Reflection matrices for the $U_{q}[sl(r|2m)^{(2)}]$ vertex model

UFSCARF-TH-06-11 Reflection matr ices for the U q [ sl ( r | 2 m ) (2) ] v ertex m o del A. Lima-San tos and W. Galleas Universidade F e der al de S˜ ao Carlos Dep artamento de F ´ ısic a C.P. 676, 13565 -905, S˜ ao Carlo s-SP, Br asil Abstract The graded reflection equa tio n is inv estigated for the U q [ sl ( r | 2 m ) (2) ] vertex model. W e hav e found four classes of diagonal solutions and t welv e class e s of no n-diagona l ones. The num ber of free para meters for s ome s olutions dep ends on the num ber of b os onic and fermionic degr ees of freedom cons idered. P A CS num ber s: 05.50 + q, 02.30 .IK , 75 .10.Jm Keywords: Reflection Equatio ns, K-matrices, Sup era lgebras June 2 008 1 In tro duction Tw o-dimensional integrable systems in statistical m echanics and qu antum field theory hav e b een a sub ject of high imp ortance in the last d ecades. In p articular, s y s tems of statistical m ec hanics with short ran ge in teractions are b eliev ed to b e conformally inv arian t at criticalit y [1] with significan t implications in t w o-dimensions [2, 3]. More r ecen tly , a v arie t y of in tegrable structures h as emerged in the context of AdS/CFT corresp ondence [4] in b oth sides of the dualit y . In the pioneer work [5], Minahan and Zarem b o show ed that the planar one-lo op matrix of anomalous dimensions for a class of gauge in v ariant op erators in the N = 4 Sup er Y ang-Mills corresp onds to the Hamiltonian of an in tegrable spin chain, wh ic h w as latter generaliz ed to all gauge in v arian t lo cal operators in [6]. F urthermore, in the string theory sid e of the corresp ondence, it w as sh o wn by Bena, Po lc hinksy and Roiban [7] that the classical str in g theory s igma mod el on AdS 5 × S 5 is also in tegrable with evidences of in tegrabilit y p ersisting at the quantum lev el [8]. In tegrabilit y in classical v ertex mo dels and qu an tum spin c hains is intimate ly conn ected with solutions of the Y ang-Baxter equation [9]. This equatio n pla ys a cen tral role in the Quant um Inv ers e Scattering Metho d wh ic h p ro vides an un ified appr oac h to construct and study physical prop erties of integ rable mo dels [10, 11]. Usually these systems are studied w ith p erio dic b oundary conditions but m ore general b oun daries can also b e included in this framew ork as w ell. Ph ysical prop erties asso ciated with the bulk of the s y s tem are n ot exp ected to b e infl u enced by b oundary cond itions in the th ermo dynamical limit. Nev ertheless, there are surface p rop erties suc h as the interfacia l tension wh ere the b oun d ary conditions are of relev ance. Moreo v er, th e conformal sp ectra of lattice mo dels at cr iticalit y can b e mo difi ed by th e effect of b oundaries [12]. In tegrable mo dels with op en b oundary conditions are also of in terest in the conte xt of AdS/CFT corresp ondence. Besides single trace op erators, the gauge theory also con tain bary- onic op erators, i.e. the so-called gian t gra vitons, corresp ond ing to D-brane excitations in the string coun terpart [13]. Su c h D-branes can app ear in sev eral circumstances [14, 15, 16] and Berenstein and V azquez h a v e sho wn that the one-lo op mixing of n on-BPS gian t gra vitons can b e d escrib ed within the p aradigm of integ rable spin c hains with op en b oundary conditions [16]. In tegrable sys tems w ith op en b oun dary conditions can also b e accommod ated within the framew ork of the Quant um Inv erse Scattering Metho d [17]. In add ition to the solution of the Y ang-Baxter equ ation go v erning th e dy n amics of the bulk there is another fundamental ingredient, the refl ection matrices [18]. These m atrices, also referred as K -matrices, r epresen t th e in teractions at the b oundaries and compatibilit y with the bulk in tegrabilit y requires these matrices to satisfy the so-called reflection equations [17, 18]. 1 A t the moment, th e study of general r egular solutions of the r eflection equations for v ertex mo dels b ased on q -deformed Lie algebras [19, 20] h as b een s uccessfully accomplished. See [21] for instance and references therein. How ev er, this same analysis for v ertex mo dels based on Lie sup er algebras are s till r estricted to diagonal solutions asso ciated w ith th e U q [ sl ( m | n )] [22, 23] and U q [ osp (2 | 2)] symm etries [24] and non-diagonal solutions related to sup er-Y angians osp ( m | n ) [25] and sl ( m | n ) [26, 27]. The aim of this pap er is to start to b ridge this gap by presen ting the most general set of solutions of th e reflection equation for the U q [ sl ( r | 2 m ) (2) ] v ertex mo del. This p ap er is organized as follo ws. In the next section w e present the R -matrix of the U q [ sl ( r | 2 m ) (2) ] verte x m o del in terms of standard W eyl m atrices. This step pav es the w a y f or the analysis of the corresp ond ing r eflection equations and in the section 3 w e pr esen t what we hop e to b e the most general set of K -matrices. Concluding remarks are discussed in the section 4, and in the app endices A and B we present sp ecial solutions asso ciated with the U q [ sl (1 | 2) (2) ] and U q [ sl (2 | 2) (2) ] cases r esp ectiv ely . 2 The U q [ sl ( r | 2 m ) (2) ] v ertex mo del Classical v ertex mo dels of statistical mechanics are no w ada ys we ll kno wn to play a fundamen tal role in the theory of tw o-dimensional int egrable systems [9]. In this sense, it tu r ns out that a R -matrix satisfying the Y ang-Baxter equation giv es r ise to the Boltzmann we igh ts of an exactly solv able v ertex m o del. The Y ang-Baxter equation consist of an op erator relation for a complex v alued matrix R : C → End ( V ⊗ V ) reading R 12 ( x ) R 13 ( xy ) R 23 ( y ) = R 23 ( y ) R 13 ( xy ) R 12 ( y ) , (1) where R ij ( x ) refers to the R -matrix acting non-trivially in the i th and j th spaces of the tensor pro du ct V ⊗ V ⊗ V and the complex v ariable x denotes the sp ectral p arameter. Here V is a fi nite dimensional Z 2 graded linear space and the tensor pro d ucts app earing in the ab ov e definitions should b e un dersto o d in the graded sense. F or instance, w e ha v e [ A ⊗ B ] αγ β δ = A α β B γ δ ( − 1) ( p α + p β ) p γ for generic matrices A and B . Th e Grassmann parities p α assume v al ues on the group Z 2 and enable us to distinguish b osonic and fermionic degrees of freedom. More sp ecificall y , th e α th degree of freedom is distinguished by the Grassmann parit y p α =  0 for α b osonic 1 for α fermionic . (2) An imp ortan t class of solutions of the Y ang-Baxter equation (1 ) is d enominated trigonometric R -matrices conta ining an additional parameter q b esides the sp ectral parameter. Usu ally suc h R - matrices ha v e their r o ots in the U q [ G ] quan tum group framew ork, wh ic h p ermit us to asso ciate 2 a fu ndamenta l trigonomet ric R -matrix to eac h Lie algebra or Lie sup eralgebra G [19, 20, 28]. In particular, explicit R -matrices were exhibited in [29, 30] f or a v ariet y of qu an tum sup eralgebras in terms of standard W eyl matrices, pr o viding in this w a y a s uitable basis for