Murphy elements from the double-row transfer matrix

Murph y elemen ts from the double-ro w transfer matrix Anastasia Doik ou 1 Univ ersit y of P atras, Departmen t of Engineering Sciences, GR-2650 0 P atras, Greece Abstract W e c onsider the double-ro w (op en) transfer matrix constructed fr om generic tensor-t yp e represent ations of v arious t yp es of Hec k e alg ebras. F or d ifferen t c hoices of b oun dary conditions for the relev an t in tegrable lattice mo del we express th e double-ro w transfer matrix solely in terms of generators of the corresp onding H ec k e algebra (tensor-t yp e realizations). W e then exp and th e op en transfer matrix and extract the associated Murphy elemen ts from the first/last te rms of the e xpansion. Suitable com binations of the Murphy elemen ts as has b een shown comm ute with the corresp onding He c k e algebra. Keyw ords: algebraic structures of in tegrable mo dels, in tegrable spin c hains (v ertex mo dels), solv able lattice mo dels 1 adoikou@upatras.gr 1 In tro ductio n There has b een muc h activit y lately asso ciated to algebraic structures underlying inte- grable lattice mo dels. On the one hand there is an immediate connection b et we en these mo dels and realizations of the braid gro up [1]– [7], giv en that spin c hain mo dels may b e constructed as tensorial represen tations of quotien ts of the bra id group called Hec k e al- gebras. On the other hand integrable latt ice mo dels provide p erhaps the most natural framew ork for the study of quan tum groups [8, 9]. The symmetry a lg ebras underlying these mo dels may b e seen as deformations of the usual Lie algebras [1, 1 0], and their defining relations emanate directly from the fundamen tal relations ruling suc h mo dels, that is the Y ang-Baxter [11] and reflection equations [1 2]. Sev eral studies hav e b een dev oted on the unco v ering o f the symmetries of op en spin c hain mo dels as well as o n connecting the asso ciated Hec k e algebras with the underlying quan tum group symmetries , and in most cases it turns out t ha t the exact symme tries – quan tum algebras– comm ute with the Hec k e algebra (see e.g. [13, 14, 4]). In the spin c hain con text the tra nsfe r matr ices may b e usually expres sed in terms of the quan tum algebra elemen ts in a univ ersal manner, i.e. indep enden t of the c hoice of represen tation of the quan tum algebra. Ho w ev er, such generic expressions in terms of Heck e algebra elemen ts are missing, with the exception of generic fo rm ulas of integrable Hamilto nians (see e.g. [3, 4] for computational details). In the presen t in v estigatio n w e prov ide generic ex pressions, of double-ro w transfer matrices [15] in terms of generators of Hec k e algebras (tensor t yp e represen ta t ions). It is worth noting that suc h generic expre ssions starting from Skly anin’s transfer matrix [1 5] offer an immediate link b et w een spin c hain lik e systems and other in tegrable lattice models suc h as P otts models and in general face t ype mo dels [16, 17, 18]. Ha ving such expressions at our disp osal w e are then able to extract