation and Drinfeld’s second r ealization. Namely , L − op erator s were sho wn to admit a unique (Gauss) decomp osition in terms of Drinfeld’s current s x ± i ( u ) , ϕ i ( u ) , ψ i ( u ). So, all these different realiza tions may b e summarized by the following picture which pr ovides an unifying algebraic scheme for quan tum affine algebras : “RLL” algebra [FR T1] Y a ng-Baxter equ ation ✛ ✲ [DiF] Curren t algebra [Dr2] Drinfeld’s presentation { x ± i,k , ϕ i,m , ψ i,n } U q ( b g ) ❅ ❅ ❅ ■ ❅ ❅ ❅ ❘ [RS],[Di F]    ✠ [Be]    ✒ [Jin] Drinfeld-Jimbo [Dr1], [Jim] 1 2 P . BASEILHAC AND K. SHIGECHI Beyond the in teres t of the alg ebraic structures inv olved, the explicit r elation b etw een the tw o different realizations (“RLL” a nd Drinfeld’s one) of U q ( b g ) has found many interesting applicatio ns in the study of quantum integrable systems and representation theory . F or qua nt um integrable systems with b oundaries , Cherednik [Cher] and later on [Sk] in tro duced an- other example o f qua dratic algebra ass o ciated with the so-called r eflection equations. In this case , given an R − matrix asso ciated with U q ( b g ) one is lo oking for a K − op erator (so metimes called a Sklyanin’s o pe r - ator) satisfying “RKRK ” algebra ic relations . Motiv ated b y the study o f related integrable systems, several examples of K − o per ators acting o n finite dimensional representations have b een constructed. How ever, a formulation of K − op era tors in terms of current algebras i.e. a “boundar y” - in reference to bounda r y in te- grable models - version of Ding-F renkel [DiF] analysis has never b een explicitly presented, nor a “bounda ry” analogue of Drinfeld’s pr esentation even in the s implest case U q ( c sl 2 ). In this paper, we a rgue that the q − Ons ager algebra T (a type of tr idiagonal alg ebra) which indep en- dent ly app eare d in the con text of orthogonal p oly nomial asso ciation schemes [T er2] and hidden sy mme- tries o f b ounda ry integrable mo dels [Bas] admits a nalogously tw o alternative realizations. One realiza- tion is g iven in ter ms of a K − op era tor sa tisfying “RKRK” defining relations for the U q ( c sl 2 ) R - matrix, and the other re alization in terms of a new type of current algebra asso cia ted with the g enerating set {W − k , W k +1 , G k +1 , ˜ G k +1 | k ∈ Z + } intro duced in [BasK]. A new algebra ic scheme follows, which extends to the family of reflection equatio n algebr as the standard scheme relating the F addeev-Res he tik hin- T akhta jan, Jimbo and Drinfeld (firs t and second) re a lizations of quantum affine algebras (see ab ove picture). Althoug h it is not considered here, the extension of our work to other classical Lie a lgebra - technically more complica ted - is a n interesting and op en problem. The pap er is orga nized as follows. In Section 2, a new curr ent a lg ebra - denoted O q ( c sl 2 ) b elow - with generator s W ± ( u ) , G ± ( u ) and formal v ariable u is intro duced. It is shown to be isomorphic to the “RKRK” algebra. A coaction map, the analogue of the copro duct for Hopf ’s algebra s, is a lso explicitly derived. In Section 3 , the new currents ar e found to b e genera ting functions in the symmetric v ariable U = ( q u 2 + q − 1 u − 2 ) / ( q + q − 1 ) whic h co efficients co inc ide with the elemen ts of the infinite dimensional algebra - denoted A q below - introduced in [BasK]. In the la st section, based on the commuting prope rties of the K − op erator with the tw o generator s of the q − Onsager algebra we establish the isomor phism b etw een T and the “ RK RK” algebra. A new alg ebraic scheme unifying these realiza tions is then prop osed. Notation . In this p ap er, R , C , Z denote t he field of r e al, c omplex n umb ers and inte gers, r esp e ctively. We denote R ∗ = R \{ 0 } , C ∗ = C \{ 0 } , Z ∗ = Z \{ 0 } and Z + for nonne gative inte gers. We intr o duc e the q − c ommutator  X , Y  q = q X Y − q − 1 Y X wher e q is t he deformation p ar ameter, assume d not to b e a ro ot of u n ity. 2. A new current algebra Let V be a finite dimensio nal space . Let the o p er ator-v alued function R : C ∗ 7→ End( V ⊗ V ) be the int ertwining op erator (quantum R − matrix) b etw een the tensor pro duct of tw o fundamental representations V = C 2 asso ciated with the a lgebra U q ( c sl 2 ). The element R ( u ) dep ends on the deforma tion parameter q and is defined by [Bax ter] R ( u ) =     uq − u − 1 q − 1 0 0 0 0 u − u − 1 q − q − 1 0 0 q − q − 1 u − u − 1 0 0 0 0 uq − u − 1 q − 1     , (2.1) A NEW CURRENT ALGEBRA AND THE REFLE CTION EQUA T ION 3 where u is calle d the s p ectr al parameter. Then R ( u ) s atisfies the quantum Y ang-Baxter eq ua tion in the space V 1 ⊗ V 2 ⊗ V 3 . Using the standar d notation R ij ( u ) ∈ End( V i ⊗ V j ), it r eads R 12 ( u/v ) R 13 ( u ) R 23 ( v ) = R 23 ( v ) R 13 ( u ) R 12 ( u/v ) ∀ u, v . (2.2) Let us now consider an extension rela ted with the reflectio n eq ua tion or bo undary quantum Y ang-Ba xter equation - which was first introduced in the con text of b oundar y q uantum inv erse scattering theor y (see [Cher],[Sk] for details) -. F or simplicity and without lo osing generality we cons ider the simplest case, i.e . the U q ( c sl 2 ) R − matrix defined a bove. Definition 2.1 (“RKRK” Reflection equation alg e br a) . Define R ( u ) to b e (2.1). B q ( c sl 2 ) is an asso cia tive algebr a with u nit 1 and gener ators K 11 ( u ) ≡ A ( u ) , K 12 ( u ) ≡ B ( u ) , K 21 ( u ) ≡ C ( u ) , K 22 ( u ) ≡ D ( u ) c onsider e d as the elements of the 2 × 2 squar e matrix K ( u ) which ob eys the defining r elatio ns ∀ u, v R 12 ( u/v ) ( K ( u ) ⊗ I I ) R 12 ( uv ) ( I I ⊗ K ( v )) = ( I I ⊗ K ( v )) R 12 ( uv ) ( K ( u ) ⊗ I I ) R 12 ( u/v ) . (2.3) It is known that given a solution K ( u ) of the reflectio n equation (2.3), one can cons tr uct b y induction other solutions us ing suitable combinations of Lax op erato rs L ( u ). This is so metimes na med as the “dressing ” pro cedure. In particular, for the simplest case one has : Prop ositio n 2.1 (see [Sk]) . Given R ( u ) define d by (2.1), let L ( u ) b e a solution of the quantum Y ang-Baxter algebr a with defining r ela tions ∀ u, v R ( u/v )( L ( u ) ⊗ I I )( I I ⊗ L ( v )) = ( I I ⊗ L ( v ))( L ( u ) ⊗ I I ) R ( u/v ) . (2.4) L et K ( u ) b e a solution of (2.3). Then, the matrix L ( u ) K ( u ) L − 1 ( u − 1 ) is a solution of the r efle ction e quation (2.3). F or instance, using the g enerating set { S ± , s 3 } of the quantum alg ebra U q ( sl 2 ) with defining relations [ s 3 , S ± ] = ± S ± and [ S + , S − ] = ( q 2 s 3 − q − 2 s 3 ) / ( q − q − 1 ) , it is known that the Lax op erato r L ( u ) =  uq 1 2 q s 3 − u − 1 q − 1 2 q − s 3 ( q − q − 1 ) S − ( q − q − 1 ) S + uq 1 2 q − s 3 − u − 1 q − 1 2 q s 3  (2.5) satisfies the quantum Y ang- Baxter algebra (2.4). In q uantum integrable lattice mo dels with b oundaries , the “dressing” pro cedure is often used. Starting from an elemen tary solution with c − num ber entries (a sso ciated with one boundar y o f the system) and dressing the K − op erato r with a product of N L − op era tors acting on differen t quan tum spaces, one reconstructs a who le s pin chain with N sites including inhomogeneities, if necessary [Sk]. In order to exhibit the new current algebr a starting from the “RKRK” reflection equa tio n algebr a, based on previous works on bo unda ry quantum in tegrable systems on the la ttice [Bas, BasK] it seems rather na tur al to write the elemen ts A ( u ), B ( u ), C ( u ), D ( u ) in terms o f new curre nts as follows. It may be important to stress that Prop osition 2.1 plays an essential role (see [Bas, BasK]) in suggesting such a decompo sition. Lemma 2.1. Supp ose q 6 = 1 , u 6 = q − 1 and k ± ∈ C ∗ . Any solution of the r efle ction e quation algebr a B q ( c sl 2 ) admits the fo l lowing de c o mp osition in terms of new elements W ± ( u ) , G ± ( u ) : A ( u ) = u q W + ( u ) − u − 1 q − 1 W − ( u ) , (2.6) D ( u ) = uq W − ( u ) − u − 1 q − 1 W + ( u ) , (2.7) B ( u ) = 1 k − ( q + q − 1 ) G + ( u ) + k + ( q + q − 1 ) ( q − q − 1 ) , (2.8) C ( u ) = 1 k + ( q + q − 1 ) G − ( u ) + k − ( q + q − 1 ) ( q − q − 1 ) . (2.9) Given the element s A ( u ) , B ( u ) , C ( u ) of this form, this de c omp osition is unique. 4 P . BASEILHAC AND K. SHIGECHI Pr o of . The reflection eq uation b eing satisfied for arbitra ry u, v ∈ C ∗ and g eneric q , in v iew o f (2.1) the elements A ( u ), B ( u ), C ( u ), D ( u ) are a priori for mal p ow er ser ies in u . With no res tr ictions, let us choose A ( u ), B ( u ), C ( u ) to b e (2.6), (2.8), (2.9), resp ectively . W e hav e to sho w that D ( u ) is uniquely defined by (2.7). T o prov e it, a ssume the set { A, B , C, D } given by (2.6)-(2.9) s atisfies the reflection equation a lgebra with (2.1). In ter ms of these elemen ts, explicitly (2.3) reads ( i ) a − c + ( B C ′ − B ′ C ) + a − a + [ A, A ′ ] = 0 , ( i ′ ) a − c + ( C B ′ − C ′ B ) + a − a + [ D , D ′ ] = 0 , ( ii ) b − b + [ A, D ′ ] + c − c + [ D , D ′ ] + c − a +  C B ′ − C ′ B  = 0 , ( ii ′ ) b − b + [ D , A ′ ] + c − c + [ A, A ′ ] + c − a +  B C ′ − B ′ C  = 0 , ( iii ) c − b +  D A ′ − D ′ A  + b − c +  AA ′ − D ′ D  + b − a + [ B , C ′ ] = 0 , ( iii ′ ) c − b +  AD ′ − A ′ D  + b − c +  D D ′ − A ′ A  + b − a + [ C, B ′ ] = 0 , ( iv ) b − b + AC ′ + c − c + D C ′ + c − a + C A ′ − a − a + C ′ A − a − c + D ′ C = 0 , ( v ) b − b + B ′ A + c − c + B ′ D + c − a + A ′ B − a − a + AB ′ − a − c + B D ′ = 0 , ( v i ) b − b + C ′ D + c − c + C ′ A + c − a + D ′ C − a − a + D C ′ − a − c + C A ′ = 0 , ( v ii ) b − b + D B ′ + c − c + AB ′ + c − a + B D ′ − a − a + B ′ D − a − c + A ′ B = 0 , ( v iii ) b − a + B D ′ + c − b + D B ′ + b − c + AB ′ − a − b + D ′ B = 0 , ( ix ) b − a + A ′ B + c − b + B ′ A + b − c + B ′ D − a − b + B A ′ = 0 , ( x ) b − a + D ′ C + c − b + C ′ D + b − c + C ′ A − a − b + C D ′ = 0 , ( xi ) b − a + C A ′ + c − b + AC ′ + b − c + D C ′ − a − b + A ′ C = 0 , ( xii ) a − b + [ B , B ′ ] = 0 , ( xiii ) a − b + [ C, C ′ ] = 0 , where a ( u ) = u q − u − 1 q − 1 , b ( u ) = u − u − 1 , c ± = q − q − 1 and w e used the shorthand notations a − = a ( u/ v ), a + = a ( u v ) and similarly for b . Also A = A ( u ), A ′ = A ( v ) and similarly for B , C and D . Now, consider another set, say { A, B , C, D } , D ( u ) = D ( u ) + f ( u ) where f ( u ) is an unkno wn function o f u and the elements of the r eflection equation alg ebra. If { A, B , C , D } is a lso a solution of the reflection equation algebr a , then f ( u ) ≡ f ( A, B , C, D ; u ) - the equations ( i ) − ( xiii ) being the co mplete set of defining relatio ns . Replacing D ( u ) in ( iv ) − ( xi ), we obtain B ( u ) f ( v ) = f ( u ) B ( v ) = C ( u ) f ( v ) = f ( u ) C ( v ) = 0 ∀ u, v . On the other ha nd, from ( i ) − ( ii i ′ ) o ne gets  A ( u ) , f ( v )  = 0. Acting with the l.h.s of ( ix ) o n f ( w ) and using pr evious e q uations it follo ws  D ( u ) , f ( w )  = 0 ∀ u, w . All these equations imply that f ( u ) ≡ 0 ∀ u .  The next step is to prov e the equiv alence betw een the (sixteen in total) indep endent equations coming from the reflection equa tion algebra (2.3 ) with (2 .1) and a clos ed system o f comm utation relations among the currents. The r elations below are a mong the main results of the pap er. Definition 2.2 (Curren t algebr a) . O q ( c sl 2 ) is an asso ciative algebr a with un it 1 , curr ent gener ators W ± ( u ) , G ± ( u ) and p ar ameter ρ ∈ C ∗ . Define the formal variables U = ( q u 2 + q − 1 u − 2 ) / ( q + q − 1 ) and V = ( q v 2 + q − 1 v − 2 ) / ( q + q − 1 ) ∀ u, v . The defining r elations ar e:  W ± ( u ) , W ± ( v )  = 0 , (2.10)  W + ( u ) , W − ( v )  +  W − ( u ) , W + ( v )  = 0 , (2.11) ( U − V )  W ± ( u ) , W ∓ ( v )  = ( q − q − 1 ) ρ ( q + q − 1 ) ( G ± ( u ) G ∓ ( v ) − G ± ( v ) G ∓ ( u )) (2.12) + 1 ( q + q − 1 )  G ± ( u ) − G ∓ ( u ) + G ∓ ( v ) − G ± ( v )  , A NEW CURRENT ALGEBRA AND THE REFLE CTION EQUA T ION 5 W ± ( u ) W ± ( v ) − W ∓ ( u ) W ∓ ( v ) + 1 ρ ( q 2 − q − 2 )  G ± ( u ) , G ∓ ( v )  (2.13) + 1 − U V U − V  W ± ( u ) W ∓ ( v ) − W ± ( v ) W ∓ ( u )  = 0 , U  G ∓ ( v ) , W ± ( u )  q − V  G ∓ ( u ) , W ± ( v )  q − ( q − q − 1 )  W ∓ ( u ) G ∓ ( v ) − W ∓ ( v ) G ∓ ( u )  (2.14) + ρ  U W ± ( u ) − V W ± ( v ) − W ∓ ( u ) + W ∓ ( v )  = 0 , U  W ∓ ( u ) , G ∓ ( v )  q − V  W ∓ ( v ) , G ∓ ( u )  q − ( q − q − 1 )  W ± ( u ) G ∓ ( v ) − W ± ( v ) G ∓ ( u )  (2.15) + ρ  U W ∓ ( u ) − V W ∓ ( v ) − W ± ( u ) + W ± ( v )  = 0 ,  G ǫ ( u ) , W ± ( v )  +  W ± ( u ) , G ǫ ( v )  = 0 , ∀ ǫ = ± , (2.16)  G ± ( u ) , G ± ( v )  = 0 , (2.17)  G + ( u ) , G − ( v )  +  G − ( u ) , G + ( v )  = 0 . (2.18) Remark 1. Ther e exists an automorphism Ω define d by: Ω( W ± ( u )) = W ∓ ( u ) , Ω( G ± ( u )) = G ∓ ( u ) . (2.19) Contrary to a ll known examples of Drinfeld cur r ents asso ciated with quantum affine Lie algebras or sup e ralgebra s, it is important to notice that the v aria bles u, v only a rise through the sy mmetric ( q x 2 ↔ q − 1 x − 2 , ∀ x ∈ u, v ) combinations U, V , r esp ectively . In view of the connections with algebr aic structures that appea r in b oundary quantum integrable mo dels [Bas, Bas2], such a fact is not surpr is ing althoug h no t obvious from (2.3). W e no w turn to the deriv ation of a ll eq uations above. Theorem 1. The map Φ : B q ( c sl 2 ) 7→ O q ( c sl 2 ) define d by (2.6-2.9) is an algebr a isomorphi sm. Pr o of . First, acco rding to Lemma 2.1 we ha ve to sho w that the map Φ defined by (2.6-2.9) is an alg ebra homomorphism from B q ( c sl 2 ) to O q ( c sl 2 ). Set ρ ≡ k + k − ( q + q − 1 ) 2 and define X 1 ≡  W + ( u ) , W + ( v )  , X 2 ≡  W − ( u ) , W − ( v )  , X 3 ≡  W + ( u ) , W − ( v )  +  W − ( u ) , W + ( v )  , X 4 ≡  G + ( u ) , G − ( v )  +  G − ( u ) , G + ( v )  , X 5 ≡ ( q + q − 1 )( U − V )  W + ( u ) , W − ( v )  − ( q − q − 1 ) ρ ( G + ( u ) G − ( v ) − G + ( v ) G − ( u )) −  G + ( u ) − G − ( u ) + G − ( v ) − G + ( v )  , where the v ariables U ≡ ( q u 2 + q − 1 u − 2 ) / ( q + q − 1 ) and similarly for V ar e introduced. In terms o f the combinations X i , it is s traightforw ar d to show that the eq uations ( i ) , ( i ′ ) , ( ii ) , ( ii ′ ) ab ov e can b e simply written, resp ectively , as ( i ) ⇔ a ( uv ) uv q 2 X 1 + a ( uv ) u − 1 v − 1 q − 2 X 2 − a ( uv ) u − 1 v X 3 − X 5 = 0 , ( i ′ ) ⇔ a ( uv ) uv q 2 X 2 + a ( uv ) u − 1 v − 1 q − 2 