the analysis of the corresp ondin g reflection equation. The U q [ sl ( r | 2 m ) (2) ] inv arian t R -matrices are give n by R ( x ) = N X α =1 α 6 = α ′ ( − 1) p α a α ( x ) ˆ e αα ⊗ ˆ e αα + b ( x ) N X α,β =1 α 6 = β ,α 6 = β ′ ˆ e αα ⊗ ˆ e β β + ¯ c ( x ) X α,β =1 α<β ,α 6 = β ′ ( − 1) p α p β ˆ e β α ⊗ ˆ e αβ + c ( x ) N X α,β =1 α>β ,α 6 = β ′ ( − 1) p α p β ˆ e β α ⊗ ˆ e αβ + N X α,β =1 ( − 1) p α d α,β ( x ) ˆ e αβ ⊗ ˆ e α ′ β ′ (3) where N = r + 2 m is the dimens ion of the grad ed space with r b osonic and 2 m fermionic degrees of freedom. Here α ′ = N + 1 − α corresp onds to the conju gated in dex of α and ˆ e αβ refers to a usual N × N W eyl m atrix with only one n on-n ull entry w ith v alue 1 at the row α and column β . In [31] it was demons tr ated that the u s e of an appropriate grading stru cture pla ys a d ecisive role in the inv estig ation of the thermo dynamic limit and finite size prop erties of integrable quantum spin chains based on sup eralgebras. In wh at follo ws we sh all adopt the grading structur e p α = ( 1 for α = 1 , . . . , m and α = r + m + 1 , . . . , r + 2 m 0 for α = m + 1 , . . . , r + m , (4) and the corresp onding Boltzmann weigh ts a α ( x ), b ( x ), c ( x ), ¯ c ( x ) and d αβ ( x ) are then giv en b y a α ( x ) = ( x − ζ )( x (1 − p α ) − q 2 x p α ) b ( x ) = q ( x − 1)( x − ζ ) c ( λ ) = (1 − q 2 )( x − ζ ) ¯ c ( x ) = x (1 − q 2 )( x − ζ ) d α,β ( x ) =                  q ( x − 1)( x − ζ ) + x ( q 2 − 1)( ζ − 1) α = β = β ′ ( x − 1)  ( x − ζ )( − 1) p α q 2 p α + x ( q 2 − 1)  α = β 6 = β ′ ( q 2 − 1)  ζ ( x − 1) ǫ α ǫ β q t α − t β − δ α,β ′ ( x − ζ )  α < β ( q 2 − 1) x  ( x − 1) ǫ α ǫ β q t α − t β − δ α,β ′ ( x − ζ )  α > β (5) where ζ = − q r − 2 m . Th e r emaining v ariables ǫ α and t α dep end strongly on the grading structure considered and they are d etermined by the relations ǫ α =            ( − 1) − p α 2 1 ≤ α < N + 1 2 1 α = N + 1 2 ( − 1) p α 2 N + 1 2 < α ≤ N (6) 3 t α =                        α +    1 2 − p α + 2 X α ≤ β < N +1 2 p β    1 ≤ α < N +1 2 N +1 2 α = N +1 2 α −    1 2 − p α + 2 X N +1 2 <β ≤ α p β    N +1 2 < α ≤ N . (7) The R -matrix (3)-(7) satisfies imp ortant symmetry relations, b esides the standard p rop erties of regularit y and unitarit y , namely PT-Symmetry: R 21 ( x ) = R st 1 st 2 12 ( x ) Crossing Symmetry: R 12 ( x ) = ρ ( x ) ρ ( x − 1 η − 1 ) V 1 R st 2 12 ( x − 1 η − 1 ) V − 1 1 , where the symbol s t k stands for the su p ertransp osition op eration in the space with ind ex k . In its turn ρ ( x ) is an app ropriate norm alizati on function giv en by ρ ( x ) = q ( x − 1)( x − ζ ) and the crossing parameter is η = ζ − 1 . A t th is stage it is con v enien t to consider the U q [ sl (2 n | 2 m ) (2) ] and the U q [ sl (2 n + 1 | 2 m ) (2) ] vertex mo dels separately and their corresp onding crossing m atrix V is an an ti-diagonal matrix with the follo wing non-null en tries V αα ′ , • U q  sl (2 n | 2 m ) (2)  : V αα ′ =                      ( − 1) 1 − p α 2 q 0 B @ α − 1+ p 1 − p α − 2 α − 1 X β =1 p β 1 C A 1 ≤ α ≤ N 2 ( − 1) 1+ p α 2 q 0 B B B B B @ α − 2 − p 1 − p α − 2 α − 1 X β =2 6 = N 2 +1 p β 1 C C C C C A N 2 + 1 ≤ α ≤ N (8) • U q  sl (2 n + 1 | 2 m ) (2)  : V αα ′ =                                ( − 1) 1 − p α 2 q 0 B @ α − 1+ p 1 − p α − 2 α − 1 X β =1 p β 1 C A 1 ≤ α ≤ N − 1 2 ( − 1) 1 − p α 2 q 0 B B B @ N 2 − 1 − p 1 − p α − 2 N − 1 2 X β =2 p β 1 C C C A α = N +1 2 ( − 1) 1+ p α 2 q 0 B @ α − 2 − p 1 − p α − 2 α − 1 X β =2 p β 1 C A N +3 2 + 1 ≤ α ≤ N . (9) In the next section we shall inv esti gate the p ossible regular solutions of th e graded reflection equation f or the U q [ sl ( r | 2 m ) (2) ] vertex mo del. 4 3 The U q [ sl ( r | 2 m ) (2) ] reflection matrices The construction of int egrable mod els with op en b ound aries w as largely impulsed b y S kly anin’s pioneer w ork [17]. In Sk lyanin’s app roac h the construction of such mo dels are based on solutions of th e s o-called reflection equations [18, 17] for a giv en int egrable bulk system. Th e reflection equations determine the b oun d ary conditions compatible with the bu lk int egrabilit y and it r eads R 21 ( x/y ) K − 2 ( x ) R 12 ( xy ) K − 1 ( y ) = K − 1 ( y ) R 21 ( xy ) K − 2 ( x ) R 12 ( x/y ) , (10) where the tens or p ro ducts app earing in (10) should b e und ersto o d in the graded sense. The matrix K − ( x ) describ es th e reflection at one of the ends of an op en chain while a similar equation should also hold for a matrix K + ( x ) describing the reflection at the opp osite b oundary . As discus sed in the previous section, the U q [ sl ( r | 2 m ) (2) ] R -matrix satisfies imp ortant symmetry relat ions suc h as the PT-symm etry and crossing symmetry . When these pr op erties are fulfilled one can follo w the sc heme devised in [23, 32] and the matrix K − ( x ) is obtained b y solving th e Eq. (10) while the matrix K + ( x ) can b e obtained from the isomorphism K − ( x ) 7→ K + ( x ) st = K − ( x − 1 η − 1 ) V st V . The pur p ose of this work is to inv estigate the general families of regular solutions of the graded reflection equation (10). Regular solutions m ean that the K -matrices hav e the general form K − ( x ) = N X α,β =1 k α,β ( x ) ˆ e αβ , (11) suc h that the condition k α,β (1) = δ αβ holds for all matrix elements. The direct su bstitution of (11) and the U q [ sl ( r | 2 m ) (2) ] R -matrix (3)-(7 ) in the graded reflec- tion equation (10), lea v e u s with a system of N 4 functional equations for the en tries k α,β ( x ). In order to solv e these equations w e sh all make use of the deriv ativ e metho d. T h us, by differentia ting the equ ation (10) with r esp ect to y and setting y = 1, w e obtain a set of algebraic equations for the matrix elemen ts k α,β ( x ). Although we ob tain a large num b er of equations only a few of them are actually indep enden t and a direct insp ection of th ose equ ations, in the lines describ ed in [21], allo ws us to find the b ranc hes of r egular solutions. In what follo ws w e sh all present our find ings for the regular solutions of the reflection equation asso ciated with the U q [ sl ( r | 2 m ) (2) ] v ertex mo d el. W e ha v e obtained four families of diagonal solutions and t we lv e families of non-diagonal ones. The sp ecial solutions asso ciated w ith the cases U q [ sl (1 | 2) (2) ] and U q [ sl (2 | 2) (2) ] are presented in the app end ices A and B resp ectiv ely . 