from the double-row transfer matr ix the so-called Murph y elemen ts, whic h comm ute with the Hec ke a lgebras (see [1 9, 20] and references therein). The o utline of this pap er is a s follow s. In t he next section w e giv e basic definitions regarding the A, B and C -t yp e Hec k e a lg ebras. W e also define the Murphy elemen ts asso ciated to each one of the aforemen tioned Hec k e algebras. In section 3 starting from the double-row transfer matrix [15] w e end up with generic form ulas expressed in terms of generators of Hec k e alg ebras (tensor r epresen tations). W e finally pro v e that the Murphy elemen ts are directly obtained fr om suitable double-ro w transfer matr ices of v arying di- mension. In the last section w e briefly discus s the findings of this study , and also prop ose p ossible directions for future in v estigations. 1 2 Hec k e alg ebras: definitio ns W e shall rev iew in this section basic definitions regarding v arious types of Hec k e algebras, and t he asso ciated Murph y elemen ts (see also [20 ]–[28]). Definition 2.1. The A -typ e He cke al g ebr a H N ( q ) is define d by the gener ators g l , l = 1 , . . . , N − 1 satisfying the fol lowing r elations: g l g l +1 g l = g l +1 g l g l +1 , (2.1) h g l , g m i = 0 , | l − m | > 1 (2.2) ( g l − q ) ( g l + q − 1 ) = 0 . (2.3) Definition 2.2. The B -typ e He cke algebr a B N ( q , Q 0 ) is define d by gener ators g l , l ∈ { 1 , . . . , N − 1 } , satisfying the He ck e r ela tion s (2.1)-(2.3) and g 0 ob eying: g 1 g 0 g 1 g 0 = g 0 g 1 g 0 g 1 , (2.4) h g 0 , g l i = 0 , l > 1 (2.5) ( g 0 − Q 0 ) ( g 0 + Q − 1 0 ) = 0 . (2.6) The algebra ab ov e is a pparen tly an extension of the Heck e algebra defined in (2.3). Also the B - t yp e Hec k e algebra is a quotient o f the affine Hec k e algebra, whic h is defined by generators g i , g 0 that satisfy (2.1)-( 2 .5). Definition 2.3. The C -typ e He cke algebr a C N ( q , Q 0 , Q N ) , is define d by the ge n er ators g l , l ∈ { 1 , . . . , N − 1 } , g 0 satisfying (2.1)-(2.6) and an extr a gener ator g N , ob eying g N g N − 1 g N g N − 1 = g N − 1 g N g N − 1 g N (2.7) h g N , g i i , 0 ≤ i ≤ N − 2 (2.8) ( g N − Q N ) ( g N + Q − 1 N ) = 0 . (2.9) There is also a quotien t of the C -t yp e Heck e alg ebra called the t wo boundary T emp erley- Lieb algebra [29]–[32], [2 , 3] with a t ypical repres en tation b eing the b oundary XXZ mo del. Definition 2.4. The two b ounda ry T emp erley-Lieb algebr a i s define d by gener ators satis- fying (2.1)-(2.9). In ad d ition to the latter r el a tions some extr a e quations ar e also satisfie d. 2 L et e i = g i − q , e 0 = g 0 − Q 0 , e N = g N − Q N then: e i e i ± 1 e i = e i , 2 ≤ i ≤ N − 1 (2.10) e 1 e 0 e 1 = κ − e 1 (2.11) e N − 1 e N e N − 1 = κ + e N − 1 . (2.12) It is w orth men tioning that by remo ving the third of the ab ov e equations we obtain the b oundary T emp erley-Lieb (blob) [30] algebra, and b y remo ving the second equation a s w ell w e end up with the usual T emp erley-Lieb algebra [29]. Definition 2.5. Define a lso the: A -typ e Murphy e l e m ents J ( A ) 1 = g 2 1 J ( A ) i = g i J ( A ) i − 1 g i , 2 ≤ i ≤ N − 1 (2.13) B -typ e Murphy elements J ( B ) 0 = g 0 J ( B ) i = g i J ( A ) i − 1 g i , 1 ≤ i ≤ N − 1 (2.14) C -typ e Murphy elements J ( C ) 0 = g − 1 1 g − 1 2 . . . g − 1 N − 1 g N g N − 1 . . . g 2 g 1 g 0 J ( C ) i = g i J ( C ) i − 1 g i , 1 ≤ i ≤ N − 1 . (2.15) It was sho wn that Murph y ele men ts are pairwise comm uting, and symmetric p olynomials in { J ( A, B ) i } comm ute with the A, B Hec k e algebras resp ectiv ely (see [19, 2 0] and refer- ences therein, see also [31]). Moreo v er, symmetric p olynomials in { J ( C ) i , ( J ( C ) i ) − 1 } a r e cen tral in C -t yp e Hec k e algebras. In the next section we shall show that the Murph y elemen ts defined ab o v e arise naturally from certain hierarch ies o f open transfer matrices. Let us p oin t out that the B -ty p e Murph y elemen ts ma y b e t ho ugh t of as represen- tations o f the so-called B -ty p e Artin braid gro up B N defined by g ene rators g i , g 0 and relations (2 .1), ( 2 .2), ( 2.4), (2.5) –it is eviden t that the B -t yp e Hec k e algebra is a quo- tien t of the Art in group B N . Suc h represen tations ar e known as the ‘a uxiliary string’ represen ta t ions σ l : B N ֒ → B N + l σ l ( g 0 ) = g l g l − 1 . . . g 1 g 0 g 1 . . . g l − 1 g l σ l ( g i ) = g i + l (2.16) 3 and hav e b een extensiv ely discussed for instance in [3, 32]. The auxiliary spin represen- tation giv es rise to ‘dynamical’ b oundary conditions prov iding extra b oundary degrees of freedom (see also relev ant discussion in [3]). It will b e useful for the fo llo wing to consider tensor type r epres en tations of the Hec k e algebra; let π : C N ( q , Q 0 , Q N ) ֒ → End( V ⊗ N ) suc h that π ( g i ) = I ⊗ I . . . ⊗ g |{z} i, i + 1 ⊗ . . . ⊗ I π ( g 0 ) = g 0 |{z} 1 ⊗ I . . . ⊗ I π ( g N ) = I ⊗ I . . . I ⊗ g N |{z} N . (2.17) It is clear that the absence of the extra generators g N , g 0 leads to represen tations of the B a nd A t yp e Hec k e algebras. F or instance in the represen t a tion o f the C - t ype Hec k e algebra fo r t he U q ( d g l N ) series ( V ≡ C N ) (see a lso [33, 4] and [34]) w e define: g = q I + X a 6 = b  e ab ⊗ e ab − q sg n ( a − b ) e aa ⊗ e bb  g 0 = − Q − 1 0 e 11 − Q 0 e N N + x + 0 e 1 N + x − 0 e N 1 + Q 0 I g N = − Q N e 11 − Q − 1 N e N N + x + N e 1 N + x − N e N 1 + Q N I (2.18) where ( e ij ) k l = δ ik δ j l . F or N = 2 in pa rticular we reco v er the w ell known XXZ represen- tation of the t w o-b oundary T emp erley-Lieb a lgebra. 3 Murph y elements from op en transfer matrices Ha ving introduced the ba sic algebraic se tting w e are now in a p osition to extract the ab o v e defined Murph y elemen ts f r om the double-row transfer ma t r ix [1 5]. P articular c hoice o f b oundary conditions en tails Murphy elemen ts asso ciated to the three differen t ty p es of Hec k e algebras defined in the previous section. In tro duce no w the Y ang -Baxter and reflection equations. The Y ang–Baxter equation is given b y [11]: ˇ R 12 ( λ 1 − λ 2 ) ˇ R 23 ( λ 1 ) ˇ R 12 ( λ 2 ) = ˇ R 23 ( λ 2 ) ˇ R 12 ( λ 1 ) ˇ R 23 ( λ 1 − λ 2 ) (3.1) acting