On boundary super algebras

On b oundary s up er algebras Anastasia Doik ou Univ ersity of Patras, Department of Engineering Sciences, GR-2650 0 , Patras, Greece E-mail : adoiko u @ upatras.gr Abstract W e examine the symmetry breaking of sup er algebras d ue to the presence of appropriate in tegrable b oundary conditions. W e in vestig ate the b oundary breaking symmetry associated to b oth reflection algebras and t wisted sup er Y angians. W e extract the generators of th e resulting b oundary symm etry as w ell as w e provide explicit expressions of the asso ciated Casimir op erators. Con te n ts 1 In tr o duction 1 2 The sup er Y angian Y ( g l (m | n)) 3 2.1 The reflection algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The tw isted sup er Y angian . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 The U q ( g l (m | n)) algebra 10 3.1 The reflection algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3.1.1 Diagonal reflection matr ices . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 Non-diagonal reflection matrices . . . . . . . . . . . . . . . . . . . . 16 3.2 The q t wisted sup er Y angian . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A App endix 19 1 In tro duc tion Symmetry breaking pro cesses are of the most fundamen ta l concepts in phys ics. It was sho wn in a series of earlier w orks (see e.g. [1]–[9]), within the con text of quan tum in te- grabilit y , that the presence of suitable b oundary conditions may break a symmetry do wn without sp oiling the in tegra bility of the system. It w as a lso shown [10] that the breaking symmetry mec ha nism due to the presence of in tegra ble b oundary conditions may b e uti- lized to provide certain cen trally extended algebras. Here, w e shall inv estigate in detail the resulting b oundary symmetries in the contex t of quan tum in tegrable mo dels asso ciated to v arious sup er-algebras. More precisely , the main aim of the pre sen t w ork is the study of the algebraic structures underlying quantum integrable systems asso ciated to certain sup er-alg ebras, Y ( g l ( m | n)) and U q ( g l (m | n), once non-trivial b oundary conditions are implemen ted. In this inv esti- gation w e shall fo cus on the relev an t algebraic con t en t, and our primary ob jectiv es will b e the study of the related exact symmetries as w ell a s the construction of the relev an t Casimir op erators. The existence of cen trally extended sup er algebras emerging from 1 these b o undary a lgebras is one of the main motiv ations for the presen t in v estigation, and will b e discussed in full detail in a for t hcoming w ork giv en that is a separate significan t topic. W e shall fo cus here o n the sup er Y angian Y ( g l (m | n)) and its q -defor med counte rpart the U q ( g l (m | n)) algebra. I t is necessary to first intro duce some useful notation a sso ciated to sup er algebras. Consider the m + n dimensional column v ectors ˆ e i , with 1 at p osition i and zero ev erywhere else, and the (m + n) × (m + n) e ij matrices: ( e ij ) k l = δ ik δ j l . Then define the gra des: [ ˆ e i ] = [ i ] , [ e ij ] = [ i ] + [ j ] . (1.1) The tensor pro duct is also graded as: ( A ij ⊗ A k l )( A mn ⊗ A pq ) = ( − 1) ([ k ]+[ l ] )([ m ]+[ n ]) A ij A mn ⊗ A k l A pq . (1.2) Define also the transp o sition A T = m+n X i,j =1 ( − 1) [ i ][ j ]+[ j ] e j i ⊗ A ij , A = m+n X i,j =1 e ij ⊗ A ij , (1.3) and the sup er- t r ace as: str A = X i ( − 1) [ i ] A ii . (1.4) It will also b e con ve nien t for our purp oses here to define the sup er-transp osition as: A t = V − 1 A T V (1.5) where the matrix V will b e defined later in the text when appropriate. Also it is con v enient for what f ollo ws to in tro duce the distinguished a nd symmetric gra ding , corresp onding apparen tly to the distinguished and symmetric Dynkin diagrams. In the distinguished grading we define: [ i ] = ( 0 , 1 ≤ i ≤ m , 1 , m + 1 ≤ i ≤ m + n . (1.6) In the g l (m | 2k) w e also define the symmetric grading as: [ i ] = ( 0 , 1 ≤ i ≤ k , m + k + 1 ≤ i ≤ m + 2k 1 , k + 1 ≤ i ≤ m + k . (1.7) 2 2 The sup er Y angian Y ( g l (m | n)) Let us first in tro duce the basic algebraic ob jects asso ciated to the Y angian Y ( g l (m | n)). The R matrix solution of the Y ang- Baxter equation [11] asso ciated to Y ( g l (m | n)) is [12, 13, 14 , 15]: R ( λ ) = λ + iP (2.1) where P is the sup er-p erm utation op erator defined a s: P = X i,j ( − 1) [ j ] e ij ⊗ e j i . (2.2) Also define ¯ R 12 ( λ ) := R t 1 12 ( λ − iρ ) , ¯ R 21 ( λ ) := R t 2 12 ( − λ − iρ ) and ¯ R 12 ( λ ) = ¯ R 21 ( λ ) ¯ λ = − λ − iρ, and ρ = n − m 2 . (2.3) The ¯ R mat r ix may b e written as ¯ R 12 ( λ ) = ¯ λ + iQ 12 (2.4) where Q is a pro jector satisfying Q 2 = 2 ρQ, P Q = Q P = Q. (2.5) Consider a lso the L -op erator expressed as L ( λ ) = λ + i P , P = X a,b e ab ⊗ P ab (2.6) with P ab ∈ g l ( m | n ). L is a solution of the equation: R 12 ( λ 1 − λ 2 ) L 1 ( λ 1 ) L 2 ( λ 2 ) = L 2 ( λ 2 ) L 1 ( λ 1 ) R 12 ( λ 1 − λ 2 ) (2.7) with R b eing the matrix ab ov e (2 .1). The algebra defined b y (2.7) is equipp ed with a co pro duct: let L ( λ ) = P i,j e ij ⊗ l ij ( λ ) ∆( L ( λ )) = L 02 ( λ ) L 01 ( λ ) ⇒ ∆( l il ( λ )) = X j l j l ( λ ) ⊗ l ij ( λ ) . (2.8) 3 Define a lso the opp osite co pro duct. Let Π b e the ‘shift o p erator’ Π : V 1 ⊗ V 2 ֒ → V 2 ⊗ V 1 ∆ ′ = Π ◦ ∆ (2.9) in pa r ticular ∆ ′ ( L ( λ )) = L 01 ( λ ) L 02 ( λ ) ⇒ ∆ ′ ( l il ( λ )) = X j ( − 1) ([ i ]+[ j ])([ j ]+[ l ]) l ij ( λ ) ⊗ l j l ( λ ) . (2.10) The L co-pro ducts are deriv ed b y iteratio n as: ∆ ( L ) = (id ⊗ ∆ ( L − 1) )∆ , ∆ ′ ( L ) = (id ⊗ ∆ ( L − 1) )∆ ′ . (2.11) Let us now define the sup er comm utator as h A, B o = AB − ( − 1) [ A ][ B ] AB . (2.12) It is easy to sho w from (2.7) that P ab satisfy the g l (m | n) algebra, whic h reads as h P ij , P k l o = 0 , k 6 = j, i 6 = l h P ij , P k i o = ( − 1) [ i ] P k j , k 6 = j h P ij , P j l o = − ( − 1) [ i ]([ j ]+[ l ])+[ j ][ l ] P il i 6 = l h P ij , P j i o = ( − 1) [ i ] ( P j j − P ii ) . (2.13) 2.1 The reflection algebra This section serv es mostly as a w arm up, although some alternativ e pro ofs for the symme- tries are provided, and explicit expressions of quadratic Casimir op erators are also giv en. Consider no w the situatio n o f a b oundary in tegrable system described b y the reflection equation [16, 17], whic h also provides t he exc hange relations of the underlying algebra, i.e. the reflection algebra R 12 ( λ 1 − λ 2 ) K 1 ( λ 1 ) R 21 ( λ 1 + λ 2 ) K 2 ( λ 2 ) = K 2 ( λ 2 ) R 12 ( λ 1 + λ 2 ) K 1 ( λ 1 ) R 21 ( λ 1 − λ 2 ) , (2.14) As show n in [17] a tensorial type represen tation of the reflection algebra is giv en by : T ( λ ) = T ( λ ) K ( λ ) ˆ T ( λ ) (2.15) 4 where we define ˆ T ( λ ) = T − 1 ( − λ ) , and T ( λ ) = ∆ ( N +1) ( L ) = L 0 N ( λ ) . . . L 01 ( λ ) (2.16) K is a c -n umber solution o f the reflection equation. The asso ciated t ransfer mat rix is defined as t ( λ ) = str { K + ( λ ) T ( λ ) } (2.17) K + is also a solution of the reflection equation, and h t ( λ ) , t ( λ ′ ) i = 0 . (2.18) In the sp ecial case where K + = K = I the transfer matrix enjo ys the full g l ( m | n ) symmetry . Here w e shall pro vide an explicit pro of based o n the linear relations satisfied b y the algebra co pro ducts and the mat r ix T . The pro o f of this statemen t go es as follows: let us first recall the co- pro duct of the g l ( m | n ) elemen ts ∆( P ab ) = I ⊗ P ab + P ab ⊗ I . (2.19) Define also the follow ing represen ta t io n π : g l ( m | n ) ֒ → End( C n + m ) such that: π ( P ab ) = P ab . The N + 1 co- pro duct satisfies the follow ing comm utation relations with the mon- o dromy matrix ( π ⊗ id ⊗ N )∆ ( N +1) ( P ab ) T ( λ ) = T ( λ ) ( π ⊗ id ⊗ N )∆ ( N +1) ( P ab ) . (2.20) The later relations may b e written in a more straightforw ard form as:  P ab ⊗ I + I ⊗ ∆ ( N ) ( P ab )  T ( λ ) = T ( λ )  P ab ⊗ I + I ⊗ ∆ ( N ) ( P ab )  . (2.21) It is also clear tha t T − 1 ( − λ ) also satisfies relations (2.21), so for K = I it is quite straigh tf o rw ard to sho w tha t (recall also that P ab = ( − 1) [ b ] e ab )  ( − 1) [ b ] e ab ⊗ I + I ⊗ ∆ ( N ) ( P ab )  T ( λ ) = T ( λ )  ( − 1) [ b ] e ab ⊗ I + I ⊗ ∆ ( N ) ( P ab )  . (2.22) If w e no w express T ( λ ) = P i, j e ij ⊗ T ij ( λ ) then the latter relations b ecome X j ( − 1) [ b ] e aj ⊗ T bj ( λ ) + X i,j ( − 1) ([ a ]+[ b ])([ i ]+[ j ]) e ij ⊗ ∆ ( N ) ( P ab ) T ij ( λ ) = X i ( − 1) [ b ]+([ a ]+[ b ])([ a ]+[ i ]) e ib ⊗ T ia ( λ ) + X i,j e ij ⊗ T ij ( λ )∆ ( N ) ( P ab ) . (2.23) 5 W e are how ev er in terested in the sup er-trace ov er the auxiliary space, so w e are dealing basically with the diagonal terms of the ab o v e equation, hence w e obtain the followin g exc hange relations: h T ii ( λ ) , ∆ ( N ) ( P ab ) i = 0 , i 6 = a, i 6 = b h T aa ( λ ) , ∆ ( N ) ( P ab ) i = ( − 1) [ b ] T ba ( λ ) , h T bb ( λ ) , ∆ ( N ) ( P ab ) i = − ( − 1) [ a ] T ba ( λ ) . (2.24) By taking now the sup er-trace (w e are considering K + = I ) w e ha ve h X i ( − 1) [ i ] T ii ( λ ) , ∆ ( N ) ( P ab ) i = h ( − 1) [ a ] T aa ( λ ) + ( − 1) [ b ] T bb , ∆ ( N ) ( P ab ) i = . . . = 0 ⇒ h t ( λ ) , ∆ ( N ) ( P ab ) i = 0 (2.25) and consequen tly: h t ( λ ) , g l (m | n) i = 0 . (2.26) Recall that here we are fo cusing on the distinguished Dynkin diagram (1.6 ) . Consider no w the non trivial situation where the K - ma t r ix has the follo wing diago nal form (see also [4]). K ( λ ) = diag(1 , . . . 