5 3.1 Diagonal solutions The diagonal solutions of the graded reflection equation (10 ) is charac terized by a K -matrix of the form K − ( x ) = N X α =1 k α,α ( x ) ˆ e αα . (12) W e ha v e foun d four families of diagonal K -matrices for the U q [ sl ( r | 2 m ) (2) ] ve rtex mo del that w e shall refer as solutions of t yp e D 1 , D 2 , D 3 and D 4 . • Solution D 1 : Solution with only one free parameter Ω. k α,α ( x ) =            Ω( x − 1 − 1) + 2 Ω( x − 1) + 2 α = 1 1 α = 2 , . . . , N − 1 x Ω(1 − xq 2 ζ ) − 2 Ω( x − q 2 ζ ) − 2 x α = N . (13) • Solution D 2 : F amily formed b y m solutions without fr ee parameters and c haracterized by the lab el p assuming discrete v alues in th e in terv al 2 ≤ p ≤ m + 1. k α,α ( x ) =      1 α = 1 , . . . , p − 1 x + κq 2 p − 3 √ ζ x − 1 + κq 2 p − 3 √ ζ α = p, . . . , N + 1 − p x 2 α = N + 2 − p, . . . , N . (14) Here and in w hat follo ws, κ is a d iscr ete parameter assu ming the v alues ± 1. • Solution D 3 : Solutions v alid for r ≥ 4 which d o es not con tain free p arameters. The discrete lab el p can assume v alues in th e inte rv al m + 2 ≤ p < N +1 2 . k α,α ( x ) =      1 α = 1 , . . . , p − 1 x + κq 4 m +1 − 2 p √ ζ x − 1 + κq 4 m +1 − 2 p √ ζ α = p, . . . , N + 1 − p x 2 α = N + 2 − p, . . . , N . (15) • Solution D 4 : Class of solution v alid only for r = 2 n ( n ≥ 1) with one free parameter Ω. k α,α ( x ) =                1 α = 1 , . . . , n + m − 1 Ω( x − 1) + 2 Ω( x − 1 − 1) + 2 α = n + m x  Ω( x − q 2 ζ − 1 ) − 2 x Ω(1 − xq 2 ζ − 1 ) − 2  α = n + m + 1 x 2 α = n + m + 2 , . . . , N . (16) 6 3.2 Non-Diagonal Solutions Here we shall f o cus on the n on-diagonal solutions of the graded reflection equ ation (10). W e hav e found t w elv e classes of non-diagonal solutions that we r efer in wh at follo ws as solutions of type N 1 to typ e N 12 . • Solution N 1 : The solution of typ e N 1 is v alid only for the U q [ sl ( r | 2) (2) ] m o del and th e K -matrix has the follo wing blo c k structure K − ( x ) =     k 1 , 1 ( x ) O 1 × r k 1 ,N ( x ) O r × 1 K 1 ( x ) O r × 1 k N , 1 ( x ) O 1 × r k N ,N ( x )     , (17) where O a × b is a a × b n ull matrix and K 1 ( x ) = x 2 − ζ 1 − ζ I r × r . (18) Here and in what follo ws I a × a denotes a a × a identit y matrix and the r emaining non-null ent ries are giv en b y k 1 , 1 ( x ) = 1 k 1 ,N ( x ) = 1 2 Ω( x 2 − 1) k N , 1 ( x ) = 2 ζ (1 − ζ ) 2 ( x 2 − 1) Ω k N ,N ( x ) = x 2 , (19) where Ω is a fr ee p arameter. • Solution N 2 : The U q [ sl ( r | 4) (2) ] vertex m o del admits the solution N 2 whose corresp onding K -matrix has the follo wing structure K − ( x ) =         k 1 , 1 ( x ) k 1 , 2 ( x ) k 2 , 1 ( x ) k 2 , 2 ( x ) O 2 × r k 1 ,N − 1 ( x ) k 1 ,N ( x ) k 2 ,N − 1 ( x ) k 2 ,N ( x ) O r × 2 K 2 ( x ) O r × 2 k N − 1 , 1 ( x ) k N − 1 , 2 ( x ) k N , 1 ( x ) k N , 2 ( x ) O 2 × r k N − 1 ,N − 1 ( x ) k N − 1 ,N ( x ) k N ,N − 1 ( x ) k N ,N ( x )         , (20) where K 2 ( x ) = k 3 , 3 ( x ) I r × r . The non-diagonal ent ries can b e written as k 1 , 2 ( x ) = Ω 1 , 2 G 1 ( x ) k 2 , 1 ( x ) = Ω 2 , 1 G 1 ( x ) k 1 ,N − 1 ( x ) = Ω 1 ,N − 1 G 1 ( x ) k N − 1 , 1 ( x ) = q 2 ζ Ω 1 , 2 Ω 2 , 1 Ω 1 ,N − 1 G 1 ( x ) k 2 ,N − 1 ( x ) = − Ω 2 , 1 Ω 1 ,N Ω 1 , 2 G 1 ( x ) H ( x ) k N − 1 , 2 ( x ) = − q 2 ζ Ω 1 , 2 Ω 2 , 1 Ω 1 ,N Ω 2 1 ,N − 1 G 1 ( x ) H ( x ) 7 k 2 ,N ( x ) = − Γ r Ω 1 ,N Ω 1 , 2 xG 2 ( x ) k N , 2 ( x ) = − q 2 ζ Γ r Ω 2 , 1 Ω 1 ,N Ω 2 1 ,N − 1 xG 2 ( x ) k N − 1 ,N ( x ) = − q 2 ζ Γ r Ω 1 ,N Ω 1 ,N − 1 xG 2 ( x ) k N ,N − 1 ( x ) = − q 2 ζ Γ r Ω 2 , 1 Ω 1 ,N Ω 1 , 2 Ω 1 ,N − 1 xG 2 ( x ) k N , 1 ( x ) = q 2 ζ Ω 2 2 , 1 Ω 1 ,N Ω 2 1 ,N − 1 G 1 ( x ) H ( x ) k 1 ,N ( x ) = Ω 1 ,N ( x − 1) , (21) and the auxiliary fu nctions are given by G 1 ( x ) =  − q 2 ζ Γ r Ω 1 ,N Ω 1 , 2 Ω 1 ,N − 1 ( x − 1) + x − q 2 ζ  x − 1 x 2 − q 2 ζ , G 2 ( x ) =  − Ω 1 , 2 Ω 1 ,N − 1 Γ r Ω 1 ,N ( x − 1) + x − q 2 ζ  x − 1 x 2 − q 2 ζ , (22) H ( x ) = Ω 1 , 2 Ω 1 ,N − 1 ( x 2 − q 2 ζ ) [ − q 2 ζ Γ r Ω 1 ,N ( x − 1) + Ω 1 , 2 Ω 1 ,N − 1 ( x − q 2 ζ )] and Γ r = Ω 2 , 1 Ω 1 ,N Ω 1 ,N − 1 − 2 1 − q 2 ζ . (23) With resp ect to th e diagonal matrix elements, we hav e the follo wing expr essions k 1 , 1 ( x ) =  q 2 ζ Ω 1 ,N Γ 2 r Ω 1 , 2 Ω 1 ,N − 1 (1 − q 2 ζ x ) − ( x − q 2 ζ )  Ω 1 , 2 Ω 1 ,N − 1 Ω 1 ,N − Ω 2 , 1 Ω 1 ,N Ω 1 ,N − 1 (1 + q 2 ζ )  + 2(1 + q 2 ζ ) 2 x − 4 q 2 ζ ( x 2 + 1) (1 − q 2 ζ )( x − 1)  ( x − 1) ( x + 1)( x 2 − q 2 ζ ) k 2 , 2 ( x ) = k 1 , 1 ( x ) + Ω 2 , 2 G 1 ( x ) k 3 , 3 ( x ) = k 1 , 1 ( x ) + Ω 3 , 3 G 1 ( x ) + ∆ 1 ( x ) k N − 1 ,N − 1 ( x ) = k 3 , 3 ( x ) + (Ω N − 1 ,N − 1 − Ω 3 , 3 ) xG 2 ( x ) + ∆ 2 ( x ) k N ,N ( x ) = k N − 1 ,N − 1 ( x ) + (Ω N ,N − Ω N − 1 ,N − 1 ) x G 2 ( x ) , (24) where ∆ 1 ( x ) = Ω 2 , 1 Ω 1 ,N − 1  x + q 2 ζ Γ r Ω 1 ,N Ω 1 , 2 Ω 1 ,N − 1  ( x − 1) 2 ( x 2 − q 2 ζ ) Ω 1 ,N ∆ 2 ( x ) = − q 2 ζ Ω 1 , 2 Ω 1 ,N − 1 Γ r Ω 1 ,N ∆ 1 ( x ) . (25) The diagonal en tries (24) d ep end on the v ariables Ω α,α whic h are relat ed to the free parameters Ω 1 , 2 , Ω 2 , 1 , Ω 1 ,N − 1 and Ω 1 ,N through the exp r essions Ω 2 , 2 = Ω 1 , 2 Ω 1 ,N − 1 Ω 1 ,N − Γ r Ω N − 1 ,N − 1 = 2 + Ω 1 , 2 Ω 1 ,N − 1 Ω 1 ,N − q 2 ζ Γ r Ω 3 , 3 = Ω 1 , 2 Ω 1 ,N − 1 Ω 1 ,N + 2 1 − q 2 ζ Ω N ,N = 2 + Ω 1 , 2 Ω 1 ,N − 1 Ω 1 ,N − q 2 ζ Γ 2 r Ω 1 ,N Ω 1 , 2 Ω 1 ,N − 1 . (26) 8 • Solution N 3 : This class of solution is v ali d for the U q [ sl ( r | 2 m ) (2) ] ve rtex mo del with m ≥ 3 and the corresp ondin g K -matrix p ossess the follo wing general form K − ( x ) =               k 1 , 1 ( x ) · · · k 1 ,m ( x ) . . . . . . . . . k m, 1 ( x ) · · · k m,m ( x ) O m × r k 1 ,r + m +1 ( x ) · · · k 1 ,N ( x ) . . . . . . . . . k m,r + m +1 ( x ) · · · k m,N ( x ) O r × m K 3 ( x ) O r × m k r + m +1 , 1 ( x ) · · · k r + m +1 ,m ( x ) . . . . . . . . . k N , 1 ( x ) · · · k N ,m ( x ) O m × r k r + m +1 ,r + m +1 ( x ) . . . k r + m +1 ,N ( x ) . . . . . . . . . k N ,r + m +1 ( x ) . . . k N ,N ( x )               , (27) where K 3 ( x ) is a d iagonal m atrix giv en by K 3 ( x ) = k m +1 ,m +1 ( x ) I r × r . (28) With resp ect to the elemen ts of the last column, we h a v e the follo wing expression k α,N ( x ) = − κ √ ζ ǫ α ǫ 1 q t α − t 1 Ω 1 ,α ′ xG ( x ) α = 2 , . . . , m and (29) α = r + m + 1 , . . . , N − 1 , where G ( x ) = x − 1. In their tu rn the entries of the first column are mainly give n by k α, 1 ( x ) = ǫ α ǫ 2 q t α − t 2 Ω 2 , 1 Ω 1 ,α ′ Ω 1 ,N − 1 G ( x ) α = 3 , . . . , m and ( 30) α = r + m + 1 , . . . , N − 1 . In the last row we hav e k N ,α ( x ) = − κ √ ζ ǫ N ǫ 2 q t N − t 2 Ω 2 , 1 Ω 1 ,α Ω 1 ,N − 1 xG ( x ) α = 2 , . . . , m and (31) α = r + m + 1 , . . . , N − 1 , while the elements of th e firs t row are k 1 ,α ( x ) = Ω 1 ,α G ( x ) for α = 2 , . . . , m and α = r + m + 1 , . . . , N − 1. Concerning the elemen ts of the secondary diagonal, they are giv en by k α,α ′ ( x ) = q t 1 − t α ′ ǫ 1 ǫ α ′ (1 − κq √ ζ ) 2 ( q + 1) 2 Ω 2 1 ,α ′ Ω 1 ,N G ( x ) H f ( x ) α = 2 , . . . , m , α 6 = α ′ α = r + m + 1 , . . . , N − 1 , (32) while the remaining entries k 1 ,N ( x ) and k N , 1 ( x ) are determined by the f ollo wing expressions k 1 ,N ( x ) = Ω 1 ,N G ( x ) H f ( x ) k N , 1 ( x ) = ǫ N − 1 ǫ 2 q t N − 1 − t 2 Ω 1 ,N Ω 2 2 , 1 Ω 2 1 ,N − 1 G ( x ) H f ( x ) (33) 9 with H f ( x ) = x − κq √ ζ 1 − κq √ ζ . The remaining matrix elemen ts k α,β ( x ) with α 6 = β are then k α,β ( x ) =                              − κ √ ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q + 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N G ( x ) α < β ′ 2 ≤ α, β ≤ N − 1 1 ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q + 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N xG ( x ) α > β ′ 2 ≤ α, β ≤ N − 1 2( − 1) m q m − 2 (1 − κq √ ζ ) (1 + κ √ ζ )( κq m − 1 √ ζ + ( − 1) m )( κq m √ ζ + ( − 1) m ) Ω 1 ,N − 1 Ω 1 ,N G ( x ) α = 2 , β = 1 − 2 ζ q m − 1 ( q + 1) 2 (1 − κq √ ζ )(1 + κ √ ζ )( κq m − 1 √ ζ + ( − 1) m )( κq m √ ζ + ( − 1) m ) Ω 1 ,N Ω 1 ,N − m +1 G ( x ) α = 1 , β = m , (34) and the p arameters Ω 1 ,α are constrained by th e relation Ω 1 ,α = − Ω 1 ,α +1 Ω 1 ,N − α Ω 1 ,N +1 − α α = 2 , . . . , m − 1 . (35) With regard to the d iagonal m atrix elemen ts, th ey are given by k α,α ( x ) =                             2 x x 2 − 1 − Ω m +1 ,m +1 x + 1  G ( x ) + ∆ 1 ( x ) + ∆ 2 ( x ) x 2 − 1 α = 1 k 1 , 1 ( x ) + Ω α,α G ( x ) 2 ≤ α ≤ m k 1 , 1 ( x ) + Ω m +1 ,m +1 G ( x ) + ∆ 1 ( x ) α = m + 1 k m +1 ,m +1 ( x ) + (Ω r + m +1 ,r + m +1 − Ω m +1 ,m +1 ) x G ( x ) + ∆ 2 ( x ) α = r + m + 1 k α − 1 ,α − 1 ( x ) + (Ω α,α − Ω α − 1 ,α − 1 ) xG ( x ) r + m + 2 ≤ α ≤ N − 1 x 2 k 1 , 1 ( x ) α = N (36) where ∆ 1 ( x ) = 2( x − 1) 2 (1 + κ √ ζ )( κq m − 1 √ ζ + ( − 1) m )( κq m √ ζ + ( − 1) m ) , ∆ 2 ( x ) = κq r − 1 p ζ ∆ 1 ( x ) . (37) In their tu rn the d iagonal parameters Ω α,α are fixed by the relations Ω α,α =                      Λ m α − 2 X k =0 ( − 1 q ) k 2 ≤ α ≤ m Λ m  q q + 1 + ( − 1) m q m − 2 1 ( q + 1) 2 1 + κ √ ζ κ √ ζ  α = m + 1 2 − κq p ζ Λ m N − α − 1 X k =0 ( − q ) k r + m + 1 ≤ α ≤ N − 1 (38) with Λ m = 2( − 1) m q m − 2 (1 + q ) 2 κ √ ζ (1 + κ √ ζ )( κq m − 1 √ ζ + ( − 1) m )( κq m √ ζ + ( − 1) m ) . (39) The class of solution N 3 has a total amount of m free parameters namely Ω 1 ,r + m +1 , . . . , Ω 1 ,N . 10 • Solution N 4 : This family of solutions is v alid only for the U q [ sl (2 | 2 m ) (2) ] v ertex mo del and the corresp onding K -matrix h as the follo wing blo ck stru ctur e K − ( x ) =       k 1 , 1 ( x ) I m × m O m × 2 O m × m O 2 × m k m +1 ,m +1 ( x ) k m +1 ,m +2 ( x ) k m +2 ,m +1 ( x ) k m +2 ,m +2 ( x ) O 2 × m O m × m O m × 2 k N ,N ( x ) I m × m       . (40) The non-null entries are giv en by k 1 , 1 ( x ) = 1 k N ,N ( x ) = x 2 k m +1 ,m +1 ( x ) = x 2 (1 − ζ ) x 2 − ζ k m +1 ,m +2 ( x ) = Ω 2 x ( x 2 − 1)(1 − ζ ) x 2 − ζ k m +2 ,m +1 ( x ) = 2 Ω x ( x 2 − 1) ζ (1 − ζ )( x 2 − ζ ) k m +2 ,m +2 ( x ) = x 2 (1 − ζ ) x 2 − ζ (41) where Ω is a free parameter. W e remark here that this solution for m = 1 consist of a particular case of th e solution giv en in th e app endix B for the U q [ sl (2 | 2) (2) ] vertex mo del. • Solution N 5 : The family N 5 is accepta ble b y the ve rtex mo del U q [ sl (3 | 2 m ) (2) ] and it is c haracterized b y a K - matrix of the form K − ( x ) =          k 1 , 1 ( x ) I m × m O m × 3 O m × m O 3 × m k m +1 ,m +1 ( x ) k m +1 ,m +2 ( x ) k m +1 ,m +3 ( x ) k m +2 ,m +1 ( x ) k m +2 ,m +2 ( x ) k m +2 ,m +3 ( x ) k m +3 ,m +1 ( x ) k m +3 ,m +2 ( x ) k m +3 ,m +3 ( x ) O 3 × m O m × m O m × 3 k N ,N ( x ) I m × m          . (42) The non-diagonal m atrix elemen ts are give n by the exp ressions k m +1 ,m +2 ( x ) = Ω m +1 ,m +2 G ( x ) k m +2 ,m +1 ( x ) = Ω m +2 ,m +1 G ( x ) k m +3 ,m +2 ( x ) = Ω m +3 ,m +2 xG ( x ) k m +2 ,m +3 ( x ) = Ω m +2 ,m +3 xG ( x ) (43) k m +1 ,m +3 ( x ) = Ω m +1 ,m +3 G ( x ) H b ( x ) k m +3 ,m +1 ( x ) = Ω m +3 ,m +1 G ( x ) H b ( x ) where H b ( x ) = q x + κ √ ζ q + κ √ ζ and G ( x ) = x − 1. In their turn the ab o v e parameters Ω α,β are constrained b y th e relations Ω m +2 ,m +3 = − iκq m − 1 Ω m +1 ,m +2 Ω m +3 ,m +2 = − iκq m − 1 Ω m +2 ,m +1 Ω m +3 ,m +1 = Ω m +1 ,m +3  Ω m +2 ,m +1 Ω m +1 ,m +2  2 (44) 11 and Ω m +2 ,m +1 = iκq m − 1 Ω m +1 ,m +2 Ω m +1 ,m +3 " √ q ( q + 1) Ω 2 m +1 ,m +2 Ω m +1 ,m +3 − 2 ( q m − 1 / 2 + iκ )( q m − 3 / 2 − iκ ) # . (45) The diagonal en tries are then giv en by k 1 , 1 ( x ) = " 2 x 2 − 1 + iκ q m − 1 ( xq 2( m − 1) − 1) x ( x + 1) Ω m +2 ,m +1 Ω m +1 ,m +3 Ω m +1 ,m +2 # G ( x ) H b ( x ) + iκq m − 1 Ω 2 m +1 ,m +2 Ω m +1 ,m +3 G ( x ) x + 1 k m +1 ,m +1 ( x ) = " 2 x 2 − 1 + iκ q m − 1 ( q 2( m − 1) + 1) ( x + 1) Ω m +2 ,m +1 Ω m +1 ,m +3 Ω m +1 ,m +2 # G ( x ) H b ( x ) + iκq m − 1 Ω 2 m +1 ,m +2 Ω m +1 ,m +3 G ( x ) x + 1 (46) k m +2 ,m +2 ( x ) = k m +1 ,m +1 ( x ) + (Ω m +2 ,m +2 − Ω m +1 ,m +1 ) G ( x ) + ∆( x ) k m +3 ,m +3 ( x ) = k m +2 ,m +2 ( x ) + (Ω m +3 ,m +3 − Ω m +2 ,m +2 ) xG ( x ) + κ p ζ ∆( x ) k N ,N ( x ) = x 2 k 1 , 1 ( x ) where ∆( x ) = − q m − 1 ( x − 1) 2 ( q m − 1 / 2 + iκ ) Ω m +2 ,m +1 Ω m +1 ,m +3 Ω m +1 ,m +2 , (47) and the p arameters Ω m +1 ,m +1 , Ω m +2 ,m +2 and Ω m +3 ,m +3 are fixed by the relations Ω m +1 ,m +1 = iκ q m − 1 Ω m +2 ,m +1 Ω m +1 ,m +3 Ω m +1 ,m +2 Ω m +2 ,m +2 = 2 iκ q m − 1 / 2 + iκ − iκq m − 1 Ω 2 m +1 ,m +2 Ω m +1 ,m +3 (48) Ω m +3 ,m +3 = 2 − iκq m − 1 Ω m +2 ,m +1 Ω m +1 ,m +3 Ω m +1 ,m +2 . The solution N 5 p ossess t w o free parameters namely Ω m +1 ,m +2 and Ω m +1 ,m +3 . • Solution N 6 : The solution N 6 is admitted f or the U q [ sl (4 | 2 m ) (2) ] mo dels with the f ollo wing K -matrix K − ( x ) =             k 1 , 1 ( x ) I m × m O m × 4 O m × m O 4 × m k m +1 ,m +1 ( x ) k m +1 ,m +2 ( x ) k m +1 ,m +3 ( x ) k m +1 ,m +4 ( x ) k m +2 ,m +1 ( x ) k m +2 ,m +2 ( x ) k m +2 ,m +3 ( x ) k m +2 ,m +4 ( x ) k m +3 ,m +1 ( x ) k m +3 ,m +2 ( x ) k m +3 ,m +3 ( x ) k m +3 ,m +4 ( x ) k m +4 ,m +1 ( x ) k m +4 ,m +2 ( x ) k m +4 ,m +3 ( x ) k m +4 ,m +4 ( x ) O 4 × m O m × m O m × 4 k N ,N ( x ) I m × m             . (49) 12 The non-diagonal elements are all group ed in the 4 × 4 cen tral blo ck matrix. With resp ect to this cen tral blo ck, the entries of the secondary d iagonal are given by k m +1 ,m +4 ( x ) = ( x − 1)Ω m +1 ,m +4 k m +2 ,m +3 ( x ) = − Ω m +2 ,m +1 Ω m +1 ,m +2 Ω m +1 ,m +4 ( x − 1) k m +3 ,m +2 ( x ) = − q 2 ζ Ω m +1 ,m +2 Γ 2 m Ω m +2 ,m +1 Ω m +1 ,m +4 ( x − 1) (50) k m +4 ,m +1 ( x ) = q 2 ζ Γ 2 m Ω m +1 ,m +4 ( x − 1) , and the remainin g n on-diagonal element s can b e w r itten as k m +1 ,m +2 ( x ) = Ω m +1 ,m +2 G 1 ( x ) k m +2 ,m +1 ( x ) = Ω m +2 ,m +1 G 1 ( x ) k m +1 ,m +3 ( x ) = Ω m +1 ,m +3 G 2 ( x ) k m +3 ,m +1 ( x ) = q 2 ζ Ω m +1 ,m +2 Ω m +1 ,m +3 Γ 2 m Ω m +2 ,m +1 Ω 2 m +1 ,m +4 G 2 ( x ) k m +2 ,m +4 ( x ) = − Ω m +2 ,m +1 Ω m +1 ,m +4 Γ m xG 1 ( x ) k m +4 ,m +2 ( x ) = − q 2 ζ Ω m +1 ,m +2 Γ m Ω m +1 ,m +4 xG 1 ( x ) k m +3 ,m +4 ( x ) = − q 2 ζ Ω m +1 ,m +2 Ω m +1 ,m +3 Γ m Ω m +1 ,m +4 Ω m +2 ,m +1 xG 2 ( x ) k m +4 ,m +3 ( x ) = − q 2 ζ Ω m +1 ,m +3 Γ m Ω m +1 ,m +4 xG 2 ( x ) , (51) where G 1 ( x ) =  ζ − q 2 x q 2 ( x − 1) + Ω m +1 ,m +3 Γ m Ω m +2 ,m +1 Ω m +1 ,m +4  q 2 ( x − 1) 2 ζ − q 2 x 2 , G 2 ( x ) =  ζ − q 2 x x − 1 + ζ Ω m +2 ,m +1 Ω m +1 ,m +4 Ω m +1 ,m +3 Γ m  ( x − 1) 2 ζ − q 2 x 2 (52) and Γ m = Ω m +1 ,m +2 Ω m +1 ,m +3 Ω m +1 ,m +4 + 2 ζ q 2 − ζ . (53) In their tu rn the d iagonal en tries are give n b y the follo wing expressions k 1 , 1 ( x ) =  ζ − q 2 x ( x + 1)( ζ − q 2 x 2 )  Ω m +1 ,m +2 Ω m +2 ,m +1 Γ m + Ω m +1 ,m +2 Ω m +1 ,m +3 ζ Ω m +1 ,m +4  q 2 Ω m +1 ,m +3 Γ m Ω m +1 ,m +4 Ω m +2 ,m +1 + ( ζ + q 2 x 2 ) x  + 2  ζ − q 2 x 2  x ( x 2 − 1) ( ζ − q 2 ) ) ( x − 1) k m +1 ,m +1 ( x ) = k 1 , 1 ( x ) − Γ m G 1 ( x ) + ∆ 1 ( x ) k m +2 ,m +2 ( x ) = k m +1 ,m +1 ( x ) + (Ω m +2 ,m +2 + Γ m ) G 1 ( x ) (54) k m +3 ,m +3 ( x ) = k m +2 ,m +2 ( x ) + ∆ 2 ( x ) k m +4 ,m +4 ( x ) = k m +3 ,m +3 ( x ) + (Ω m +4 ,m +4 − Ω m +3 ,m +3 ) xG 2 ( x ) k N ,N ( x ) = x 2 k 1 , 1 ( x ) 13 where ∆ 1 ( x ) = Γ m  Γ m Ω m +1 ,m +3 q 2 x + ζ Ω m +1 ,m +4 Ω m +2 ,m +1  Ω m +1 ,m +4 Ω m +2 ,m +1 ( x − 1) 