Reflection matrices for the $U_{q}[osp(r|2m)^{(1)}]$ vertex model

UFSCARF-TH-09-11 Reflection matr ices for the U q [ osp ( r | 2 m ) (1) ] vertex mo del A. Lima-San tos Universidade F e der al de S˜ ao Carlos Dep artamento d e F ´ ısic a C.P. 676, 13565 -905, S˜ ao Carlos-SP, Br a sil Abstract The graded reflection equa tion is inv estigated for the U q [ osp ( r | 2 m ) (1) ] vertex mo del. W e hav e f ound four classes of diagonal solutions with at the most one free parameter and t w elv e classes of non-diagonal ones with the num ber of free para meters dep ending on the num ber of bo sonic ( r ) and fermionic (2 m ) degrees of freedom. P A CS n um bers: 05.50 +q, 02.30.IK, 75.10.Jm Keywords: Reflection Equations , K-matrices, Su pe r algebra s Octob er 28, 2018 1 In tro d uction In tegrabilit y in classic al verte x mo d els and quan tum spin c hains is in timately connected with solu- tions of the Y ang-Baxte r equation [1]. This equation p la ys a cent ral role in the Quant um In verse Scattering Metho d wh ic h pr o vides an unified approac h to construct and stud y ph ysical prop erties of integrable mo d els [2, 3]. Usually th ese systems are stud ied with p erio dic b oundary conditions but m ore general b ound aries can also b e includ ed in this framework as w ell. Ph ys ical p rop erties asso ciated with the bulk of the s y s tem are n ot exp ected to b e influen ced by b oundary cond itions in th e thermo d ynamical limit. Nev ertheless, there are su r face prop erties su c h as the interfaci al tension where the b oundary conditions are of relev ance. Moreo ver, the conformal sp ectra o f lattice mo dels at criticalit y can b e mo dified by the effect of b ound aries [4]. In tegrable systems with op en b oundary conditions can also b e accommodated within the framew ork of the Quan tum Inv erse Scattering Metho d [5]. In addition to the solution of the Y ang- Baxter equation go v ern ing the dyn amics of the bu lk there is another f u ndamenta l ingredient, the reflection matrices [6]. These matrices, also referred as K -matrices, r epresen t the interact ions at the b oun daries and compatibilit y w ith the bulk in tegrabilit y requir es th ese matrices to satisfy the so-calle d reflection equations [5, 6]. A t the moment, the study of general regular solutions of the reflection equations f or vertex mo dels based on q -deformed Lie algebras [7 , 8] has b een successfully accomplished. S ee [9] for instance and references ther ein. Ho wev er, this same analysis for ve rtex mo dels b ased on Lie su - p eralgebras are still restricted to diagonal solutions asso ciated with the U q [ sl ( m | n )] [10 , 11] and U q [ osp (2 | 2)] symmetries [12] and n on-diagonal s olutions related to sup er-Y angians osp ( m | n ) [13] and sl ( m | n ) [14, 15]. The aim of this pap er is to touc h again in the classification of the solutions of the refl ection equations based in the L ie sup eralgebras already initialized in [16] with the U q [ sl ( r | 2 m ) (2) ] v ertex mo del. Here we will presen t the most general set of solutions of the reflection equation for the U q [ osp ( r | 2 m ) (1) ] ve rtex mo del, kee p ing the structure pr esen ted in[16]. This pap er is organized as follo ws. In the next sec tion w e present the U q [ osp ( r | 2 m ) (1) ] vertex mo del. T h is sup plies the w ay for the analysis of the corresp onding reflection