1 | {z } m 1 , − 1 , . . . − 1 | {z } m 2 + n 2 , 1 , . . . , 1 | {z } n 1 ) (2.27) suc h that m = m 1 + m 2 , n = n 1 + n 2 . In general, an y solution [4] may b e written in the form K ( λ ) = iξ + λ E , E 2 = I . E may b e diagonalized into (2.27 ) and tha t is wh y we mak e this con v enien t c hoice for the K matrix (2.27), w e also c hose for simplicit y ξ = 0. Let us first extract the non-lo cal c harges fo r any generic K -matrix of the fo rm: K ( λ ) = K + 1 λ ξ 1 + 1 λ 2 ξ 2 + O ( 1 λ 3 ) , (2.28) (see also [18] f or a brief discussion o n the symmetry). F r om the asymptotic b eha vior of the dynamical T w e then obtain: T ( λ → ∞ ) ∼ K + i λ Q (0) − 1 λ 2 Q (1) + . . . (2.29) The first or der quan tity provides the generators of the remaining b oundary symmetry , Q (0) = ∆ ( N ) ( P ) K + K ∆ ( N ) ( P ) + ξ 1 (2.30) 6 and for the sp ecial c hoice o f K - matrix ( 2.27) w e conclude Q (0) = m 1 X i,j =1 e ij ⊗ ∆ ( N ) ( P ij ) + m+n X i,j =m+n 2 e ij ⊗ ∆ ( N ) ( P ij ) − m+n 2 X i,j =1m 1 +1 e ij ⊗ ∆ ( N ) ( P ij ) + m 1 X i =1 m+n X j =m+n 2 +1 e ij ⊗ ∆ ( N ) ( P ij ) + m+n X i =m+n 2 +1 m 1 X j =1 e ij ⊗ ∆ ( N ) ( P ij ) . (2.31) More precisely , t he elemen ts: P ij , i, j ∈ (1 , m 1 ) ∪ (m + n 2 + 1 , m + n) form the g l (m 1 | n 1 ) P ij , i, j ∈ (m 1 + 1 , m + n 2 ) form the g l (m 2 | n 2 ) . (2.32) These are exactly the generators that, in the fundamental represen tation, comm ute with the K matrix (2.27). That is the g l (m | n) symmetry breaks do wn to g l (m 1 | n 1 ) ⊗ g l (m 2 | n 2 ). Since the K - matrix comm utes with a ll the ab ov e generators follow ing the pro cedure ab ov e w e can sho w relations (2.21) but only with the generators (2.32) and finally: h t ( λ ) , g l (m 1 | n 1 ) ⊗ g l (m 2 | n 2 ) i = 0 . (2.33) W e shall fo cus now for simplicit y on the ‘o ne-particle’ represen tatio n N = 1. The trace of the second order quantit y provides t he quadratic quadratic Casimir asso ciated t o the g l (m | n) in the case where K = 1. When K is of the diagonal form (2.27) the Casimir (see also [19, 20]) is asso ciated to g l (m 1 | n 1 ) ⊗ g l (m 2 | n 2 ). More sp ecifically , for K ∝ I (set N = 1): Q (1) = 2 P 2 and C = str Q (1) = 2 m+n X i,j =1 ( − 1) [ j ] P ij P j i (2.34) and for K giv en by the diagonal matrix (2.28) Q (1) = P K P + K P 2 − i P ξ 1 − iξ 1 P − ξ 2 and C = str Q (1) . (2.35) 7 F or the sp ecial ch oice of K -matrix (2.27) w e hav e: C = m 1 X i =1  m 1 X j =1 ( − 1) [ j ] P ij P j i + m+n X i =m+n 2 ( − 1) [ j ] P ij P j i  + m 1 X j =1  m 1 X i =1 ( − 1) [ j ] P ij P j i + m+n X i =m+n 2 ( − 1) [ j ] P ij P j i  − m+n 2 X i,j =m 1 +1 ( − 1) [ j ] P ij P j i . (2 .3 6) Higher Casimir o p erators ma y b e extracted by considering the higher order terms in the expansion of the transfer matrix in p ow ers of 1 λ . More precisely , let us fo cus on the N = 1 case and see more precisely ho w one obtains the higher Casimir op erators fr o m the expansion of the transfer matrix t ( λ ) = P 2 N k =1 t ( k − 1) λ k . Recall the N = 1 represen tation of the reflection algebra T ( λ ) = L ( λ ) k ˆ L ( λ ) = (1 + i λ P ) k (1 + i λ P − 1 λ 2 P 2 − i λ 3 P 3 + 1 λ 4 P 4 . . . ) = k + i λ ( P k + k P ) − 1 λ 2 ( P k P + k P 2 ) − 1 λ 2 ( P k P 2 + k P 3 ) . . . (2.37) where k is diagonal then t ( k − 1) ∝ X a,b ( P ab k bb P k − 1 ba + k aa P k aa ) (2.38) All t ( k ) are the hig her Casimir quantities and b y construction they comm ute with eac h other and they comm ute as sho wn earlier with the exact symmetry o f the system. De- p ending on the rank of the considered algebra the expansion of t ( λ ) should truncate at some p oint; note that expressions (2.37), (2.38) are generic and hold for any g l (m | n). The sp ectra of all Casimir op erato rs asso ciat ed to a sp ecific algebra may b e deriv ed via the Bethe ansatz metho dolog y . In particular, the sp ectrum fo r generic represen ta tions of sup er symmetric algebras is known (see e.g. [21]). By appropriately expanding the eigen- v alues in p ow ers of 1 λ w e ma y iden tify the sp ectrum of eac h one of t he r elev an t Casimir op erators. 2.2 The t wisted sup er Y angian Note that we fo cus here in the g l (m | 2k) case and the symmetric Dynkin diag r a mm ( 1.7). Let us first define some basic no t a tion useful or our purp oses here. Consider the mat r ix V = X i f i e i ¯ i , where , ¯ i = m + 2k − i + 1 . (2.39) 8 More precisely we shall consider here the following anti-diagonal matrix: V = an tidiag(1 , . . . , 1 | {z } m+k , − 1 , · · · − 1 | {z } k ) . (2.40) The t wisted sup er Y angian defined b y [22 , 23, 2 4] (fo r more details on the phys ical meaning o f reflection algebra and t wisted Y a ng ian see [25, 4]): R 12 ( λ 1 − λ 2 ) ¯ K 1 ( λ 1 ) ¯ R 12 ( λ 1 − λ 2 ) ¯ K 2 ( λ 2 ) = ¯ K 2 ( λ 2 ) ¯ R 12 ( λ 1 − λ 2 ) ¯ K 1 ( λ 1 ) R 12 ( λ 1 − λ 2 ) (2.41) the matr ix ¯ R 12 is defined in (2.3). Define also ˆ L 0 n ( λ ) = L t 0 0 n ( − λ − iρ ) ∝ 1 + i λ ˆ P 0 n , where ˆ P 0 n = ρ − P t 0 0 n . (2.42) Consider now the generic tensorial represen tation of the t wisted sup er Y angian: ¯ T ( λ ) = T ( λ ) K ( λ ) T t 0 ( − λ − iρ ) . (2.43) K is a c -n umber solution o f the tw isted Y angian (2.41). As in the previous section express the a b o v e tensor represen tation in p o w ers of 1 λ : ¯ T ( λ ) = 1 + i λ ( ¯ Q (0) + N ρ ) − 1 λ 2 ¯ Q (1) + . . . (2.44) where ¯ Q (0) = ∆ ( N ) ( P ) − ∆ ( N ) ( P t 0 ) . (2.45) It is clear