2 x ( ζ − q 2 x 2 ) ∆ 2 ( x ) = Ω m +1 ,m +2  ζ Ω 2 m +2 ,m +1 Ω 2 m +1 ,m +4 − q 2 Ω 2 m +1 ,m +3 Γ 2 m  Γ m Ω 2 m +1 ,m +4 Ω m +2 ,m +1 x ( ζ − q 2 )( x − 1) ζ ( ζ − q 2 x 2 ) . (55) The v ariables Ω α,α are giv en in terms of the free parameters Ω m +1 ,m +2 , Ω m +2 ,m +1 , Ω m +1 ,m +3 and Ω m +1 ,m +4 through the r elations Ω m +2 ,m +2 = 2 ζ ζ − q 2 − Ω m +1 ,m +2 Ω m +2 ,m +1 Γ m Ω m +3 ,m +3 = 2 ζ ζ − q 2 − q 2 ζ Γ m Ω m +1 ,m +2 Ω 2 m +1 ,m +3 Ω m +2 ,m +1 Ω 2 m +1 ,m +4 (56) Ω m +4 ,m +4 = 2 ζ ζ − q 2 − q 2 ζ Ω m +1 ,m +2 Ω m +1 ,m +3 Ω m +1 ,m +4 . • Solution N 7 : The v ertex mo del U q [ sl (2 n | 2 m ) (2) ] admits the solution N 7 for n ≥ 3, whose K -matrix has the follo wing b lo c k structure K − ( x ) =          k 1 , 1 ( x ) I m × m O m × 2 n O m × m O 2 n × m k m +1 ,m +1 ( x ) · · · k m +1 , 2 n + m ( x ) . . . . . . . . . k 2 n + m,m +1 ( x ) · · · k 2 n + m, 2 n + m ( x ) O 2 n × m O m × m O m × 2 n k N ,N ( x ) I m × m          . (57) The central blo ck matrix cluster all non-diagonal element s differen t from zero. Concerning that cen tral blo ck, w e hav e the follo wing expressions determining en tries of the b ord ers, k α, 2 n + m ( x ) = κ √ ζ q t α − t m +1 Ω m +1 ,α ′ xG ( x ) α = m + 2 , . . . , 2 n + m − 1 k 2 n + m,α ( x ) = κ √ ζ q t 2 n + m − t m +2 Ω m +2 ,m +1 Ω m +1 ,α Ω m +1 , 2 n + m − 1 xG ( x ) α = m + 2 , . . . , 2 n + m − 1 k α,m +1 ( x ) = q t α − t m +2 Ω m +2 ,m +1 Ω m +1 ,α ′ Ω m +1 , 2 n + m − 1 G ( x ) α = m + 3 , . . . , 2 n + m − 1 k m +1 ,α ( x ) = Ω m +1 ,α G ( x ) α = m + 2 , . . . , 2 n + m − 1 (58) with G ( x ) = x − 1. The entries of the secondary diagonal are given by k α,α ′ ( x ) =              Ω m +1 , 2 n + m G ( x ) H b ( x ) α = m + 1 q 2 n − 2 ζ q t m +1 − t α ′  q + κ √ ζ q + 1  2 Ω 2 m +1 ,α ′ Ω m +1 , 2 n + m G ( x ) H b ( x ) α = m + 2 , . . . , 2 n + m − 1 q t 2 n + m − 1 − t m +2 Ω 2 m +2 ,m +1 Ω m +1 , 2 n + m Ω 2 m +1 , 2 n + m − 1 G ( x ) H b ( x ) α = 2 n + m (59) 14 with H b ( x ) = xq + κ √ ζ q + κ √ ζ , and the r emaining non-diagonal elemen ts are determined by the exp r ession k α,β ( x ) =                                κ √ ζ q t α − t m +1  q + κ √ ζ q + 1  Ω m +1 ,α ′ Ω m +1 ,β β m +1 , 2 n + m G ( x ) α < β ′ , m + 1 < α, β < 2 n + m 1 ζ q t α − t m +1  q + κ √ ζ q + 1  Ω m +1 ,α ′ Ω m +1 ,β Ω m +1 , 2 n + m xG ( x ) α > β ′ , m + 1 < α, β < 2 n + m 2 κ √ ζ (1 − κ √ ζ ) ( − 1) n ζ (1 + q ) 2 ( q n − ( − 1) n κ √ ζ )( q n − 1 − ( − 1) n κ √ ζ )( q + κ √ ζ ) Ω m +1 , 2 n + m Ω m +1 ,m + n +1 G ( x ) α = m + 1 , β = m + n − 2 κ √ ζ (1 − κ √ ζ ) q ( q + κ √ ζ ) ( q n − ( − 1) n κ √ ζ )( q n − 1 − ( − 1) n κ √ ζ ) Ω m +1 ,m +2 n − 1 Ω m +1 ,m +2 n G ( x ) α = m + 2 , β = m + 1 . (60) In their tu rn the d iagonal en tries k α,α ( x ) are giv en b y k α,α ( x ) =                             x − κ √ ζ 1 − κ √ ζ   xq n − ( − 1) n κ √ ζ q n − ( − 1) n κ √ ζ   xq n − 1 − ( − 1) n κ √ ζ q n − 1 − ( − 1) n κ √ ζ  2 x ( x + 1) α = 1 k 1 , 1 ( x ) + Γ n ( x ) α = m + 1 k m +1 ,m +1 ( x ) + (Ω α,α − Ω m +1 ,m +1 ) G ( x ) α = m + 2 , . . . , m + n k n + m,n + m ( x ) α = n + m + 1 k n + m,n + m ( x ) + (Ω α,α − Ω n + m,n + m ) x G ( x ) α = n + m + 2 , . . . , 2 n + m x 2 k 1 , 1 ( x ) α = N (61) where Γ n ( x ) = − 2 ζ ( q x + κ √ ζ ) (1 − κ √ ζ )( q n − ( − 1) n κ √ ζ )( q n − 1 − ( − 1) n κ √ ζ ) G ( x ) x . (62) The diagonal parameters Ω α,α are then fixed by the relations Ω α,α =                        − 2 ζ ( q + κ √ ζ ) (1 − κ √ ζ )( q n − ( − 1) n κ √ ζ )( q n − 1 − ( − 1) n κ √ ζ ) α = m + 1 Ω m +1 ,m +1 + ∆ n α − m − 2 X k =0 ( − q ) k α = m + 2 , . . . , n + m Ω n + m,n + m α = n + m + 1 Ω n + m,n + m + iκ ( − 1) n q m − 1 ∆ n α − n − m − 2 X k =0 ( − q ) k α = n + m + 2 , . . . , 2 n + m (63) and the auxiliary p arameter ∆ n is giv en by ∆ n = 2 ζ (1 + q ) 2 (1 − κ √ ζ )( q n − ( − 1) n κ √ ζ )( q n − 1 − ( − 1) n κ √ ζ ) . (64) Besides the ab o v e relations the follo wing constraints should also holds Ω m +1 ,m + α = − Ω m +1 ,α + m +1 Ω m +1 , 2 n + m − α Ω m +1 , 2 n + m +1 − α α = 2 , . . . , n − 1 , (65) and Ω m +1 ,m + n +1 , . . . , Ω m +1 , 2 n + m are regarded as free p arameters. 15 • Solution N 8 : F or n ≥ 2 th e v ertex mo del U q [ sl (2) (2 n + 1 | 2 m )] adm its the f amily of solutions N 8 whose K -matrix is of th e form K − ( x ) =          k 1 , 1 ( x ) I m × m O m × (2 n +1) O m × m O (2 n +1) × m k m +1 ,m +1 ( x ) · · · k m +1 , 2 n + m +1 ( x ) . . . . . . . . . k 2 n + m +1 ,m +1 ( x ) · · · k 2 n + m +1+ , 2 n + m +1 ( x ) O (2 n +1) × m O m × m O m × (2 n +1) k N ,N ( x ) I m × m          . (66) In th e cen tral b lo c k matrix w e fin d all non-diago nal elemen ts differen t fr om zero similarly to th e structure of the solution N 7 . The b orders of th e cent ral blo c k are then determined by the f ollo wing expressions k α, 2 n + m +1 ( x ) = κ √ ζ q t α − t m +1 Ω m +1 ,α ′ xG ( x ) α = m + 2 , . . . , 2 n + m k 2 n + m +1 ,α ( x ) = κ √ ζ q t 2 n + m +1 − t m +2 Ω m +2 ,m +1 Ω m +1 ,α Ω m +1 , 2 n + m xG ( x ) α = m + 2 , . . . , 2 n + m k α,m +1 ( x ) = q t α − t m +2 Ω m +2 ,m +1 Ω m +1 ,α ′ Ω m +1 , 2 n + m G ( x ) α = m + 3 , . . . , 2 n + m k m +1 ,α ( x ) = Ω m +1 ,α G ( x ) α = m + 2 , . . . , 2 n + m (67) recalling that κ = ± 1 and G ( x ) = x − 1. The secondary diagonal elemen ts are giv en by k α,α ′ ( x ) =              Ω m +1 , 2 n + m +1 G ( x ) H b ( x ) α = m + 1 q 2 n − 1 ζ q t m +1 − t α ′  q + κ √ ζ q + 1  2 Ω 2 m +1 ,α ′ Ω m +1 , 2 n + m +1 G ( x ) H b ( x ) α = m + 2 , . . . , 2 n + m q t 2 n + m − t m +2 Ω 2 m +2 ,m +1 Ω m +1 , 2 n + m +1 Ω 2 m +1 , 2 n + m G ( x ) H b ( x ) α = 2 n + m + 1 (68) where H b ( x ) = xq + κ √ ζ q + κ √ ζ . The remaining non-diagonal en tries are determined by k α,β ( x ) =                                              κ √ ζ q t α − t m +1  q + κ √ ζ q + 1  Ω m +1 ,α ′ Ω m +1 ,β Ω m +1 , 2 n + m +1 G ( x ) α < β ′ , m + 2 ≤ α, β ≤ 2 n + m 1 ζ q t α − t m +1  q + κ √ ζ q + 1  Ω m +1 ,α ′ Ω m +1 ,β Ω m +1 , 2 n + m +1 xG ( x ) α > β ′ , m + 2 ≤ α, β ≤ 2 n + m ( − 1) n q 2 m − 2 n  q + κ √ ζ q + 1  2 Ω m +1 ,n + m Ω m +1 ,n + m +2 Ω m +1 , 2 n + m Ω 2 m +1 , 2 n + m +1 G ( x ) α = m + 2 , β = m + 1 " iκ ( − 1) n q m − 1 ( q + 1) ( q m − 1 2 − iκ ( − 1) n ) 2 Ω 2 m +1 ,n + m +1 Ω m +1 ,n + m +2 − 2 κ ( − 1) n ( q + 1) 2 √ ζ ( q m − 1 2 − iκ ( − 1) n ) 2 (1 − κ √ ζ )( q + κ √ ζ ) Ω m +1 , 2 n + m +1 Ω m +1 ,m + n +2 # G ( x ) α = m + 1 , β = m + n , (69) 16 while the diagonal matrix element s are giv en b y the relations k α,α ( x ) =                                                    " 2( x − κ √ ζ )( xq m − 1 2 − iκ ( − 1) n ) 2 x ( x 2 − 1)(1 − κ √ ζ )( q m − 1 2 − iκ ( − 1) n ) 2 − iq m − 1 (1 + q 2 m − 1 x )( q + κ √ ζ )( x − κ √ ζ ) √ ζ x ( x + 1)( q + 1)( q m − 1 2 − iκ ( − 1) n ) 2 # G ( x ) α = 1 k 1 , 1 ( x ) + Γ( x ) α = m + 1 k m +1 ,m +1 ( x ) + (Ω α,α − Ω m +1 ,m +1 ) G ( x ) α = m + 2 , . . . , m + n k m +1 ,m +1 ( x ) + (Ω n + m +1 ,n + m +1 − Ω m +1 ,m +1 ) G ( x ) + ∆( x ) α = n + m + 1 k n + m +1 ,n + m +1 ( x ) + (Ω n + m +2 ,n + m +2 − Ω n + m +1 ,n + m +1 ) x G ( x ) + κ √ ζ ∆ ( x ) α = n + m + 2 k α − 1 ,α − 1 ( x ) + (Ω α,α − Ω α − 1 ,α − 1 ) x G ( x ) α = n + m + 3 , . . . , 2 n + m + 1 x 2 k 1 , 1 ( x ) α = N . (70) The auxiliary functions ∆( x ) and Γ ( x ) are ∆( x ) = q 2 m − n − 1 ( q + κ √ ζ ) ( q + 1) 2 Ω m +1 ,n + m Ω m +1 ,n + m +2 Ω m +1 , 2 n + m +1 ( x − 1) 2 (71) Γ( x ) = ( − 1) n +1 ( xq + κ √ ζ )( q + κ √ ζ ) ( κ √ ζ x ( q + 1) 2 ) Ω m +1 ,n + m Ω m +1 ,n + m +2 Ω m +1 , 2 n + m +1 G ( x ) , (72) and the p arameters Ω m +1 ,m + α are constrained by th e recurr ence formula Ω m +1 ,α + m = − Ω m +1 ,α + m +1 Ω m +1 , 2 n + m +1 − α Ω m +1 , 2 n + m +2 − α α = 2 , . . . , n − 1 . (73) In their tu rn the d iagonal parameters Ω α,α are fixed by Ω α,α =                                                            2 iκ ( q + κ √ ζ ) ( q n − m + 1 2 + iκ )( q m − 1 2 − iκ ( − 1) n ) 2 − q 2 m − n − 3 2 ( q + κ √ ζ ) 2 ( q + 1)( q m − 1 2 − iκ ( − 1) n ) 2 Ω 2 m +1 ,n + m +1 Ω m +1 , 2 n + m +1 α = m + 1 Ω m +1 ,m +1 + Q n,m α − m − 2 X k =0 ( − q ) k α = m + 2 , . . . , n + m ( q m − 1 − iκq 2 m − n − 1 2 )( q n − ( − 1) n ) ( q + 1)( q m − 1 2 − iκ ( − 1) n ) Ω 2 m +1 ,n + m +1 Ω m +1 , 2 n + m +1 − 2 iκ ( − 1) n ( q m − 1 2 − iκ ( − 1) n ) α = n + m + 1 Ω n + m +1 ,n + m +1 − q 2 m − n − 1  q + κ √ ζ q + 1  2 Ω m +1 ,n + m Ω m +1 ,n + m +2 Ω m +1 , 2 n + m +1 + q 2 m − n − 3 2 q + κ √ ζ q + 1 Ω 2 m +1 ,n + m +1 Ω m +1 , 2 n + m +1 α = n + m + 2 Ω n + m +2 ,n + m +2 + ( − 1) n κq 2 m − n − 1 p ζ Q n,m α − n − m − 3 X k =0 ( − q ) k α = n + m + 3 , . . . , 2 n + m + 1 (74) where Q n,m = 2( q + 1) 2 ( κ √ ζ − 1)( q m − 1 2 − iκ ( − 1) n ) 2 + iq m − 1 ( q + 1)( q + κ √ ζ ) √ ζ ( q m − 1 2 − iκ ( − 1) n ) 2 Ω 2 m +1 ,n + m +1 Ω m +1 , 2 n + m +1 . (75) 17 This solution has altogether n + 1 free parameters corresp onding to the set of v ariables Ω m +1 ,n + m +1 , . . . , Ω m +1 , 2 n + m +1 . • Solution N 9 : The family N 9 consist of a solution of the reflection equation where all ent ries of the K -matrix are non-null. T his solution is admitted only b y the U q [ sl (1 | 2 m ) (2) ] ve rtex mo del. The asso ciated K -matrix is of the general form (11) and the matrix elemen ts of the b ord ers are mainly giv en by k α,N ( x ) = − κ √ ζ ǫ α ǫ 1 q t α − t 1 Ω 1 ,α ′ xG ( x ) α = 2 , . . . , N − 1 k α, 1 ( x ) = ǫ α ǫ 2 q t α − t 2 Ω 2 , 1 Ω 1 ,α ′ Ω 1 ,N − 1 G ( x ) α = 3 , . . . , N − 1 k N ,α ( x ) = − κ √ ζ ǫ N ǫ 2 q t N − t 2 Ω 2 , 1 Ω 1 ,α Ω 1 ,N − 1 xG ( x ) α = 2 , . . . , N − 1 k 1 ,α ( x ) = Ω 1 ,α G ( x ) α = 2 , . . . , N − 1 . (76) The secondary diagonal is c haracterized by entries of the form k α,α ′ ( x ) =                  Ω 1 ,N G ( x ) H f ( x ) α = 1 q t 1 − t α ′ ǫ 1 ǫ α ′  1 − κq √ ζ q + 1  2 Ω 2 1 ,α ′ Ω 1 ,N G ( x ) H f ( x ) α = 2 , . . . , m and α = m + 2 , . . . , N − 1 ǫ N − 1 ǫ 2 q t N − 1 − t 2 Ω 1 ,N Ω 2 2 , 1 Ω 2 1 ,N − 1 G ( x ) H f ( x ) α = N (77) where G ( x ) = x − 1 and H f ( x ) = x − κq √ ζ 1 − κq √ ζ , and the remaining n on-diagonal elemen ts are giv en b y k α,β ( x ) =                                      − κ √ ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q + 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N G ( x ) α < β ′ , 2 ≤ α, β ≤ N − 1 1 ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q + 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N xG ( x ) α > β ′ , 2 ≤ α, β ≤ N − 1 i √ q ( q − 1) Ω 2 1 ,m +1 Ω 1 ,m +2 G ( x ) α = 1 , β = m − i 2 Ω 2 1 ,m +1  q 3 2 + iκ ( − 1) m  ( q m − 1 2 + iκ )( κq √ ζ − 1) ( q 2 − 1)  √ q − iκ ( − 1) m  G ( x ) α = 1 , β = N ( − 1) m q 2 m − 2  q κ √ ζ − 1 q + 1  2 Ω 1 ,m Ω 1 , 2 m Ω 1 ,m +2 Ω 2 1 , 2 m +1 G ( x ) α = 2 , β = 1 . (78) 18 In their tu rn the d iagonal en tries k α,α ( x ) are giv en b y th e follo wing expr ession k α,α ( x ) =                       2 x x 2 − 1 − Ω m +1 ,m +1 x + 1  G ( x ) +  1 + κ √ ζ x 2 − 1  Γ( x ) α = 1 k 1 , 1 ( x ) + Ω α,α G ( x ) α = 2 , . . . , m k 1 , 1 ( x ) + Ω m +1 ,m +1 G ( x ) + Γ( x ) α = m + 1 k m +1 ,m +1 ( x ) + (Ω m +2 ,m +2 − Ω m +1 ,m +1 ) x G ( x ) + iκq 1 2 − m Γ( x ) α = m + 2 k α − 1 ,α − 1 ( x ) + (Ω α,α − Ω α − 1 ,α − 1 ) x G ( x ) α = m + 3 , . . . , N (79) where the auxiliary fu nction Γ( x ) is giv en b y Γ( x ) = q m ( x − 1) 2 ( κq √ ζ − 1) ( q + 1) 2 Ω 1 ,m Ω 1 ,m +2 Ω 1 , 2 m +1 . (80) The parameters Ω 1 ,α are constrained by th e recurr ence relation Ω 1 ,α = − Ω 1 ,α +1 Ω 1 , 2 m +1 − α Ω 1 , 2 m +2 − α α = 2 , . . . , m − 1 (81) while Ω α,α are fixed by Ω α,α =                          Q m α − 2 X k =0 ( − 1 q ) k α = 2 , . . . , m Ω m,m + δ 1 α = m + 1 Ω m,m + δ 2 α = m + 2 Ω m +2 ,m +2 + ( − 1) m q m − 1 κ p ζ Q m α − m − 3 X k =0 ( − 1 q ) k α = m + 3 , . . . , N − 1 2 α = N (82) where Q m = 2 iκ ( − 1) m q m − 1 ( q + 1)  √ q − iκ ( − 1) m  ( q m − 1 2 + iκ )( q 3 2 + iκ ( − 1) m ) , δ 1 = − 2 ( q m + 1 2 + 2 iκq + iκ ) ( √ q + iκ ( − 1) m )( q 3 2 + iκ ( − 1) m )( q m − 1 2 + iκ ) , (83) δ 2 = 2 ( q m + 1 2 − iκ )  √ q − iκ ( − 1) m  ( q 3 2 + iκ ( − 1) m )( q m − 1 2 + iκ ) . In this s olution th e hav e a total amount of m fr ee p arameters, n amely Ω 1 ,m +1 , . . . , Ω 1 , 2 m . • Solution N 10 : The series of solutions N 10 is v alid for the U q [ sl (2 | 2 m ) (2) ] mo del and the corresp onding K - matrix also p ossess all entries d ifferen t from zero. In the first and last columns, the matrix elemen ts are mainly give n by k α, 1 ( x ) = ǫ α ǫ 2 q t α − t 2 Ω 2 , 1 Ω 1 ,α ′ Ω 1 ,N − 1 G ( x ) α = 3 , . . . , N − 1 k α,N ( x ) = − κ √ ζ ǫ α ǫ 1 q t α − t 1 Ω 1 ,α ′ xG ( x ) α = 2 , . . . , N − 1 (84) 19 while the ones in th e fir st and last rows are resp ecti v ely k 1 ,α ( x ) = Ω 1 ,α G ( x ) α = 2 , . . . , N − 1 k N ,α ( x ) = − κ √ ζ ǫ N ǫ 2 q t N − t 2 Ω 2 , 1 Ω 1 ,α Ω 1 ,N − 1 xG ( x ) α = 2 , . . . , N − 1 (85) with G ( x ) = x − 1. In the secondary diagonal we hav e th e follo wing exp r ession determining the matrix elemen ts k α,α ′ ( x ) =                          Ω 1 ,N G ( x ) H f ( x ) α = 1 q t 1 − t α ′ ǫ 1 ǫ α ′  1 − κq √ ζ q + 1  2 Ω 2 1 ,α ′ Ω 1 ,N G ( x ) H f ( x ) α = 2 , . . . , m and α = m + 3 , . . . , N − 1 q t 1 − t α ′ ǫ 1 ǫ α ′ (1 − κq √ ζ )( q + κ √ ζ ) ( q 2 − 1) Ω 2 1 ,α ′ Ω 1 ,N G ( x ) H b ( x ) α = m + 1 , m + 2 ǫ N − 1 ǫ 2 q t N − 1 − t 2 Ω 1 ,N Ω 2 2 , 1 Ω 2 1 ,N − 1 G ( x ) H f ( x ) α = N (86) recalling that H b ( x ) = q x + κ √ ζ q + κ √ ζ and H f ( x ) = x − κq √ ζ 1 − κq √ ζ . (87) In their tu rn the other non-diagonal entries satisfy the r elation k α,β ( x ) =                                      − κ √ ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q + 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N G ( x ) α < β ′ , 2 ≤ α, β ≤ N − 1 1 ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q + 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N xG ( x ) α > β ′ , 2 ≤ α, β ≤ N − 1 − 1 ζ q t 2 − t 1  1 − κq √ ζ q + 1  2 Ω 1 , 2 Ω 2 1 ,N − 1 Ω 2 1 ,N G ( x ) α = 2 , β = 1 i q + 1 q − 1 Ω 1 ,m +1 Ω 1 ,m +2 Ω 1 ,m +3 G ( x ) α = 1 , β = m − 2( q 2 − 1) √ ζ (1 + iκ ( − 1) m )( q + iκ ( − 1) m )(1 − κq √ ζ )(1 + κ √ ζ ) Ω 1 , 2 m +2 Ω 1 ,m +2 G ( x ) α = 1 , β = m + 1 , (88) and the p arameters Ω 1 ,α are required to satisfy the recurrence form ula Ω 1 ,α = − Ω 1 ,α +1 Ω 1 ,N − α Ω 1 ,N − α +1 α = 2 , . . . , m − 1 . (89) Considering no w the diagonal entrie s, they are giv