equations and in the section 3 w e present four classes of d iagonal solutions. In the section 4 we p resen t t welv e classes of non-diagonal solutions what we hop e to b e the most general set of K -matrices for the v ertex mo d el here considered. Concludin g remarks are discussed in the section 5, and in the app endices A and B we pr esen t sp ecial solutions asso ciated with the U q [ osp (1 | 2) (1) ] and U q [ osp (2 | 2) (1) ] cases resp ectiv ely . Finally , in the app endix C we d escrib e th e main steps of our construction. 1 2 The U q [ osp ( r | 2 m ) (1) ] v ertex mo del Classical vertex mo dels of statistical mec h anics are now ad ays well kno w n to pla y a fund amen tal role in the th eory of t w o-dimens ional in tegrable systems [1]. In th is sense, it turns out that a R -matrix satisfying th e Y ang-Baxte r equ ation giv es r ise to the Boltzmann weig h ts of an exactly solv able v ertex mo del. The Y ang-Baxt er equation consist of an op erator relation for a complex v alued matrix R : C → End ( V ⊗ V ) r eading R 12 ( x ) R 13 ( xy ) R 23 ( y ) = R 23 ( y ) R 13 ( xy ) R 12 ( x ) , (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 finite dimensional Z 2 graded linear space and the tensor prod ucts app earing in the ab o ve definitions should b e un dersto o d in the graded sense. F or instance, we ha v e [ A ⊗ B ] il j k = A i j B l k ( − 1) ( p i + p j ) p l for generic matrices A and B . Th e Grassmann parities p i assume v alues on the group Z 2 and enable us to distin gu ish b osonic and fermionic degrees of freedom. More sp ecifically , the α th degree of freedom is distinguished b y the Grassmann parit y p i = 0(1) for i b osonic (fermionic). An imp ortant class of solutions of the Y ang-Baxter equation (1) is denominated trigonometric R -matrices conta in ing an additional parameter q b esides the sp ectral p arameter. Usually suc h R - matrices h a ve their ro ots in the U q [ G ] quant um group framework, which p ermit us to asso ciate a fun d amen tal trigonometric R -matrix to eac h Lie algebra or Lie sup eralgebra G [7, 8, 17]. In particular, explicit R -matrices were exhibited in [18, 19] for a v ariet y of qu an tum su p eralgebras in terms of standard W eyl matrices, p ro vidin g in this w a y a su itable basis f or the analysis of the corresp ondin g reflection equation. The U q [ osp ( r | 2 m ) (1) ] inv arian t R -matrices are give n by R ( x ) = N X i =1 i 6 = i ′ ( − 1) p i a i ( x ) ˆ e ii ⊗ ˆ e ii + b ( x ) N X i,j =1 i 6 = j,i 6 = j ′ ˆ e ii ⊗ ˆ e j j + ¯ c ( x ) X i,j =1 ij,i 6 = j ′ ( − 1) p i p j ˆ e j i ⊗ ˆ e ij + N X i,j =1 ( − 1) p i d i,j ( x ) ˆ e ij ⊗ ˆ e i ′ j ′ (2) where N = r + 2 m is the dimension of the graded sp ace with r b osonic and 2 m f er m ionic degrees of freedom. Here i ′ = N + 1 − i corresp ond s to the conjugated index of i and ˆ e ij refers to a u sual N × N W eyl matrix with only one non-null entry with v alue 1 at the ro w i and column j . 