that the elemen ts ¯ Q ab form the osp (m | 2n) algebra, and this corresp o nds essen- tially to a folding o f the g l (m | 2 n) to osp (m | 2n). Suc h a folding o ccurs in the corresp onding symmetric Dynkin diagram. Henceforth, we shall consider the simplest solution K ∝ I , although a f ull classification is presen ted in [4]. Based on the same logic as in the previous paragraph w e may extract the corresp onding exact symmetry . W e ha ve in this case: ( π ⊗ id ⊗ N )∆ ( N +1) ( ¯ Q (0) ab ) ¯ T ( λ ) = ¯ T ( λ ) ( π ⊗ id ⊗ N )∆ ( N +1) ( ¯ Q (0) ab ) , (2.46) whic h leads to h t ( λ ) , osp (m | 2n) i = 0 , (2.47) 9 so the exact symmetry of t he considered tra nsfer matrix is indeed osp ( m | 2n). The quadratic Casimir o p erator asso ciated to osp (m | n) emerges fro m the sup er-trace of ¯ Q (1) ( N = 1): ¯ Q (1) = P ˆ P C = str ¯ Q (1) = X i,j ( − 1) [ j ] P ij ˆ P j i . (2.48) 3 The U q ( g l (m | n)) algebra W e come no w to the q deformed situation. The R -matrix asso ciated t o the U q ( \ g l ( m | n)) algebra is g iv en b y t he following expressions [26]: R ( λ ) = m+n X i =1 a i ( λ ) e ii ⊗ e ii + b ( λ ) m+n X i 6 = j =1 e ii ⊗ e j j + m + n X i 6 = j =1 c ij ( λ ) e ij ⊗ e j i , (3.1) where we define a j ( λ ) = sinh( λ + iµ − 2 iµ [ j ]) , b ( λ ) = sinh λ, c ij ( λ ) = sinh( iµ ) e sig n ( j − i ) λ ( − 1) [ j ] . (3.2) Let us now introduce the sup er symmetric La x op erator a sso ciated to U q ( \ g l ( m | n)) L ( λ ) = e λ L + − e − λ L − (3.3) and L satisfies the fundamen t al algebraic relation (2 .7) with the R matrix giv en in (3 .1). The elemen ts L ± satisfy [27, 28 ] R ± 12 L + 1 L + = L + 2 L + 1 R ± 12 , R ± 12 L − 1 L − = L − 2 L − 1 R ± 12 , R ± 12 L ± 1 L ∓ 2 = L ∓ 2 L ± 1 R ± 12 (3.4) L ± are express ed as: L + = X i ≤ j e ij ⊗ l + ij , L − = X i ≥ j e ij ⊗ l − ij (3.5) definitions of l ± ij , ˆ l ± ij in terms of the U q ( g l (m | n)) algebra generators in the Chev alley- Serre basis are giv en in App endix A. The equations a b o v e (3.4) pro vide all the exc hange relations of the U q ( g l (m | n)) algebra. This is in fact the so called FR T realization of the U q ( g l (m | n)) alg ebra ( see e.g. [27 ]). 10 3.1 The reflection algebra Our main ob jective here is to extract the exact symmetry of the op en transfer matrix asso ciated to U q ( g l ( d m | n)). W e shall fo cus in this section on the distinguished Dynkin diagram. The op en transfer matrix is g iv en by (2.17), and from now on w e consider for our purp o ses here the left b oundary to b e the tr ivial solutio n K + = M = m+n X k =1 q n+m − 2 k +1 q − 2[ k ]+4 P k i =1 [ i ] e k k . (3.6) The elemen ts one extracts from the asymptotic expansion of T by k eeping the leading con tribution, T ± ab are the b oundary non-lo cal c harges, whic h form the b oundary sup er algebra with exc hange relations dictated b y: R ± 12 T ± 1 R ± 21 T ± 2 = T ± 2 R ± 21 T ± 1 R ± 12 . (3.7) In general, it may b e sho wn that the b oundary sup er algebra is an exact symmetry of the double row transfer matrix. Indeed b y in tro ducing the elemen t τ ± = str ( e ab T ± ) = ( − 1) [ a ] T ± ba (3.8) it is quite straigh tforw ar d to show along t he lines describ ed in [8] that [ τ ± , t ( λ )] = 0 ⇒  T ± ab , t ( λ )  = 0 . (3.9) Let T ± = X ab e ab ⊗ T ± ab , ˆ T ± = X ab e ab ⊗ ˆ T ± ab and T ± ab = ∆ ( N ) ( l ± ab ) , ˆ T ± ab = ∆ ( N ) ( ˆ l ± ab ) , (3.10) see a lso App endix A for definitions of l ± ij , ˆ l ± ij . The b oundary non-lo cal c har g es emanating from T ± are of the explicit form: T ± ad = X b,c ( − 1) ([ a ]+[ b ])([ b ]+[ d ]) T ± ab K + bc ˆ T ± cd . (3.11) It is clear that differen t choice o f K -matrix leads t o differen t non- lo cal charges and con- sequen tly to differen t symmetry . Also, the quadratic Casimir op erators are o btained by C ± = str 0 ( M 0 T ± 0 ) = X a,b,c ( − 1) [ b ] M aa T ± ab K ± bc ˆ T ± ca . (3.12) 11 Explicit expressions f or the quadratic Casimir o p erators will b e giv en b elow for particular simple examples. Note that higher Casimir op erators ma y b e extracted from t he higher order terms in the expansion of the transfer matrix in p ow ers of e ± 2 λ . 3.1.1 Diagonal reflection matrices W e shall distinguish tw o main cases of diagonal matrices, and w e shall examine the cor- resp onding exact symmetry . First consider the simplest b oundary conditions describ ed b y K ∝ I , we shall explicitly show that the asso ciated symmetry is t he U q ( g l (m | n)). O ne could just notice that the resulted T ± form essen tially the U q ( g l (m | n)), but t his is not quite obv io us. Then via (3.9) o ne can sho w the exact symmetry of the transfer matrix. Let us ho w ev er here follow a more straigh tf o rw ard and clean approac h regarding the sym- metry , that is we shall show that the op en transfer matr ix comm utes with eac h one of the algebra generators. Our pro ofs hold f o r generic no n-trivial integrable b oundary con- ditions as will b e