en by k α,α ( x ) =                       2 x x 2 − 1 − Ω m +1 ,m +1 x + 1  G ( x ) α = 1 k 1 , 1 ( x ) + Ω α,α G ( x ) α = 2 , . . . , m + 1 k m +1 ,m +1 ( x ) α = m + 2 k m +1 ,m +1 ( x ) + (Ω α,α − Ω m +1 ,m +1 ) x G ( x ) α = m + 3 , . . . , N − 1 x 2 k 1 , 1 ( x ) α = N (90) 20 where the p arameters Ω α,α are determined by the expressions Ω α,α =                  ∆ m α − 2 X k =0 ( − 1 q ) k α = 2 , . . . , m ∆ m  q q + 1 + ( − 1) m 1 q m − 2 q − 1 ( q + 1) 2  α = m + 1 , m + 2 Ω m +1 ,m +1 + iκ ∆ m  ( − 1) m +1 q ( q − 1) ( q + 1) 2 + ( − 1) α q m +4 − α q + 1  α = m + 3 , . . . , N − 1 (91) with ∆ m = 2 q − 1 ( q + 1) 2 (1 − iκ ( − 1) m )( q + iκ ( − 1) m )(1 + κ √ ζ ) . (92) This solution has altoget her m + 1 f r ee parameters, namely Ω 1 ,m +2 , . . . , Ω 1 ,N . • Solution N 11 : The class of solutions N 11 is v alid for the v ertex mo del U q [ sl (2 n | 2) (2) ] and the corresp onding K - matrix con tains only non-null entries. Th e b order elemen ts are mainly giv en b y the follo wing expressions k α,N ( x ) = − κ √ ζ ǫ α ǫ 1 q t α − t 1 Ω 1 ,α ′ xG ( x ) α = 2 , . . . , N − 1 k α, 1 ( x ) = ǫ α ǫ 2 q t α − t 2 Ω 2 , 1 Ω 1 ,α ′ Ω 1 ,N − 1 G ( x ) α = 3 , . . . , N − 1 k N ,α ( x ) = − κ √ ζ ǫ N ǫ 2 q t N − t 2 Ω 2 , 1 Ω 1 ,α Ω 1 ,N − 1 xG ( x ) α = 2 , . . . , N − 1 k 1 ,α ( x ) = Ω 1 ,α G ( x ) α = 2 , . . . , N − 1 (93) with G ( x ) = x − 1. The secondary diagonal is constituted by elements k α,α ′ ( x ) giv en by k α,α ′ ( x ) =              Ω 1 ,N G ( x ) H f ( x ) α = 1 ǫ 1 ǫ α ′ q t 1 − t α ′ (1 − κq √ ζ )( q + κ √ ζ ) ( q 2 − 1) Ω 2 1 ,α ′ Ω 1 ,N G ( x ) H b ( x ) α = 2 , . . . , N − 1 ǫ N − 1 ǫ 2 q t N − 1 − t 2 Ω 1 ,N Ω 2 2 , 1 Ω 2 1 ,N − 1 G ( x ) H f ( x ) α = N , (94 ) where the fun ctions H b ( x ) and H f ( x ) we re already giv en in (87). The remaining non-diagonal en tries are determined by th e expression k α,β ( x ) =                          κ √ ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q − 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N G ( x ) α < β ′ , 2 ≤ α, β ≤ N − 1 1 ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q − 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N xG ( x ) α > β ′ , 2 ≤ α, β ≤ N − 1 ( − 1) n +1 ( q n + iκ ) 2 ζ ( q 2 − 1) Ω 1 ,n +1 Ω 1 ,n +2 Ω 1 ,N − 1 Ω 2 1 ,N G ( x ) α = 2 , β = 1 2 κ ( q 2 − 1)( − q ) n − 1 (1 + iκ ( − 1) n )( q + iκ ( − 1) n )( q n − 1 + iκ )( q n + iκ ) Ω 1 ,N Ω 1 ,n +2 G ( x ) α = 1 , β = n + 1 , (95) 21 and the p arameters Ω 1 ,α are required to satisfy Ω 1 ,α = − Ω 1 ,α +1 Ω 1 ,N − α Ω 1 ,N +1 − α α = 2 , . . . , n. (96) With resp ect to the diagonal en tries, they are giv en by k α,α ( x ) =                       2 x x 2 − 1 − Ω n +1 ,n +1 x + 1  G ( x ) α = 1 k 1 , 1 ( x ) + Ω α,α G ( x ) α = 2 , . . . , n + 1 k n +1 ,n +1 ( x ) α = n + 2 k n +1 ,n +1 ( x ) + (Ω α,α − Ω n +1 ,n +1 ) xG ( x ) α = n + 3 , . . . , N − 1 x 2 k 1 , 1 ( x ) α = N (97) and the p arameters Ω α,α are fixed by the r elations Ω α,α =              − 2( − 1) α + n  q α − 1 + q α − 2 − ( − 1) α ( q − 1)  (1 − iκ ( − 1) n )( q + iκ ( − 1) n )( iκ + q n − 1 ) α = 2 , . . . , n + 1 Ω n +1 ,n +1 α = n + 2 Ω n +1 ,n +1 + ∆ n α − n − 3 X k =0 ( − q ) k α = n + 3 , . . . , N − 1 (98) where ∆ n = 2( − 1) n +1 ( q + 1) 2 (1 + iκ ( − 1) n )( q + iκ ( − 1) n )( q n − 1 + iκ ) . (99) The v ariables Ω 1 ,n +2 , . . . , Ω 1 ,N giv e us a total amount of n + 1 free p arameters. • Solution N 12 : The solution N 12 also do es not conta in n ull entries and it is v alid f or the U q [ sl (2 n + 1 | 2) (2) ] ve r- tex mo d el. Consider in g first the non-diagonal entrie s, we hav e the f ollo wing expression determining b order elemen ts, k α,N ( x ) = − κ √ ζ ǫ α ǫ 1 q t α − t 1 Ω 1 ,α ′ xG ( x ) α = 2 , . . . , N − 1 k α, 1 ( x ) = ǫ α ǫ 2 q t α − t 2 Ω 2 , 1 Ω 1 ,α ′ Ω 1 ,N − 1 G ( x ) α = 3 , . . . , N − 1 k N ,α ( x ) = − κ √ ζ ǫ N ǫ 2 q t N − t 2 Ω 2 , 1 Ω 1 ,α Ω 1 ,N − 1 xG ( x ) α = 2 , . . . , N − 1 k 1 ,α ( x ) = Ω 1 ,α G ( x ) α = 2 , . . . , N − 1 , (100 ) and the follo wing one for th e en tries of the secondary d iagonal k α,α ′ ( x ) =              Ω 1 ,N G ( x ) H f ( x ) α = 1 ǫ 1 ǫ α ′ q t 1 − t α ′ (1 − κq √ ζ )( q + κ √ ζ ) ( q 2 − 1) Ω 2 1 ,α ′ Ω 1 ,N G ( x ) H b ( x ) α = 2 , . . . , N − 1 , α 6 = N − 1 2 ǫ N − 1 ǫ 2 q t N − 1 − t 2 Ω 1 ,N Ω 2 2 , 1 Ω 2 1 ,N − 1 G ( x ) H f ( x ) α = N , (101) 22 with H b ( x ) and H f ( x ) giv en in (87). The remaining non-diagonal en tries are determined by k α,β ( x ) =                                      κ √ ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q − 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N G ( x ) α < β ′ , 2 ≤ α, β ≤ N − 1 1 ζ ǫ α ǫ 1 q t α − t 1  1 − κq √ ζ q − 1  Ω 1 ,α ′ Ω 1 ,β Ω 1 ,N xG ( x ) α > β ′ , 2 ≤ α, β ≤ N − 1 ( − 1) n (1 − κq √ ζ ) 2 ζ ( q 2 − 1) Ω 1 ,n +1 Ω 1 ,n +3 Ω 1 ,N − 1 Ω 2 1 ,N G ( x ) α = 2 , β = 1 " 2 κ ( − 1) n q n − 1 2 ( q − 1)( √ q − iκ ( − 1) n ) (1 − κ √ ζ )(1 − κq √ ζ )( √ q + iκ ( − 1) n ) Ω 1 ,N Ω 1 ,n +3 − iκ ( − 1) n ( √ q − iκ ( − 1) n ) ( √ q + iκ ( − 1) n ) Ω 2 1 ,n +2 Ω 1 ,n +3 # G ( x ) α = 1 , β = n + 1 , (102) and the follo wing recurrence f orm ula should also holds Ω 1 ,α = − Ω 1 ,α +1 Ω 1 ,N − α Ω 1 ,N +1 − α α = 2 , . . . , n. (10 3) With regard to the d iagonal entries, they are giv en by k α,α ( x ) =                             2 x ( x 2 − 1) − Ω n +2 ,n +2 ( x + 1)  G ( x ) + (1 + κ √ ζ ) ( x 2 − 1) ∆( x ) α = 1 k 1 , 1 ( x ) + Ω α,α G ( x ) α = 2 , . . . , n + 1 k 1 , 1 ( x ) + Ω n +2 ,n +2 G ( x ) + ∆( x ) α = n + 2 k n +2 ,n +2 ( x ) + (Ω n +3 ,n +3 − Ω n +2 ,n +2 ) x G ( x ) + κ p ζ ∆ ( x ) α = n + 3 k n +3 ,n +3 ( x ) + (Ω α,α − Ω n +3 ,n +3 ) x G ( x ) α = n + 4 , . . . , N − 1 x 2 k 1 , 1 ( x ) α = N (104) where ∆( x ) = iq 1 − n (1 − κq √ ζ ) ( q 2 − 1) Ω 1 ,n +1 Ω 1 ,n +3 Ω 1 ,N ( x − 1) G ( x ) . (105) In their tu rn the p arameters Ω α,α are fixed by the follo wing expr ession Ω α,α =                              ( − 1) α Q n  q α − 2 ( q + 1) + ( − 1) α (1 − q )  α = 2 , . . . , n + 1 2 q + 1 + 2 iκ √ q ( − 1) n ( q − 1) 2  q n − 1 + ( − 1) n  ( q n − 1 2 + iκ )( q 2 − 1)( √ q + iκ ( − 1) n ) − i ( q 3 + 1)( q n − 1 2 − iκ )( q n + 1 2 + iκ ) q n − 1 2 ( q 2 − 1)( q 3 2 − iκ ( − 1) n )( √ q + iκ ( − 1) n ) Ω 2 1 ,n +2 Ω 1 ,N α = n + 2 2 − iκ √ q Q n  q n − q n − 1 + ( − 1) n +1 ( q + 1)  α = n + 3 Ω n +3 ,n +3 − iκ ( − 1) n √ q ( q + 1) 2 Q n α − n − 4 X k =0 ( − q ) k α = n + 4 , . . . , N − 1 (106) and the auxiliary p arameter Q n is giv en by Q n = − 2 (1 − κ √ ζ )( √ q + iκ ( − 1) n ) 2 + i (1 − κq √ ζ ) q n − 1 2 ( q − 1)( √ q + iκ ( − 1) n ) 2 Ω 2 1 ,n +2 Ω 1 ,N . (107) This solution p ossess n + 2 f r ee parameters, namely Ω 1 ,n +2 , . . . , Ω 1 ,N . 