X 1 − a ( uv ) uv − 1 X 3 + q − q − 1 ρ X 4 + X 5 = 0 , ( ii ) ⇔  b ( u/v ) b ( u v ) uv − 1 − ( q − q − 1 ) 2 u − 1 v − 1 q − 2  X 1 +  b ( u/v ) b ( u v ) u − 1 v − ( q − q − 1 ) 2 uv q 2  X 2 −  b ( u/v ) b ( u v ) u − 1 v − 1 q − 2 − ( q − q − 1 ) 2 uv − 1  X 3 − a ( uv ) ( q − q − 1 ) ρ X 4 − a ( uv ) X 5 = 0 , 6 P . BASEILHAC AND K. SHIGECHI ( ii ′ ) ⇔  b ( u/v ) b ( u v ) u − 1 v − ( q − q − 1 ) 2 uv q 2  X 1 +  b ( u/v ) b ( u v ) uv − 1 − ( q − q − 1 ) 2 u − 1 v − 1 q − 2  X 2 −  b ( u/v ) b ( u v ) uv q 2 − ( q − q − 1 ) 2 u − 1 v  X 3 − a ( uv ) X 5 = 0 . Simplifying these expressions, in par ticular it follows a ( uv )( i ) − ( ii ′ ) ⇔ v 2 q 2 X 1 + v − 2 q − 2 X 2 − X 3 = 0 , a ( uv )( i ′ ) − ( ii ) ⇔ v 2 q 2 X 2 + v − 2 q − 2 X 1 − X 3 = 0 . Considering b oth e quations for v arbitrar y , it implies X 1 = X 2 . Then it is impo r tant to notice that the combinations X i | u ↔ v = − X i for i = 1 , 2 , 3. As now X 3 = ( v 2 q 2 + v − 2 q − 2 ) X 1 and u is arbitra ry , it immediately follows X 3 ≡ X 1 ≡ X 2 ≡ 0. Plugged in to ( ii ), ( ii ′ ) w e o btain X 4 ≡ X 5 ≡ 0. In terms of the currents, these equalities lead to the commutation rela tions (2.10), (2.11), (2.12), (2.18). As a consequence of these relations, after some straightforward calculations one finds that the equations ( iii ) , ( iii ′ ) drastically s implify in to the r elations (2.13). Let us now co nsider the equations ( iv ) , ( v i ) , ( x ) , ( xi ) a b ove. P ro ceeding similarly , let us introduce Y 1 ≡ ( q + q − 1 )  U  C ( v ) , W + ( u )  q − V  C ( u ) , W + ( v )  q + ( q − q − 1 )  W − ( v ) C ( u ) − W − ( u ) C ( v )  , Y 2 ≡ ( q + q − 1 )  U  W − ( u ) , C ( v )  q − V  W − ( v ) , C ( u )  q + ( q − q − 1 )  W + ( v ) C ( u ) − W + ( u ) C ( v )  , Y 3 ≡  C ( u ) , W + ( v )  +  W + ( u ) , C ( v )  , Y 4 ≡  C ( u ) , W − ( v )  +  W − ( u ) , C ( v )  . In terms o f these combinations, the equa tions ( iv ) , ( v i ) , ( x ) , ( xi ) b ecome, resp ectively , ( iv ) ⇔ u  q Y 1 + q ( v 2 + v − 2 ) Y 3 + ( q − q − 1 ) Y 4 )  + u − 1  q − 1 Y 2 − q − 1 ( v 2 + v − 2 ) Y 4 + ( q − q − 1 ) Y 3 )  = 0 , ( v i ) ⇔ u  q Y 2 − q ( v 2 + v − 2 ) Y 4 + q 2 ( q − q − 1 ) Y 3 )  + u − 1  q − 1 Y 1 + q − 1 ( v 2 + v − 2 ) Y 3 + q − 2 ( q − q − 1 ) Y 4 )  = 0 , ( x ) ⇔ v  Y 2 − q ( q + q − 1 ) U Y 4 + ( q 2 − q − 2 ) Y 3 )  + v − 1  Y 1 + q − 1 ( q + q − 1 ) U Y 3 + ( q 2 − q − 2 ) Y 4 )  = 0 , ( xi ) ⇔ v  Y 1 + q ( q + q − 1 ) U Y 3 )  + v − 1  Y 2 − q − 1 ( q + q − 1 ) U Y 4 )  = 0 . The v ariables u , v and deforma tion para meter q being arbitrar y , co mpatibilit y of these equations implies Y 1 ≡ Y 2 ≡ Y 3 ≡ Y 4 ≡ 0. Replacing the explicit expression of C ( u ) into Y i , one ends up with the commu- tation relations (2.14), (2.15), (2.16) for the curr ent G − ( u ). Similar ana lysis for the remaining equations ( v ) , ( v ii ) , ( v ii i ) , ( ix ) imply (2.14), (2.1 5), (2.16) for G + ( u ). Finally , from ( xii ) , ( xiii ) we immediately obtain (2.17). Surjectivity of the map b eing shown, the injectivity of the homomorphism follows from the fact that Φ is inv ertible fo r u gener ic. This co mpletes the pro o f.  Quantum affine a lg ebras are kno wn to b e Hopf a lgebras, thanks to the existence of a copro duct, counit and an tip o de actions. Althoug h the explicit Hopf a lgebra isomorphism b etw een Drinfeld’s new re alization (currents) and Drinfeld-Jimbo cons truction is still an o pe n problem, sev era l r esults ar e alr eady known (se e for instance [DiF]). F or the new current a lgebra (2.10)-(2.18), it is also impor ta nt to exhibit analogous prop erties. Actually , solely using the re s ults o f [Sk] a coaction map [Cha ] can b e easily identified. A NEW CURRENT ALGEBRA AND THE REFLE CTION EQUA T ION 7 Prop ositio n 2.2. F o r any k ± , w ∈ C ∗ , ther e exists an algebr a homomorphism δ w : O q ( c sl 2 ) 7→ U q ( sl 2 ) × O q ( c sl 2 ) such that δ w ( W ± ( u )) =  ( q − q − 1 ) 2 S ± S ∓ − q ( q ± 2 s 3 − q ∓ 2 s 3 )  ⊗ W ∓ ( u ) − ( w 2 + w − 2 ) I I ⊗ W ± ( u ) + ( q − q − 1 ) k + k − ( q + q − 1 )  k + w ± 1 q ± 1 / 2 S + q ± s 3 ⊗ G + ( u ) + k − w ∓ 1 q ∓ 1 / 2 S − q ± s 3 ⊗ G − ( u )  +( q + q − 1 )  ( k + w ± 1 q ± 1 / 2 S + q ± s 3 + k − w ∓ 1 q ∓ 1 / 2 S − q ± s 3 ) ⊗ I I + q ± 2 s 3 ⊗ U W ± ( u )  , δ w ( G ± ( u )) = k ∓ k ± ( q − q − 1 ) 2 S 2 ∓ ⊗ G ∓ ( u ) − ( w 2 q ± 2 s 3 + w − 2 q ∓ 2 s 3 ) ⊗ G ± ( u ) + I I ⊗ ( q + q − 1 ) U G ± ( u ) + ( q + q − 1 ) 2 ( q − q − 1 )  k ∓ w ± 1 q ∓ 1 / 2 S ∓ q s 3 ⊗ ( U W + ( u ) − W − ( u )) + k ∓ w ∓ 1 q ± 1 / 2 S ∓ q − s 3 ⊗ ( U W − ( u ) − W + ( u ))  + k + k − ( q + q − 1 ) 2 ( q − q − 1 )  ( q + q − 1 ) U + k ∓ k ± ( q − q − 1 ) 2 S 2 ∓ − ( w 2 q ± 2 s 3 + w − 2 q ∓ 2 s 3 + 1)  ⊗ I I . Pr o of . According to [Sk] (see P rop osition 2.1) and the Lax op er ator (2.5), L ( uw ) K ( u ) L ( uw − 1 ) is a solution ∀ w of (2.3). Expanding this expression using (2.6)-(2.9), the new entries of L ( uw ) K ( u ) L ( uw − 1 ) are found to take the form (2.6)-(2.9) r eplacing W ± ( u ) → δ w ( W ± ( u )), G ± ( u ) → δ w ( G ± ( u )). F or more details, w e refer the reader to [Ba sK] wher e s imila r calculations have be e n perfor med.  3. Another present a tion In [Ba sK], an infinite dimensional algebra denoted be low A q was prop osed in or der to so lve b ounda ry int egr able systems with hidden s ymmetries related with a co ideal subalgebr a of U q ( c sl 2 ). How ever, its defining relations were essentially co njectur ed ba sed on the comm utation relations a nd pr op erties o f cer tain op erator s acting on ir reducible finite dimensional tensor product of ev aluation repres e n tations . The aim o f this Section is to construct an analog ue of Drinfeld’s presentation for the cur r ent algebra (2 .10)-(2.18). As a conseq uence, it provides a r igoro us deriv ation of the relations co njectured in [BasK]. Definition 3.1 ([BasK]) . A q is an asso ciative algebr a with p ar ameter ρ ∈ C ∗ , unit 1 and gener ators {W − k , W k +1 , G k +1 , ˜ G k +1 | k ∈ Z + } satisfying the fol lowing r elations:  W 0 , W k +1  =  W − k , W 1  = 1 ( q + q − 1 )  ˜ G k +1 − G k +1  , (3.1)  W 0 , G k +1  q =  ˜ G k +1 , W 0  q = ρ W − k − 1 − ρ W k +1 , (3.2)  G k +1 , W 1  q =  W 1 , ˜ G k +1  q = ρ W k +2 − ρ W − k , (3.3)  W − k , W − l  = 0 ,  W k +1 , W l +1  = 0 , (3.4)  W − k , W l +1  +  W k +1 , W − l  = 0 , (3.5)  W − k , G l +1  +  G k +1 , W − l  = 0 , (3.6)  W − k , ˜ G l +1  +  ˜ G k +1 , W − l  = 0 , (3.7)  W k +1 , G l +1  +  G k +1 , W l +1  = 0 , (3.8)  W k +1 , ˜ G l +1  +  ˜ G k +1 , W l +1  = 0 , (3.9)  G k +1 , G l +1  = 0 ,  ˜ G k +1 , ˜ G l +1  = 0 , (3.10)  ˜ G k +1 , G l +1  +  G k +1 , ˜ G l +1  = 0 . (3.11) 8 P . BASEILHAC AND K. SHIGECHI A natur al ordering of A q arises from the study of the comm utation relations above. Indeed, sta rting fro m monomials of lo west k = 0 , 1 , ... and using (3.1) p o s sible definitions of G 1 , ˜ G 1 are s uch tha t d[ G 1 ] = d[ ˜ G 1 ] ≤ 2, where d denotes the degree of the monomials in the elements W 0 , W 1 . By induction, fro m (3.2), (3.3) with (3.1) one immediately gets: Corollary 3.1 . The elements of A q ar e monomials in W 0 , W 1 of de gr e e: d[ W − k ] = d[ W k +1 ] ≤ 2 k + 1 and d[ G k +1 ] = d[ ˜ G k +1 ] ≤ 2 k + 2 , k ∈ Z + . (3.12) Note that writing explicitly a ll highe r elements of A q in terms of W 0 , W 1 is es s ent ially related with the construction of a P oincar e-Birkoff-Witt basis for the algebra cons ide r ed in the next Section, a problem that will be consider ed elsewher e. Remark 2. A c c or ding to the or dering (3. 12 ), the elements G 1 , ˜ G 1 ∈ A q ar e uniquely determine d: G 1 =  W 1 , W 0  q + α and ˜ G 1 =  W 0 , W 1  q + α ∀ α ∈ C . (3.13) F or the deriv ation of the s econd theor em, s everal other equa lities will b e require d which can all b e deduced from the r e lations above and (3.1 3). Indeed, let us show the follo wing. Prop ositio n 3. 1. If (3.1)- (3.11) ar e satisfie d, t hen the fol lowing r elations hold:  W − k − 1 , W l +1  −  W − k , W l +2  = q − q − 1 ρ ( q + q − 1 )  G k +1 ˜ G l +1 − G l +1 ˜ G k +1  , (3.14) −W − k W 0 + W k +1 W 1 − W − k − 1 W 1 + W 0 W k +2 − 1 ρ ( q 2 − q − 2 )  G k +1 , ˜ G 1  = 0 , (3.15) W − k − 1 W − l − W k +2 W l +1 − W − k W − l − 1 + W k +1 W l +2 (3.16) + W − k W l +1 − W − l W k +1 − W − k − 1 W l +2 + W − l − 1 W k +2 + 1 ρ ( q 2 − q − 2 )  G k +2 , ˜ G l +1  −  G k +1 , ˜ G l +2  = 0 ,  G l +1 , W k +2  q −  G k +1 , W l +2  q − ( q − q − 1 )  W − k G l +1 − W − l G k +1  = 0 , (3.17)  W − k − 1 , G l +1  q −  W − l − 1 , G k +1  q − ( q − q − 1 )  W k +1 G l +1 − W l +1 G k +1  = 0 , (3.18)  ˜ G l +1 , W − k − 1  q −  ˜ G k +1 , W − l − 1  q − ( q − q − 1 )  W k +1 ˜ G l +1 − W l +1 ˜ G k +1  = 0 , (3.19)  W k +2 , ˜ G l +1  q −  W l +2 , ˜ G k +1  q − ( q − q − 1 )  W − k ˜ G l +1 − W − l ˜ G k +1  = 0 . (3.20) Pr o of . T o show (3.14), let us consider the first commutator. Expand it using (3.2). Com bining W 0 and W l +1 using (3.1), one finds:  W − k − 1 , W l +1  = q ρ ( q + q − 1 )  ˜ G l +1 G k +1 − ˜ G k +1 G l +1  + q − 1 ρ ( q + q − 1 )  G l +1 ˜ G k +1 − G k +1 ˜ G l +1  +  W − l − 1 , W k +1  . Then, using (3 .5) and (3.11) one obtains (3.1 4). Consider now (3 .15). Introduce (3.13) in the last co mm utator , and ex pand using (3.2) and (3.3). Collecting terms and simplifying, o ne obtains (3.15). Equation (3.16), a lthough tec hnically slightly mor e co mplicated, is de r ived along the same line . T o show (3.1 7)-(3.20), the same pro cedure will b e used so we o nly explain (3.17). Consider the t wo commutators and expand using (3.3). Then, using (3.8) a nd (3.11), o ne verifies that (3 .17) is indeed satisfied.  