more transparen t subsequen tly . In fact, in this a nd the previous section w e prepare somehow the general algebraic setting so tha t w e may extract the symmetry for generic integrable b oundary conditions. Let us now set the basic algebraic mac hinery necessary for the pro ofs that follow. Our aim now is to sho w tha t the double row tra nsfer matrix with trivial b o undary conditions that is K = I , K ( L ) = M enjo ys the f ull U q ( g l (m | n)) symmetry . W e shall show in particular that t he o p en tra nsfer mat r ix commu tes with each o ne of the generators of U q ( g l (m | n)). Let us o utline the pro of ; first it is quite easy to show from the fo rm of the co pro duct fo r ǫ i and following the log ic describ ed in the previous section see equation (2.21) that [∆ ( N ) ( ǫ i ) , t ( λ )] = 0 . (3.13) W e may now sho w the commutation b etw een t he tra nsfer matrix and the other elemen ts of the sup er algebra. Intro duce the represen tation π : U q ( g l (m | n)) ֒ → End( C N ) π ( e i ) = ε i , π ( f i ) = φ i , π ( q h i 2 ) = k i . (3.14) Consider a lso the following more con v enien t notation ∆ ( N ) ( q ± h i 2 ) ≡ ( K i N ) ± 1 , ∆ ( N ) ( e i ) ≡ E i N , ∆ ( N ) ( f i ) ≡ F i N . (3.15) 12 W e shall no w mak e use of the generic relations, whic h clearly T satisfie s due to the particular c hoice of b oundary conditions (see also [7]): ( π ⊗ id ⊗ N )∆ ′ ( N +1) ( Y ) T ( λ ) = T ( λ ) ( π ⊗ id ⊗ N )∆ ′ ( N +1) ( Y ) , Y ∈ U q ( g l (m | n)) . (3.16) Let us restrict our pro of to e i although the same logic fo llo ws for pr oving the com- m utation of the tra nsfer matrix with f i . In addition to the algebra exc hange relat io ns, presen ted in App endix A, w e shall need for our pro of the fo llo wing relations: M e i = q a ii e i M , M f i = q − a ii f i M ( k i 0 K i N ) ± 1 T 0 ( λ ) = T 0 ( λ ) ( k i 0 K i N ) ± 1 . (3.17) It is conv enien t to rewrite the relation ab ov e for e i as f o llo ws: ( k i 0 E i N + ε i 0 ( K i N ) − 1 ) T 0 ( λ ) = T 0 ( λ )( k i 0 E i N + ε i 0 ( K i N ) − 1 ) . . . ⇒ E i N M 0 T 0 ( λ ) + q α ii 2 ε i 0 M 0 T 0 ( λ )( k i 0 ) − 1 ( K i N ) − 1 = ( k i 0 ) − 1 M 0 T 0 ( λ ) k i 0 E i N + ( k i 0 ) − 1 M 0 T 0 ( λ ) ε i 0 ( K i N ) − 1 , (3.18) where the subscript 0 denotes the auxiliary space, whereas the N co pro ducts lea ve exclu- siv ely on the quan tum space, recall also that ε i ∝ e ii +1 . By fo cusing only on the diago na l con tributions o v er the auxiliary space, after some algebraic manipulations, a nd b earing in mind (3.18), w e end up to the follo wing exc ha nge relations (recall M is dia g onal): [ E i N , M aa T aa ( λ )] = 0 , a 6 = i, i + 1 [ E i N , M ii T ii ( λ )] = − ρ i M i +1 i +1 T i +1 i ( λ )( K i N ) − 1 [ E i N , M i +1 i +1 T i +1 i +1 ( λ )] = ( − 1) [ i ]+[ i +1] ρ i M i +1 i +1 T i +1 i ( λ )( K i N ) − 1 (3.19) ρ i is a scalar dep ending on a ii and the g rading. Then based on the r elat io ns ab ov e w e ha ve [ E i N , N X a =1 ( − 1) [ a ] M aa T aa ( λ )] = 0 ⇒ [ E i N , t ( λ )] = 0 . (3.20) Similarly one may sho w the comm ut a tivit y of t he transfer matrix with F i N , hence [∆ ( N ) ( x ) , t ( λ )] = 0 , x ∈ U q ( g l (m | n)) , (3.21) 13 and this concludes our pro of on the exact symmetry of the double r ow transfer ma t rix for the particular choice of b oundary conditions. It is clear that t he pr o of ab ov e can b e easily applied to the usual non sup er symmetric deformed algebra. W e hav e not seen, as far as w e know, suc h an explicit and elegan t pro of elsewhere not ev en in the non sup er symmetric case. In [7] the pro of is explicit, but rather tedious, whereas in [2] one has to realize that the emanating non lo cal c harges fo rm the U q ( g l ( N )), and t his is not quite ob vious. Note that T ± ab are quadratic com bina t io ns of the algebra g enerators, a nd in fact the corresp onding sup er-trace provides the asso ciated Casimir, whic h a gain giv es a hint ab out the asso ciated exact symmetry . In the case of trivial b o undary conditions ( K ∝ I ) there is a discussion on the symmetry in [29], but is restricted o nly in this particular case, whereas our pro o f holds for generic non-trivial integrable b oundary condition (see also next section). Let us no w give explicit expressions of the quadrat ic q -Casimir for t he simplest case (see a lso [30]), that is the U q ( g l (1 | 1)) situation. In general, the quadratic Casimir op erator is given by ( 3 .12), but now K ± ab = δ ab C + ∝ ∆ ( N ) ( q 2 ǫ 1 ) − ∆ ( N ) ( q 2 ǫ 2 ) − ( q − q − 1 ) 2 ∆ ( N ) ( q ǫ 1 + ǫ 2 2 )∆ ( N ) ( f 1 )∆ ( N ) ( q ǫ 1 + ǫ 2 2 )∆ ( N ) ( e 1 ) C − ∝ ∆ ( N ) ( q − ǫ 1 ) − ∆ ( N ) ( − q 2 ǫ 2 ) + ( q − q − 1 ) 2 ∆ ( N ) ( e 1 )∆ ( N ) ( q − ǫ 1 + ǫ 2 2 )∆ ( N ) ( f 1 )∆ ( N ) ( q − ǫ 1 + ǫ 2 2 ) . (3.22) Consider dia gonal no n- trivial solutions o f the reflection equation (see also a brief dis- cussion o n t he symmetry in [21]): K ( λ ) = diag( a ( λ ) , . . . , a ( λ ) , | {z } α b ( λ ) , . . . , b ( λ ) | {z } N − α ) . (3.23) In this case it is clear that the the K matrix satisfies (recall (3.14)) [ π ( x ) , K ( λ )] = 0 x ∈ U q ( g l (m − α | n)) ⊗ U q ( g l ( α )) , if α bo sonic x ∈ U q ( g l (m | ˜ α )) ⊗ U q ( g l (n − ˜ α )) if α = m + ˜ α fermionic (3.24) 14 and consequen tly , ( π ⊗ id ⊗ N )∆ ′ ( N +1) ( x ) T ( λ ) = T ( λ ) ( π ⊗ id ⊗ N )∆ ′ ( N +1) ( x ) x ∈ U q ( g l (m − α | n)) ⊗ U q ( g l ( α )) , if α bo sonic x ∈ U q ( g l (m | ˜ α )) ⊗ U q ( g l (n − ˜ α )) if α = m + ˜ α fermionic . (3.25) Th us following the logic described in the case where K ∝ I we show that: h t ( λ ) , U q ( g l ( α | n)) ⊗ U q ( g l (m − α )) i , if α bosonic h t ( λ ) , U q ( g l (m | ˜ α )) ⊗ U q ( g l (n − ˜ α )) i if α = m + ˜ α fermionic . (3.26) It is clear that in the limit q → 1 relev an t isotro pic results are recov ered a sso ciated to the Y ( g l (m | n) ) . Consider the simplest non trivial cases, that is the U q ( g l (2 | 1)) and U q ( g l (2 | 2)) algebras with diagonal K matrices (3.23) with α = 2. According to the preceding discussion the U q ( g l (2 | 1)) and U q ( g l (2 | 2)) symmetries break down to U q ( g l (2)) ⊗ u (1) and U q ( g l (2)) ⊗ U q ( g l (2)) r esp ective ly . It is w o rth presen ting the asso ciated Casimir op erators for the to w particular examples. Consider first the U q ( g l (2 | 1)) case, then fr o m the λ → ±∞ asymptotic b eha vior o f the op en transfer matrix w e obta in: C + = q 2 ∆ ( N ) ( q ǫ 1 + ǫ 2 )  q ∆ ( N ) ( q ǫ 1 − ǫ 2 ) + q − 1 ∆ ( N ) ( q − ǫ 1 + ǫ 2 ) + ( q − q − 1 ) 2 ∆ ( N ) ( f 1 )∆ ( N ) ( e 1 )  C − = q 2 ∆ ( N ) ( q − 2 ǫ 3 ) . (3.27) Notice that all ǫ i , i = 1 , 2 , 3 elemen ts comm ute with the tr a nsfer matrix and also the par enthes is in the first line of (3.27) is essen tially the ty pical U q ( sl 2 ) quadratic Casimir. C − is basically a u (1)) type quan tity . It is th us clear that C + is the quadratic Casimir asso ciated to U q ( g l 2 ); indeed in this case the U q ( g l (2 | 1)) symmetry breaks do wn to U q ( g l (2)) ⊗ u (1). In general the implemen tation of b oundary conditions describ ed b y the diagonal matrix (3.23 ) with α = m ob viously breaks the super symmetry to U q ( g l (m)) ⊗ U q ( g l (n)), so in fact the sup er algebra reduces to t wo non sup er symmetric quan tum algebras. Also, C + is the Casimir asso ciated U q ( g l (m)) to whereas C − is the Casimir asso ciated to U q ( g l (n)). 15 This will b ecome more transparen t when examining the U q ( g l (2 | 2)) case with b oundary conditions describ ed by K ( 3 .23) and α = 2. The asso ciated Casimir op erators are then giv en by C + = q 2 ∆ ( N ) ( q ǫ 1 + ǫ 2 )  q ∆ ( N ) ( q ǫ 1 − ǫ 2 ) + q − 1 ∆ ( N ) ( q − ǫ 1 + ǫ 2 ) + ( q − q − 1 ) 2 ∆ ( N ) ( f 1 )∆ ( N ) ( e 1 )  C − = q 2 ∆ ( N ) ( q − ǫ 3 − ǫ 4 )  q ∆ ( N ) ( q ǫ 4 − ǫ 3 ) + q − 1 ∆ ( N ) ( q − ǫ 4 + ǫ 3 ) + ( q − q − 1 ) 2 ∆ ( N ) ( f 3 )∆ ( N ) ( e 3 )  . (3.28) As exp ected in t his case, since now the symmetry is broke n to U q ( g l 2 ) ⊗ U q ( g l 2 ), the C + Casimir is asso ciated to the one U q ( g l 2 ) symmetry whereas the C − is a sso ciated to the other U q ( g l 2 ). 3.1.2 Non-diagonal reflection matric es Let us finally consider non-diagonal reflection matrices. A new class of non-diago nal reflection matrices asso ciated to U q ( g l m | n) w as recently deriv ed in [31]. Sp ecifically , first define the conjuga t e index ¯ a suc h that: [ a ] = [¯ a ], a nd more sp ecifically: ¯ a = 2k + m + 1 − a ; Symmetric diagra m ¯ a = m + 1 − a, a b o sonic ; ¯ a = 2m + n + 1 − a, a fermionic; Distinguished diagram. (3.29) Then the non diagonal matrices read as: Symmetric Dynkin diagram: K aa ( λ ) = e 2 λ cosh iµm − cosh 2 iµζ , K ¯ a ¯ a ( λ ) = e − 2 λ cosh iµm − cosh 2 iµζ , K a ¯ a ( λ ) = ic a sinh 2 λ, K ¯ aa ( λ ) = ic ¯ a sinh 2 λ, 1 ≤ a ≤ L K aa ( λ ) = K ¯ a ¯ a ( λ ) = cosh(2 λ + imµ ) − cosh 2 iµζ , K a ¯ a ( λ ) = K ¯ aa ( λ ) = 0 , L < a ≤ m + 2 k 2 ; 1 ≤ L ≤ m + 2 k 2 K AA = cosh(2 λ + imµ ) − cosh 2 iµζ , A = m + 2 k + 1 2 if m o dd . (3.30) 16 m, ζ are free b oundary parameters. Let us first fo cus on the solution with the minimal n umber of non-zero en tries, and let us consider the symmetric case. The elemen ts T ± ij , i, j ∈ { 2 , . . . , 2k + m − 1 } form essen- tially the U q ( g l (m | 2(k − 1))) algebra, and it can b e sho wn, based on the logic described in the previous section, that the transfer ma t rix commutes with U q ( g l (m | 2(k − 1))) plus the elemen ts T ± 1 j ∪ T ± j 1 ∪ T ± N j ∪ T ± j N ( N = m + 2 k). Let us now deal with t he generic situation de- scrib ed b y the solutions presen ted ab ov e (3 .30). First define: U = T ± Aj ∪ T ± j A ∪ T ± ¯ Aj ∪ T ± j ¯ A = 0 , A ∈ { 1 , . . . , L } . Then it is shown for the generic non-diag onal K - matrix: Symmetric Dynkin diagram h t ( λ ) , U q ( g l (m | 2(k − L ))) ⊗ U i = 0 b osonic indices , h t ( λ ) , U q ( g l (m − 2 l )) ⊗ U i = 0 L = k + l , fermionic