on V ⊗ 3 , and as usual ˇ R 12 = ˇ R ⊗ I , ˇ R 23 = I ⊗ ˇ R . The reflection equation is a lso defined a s [12] ˇ R 12 ( λ 1 − λ 2 ) K 1 ( λ 1 ) ˇ R 12 ( λ 1 + λ 2 ) K 1 ( λ 2 ) = K 1 ( λ 2 ) ˇ R 12 ( λ 1 + λ 2 ) K 1 ( λ 1 ) ˇ R 12 ( λ 1 − λ 2 ) (3.2) 4 acting o n V ⊗ 2 , and as customary K 1 = K ⊗ I , K 2 = I ⊗ K . Notice the structural similarit y b et w een the Y ang-Baxter and reflection equation and the Hec k e algebras ab o v e, which suggests tha t represe n tations of B N ( q , Q ) should pro vide candidate solutions of the Y ang- Baxter and reflection equations. T o construct a spin c hain lik e system with tw o non-trivial b oundaries w e shall need to cons ider one more reflection eq uation asso ciated to the o t her end of the N site spin chain, i.e. ˇ R N − 1 N ( λ 1 − λ 2 ) ¯ K N ( λ 1 ) ˇ R N − 1 N ( λ 1 + λ 2 ) ¯ K N ( λ 2 ) = ¯ K N ( λ 2 ) ˇ R N − 1 N ( λ 1 + λ 2 ) ¯ K N ( λ 1 ) ˇ R N − 1 N ( λ 1 − λ 2 ) . (3.3) Consider solutio ns of the Y ang- Baxter and reflection equations in t erms of the gener- ators of the C -t yp e Hec k e algebra: g 0 , g 1 , . . . g N − 1 , g N , ˇ R i i + 1 ( λ ) = e λ g i − e − λ g − 1 i , i ∈ { 1 , . . . , N − 1 } K 1 ( λ ) = e 2 λ g 0 + c − − e − 2 λ g − 1 0 ¯ K N ( λ ) = e 2 λ g N + c + − e − 2 λ g − 1 N (3.4) the b oundary parameters are incorp orated in g 0 , g N . Note also that ˇ R and K ma t r ice s are unita ry i.e. ˇ R 12 ( λ ) ˇ R 12 ( − λ ) ∝ I , K 1 ( λ ) K 1 ( − λ ) ∝ I , ¯ K N ( λ ) ¯ K N ( − λ ) ∝ I . (3.5) Recall that R ij = P ij ˇ R ij , where P is the p erm utation op erator, and the R matrix in general satisfies R t 1 12 ( λ ) M 1 R 12 ( − λ − 2 ρ ) t 2 M − 1 1 ∝ I (3.6) with h M 1 M 2 , R 12 ( λ ) i = 0 , M t = M , (3.7) ρ is the crossing parameter, and for instance in the U q ( d g l N ) case ρ = N 2 . The latter prop ert y (3.6) together with unitarit y and the use of reflection equation are essen tial in pro ving the in tegrabilit y of an op en in tegrable lattice mo del [15]. Note that M is mo dified according to the c hoice of represen tation (see [3, 4]). F or instance M for the U q ( d g l N ) series [33] is giv en by the diagonal N × N matrix ( see also [4]): M = q N − 2 j +1 δ ij . (3.8) Henceforth w e shall fo cus on tensorial represen tations of Hec k e algebras of the type (2.17), although still we do not c ho ose a n y particular r epres en tation – i.e. the f orm of g , g 0 , g N in (2 .1 7) is not sp ecified, is ke pt generic–, so the subsequen t pro positions and pro ofs are quite g ene ric. Also fo r simplicit y w e set π ( g i ) ≡ g i . 5 With the ab ov e general se tting at our disp osal w e ma y no w sho w the follow ing propo - sitions: Prop osition 1 : T ensor r epr esentations of the Murphy eleme n ts asso ciate d to the B -typ e He cke alg ebr a ar e obtaine d fr om the hier ar chy of d o uble-r ow tr ansfe r matric es with one non-trivial b oundary : t ( n ) ( λ ) = tr 0 n M 0 R 0 n ( λ + λ 0 ) R 0 n − 1 ( λ ) . . . R 01 ( λ ) K 0 ( λ ) R 10 ( λ ) . . . R n 0 ( λ − λ 0 ) o 1 ≤ n ≤ N , t ( n ) ( λ ) ∈ End( V ⊗ n ) (3.9) pr ovide d that : tr 0 { M 0 ˇ R n 0 (2 λ 0 ) } ∝ I . (3.10) Pr o of : Notice t he presence o f the inhomogeneity λ 0 at the n th site. In g eneral w e could ha v e set inhomogeneities ev erywhere, but for our purp oses here it is sufficien t to consider only λ 0 . Consider also that λ = λ 0 , with λ 0 b eing a free parameter, t hen the transfer matrix b ecomes t ( n ) ( λ 0 ) = tr 0 { M 0 ˇ R n 0 (2 λ 0 ) } ˇ R n − 1 n ( λ 0 ) . . . ˇ R 12 ( λ 0 ) K 1 ( λ 0 ) ˇ R 12 ( λ 0 ) . . . ˇ R n − 1 n ( λ 0 ) . (3.11) Although ( 3 .10) is a requiremen t in our pro of it is relativ ely easy to show for instance that for the U q ( d g l N ) series [33], (3.10) is v alid. T aking into accoun t (3.10) w e hav e t ( n ) ( λ 0 ) ∝ ˇ R n − 1 n ( λ 0 ) ˇ R n − 2 n − 1 ( λ 0 ) . . . ˇ R 12 ( λ 0 ) K 1 ( λ 0 ) ˇ R 12 ( λ 0 ) . . . ˇ R n − 1 n ( λ 0 ) (3.12) and b earing in mind (3 .4) w e conclude: t ( n ) ( λ 0 ) ∝ ( g n − 1 − e − 2 λ 0 g − 1 n − 1 ) . . . ( g 1 − e − 2 λ 0 g − 1 1 )( g 0 + e − 2 λ 0 c − − e − 4 λ 0 g − 1 0 ) × ( g 1 − e − 2 λ 0 g − 1 1 ) . . . ( g n − 1 − e − 2 λ 0 g − 1 n − 1 ) . (3.13) The op en s pin c hain transfer matrix is ev en tually expressed solely in terms o f the B -ty p e Hec k e algebra g enerato r s. And if w e expand the transfer matrix in p o w ers of e − 2 λ 0 w e end up with: t ( n ) ( λ 0 ) ∝ g n − 1 g n − 2 . . . g 1 g 0 g 1 . . . g n − 2 g n − 1 + . . . − e − 4 nλ 0 g − 1 n − 1 g − 1 n − 2 . . . g − 1 1 g − 1 0 g − 1 1 . . . g − 1 n − 2 g − 1 n − 1 . (3.14) The first and last term of the expansion ab o v e are clearly the Murph y elemen t J ( B ) n − 1 and its opp osite resp ectiv ely . 6 Corollary : T ensor r epr es e ntations of the A -typ e Murphy elements ar e obtaine d fr om the hi e r ar chy o f tr ansfer matric es (3.9) fo r K − ∝ I . Pr o of : the pro of of this statemen t is straightforw a r d; in this case eviden tly g 0 ∝ I . Consider no w the matrices K − ( λ ) and K + = K t ( − λ − iρ ) where K − , K are solutions of the reflection equation (3.2). Consider also the dynamical t yp e solutions of the r efle ction equations (3.2) a nd (3.3) resp ectiv ely: K − 0 ( λ ) = R 0 N ( λ + N δ ) . . . R 02 ( λ + 2 δ ) R 01 ( λ + δ ) K − 0 ( λ ) R 10 ( λ − δ ) . . . R N 0 ( λ − N δ ) K + 0 ( λ ) = R 10 ( ˜ λ + δ ) R 02 ( ˜ λ − 2 δ ) . . . R N 0 ( ˜ λ − N δ ) K + 0 ( λ ) R 0 N ( ˜ λ + N δ ) . . . R 01 ( ˜ λ − δ ) ˜ λ = − λ − iρ. (3.15) Then it can b e show n that: Prop osition 2 : T e n sor r ep r e s e ntations of the Murphy elements ( J ( C ) N − 1 ) ± 1 , ( J ( C ) 0 ) ± 1 asso- ciate d to the C -typ e He cke algebr a ar e obtaine d fr om the fol lowing op en tr ansfer m atric es with two non-trivial b oundaries t ( − ) ( λ ) = tr 0 n M 0 K + 0 ( λ ) K − 0 ( λ ) o , t (+) ( λ ) = tr 0 n M 0 K + 0 ( λ ) K − 0 ( λ ) o t ( ± ) ( λ ) ∈ End( V ⊗ N ) (3.16) pr ovide d that : tr 0 { M 0 K + 0 ( λ ) ˇ R N 0 (2 λ ) } ∝ ¯ K N ( λ ) , tr 0 { K − 0 ( λ − iρ ) M 0 ˇ R 10 ( − 2 λ ) } ∝ K 1 ( λ ) . (3.17) Pr o of : Notice the main difference with the previous case, N the length of the spin c hain is now fixed, whereas previously the length of the chain w as v ariable. The presence of the second non-trivial b oundary