A NEW CURRENT ALGEBRA AND THE REFLE CTION EQUA T ION 9 By a nalogy with Drinfeld’s constructio n, w e are no w lo oking for an infinite dimensional set o f elemen ts of a n a lgebra in terms of whic h the current s W ± ( u ), G ± ( u ) can b e expanded. According to the structure of the equations (2.10)-(2.18) defining the current algebr a - in particular the dep endence in the formal v ariable U, V - w e obtain the second ma in r e sult of the pap er. Theorem 2. Define the formal variable U = ( qu 2 + q − 1 u − 2 ) / ( q + q − 1 ) . L et Ψ : O q ( c sl 2 ) 7→ A q b e the map define d by W + ( u ) = X k ∈ Z + W − k U − k − 1 , W − ( u ) = X k ∈ Z + W k +1 U − k − 1 , (3.21) G + ( u ) = X k ∈ Z + G k +1 U − k − 1 , G − ( u ) = X k ∈ Z + ˜ G k +1 U − k − 1 . (3.22) Then, Ψ is an algebr a isomorphism b etwe en O q ( c sl 2 ) and A q . Pr o of . Plugg ing (3.21), (3.22) into (2.1 0)-(2.18), expa nding a nd identifying ter ms o f sa me order in U − k V − l one finds all defining r elations (3 .1)-(3.1 1), together with the set of higher relations (3 .14)-(3.20). F r om Prop ositio n 3.1, it follows that the sixtee n indep endent algebr aic rela tions (3.1)-(3.1 1) are sufficien t i.e. the map is surjective. T he curr e nts being ana lytic in the v ariable U ∈ C , accor ding to Cauch y’s theorem an y element of A q is uniquely determined from the cur rents using contour integrals. The injectivit y of the map follows, which completes the proo f.  It is imp o rtant to stress that in [BasK], commutation relations among the so - called tr ansfer matrix were used to derive some o f the relations (3 .1)-(3.1 1). How ever, the deriv ation descr ib ed above uses so lely the reflection equation algebra . Conseq uent ly , this theorem establishes a rigoro us pro of of the relations conjectured in [Bas K]. In addition, for the ca se of the r eflection equation alg ebra with the U q ( c sl 2 ) R -ma trix it shows that the presentation {W − k , W k +1 , G k +1 , ˜ G k +1 | k ∈ Z + } is the “b oundary” analo gue of Drinfeld’s one. 4. inter t winer o f the q − Onsager (tridia gonal) algebra and the reflection equa tion The purpos e of this Section is to exhibit an int ertwiner K ( u ) of the q − Ons ager algebra, to show its uniqueness and that it coincides exactly with the solution K ( u ) of the reflectio n equation (2.3). The final aim is actually to establish the isomo rphism b etw een the new curr e n t algebr a and the q − Onsag er algebra. Although the reader may b e familiar with the ideas of [J im], it will b e useful to fir st reca ll s o me well-kno wn results. Indeed, the pro cedur e we follow to cons truct the intert winer is analog ous to the one describ ed in [Jim]. In the cont ext of quan tum in tegr able sys tems, note that for finite dimensional representations int ertwiners ha ve already been obtained along the same line in [MN, DeMS, Nep, DeG, DeM]. a. The R − matrix as an in tert winer of U q ( c sl 2 ) [Jim]. In [Jim], Jim b o pointed out that in tertwiners R of quan tum lo o p algebr as lead to tr igonometric solutions of the quantum Y ang-Baxter equa tion (2.4). Any tensor pro duct of tw o ev aluation representations with gener ic ev aluation parameter s u and v being indecomp osable, by Sch ur’s lemma the solution R is unique up to an ov erall s calar factor. In particular, considering the q uantum affine algebr a U q ( c sl 2 ) the construction of the solution R ( u ) given b y (2.1) go es as follow. First, we need to r ecall the realiza tion o f the quantum affine alg ebra U q ( c sl 2 ) in the Chev alley presenatio n { H j , E j , F j } , j ∈ { 0 , 1 } (see e.g [Cha]): 10 P . BASEILHAC AND K. SHIGECHI Definition 4.1. Define the ext ende d Cartan m atr ix { a ij } ( a ii = 2 , a ij = − 2 fo r i 6 = j ). The qu ant um affine algebr a U q ( c sl 2 ) is gener a te d by the elements { H j , E j , F j } , j ∈ { 0 , 1 } which satisfy t he defining r el ations [ H i , H j ] = 0 , [ H i , E j ] = a ij E j , [ H i , F j ] = − a ij F j , [ E i , F j ] = δ ij q H i − q − H i q − q − 1 to gether with the q − Serr e r elations [ E i , [ E i , [ E i , E j ] q ] q − 1 ] = 0 , and [ F i , [ F i , [ F i , F j ] q ] q − 1 ] = 0 . (4.1) The sum K = H 0 + H 1 is the c entr al element of the algeb r a. The Hopf algebr a structur e is ensur e d by the existenc e of a c omultiplic ation ∆ : U q ( c sl 2 ) 7→ U q ( c sl 2 ) ⊗ U q ( c sl 2 ) , antip o de S : U q ( c sl 2 ) 7→ U q ( c sl 2 ) and a c ounit E : U q ( c sl 2 ) 7→ C with ∆( E i ) = E i ⊗ q − H i / 2 + q H i / 2 ⊗ E i , ∆( F i ) = F i ⊗ q − H i / 2 + q H i / 2 ⊗ F i , ∆( H i ) = H i ⊗ I I + I I ⊗ H i , (4.2) S ( E i ) = − E i q − H i , S ( F i ) = − q H i F i , S ( H i ) = − H i S ( I I ) = 1 and E ( E i ) = E ( F i ) = E ( H i ) = 0 , E ( I I ) = 1 . Note that the o pp os ite copro duct ∆ ′ can b e similar ly defined with ∆ ′ ≡ σ ◦ ∆ wher e the p ermutation map σ ( x ⊗ y ) = y ⊗ x for a ll x, y ∈ U q ( c sl 2 ) is used. Then, b y definition the intert winer R ( u/v ) : V u ⊗V v 7→ V v ⊗V u betw een t wo fundamental U q ( c sl 2 ) − ev aluation representations ob eys R ( u/v )( π u × π v )  ∆( x )  = ( π u × π v )  ∆ ′ ( x )  R ( u/v ) ∀ x ∈ U q ( c sl 2 ) , (4.3) where the (ev aluation) endomor phism π u : U q ( c sl 2 ) 7→ End( V u ) is chosen s uch that ( V ≡ C 2 ) π u [ E 1 ] = uq 1 / 2 σ + , π u [ E 0 ] = uq 1 / 2 σ − , π u [ F 1 ] = u − 1 q − 1 / 2 σ − , π u [ F 0 ] = u − 1 q − 1 / 2 σ + , π u [ q H 1 ] = q σ 3 , π u [ q H 0 ] = q − σ 3 (4.4) in terms o f the Pauli matric es σ ± , σ 3 : σ + =  0 1 0 0  , σ − =  0 0 1 0  , σ 3 =  1 0 0 − 1  . As o ne can easily c heck, the matrix R ( u ) given by (2.1) indeed satisfies the required conditions (4.3). The tensor pr o duct V u ⊗ V v being indecomp osa ble with respec t to U q ( c sl 2 ) ev aluation representations for generic ev aluation para meters u, v , the intert winer R ( u ) is unique (up to a n ov era ll scala r factor). As a consequence, it automatically s atisfies the Y ang-Baxter equation (2.4 ) which may b e depicted by the following commutativ e diagram setting w = 1: (4.5) V u ⊗ V v ⊗ V w R ( u/v ) ⊗ id − − − − − − − → V v ⊗ V u ⊗ V w id ⊗ R ( u/w ) − − − − − − − − → V v ⊗ V w ⊗ V u   y id ⊗ R ( v /w ) R ( v/w ) ⊗ i d   y V u ⊗ V w ⊗ V v R ( u/w ) ⊗ id − − − − − − − − → V w ⊗ V u ⊗ V v id ⊗ R ( u/v ) − − − − − − − → V w ⊗ V v ⊗ V u A NEW CURRENT ALGEBRA AND THE REFLE CTION EQUA T ION 11 b. The K − matrix as an intert winer of T . T ridiag onal algebras hav e been intro duced and studied in [T er1, ITT er, T er2], where they firs t app eared in the context of P − and Q − poly no mial asso ciation sc hemes. A tr idia gonal alge br a is an asso ciative algebr a with unit which consists of tw o ge ner ators A and A ∗ called the standar d generator s. In genera l, the defining relations dep end on five scalars ρ, ρ ∗ , γ , γ ∗ and β . In the following, w e will focus on the re duc e d parameter sequence γ = 0 , γ ∗ = 0 , β = q 2 + q − 2 and ρ = ρ ∗ which ex hibits all int er e sting prop er ties that can b e extended to more general parameter seq uences. W e call the co rresp onding alg ebra the q − Ons ager a lgebra denoted T , in view of its closed relationship with the O nsager algebra [O ns] and the Dolan-Grady r e lations [DoG]. In particular, the isomorphism betw een the Onsager and Dolan- Grady algebraic structures ha s b e e n studied in [P e, AMPT, Dav] and shown ex plicitly in [DaRo]. Definition 4.2 (see also [T er2]) . The q − Onsager algebr a T is the asso ciative algebr a with unit and standar d gener ators A , A ∗ subje ct to the fol lowing r ela tions [ A , [ A , [ A , A ∗ ] q ] q − 1 ] = ρ [ A , A ∗ ] , [ A ∗ , [ A ∗ , [ A ∗ , A ] q ] q − 1 ] = ρ [ A ∗ , A ] . (4.6) Remark 3. F or ρ = 0 the r elations (4.6 ) r e duc e to the q − S erre r elations of U q ( c sl 2 ) . F or q = 1 , ρ = 16 t hey c oincid e with t he Dolan-Gr ady r elations [DoG] . By a nalogy with the co nstruction describ e d ab ov e for the R − matrix and along the lines describ ed in [DeM, DeG], an intert winer for T can b e easily constr ucted. Befo re, we need to intro duce the concept of como dule algebra us ing the analogue of the Hopf ’s algebra co pro duct action calle d the coac tio n ma p. Definition 4.3 ([Cha]) . Given a Hopf algebr a H with c omultiplic ation ∆ and c ounit E , I is c al le d a left H− c omo dule if ther e exists a left c o action map δ : I → H ⊗ I su ch that (∆ × id ) ◦ δ = ( id × δ ) ◦ δ , ( E × id ) ◦ δ ∼ = id . (4.7) Rig ht H− c omo dules ar e define d similarly. Prop ositio n 4. 1 (see also [Bas]) . L et k ± ∈ C ∗ and set ρ ≡ k + k − ( q + q − 1 ) 2 . The q-Onsager algebr a T is a left U q ( c sl 2 ) − c omo dule algebr a with c o action m ap δ : T → U q ( c sl 2 ) ⊗ T su ch that δ ( A ) = ( k + E 1 q H 1 / 2 + k − F 1 q H 1 / 2 ) ⊗ 1 + q H 1 ⊗ A , δ ( A ∗ ) = ( k − E 0 q H 0 / 2 + k + F 0 q H 0 / 2 ) ⊗ 1 + q H 0 ⊗ A ∗ . (4.8) Pr o of . The verification of the como dule algebra axioms (4.7) is immediate using (4.2). One also ha s to chec k that δ is an algebra homomo r phism i.e δ ( A ) , δ ( A ∗ ) satisfy (4.6). This ca lculation is rather long but straightforward, s o we omit the details (see also [Bas2, Bas3]).  Having identified such a co a ction map, we are now in p osition to consider an in tertwiner relating repr e- sentations of T , a key ingr edient in relating the q − Onsager a lgebra and the reflection equation algebra. Prop ositio n 4.2. L et π u : U q ( c sl 2 ) 7→ End( V u ) b e t he evaluation endomo rphism for V ≡ C 2 . L et W denote a ve ctor sp ac e over C on which the elements of T act. Ther e ex ists an int ertwinner K ( u ) : V u ⊗ W 7→ V u − 1 ⊗ W such that K ( u )( π u × id )  δ ( a )  = ( π u − 1 × id )  δ ( a )  K ( u ) , ∀ a ∈ T . (4.9) It is u nique (u p to an over a l l sc a lar factor), and it satisfies the r efle c tion e quation (2.3). 