indices . (3.31 ) The generic Casimir is giv en b y (3.1 2), where now the o nly non zero entries are giv en by : K aa , K a ¯ a It is clear that different choice s of non-diagonal reflection matrices lead to distinct preserving symmetries. This ma y p erhaps b e utilized together with the contraction pro - cess presen ted in [1 0] to offer a n algebraic description r ega rding of the underlying a lg ebra emerging in the AdS/CFT con t ext. Recall that this t yp e of con tractions leads to cen trally extended algebras, and as in kno wn in the con text of AdS/CFT w e deal basically with a cen trally extended g l (2 | 2) algebra [32]. The explicit form of the b oundary non lo cal c harg es T ab (3.11) in addition to the existence of some familiar symmetry is essen tial, and may b e for instance utilized for deriving r eflection matrices asso ciated t o higher represen tations of U q ( g l (m | n)) ( see e.g. [33]). In fact, the logic w e follow here is rather tw o-fold: o n the one hand w e try t o extract a familiar symmetry algebra if a n y , and follo wing the pro cess o f section 3.1.1 to deriv e the exact symmetry . On the other hand fo r generic K that ma y break all familiar symmetries w e sho w via the reflection equation tha t the b oundary non-lo cal charges form an algebra, and via (3.9) we show that pro vide an extra symmetry for the op en transfer matrix. Moreo v er, the know ledge o f the explicit f o rm of the b o undary non- lo cal c har g es is o f gr eat significance given that they may b e used, as already men t ioned, fo r deriving 17 reflection matrices for arbitrary represen tations [33]. 3.2 The q t wisted sup er Y angian T o complete our analysis on the b oundary sup er symmetric algebras w e shall now briefly discuss the q tw isted Y angia n. A more detailed analysis together with the classification of the corresp onding c -n umber solutions will b e pursued elsewhere. As in the case of t he t wisted Y angia n we fo cus on the symmetric Dynkin diagra m (1.7) and in tro duce V = X i f i e i ¯ i : V T V = M (3.32) and define the sup er transp osition as in t he ratio nal case. Also define the following matrices: ¯ R 12 ( λ ) := R t 1 21 ( − λ − iρ ) , ¯ R 21 ( λ ) := R t 2 12 ( − λ − iρ ) . (3.33) Recall t ha t in the isotropic case R 12 = R 21 . Then the q -Twisted Y angian is defined by: R 12 ( λ 1 − λ 2 ) K 1 ( λ 1 ) ¯ R 21 ( λ 1 + λ 2 ) K 2 ( λ 2 ) = K 2 ( λ 2 ) ¯ R 12 ( λ 1 + λ 2 ) K 1 ( λ 1 ) R 21 ( λ 1 − λ 2 ) . (3.34) The non- lo cal c harg es ar e deriv ed from the asymptotic b ehav ior o f the tensor repre- sen tation o f the q -tw isted Y angian: T ± ab = X k ( − 1) ([ a ]+[ k ])([ k ]+[ b ]) T + ak ˆ T + k b . (3.35) As in the non sup er symmetric case the non lo cal c har g es deriv ed ab ov e satisfy exc hange relations o f the ty p e (see also e.g. [9]) R ± 12 T ± 1 ¯ R ± 21 T ± 2 = T ± 2 ¯ R ± 12 T ± 1 R ± 21 (3.36) it turns out that they do not pro vide an exact symmetry of the transfer ma t r ix [9], as opp osed to the isotropic limit describ ed in section 2 .2. And a lthough in the contex t o f discrete in t egrable mo dels describ ed b y the double row transfer matrix the b oundary non- lo cal c harges do not form an exact symmetry one ma y show that in the corresp onding field theoretical con text they do provide exact symmetries (see e.g. [3]). Suc h inv estigations regarding sup er symmetric field theories will b e left how ev er for f uture in v estigations. No te finally that the classification of solutions of t he q tw isted Y a ngian for the U q ( g l (m | n)) is still an op en question, which w e hop e t o address in a fo rthcoming publication. 18 A App endix W e shall recall here some details regarding the U q ( g l (m | n)) algebra. The algebra is defined b y generators ǫ i , e j f j , i = 1 , . . . , N , j = 1 , . . . , N − 1, a nd the exc hange relations of the U q ( g l (m | n)) alg ebra a re give n b elow : q ǫ i q − ǫ i = q − ǫ i q ǫ i = 1 q ǫ i e j = q ( − 1) [ j ] δ ij − ( − 1) [ j +1] δ ij +1 e j q ǫ i q ǫ i f j = q − ( − 1) [ j ] δ ij +( − 1) [ j +1] δ ij +1 f j q ǫ i e i f j − ( − 1) ([ i ]+[ i +1])([ j ]+[ j +1]) f j e i = δ ij q ǫ i − ǫ i +1 − q − ǫ i + ǫ i +1 q − q − 1 x i x j = ( − 1) ([ i ]+[ i +1])([ j ]+[ j +1]) x j x i , x i ∈ { e i , f i } , (A.1) and q Chev alla y-Serre relations: x 2 i x i ± 1 − ( q + q − 1 ) x i x i ± 1 x i + x i ± 1 x 2 i = 0 , x i ∈ { e i , f i } i 6 = m . (A.2) No w set h i = ǫ i then the U q ( sl (m | n)) algebra is defined by generators e i , f i , h i . Let a ij the elemen ts of the related Cartan N × N matrix, whic h for instance for the distinguished Dynkin diag r a m is: a =               2 − 1 − 1 2 − 1 . . . − 1 0 1 . . . 