fixes someho w the length of the chain and this is already eviden t when defining the C -t yp e Murph y elemen ts. W e start with t he t ( − ) matrix, w e set λ = N δ then the double-row transfer matrix b ecomes : t ( − ) ( N δ ) = tr 0 { M 0 K + 0 ( N δ ) ˇ R N 0 (2 N δ ) } × ˇ R N − 1 N ((2 N − 1) δ ) . . . ˇ R 12 (( N + 1) δ ) K 1 ( N δ ) ˇ R 12 (( N − 1) δ ) . . . ˇ R N − 1 N ( δ ) (3.18) It can b e explicitly c hec k ed, that conditions (3.17) are v alid for instance for the U q ( d g l N ) series. Bearing in mind ( 3.17) w e conclude t ( − ) ( N δ ) ∝ ( g N + c + e − 2 N δ − e − 4 N δ g − 1 N )( g N − 1 − e − 2(2 N − 1) δ g − 1 N − 1 ) . . . ( g 1 − e − 2( N +1) δ g − 1 1 ) 7 × ( g 0 + e − 2 N δ c − − e − 4 N δ g − 1 0 )( g 1 − e − 2( N − 1) δ g − 1 1 ) . . . ( g N − 1 − e − 2 δ g − 1 N − 1 ) . (3.19) In this cas e the double-ro w t r ansfer matrix is express ed in terms of C - t ype Hec k e algebra generators, a nd b y expanding in p o w ers of e − δ w e get: t ( − ) ( N δ ) ∝ g N g N − 1 g N − 2 . . . g 1 g 0 g 1 . . . g N − 2 g N − 1 + (higher order terms) (3.20) the first term of the expansion ab ov e is the Murph y elemen t J ( C ) N − 1 . Similarly for the t (+) matrix we set − λ − iρ = − δ then: t (+) ( − δ + iρ ) = ˇ R 12 ( − 3 δ ) . . . ˇ R N − 1 N ( − ( N + 1) δ ) K N ( δ ) ˇ R N − 1 N (( N − 1) δ ) . . . ˇ R 12 ( δ ) × tr 0 n ˇ R 10 ( − 2 δ ) K 0 ( δ − iρ ) M 0 o . (3.21) Bearing in mind the expressions of K and ˇ R matrices in terms of the Hec k e algebra generators, a nd (3.17) w e may rewrite the t (+) as: t (+) ( − δ + iρ ) = ( − 1) N ( g − 1 1 − e − 6 δ g 1 ) . . . ( g − 1 N − 1 − e − 2( N +1) δ g N − 1 )( g N + c + e − 2 δ − e − 4 δ g − 1 N ) × ( g N − 1 − e − 2( N − 1) δ g − 1 N − 1 ) . . . ( g 1 − e − 2 δ g − 1 1 )( g 0 + c − e − 2 δ − e − 4 δ g − 1 0 ) (3.22) and fina lly by expanding t (+) w e conclude t (+) ( N δ ) ∝ g − 1 1 g − 1 2 . . . g − 1 N − 1 g N g N − 1 . . . g 1 g 0 + (higher order terms) . (3.23) Notice that the zero order term in the expansion (3.23) is the elemen t J ( C ) 0 . The opposite Murph y elemen ts ( J ( C ) N − 1 ) − 1 , ( J ( C ) 0 ) − 1 can b e obtained fro m t ( − ) , t (+) at λ = − N δ and − λ − iρ = δ resp ectiv ely as the zero order terms in the corresp onding expansions. Notice that in this case we are a ble to extract only the J ( C ) N − 1 , J ( C ) 0 elemen ts and their opp osites con trary to the previous case, where all t he Murph y elemen ts were extracted from the transfer matrices (3.9). Some commen ts on this intricate issue will b e presen ted in the discussion section b elo w, how ev er a more detailed inv estigation will b e pursued els ewhere. It is finally clear from the expressions ab o v e that fo r g N ∝ I the results of Prop osition 1 are r eco ve red. Assuming the expansion around λ = δ = 0 one obtains lo cal in tegrals of motion ( w e refer the in terested reader to [3, 4] fo r a more detailed discussion). F o r instance the first deriv a tiv e of (3.1 3 ) with resp ect to λ (at λ = 0) giv es the w ell known Hamiltonians discusse d also e.g. in [3, 4], express ed as a sum o f the Hec k e elemen ts (b etter set g i = e i + q , g 0 = e 0 + Q 0 , g N = e N + Q N ). Higher terms in suc h an expansion pro vide naturally higher Hamilto nians. 