12 P . BASEILHAC AND K. SHIGECHI Pr o of . First, let us ident ify one solution of (4.9). By definition, V u is a t wo-dimensional vector space . Then K ( u ) is a 2 × 2 matrix, which en tries a re formal power series in the v ariable u in view of (4.4) and (4.8). Define K ( u ) =  A ( u ) B ( u ) C ( u ) D ( u )  . Replacing K ( u ) in (4 .9), we find that the en tries m ust satisfy the follo wing system of eq uations  A , A ( u )  = q − 1 u − 1  k − B ( u ) − k + C ( u )  ,  A , D ( u )  = − q u  k − B ( u ) − k + C ( u )  , (4.10)  A , B ( u )  q = k +  uA ( u ) − u − 1 D ( u )  ,  A , C ( u )  q − 1 = − k −  uA ( u ) − u − 1 D ( u )  and similar relations for A ∗ , provided one substitutes q → q − 1 , u → u − 1 in (4.1 0). Then, using (3.13) in (3.2) for k = 0 it is easy to notice that the defining relations (4.6) are nothing but (3.4) for k = 0 , l = 1, provided we consider the follo wing homomorphism A 7→ W 0 , A ∗ 7→ W 1 . (4.11) Now, ident ify the entries o f K ( u ) with (2.6)-(2.9). E xpanding a nd using the defining rela tio ns (3.1)-(3.3) of the a lg ebra A q , it is e asy to chec k (4.1 0) as w ell as all other re lations for A ∗ . So, at least one solution K ( u ) exists and it is wr itten in terms of elements o f A q . F or generic u , the tensor pro duct End( V u ) ⊗ W is not decomp osable with resp ect to T representations. By Sc hur’s lemma, this means that giv en W , the solution to the in tertwining relation (4.9) is unique (up to a n o verall s calar factor). It remains to show that K ( u ) satisfying (4.9) is automatica lly a s olution of the reflection equation algebra (2 .3). T o this end, let us reca ll that K ( u ) : V u ⊗ W 7→ V u − 1 ⊗ W a nd R ( u /v ) : V u ⊗ V v 7→ V v ⊗ V u . Then, the pro of tha t this solution K ( u ) sa tisfies the reflection equation (2.3) follo ws from the commut ativity of the follo wing dia gram (up to an o verall scala r facto r ): V u ⊗ V v ⊗ W id ⊗ K ( v ) − − − − − − → V u ⊗ V v − 1 ⊗ W R ( uv ) ⊗ id − − − − − − − → V v − 1 ⊗ V u ⊗ W   y R ( u/v ) ⊗ id id ⊗ K ( u )   y V v ⊗ V u ⊗ W V v − 1 ⊗ V u − 1 ⊗ W   y id ⊗ K ( u ) R ( u/v ) ⊗ id   y V v ⊗ V u − 1 ⊗ W R ( uv ) ⊗ id − − − − − − − → V u − 1 ⊗ V v ⊗ W id ⊗ K ( v ) − − − − − − → V u − 1 ⊗ V v − 1 ⊗ W  Combining previous results, w e obtain the third main result of the pap er: Theorem 3. The q − Onsager algebr a T and the cur re nt algebr a O q ( c sl 2 ) ar e isomorphic. Pr o of . According to Prop ositio n 4.2, K ( u ) with (2.6)-(2.9) is the unique in tertwiner of T sa tisfying (4.9). Also, it satisfies the re flec tion eq uation algebra (2 .3). So, K ( u ) establishes the isomo rphism betw een T and the reflection equation alg ebra (2 .3) fo r the U q ( c sl 2 ) R − matrix. Theorem 1 then establishes the isomorphism betw een the reflection equation algebra (2.3) and O q ( c sl 2 ), whic h supp orts the claim.  Although the isomorphism betw een T and O q ( c sl 2 ) ∼ = A q is now establis hed, an int er e sting problem remains to constr uct an explicit homo morphism from A q to T , i.e. to write a ll hig her elements of A q solely in terms of W 0 , W 1 . This pro ble m will be cons ide r ed elsewhere. A NEW CURRENT ALGEBRA AND THE REFLE CTION EQUA T ION 13 T o conclude, the q − Onsage r algebra T a dmits tw o different r e a lizations: one [se e P rop osition 4.2] in terms of the reflection equatio n algebr a for the U q ( c sl 2 ) R − ma trix and another one in terms [see Theorems 1, 2, 3] of the current a lgebra O q ( c sl 2 ) ∼ = A q . Previous results ar e resumed by the picture b elow. “RKRK” algebra [Cher, Sk] Reflection equation for U q ( c sl 2 ) ✛ ✲ Theorems 1,2 Curren t algebra (Def. 2.2) Presen tation {W − k , W k +1 , G k +1 , ˜ G k +1 } [BasK] ❅ ❅ ❅ ■ ❅ ❅ ❅ ❘ O q ( c sl 2 ) Prop osi tion 4.2    ✒ Theorem 3 q − Onsager algebra T [T er2] Figure 1. An algebraic sc heme for O q ( c sl 2 ) Ac kno wledgem en ts: Part of this work has b een supp or ted by the ANR Research pro ject “ Boundary inte gr able mo dels: algebr aic structur es and c orr elation fu n ctions ”, contract n umber JC05 -5274 9. P .B tha nk s S. Pakuliak for detailed explanations and no tes a b out Dr infeld’s constructio n at the early sta ge of this w ork , as w ell as P . T erwilliger for reading the ma n uscr ipt and helpful comments. References [AMPT] H. Au-Y ang, B.M. McCo y , J.H.H. P erk and S. T ang, Solvable mo dels in statist ic al me chanics and R iemann surfac es of ge nus gr e ater than one in Algebraic Analysis, V ol. 1, M. 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