23 4 Concluding Remarks In this w ork w e ha v e p resen ted the general set of regular solutions of th e graded reflection equation for the U q [ sl ( r | 2 m ) (2) ] v ertex mo del. Our fi ndings can b e su mmarized into four classes of diagonal solutions and t w elv e classes of non-diagonal ones. These r esults pa v e th e w a y to construct, solve and study physic al prop erties of th e u nderlying quan tum spin c hains with op en b oun daries, generalizing the previous efforts mad e for the case of p erio dic b ou n dary conditions [29, 30]. Although w e exp ect that the Algebraic Bethe An satz solution of th e mo dels constructed from the diago nal solutions p resen ted here can b e obtained by ad ap tin g the r esults of [33], the algebraic-functional m etho d presente d in [34] may b e a p ossib ilit y to treat the n on -d iagonal cases. F or further r esearc h, an int eresting p ossibilit y w ould b e the in v estigatio n of soliton non- preserving b oun dary conditions [26, 35] for quantum spin chains b ased on q -deformed Lie algebras and sup eralgebras, wh ic h could also b e p erformed b y adapting the metho d d escrib ed in [21]. W e exp ect the results presente d here to motiv ate furth er d ev elopmen ts on the su b ject of int egrable op en b ound aries for v ertex mo dels based on q -deformed Lie sup eralgebras. In particular, the classification of the solutions of the graded reflection equation for others q -deformed Lie sup eralgebras, wh ic h w e hop e to rep ort in a future w ork. 5 Ac kno wledgme n ts The work of A. Lima-San tos is partially supp orted by the Brazilian r esearch coun cils CNPq and F APESP . W. Galleas thanks the agency F APES P for the finan cial sup p ort. References [1] A.M. Poly ako v, JETP L e tt. 12, (19 70) 381. [2] A.A. Bela vin, A.M. Pol y ak o v and A.B. Zamolo dc hiko v, Nucl. 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Bogoliub o v, Q uantum Inverse Sc attering Metho d and Cor- r elation F unctions , C ambridge Univ. Pr ess, Cambridge, (1993). [12] J.L. Cardy , N ucl. P hys. B 275, (1986) 200. [13] E. Witt en, JHEP 9807, (1998) 006 ; J. McGreevy , L. Susskind and N. T om bas, JHEP 000 6, (2000) 008. [14] B. Ch en , X.J. W ang and Y.S. W u, Phys. L ett. B591, (2004) 170. [15] O. DeW olfe and N. Mann, JHEP 0404, (2004) 035. [16] D. Berenstein and S .E. V azquez, JHEP 0506, (200 5) 059. [17] E.K. Sklyanin, J. Phys. A: Math. Gen. 21, (19 88) 2375. [18] I.V. Ch erednik, The or. M ath. Phys. 61, (1984) 977. [19] V.V. Bazhano v, Phys. L ett. B 159, (1985) 321. [20] M. Jimb o, Comm. Math. Phys. 102, (1986) 247. [21] R. Malara and A. Lima-Sant os, J. Stat. Me ch.: The or. Exp., (2006) P09013. [22] G.L. Li, R.H. Y u e and B.Y. Hou, Nucl. Phys. B 586, (2000) 711 ; A. Gonzal ez-Ruiz, N ucl. Phys. B 424, (1994) 468 . [23] A.J. Brac k en, X.Y. Ge, Y.Z. Z h ang and H.Q. Zhou, Nuc l. Phys. B 516, (1998) 588 . 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Doik ou, J. Phys. A: Math. Gen. 33, (2000) 8797. [36] A. Lima-Santo s, Nucl. Phys. B 558, (1999) 637. 26 App endix A : T h e U q [ sl (1 | 2) (2) ] case The reflection equation asso ciated with the U q [ sl (1 | 2) (2) ] vertex mo del adm its more general solutions than the corresp ondin g ones obtained from the general series presente d in the section 3. In this case the refl ection matrices were previously stud ied in [36] and we ha v e obtained the follo wing solutions K − ( x ) =     Ω( x − 1 − 1)+2 Ω( x − 1)+2 0 0 0 1 0 0 0 Ω x ( xq +1) − 2 x Ω( x + q ) − 2 x     (A.1) K − ( x ) =     1 0 Ω( x 2 − 1) 2 0 x 2 q +1 q +1 0 − 2 q ( x 2 − 1) Ω( q + 1) 2 0 x 2     (A.2) where Ω is a fr ee p arameter. In addition to the solutions (A.1) and (A.2) we also hav e a solution in the general form K − ( x ) =     k 1 , 1 ( x ) k 1 , 2 ( x ) k 1 , 3 ( x ) k 2 , 1 ( x ) k 2 , 2 ( x ) k 2 , 3 ( x ) k 3 , 1 ( x ) k 3 , 2 ( x ) k 3 , 3 ( x )     . (A.3) The diagonal en tries are th en giv en by k 1 , 1 ( x ) = 2( x + q )( x − 1) ( x 2 + q )( x 2 − 1) − (Ω 1 , 2 + i √ q Ω 2 , 3 )  q ( x − 1)Ω 2 , 3 − i √ q ( x + q )Ω 1 , 2  ( q − 1)( x + 1)( q + x 2 ) ( x − 1) Ω 1 , 3 −  ( q + 1)( √ q Ω 2 , 3 + ix Ω 1 , 2 )Ω 1 , 3 Ω 2 , 1 − 2 iq Ω 1 , 2 Ω 2 , 3  √ q ( x + 1)( q + x 2 ) ( x − 1) Ω 2 1 , 2 k 2 , 2 ( x ) = − xk 1 , 1 ( x ) + 2 x ( q + x 2 )  q + x + i √ q ( x − 1) Ω 2 , 3 Ω 1 , 2  −  i √ q ( x 2 − 1) + x ( q + 1) Ω 2 , 3 Ω 1 , 2  Ω 2 , 1 Ω 1 , 2 Ω 1 , 3 ( x − 1) ( q + x 2 ) k 3 , 3 ( x ) = x 2 k 1 , 1 ( x ) + " Ω 3 , 3 − 2 + i √ q ( x − 1)( x − q ) ( q − 1)( q + x 2 ) Ω 2 1 , 2 − Ω 2 2 , 3 Ω 1 , 3 # x ( x − 1) , (A.4) and the remainin g elements can b e written as k 1 , 2 ( x ) = ( q + x )Ω 1 , 2 + i ( x − 1) √ q Ω 2 , 3 q + x 2 ( x − 1) k 1 , 3 ( x ) = Ω 1 , 3 ( x − 1) k 2 , 1 ( x ) = ( q + x )Ω 2 , 1 + i ( x − 1) √ q Ω 3 , 2 q + x 2 ( x − 1) k 2 , 3 ( x ) = ( q + x )Ω 2 , 3 + i ( x − 1) √ q Ω 1 , 2 q + x 2 x ( x − 1) k 3 , 2 ( x ) = ( q + x )Ω 3 , 2 + i ( x − 1) √ q Ω 2 , 1 q + x 2 x ( x − 1) k 3 , 1 ( x ) = Ω 3 , 1 ( x − 1) . (A.5) This solution has altogether three free parameters Ω 1 , 2 , Ω 1 , 3 and Ω 2 , 1 and the remainin g v ariables Ω α,β are determined by Ω 2 , 3 = − 2 q − 1 q + 1 Ω 1 , 3 Ω 1 , 2 − i q − 1 √ q Ω 2 , 1 Ω 2 1 , 3 Ω 2 1 , 2 Ω 3 , 2 = Ω 2 , 3 Ω 2 , 1 Ω 1 , 2 27 Ω 3 , 3 = 2 + i √ q q − 1 Ω 2 2 , 3 − Ω 2 1 , 2 Ω 1 , 3 Ω 3 , 1 = Ω 1 , 3 Ω 2 2 , 1 Ω 2 1 , 2 . (A.6) App endix B : The U q [ sl (2 | 2) (2) ] case The set of K -matrices asso ciated with the U q [ sl (2 | 2) (2) ] ve rtex mo del includ es b oth diagonal and non-diagonal s olutions. The solutions intrinsicall y diagonal conta in only one free p arameter Ω and they are give n b y K − ( x ) =        x − 1 − Ω( x − 1)+2 Ω( x − 1) − 2 x Ω( x + q 2 ) − 2 x Ω( xq 2 +1) − 2 x        (B.1) and K − ( x ) =        x − 1 − Ω( x − 1)+2 Ω( x − 1) − 2 x − Ω( x − 1)+2 Ω( x − 1) − 2 x − h Ω( x − 1)+2 Ω( x − 1) − 2 x i h Ω( xq 2 +1) − 2 Ω( x + q 2 ) − 2 x i x        . (B.2) W e h a v e also foun d th e follo wing n on-diagonal solutions K − ( x ) =        1 0 0 Ω 2 2 ( x 2 − 1) 0 Ω 1 2 ( x 2 − 1) + 1 0 0 0 0 − Ω 1 2 ( x 2 − 1) + x 2 0 Ω 1 (Ω 1 − 2) 2Ω 2 ( x 2 − 1) 0 0 x 2        (B.3) K − ( x ) =        − Ω 1 ( x 2 − 1) − 2 x 2 2 x 0 0 0 0 x Ω 2 2 ( x 2 − 1) 0 0 Ω 1 (Ω 1 − 2) 2Ω 2 ( x 2 − 1) x 0 0 0 0 x 2 [Ω 1 ( x 2 − 1) + 2]        (B.4) con taining t w o fr ee parameters Ω 1 and Ω 2 , and one solution in the f orm K − ( x ) =        k 1 , 1 ( x ) k 1 , 2 ( x ) k 1 , 3 ( x ) k 1 , 4 ( x ) k 2 , 1 ( x ) k 2 , 2 ( x ) k 2 , 3 ( x ) k 2 , 4 ( x ) k 3 , 1 ( x ) k 3 , 2 ( x ) k 3 , 3 ( x ) k 3 , 4 ( x ) k 4 , 1 ( x ) k 4 , 2 ( x ) k 4 , 3 ( x ) k 4 , 4 ( x )        . (B.5) Concerning the solution (B.5), the d iagonal en tries are giv en by k 1 , 1 ( x ) =  xq − iκ q − iκ  k 2 , 2 ( x ) = x  x + iκq 1 + iκq  k 3 , 3 ( x ) = x  x + iκq 1 + iκq  k 4 , 4 ( x ) = x 2  xq − iκ q − iκ  (B.6) 28 while the non-diagonal entries are giv en b y th e follo wing expr essions, k 1 , 2 ( x ) = κ Ω 2 2Ω 1 q 2 − 1 q 2 + 1 ( x 2 − 1) k 2 , 1 ( x ) = − κ Ω 1 2Ω 2 q + iκ q − iκ ( x 2 − 1) k 1 , 3 ( x ) = Ω 1 2 ( x 2 − 1) k 2 , 3 ( x ) = − κ Ω 2 1 2Ω 2 ( q x + iκ )( q + iκ ) q 2 − 1 ( x 2 − 1) k 1 , 4 ( x ) = Ω 2 2 x − iκq 1 − iκq ( x 2 − 1) k 2 , 4 ( x ) = − κ Ω 1 2 x ( x 2 − 1) k 3 , 1 ( x ) = − 1 2Ω 1 q 2 − 1 ( q − iκ ) 2 ( x 2 − 1) k 4 , 1 ( x ) = 1 2Ω 2 x − iκq 1 − iκq  q + iκ q − iκ  2 ( x 2 − 1) k 3 , 2 ( x ) = − κ Ω 2 2Ω 2 1 q x + iκ q + iκ q 2 − 1 ( q − iκ ) 2 ( x 2 − 1) k 4 , 2 ( x ) = κ 1 2Ω 1 q 2 − 1 ( q − iκ ) 2 x ( x 2 − 1) k 3 , 4 ( x ) = − Ω 2 2Ω 1 q 2 − 1 q 2 + 1 x ( x 2 − 1) k 4 , 3 ( x ) = Ω 1 2Ω 2 q + iκ q − iκ x ( x 2 − 1) , (B.7) where κ = ± 1 and Ω 1 and Ω 2 are free parameters. 29