8 4 Discuss ion W e ha v e b een able to extract all A and B -t yp e Murph y elemen ts from suitable hierarc hies of op en transfer matrices (3.9). F or the momen t we ha v e b een able to only iden tify the C -type elemen ts J ( C ) N − 1 , J ( C ) 0 and their opp osites from the op en transfer matrices t ( ± ) (3.16). Not e tha t w e mainly fo cused on a big class of g eneric tensor t yp e (spin chain lik e) represen tations (2.17) of Heck e algebras, ho w eve r w e did not c ho ose any sp ecial represen ta t ion i.e. w e did not consider an y particular form fo r the quantities g , g 0 , g N app earing in (2.17), hence our results a re generic. W e assume that a generic c hoice o f an in tegr a ble spin chain with t w o suitable dynami- cal K ± ( n ) reflection matrices in v olving an a ppropriate sequence of inhomogeneities, w ould giv e all the C -type Murph y elemen ts. In other w ords the pro cedure described a b ov e may b e seen as a con v enien t pr escription t hat pro vides relativ ely easily the Murph y elemen ts. The idea ho w ev er is to searc h for a more systematic approac h to tac kle this pro blem. Consider a generic transfer matrix of the fo rm t ( n ) ( λ ) = tr 0 n M 0 K +( n ) 0 ( λ ) K − ( n ) 0 ( λ ) o (4.1) where we define K +( n ) 0 ( λ ) = R 0 n +1 ( ˜ λ − ( n + 1) δ ) R 0 n +2 ( ˜ λ − ( n + 2) δ ) . . . R 0 N ( ˜ λ − N δ ) × K + 0 ( λ ) R N 0 ( ˜ λ + N δ ) . . . R n +10 ( ˜ λ + ( n + 1) δ ) K − ( n ) 0 ( λ ) = R 0 n ( λ + nδ ) R 0 n − 1 ( λ + ( n − 1) δ ) . . . R 01 ( λ + δ ) × K − 0 ( λ ) R 10 ( λ − δ ) . . . R n 0 ( λ − nδ ) (4.2) recall ˜ λ = − λ − iρ . This generic t yp e of tra nsfer matrices will presumably pro vide all the Murph y elemen ts of C -type defined in (2.15). So as in the B -type Hec k e case we b etter deal with an hierarch y of op en transfer matrices of v arying length or more precisely of mo dified ‘dynamics’ as far as t he b oundaries ar e concerned. Of course one has t o tak e sp ecial care when choo sing the suitable inhomogeneit y to expand around as w ell as whe n taking t he trace o v er the auxiliary space, giv en that certain quite complicated identities in v olving dynamical K -matrices a re needed. These how ev er are rather tec hnically inv olv ed issues and will b e left for future in v estigations In the case o f tw o non-trivial b oundary XXZ chain the Murph y elemen ts, could b e expresse d in terms of the c harges in in v o lut io n, and as suc h should b e also expres sed in terms of the ab elian part of the q -Onsager algebra derive d in [35, 36]. More precisely the question ra ised is whether the relev ant Murph y elemen ts can b e ex pressed in terms of the fundamen tal ob jects, t he so-called b oundary no n- lo cal charges (see e.g. [14, 35, 37, 38]), that generate t he q - O nsager algebra [35, 36]. 9 In general for the U q ( d g l N ) series the Murph y elemen ts consist an ab elian algebra. 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