1 − 2 1 1 − 2               (A.3) the zero diagona l elemen t o ccurs in the m p osition. Define also: [ h ] q = q h − q − h q − q − 1 (A.4) then the U q ( sl (m | n)) sup er algebra for t he distinguished D ynkin diagra m reads as: [ e i , f i ] = [ h i ] q , i < m , { e m , f m } = − [ h m ] q , [ e i , f i ] = − [ h i ] q , i > m [ h j , h k ] = 0 , [ h i , e j ] = a ij e j , [ h i , f j ] = − a ij f j (A.5) 19 plus the Chev alley-Serre relations ( A.2). The a lg ebra ab ov e is equipped with a non-trivial co-pro duct: ∆( ǫ i ) = ǫ i ⊗ I + I ⊗ ǫ i ∆( x ) = q − h i 2 ⊗ x + x ⊗ q h i 2 , x ∈ { e i , f i } . (A.6) There is also an isomorphism b et w een the FR T r epresen t a tion of the algebra and the Chev alley -Serre basis. Recall that L ( λ ) = L + − e − 2 λ L − , ˆ L ( λ ) = ˆ L + − O ( e − 2 λ ) , λ → ∞ ˆ L ( λ ) = ˆ L − − O ( e 2 λ ) , λ → −∞ . (A.7) Recall f o r the reflection algebra ˆ L ( λ ) = L − 1 ( − λ ) a nd a lso L ± = X i,j e ij ⊗ l ± ij , ˆ L ± = X i,j e ij ⊗ ˆ l ± ij . (A.8) Then w e ha ve the follow ing iden tifications [27, 3 4]: l + ii = ( − 1) [ i ] q ǫ i , l + ii +1 = ( − 1) [ i +1] ( q − q − 1 ) q ǫ i + ǫ i +1 2 f i , l + i +1 i = 0 l − ii = ( − 1) [ i ] q − ǫ i , l − i +1 i = − ( − 1) [ i ] ( q − q − 1 ) e i q − ǫ i + ǫ i +1 2 , l − ii +1 = 0 ˆ l + ii = ( − 1) [ i ] q ǫ i , ˆ l + i +1 i = ( − 1) [ i +1] ( q − q − 1 ) q − a ii 2 q ǫ i + ǫ i +1 2 e i , ˆ l + ii +1 = 0 ˆ l − ii = ( − 1) [ i ] q − ǫ i , ˆ l − i +1 i = − ( − 1) [ i ] ( q − q − 1 ) q a ii 2 f i q − ǫ i + ǫ i +1 2 , ˆ l − i +1 i = 0 (A.9) l + ij , ˆ l − ij , i < j a nd l − ij , ˆ l + ij , i > j are a lso non zero, a nd a r e expressed as combinations of the U q ( g l (m | n)) g enerato r s, but ar e omitted here for brevit y , see e.g expressions in [35, 7] for the U q ( g l (n)) case. References [1] A. Doik ou and R.I. Nep omec hie, Nucl. Phys . B521 (1998) 547, hep-th/98031 18 . [2] A. Doik ou and R.I. Nep omec hie, Nucl. Phys . B530 (1998) 641, hep-th/98070 65 . 20 [3] G.W. Delius and N.J. Mac k a y , Comm un. Math. Ph ys. 233 (2 003) 17 3, hep-th/0112 023 . [4] D. Arnaudon, J. Av an, N. Cramp e, A. D oik ou, L. F rappat and E. R a goucy , J. Stat. Mec h. 0408 (2004) P005, math-ph/04060 21 . [5] A. Doik ou, J. Stat. Mec h. (20 0 5) P120 0 5, math-ph/0402067 . [6] A. Doik ou, J. Math. Ph ys. 46 (20 05) 05350 4, hep-th/040327 7 . [7] A. Doik ou, Nucl.Ph ys. B725 (2005) 493, math-ph/0409060 . [8] A. Doik ou, SIGMA 3 (2007) 009, math-ph/0606040 . [9] N. Cramp e a nd A. D oik ou, J. Math. Phys . 48 (2007) 023511 , math-ph/0 611030 . [10] A. Doik o u a nd K . Sfetsos, J. Phys . A42 (2009) 47520 4, arXiv:0904.34 37 . [11] R.J. Baxter, Ann. Phys . 70 (1972 ) 193; J. Stat. Ph ys. 8 (19 7 3) 25; Exactly solve d mo dels in statistic al me chanic s (Academic Press, 19 8 2). [12] P . Kulish, J. So v. Math. 35 (1 986) 2648. [13] M. Nazarov, Lett. Math. Ph ys. 21 (19 91) 1 2 3. [14] C.N. Y ang, Ph ys. R ev. Lett. 19 ( 1 967) 1312. [15] H. Saleur, Nucl. Ph ys. B578 (2000 ) 552. [16] I.V. Cherednik, Theor. Math. Ph ys. 61 (1984) 9 77. [17] E.K. Skly anin, J. Phys . A21 (1988) 2375. [18] E. Ragoucy and Sa t ta, JHEP 070 9:001 (200 7), arXiv:0706.33 27 . [19] A. Molev and V. Retakh, Int. Mathem. Researc h Notices ( 2004) 611, math/0309461 . [20] D. Arnaudon, C. Chryssomalak os and L. F rappat, J. Math. Ph ys. 36 (1995) 5262 , q-alg/95030 21 . [21] S. Belliard and E. Rag o ucy , J. Phy s. A42 (20 09) 20 5203, arXiv:0902.0321 . [22] G.I. Olshanski, Twisted Y a ng ians and infinite-dimensional classical Lie algebras in ‘Quan tum Groups’ (P .P . Kulish, Ed.), Lecture notes in Math. 1510 , Springer (1992 ) 103; A.I. Molev, M. Nazarov and G .I. Olshanski, Russ. Math. Survey s 51 (1 996) 206, hep-th/9409 025 . 21 [23] A.I. Molev, E. R a goucy and P . Sorba, Rev. Math. Phy s. 15 (2003) 789 , math/020814 0 . A.I. Molev, Han db o ok of Algebr a , V o l. 3, (M. Hazewink el, Ed.), Elsevier, (2003), pp. 907. [24] C. Briot and E. Ra g oucy , J. Math. Ph ys. 44 (2003 ) 1052. [25] A. Doik o u, J. Phy s. A33 (2 0 00) 8 7 97, hep-th/0006197 . [26] J.H.H. P erk a nd C.L. Sc hultz, Ph ys. Lett. A84 ( 1 981) 407; C.L. Sch ultz, Ph ysica A 122 (1983) 71. [27] L.A. T akh ta j a n, Quamtum Gr oups , In tro duction to Qua n tum Gr o ups and In tergable Massiv e mo dels of Quan t um Field Theory , eds, M.-L. Ge and B.-H. Zhao , Nank ai Lectures o n Mathematical Phy sics, W orld Scien tific, 199 0, p.p. 69; N.Y u. Reshetikhin, L.A. T akhta ja n and L. D. F addeev, Leningrad Math. J.1 (19 9 0) 193. [28] E.K. Skly anin, L.A. T akhta jan and L .D . F addeev, Theoret. Math. Phy s. 40 N2 (197 9 ) 194. [29] R. Y ue, H. F an and B. Hou, Nucl. Ph ys. B46 2 (1996) 167, cond-mat/9603022 . [30] R.B. Zhang, Lett. Mat h. Phy s. 33 (1995) 263 . [31] A. Doik o u a nd N. Karaisk o s, J. Stat. Mec h. (2009) L09004, arXiv:0907 .3408 . [32] N. Beisert, Adv. Theor. Math. Ph ys. 12 ( 2 008) 945, hep-th/0511082 . [33] R.I. Nep omec hie, Lett. Math. Ph ys. 62 (2002) 8 3, hep-th/0204181 ; R.I. Nep omec hie and G.W. Delius, J. Phy s. A35 (2002) L3 4 1, hep-th/0204076 . [34] J. Ding a nd I.B. F renke l, Comm un. Math. Phys . 156 (1993) 27 7. [35] M. Jim b o, Lett. Math. Phy s. 10 (1985) 63 , and Lett. Math. Ph ys. 11 (1986) 247. 22