2 In [20] it w as demonstrated that the u se of an ap p ropriate grading s tructure p la ys a decisiv e role in the inv estigation of th e thermo dyn amic limit and finite size prop erties of integrable qu an tu m spin chains based on su p eralgebras. I n wh at follo w s we sh all adopt the grading str u cture p i =    0 for i = 1 , ..., m and i = r + m + 1 , ..., N 1 for i = m + 1 , ..., r + m , (3) and the corr esp onding Boltzmann we igh ts a i ( x ), b ( x ), c ( x ), ¯ c ( x ) and d ij ( x ) are then giv en by a i ( x ) = ( x − ζ )( x (1 − p i ) − q 2 x p i ) , b ( x ) = q ( x − 1) ( x − ζ ) , c ( x ) = (1 − q 2 )( x − ζ ) , ¯ c ( x ) = x (1 − q 2 )( x − ζ ) (4) and d i,j ( x ) =                  q ( x − 1)( x − ζ ) + x ( q 2 − 1)( ζ − 1) , ( i = j = j ′ ) ( x − 1)[( x − ζ )( − 1) p i q 2 p i + x ( q 2 − 1)] , ( i = j 6 = j ′ ) ( q 2 − 1)[ ζ ( x − 1) θ i q t i θ j q t j − δ i,j ′ ( x − ζ )] , ( i < j ) ( q 2 − 1) x [( x − 1) θ i q t i θ j q t j − δ i,j ′ ( x − ζ )] , ( i > j ) (5) where ζ = q r − 2 m − 2 . The remaining v ariables θ i and t i dep end strongly on the grad in g str ucture considered and they are determined b y the relations θ i =          ( − 1) − p i 2 , 1 ≤ i < N +1 2 1 , i = N +1 2 ( − 1) p i 2 , N +1 2 < i ≤ N (6) t i =                    i + [ 1 2 − p i + 2 X i ≤ j < N +1 2 p j ] , 1 ≤ i < N + 1 2 N + 1 2 , i = N + 1 2 i − [ 1 2 − p i + 2 X N +1 2 i ′ (21) where F ( x ) = b 2 ( x ) − a 1 ( x ) d 1 , 1 ( x ) b 2 ( x ) d 1 , 2 ( x ) d 2 , 1 ( x ) − c 1 ( x ) c 2 ( x ) d 2 1 , 1 ( x ) k 1 ,N ( x ) β 1 ,N (22) 6 and those matrix elemen ts with the b osonic lab els i 6 = j = m + 1 , ..., m + r are write in terms of k m +1 ,N − m ( x ) k i,j ( x ) =          G ( x )  β i,j c 1 ( x ) d m +1 ,m +1 ( x ) + β j ′ ,i ′ b ( x ) d i,j ′ ( x )  , j < i ′ β i,i ′ k m +1 ,N − m ( x ) β m +1 ,N − m , i = j ′ G ( x )  β i,j c 2 ( x ) d m +1 ,m +1 ( x ) + β j ′ ,i ′ b ( x ) d i,j ′ ( x )  , j > i ′ (23) where G ( x ) = b 2 ( x ) − a m +1 ( x ) d m +1 ,m +1 ( x ) b 2 ( x ) d 1 , 2 ( x ) d 2 , 1 ( x ) − c 1 ( x ) c 2 ( x ) d 2 m +1 ,m +1 ( x ) k m +1 ,N − m ( x ) β m +1 ,N − m . (24) In general, the non-diagonal elemen ts k i,i ( x ) ha ve the stru cture k i,j ( x ) =                β i,j xG ( x ) , i > j ′ β i,j xG ( x ) , i > j ′ β i,j G ( x ) H f ( x ) , i = j ′ ( fermionic) β i,j G ( x ) H b ( x ) , i = j ′ ( b osonic) (25) Here and in w hat follo ws G ( x ) is an arbitrary fun ction satisfying the regular condition k i,j (1) = δ ij and β i,j = d dx [ k i,j ( x )] x =1 , H f ( x ) = x − ǫq √ ζ 1 − ǫq √ ζ , H b ( x ) = q x + ǫ √ ζ q + ǫ √ ζ . (2 6) Moreo v er, the diago nal en tries k i,i ( x ) satisfy defined recurrence r elations whic h dep end on the b osonic and fermionic degree of freedom. Ho wev er, for the U q [ osp ( r | 2 m ) (1) ] mo del we ha v e found K -matrix solutions with b oth degree of freedom only for the cases with m = 1 and for the cases with r = 1 an d r = 2. In this section we shall f o cus on the non-diagonal solutions of the graded reflection equation (12). W e h a ve found t welv e classes of n on-diagonal solutions that we refer in what follo w s as solutions of t yp e M 1 to t yp e M 12 . Explicitly , w e ha ve thr ee classes of solutions: solutions with only fermionic degree of freedom, named fer m ionic K -matrices; five classes of solutions with only b osonic degree freedom, named b osonic K -matrices and four solutions with b oth d egree of freedom, named complete K -matrices. 4.1 F ermionic K-matrices Here we s hall f o cus on th e n on-diagonal s olutions of the graded r efl ection equation (12) turning off the b osonic degree of freedom i.e. k i,j ( x ) = 0 for i 6 = j = m + 1 , ..., m + r . W e ha v e foun d three classes of non-diagonal solutions that we r efer in what follo ws as solutions of t yp e M 1 to type M 3 : 7 4.1.1 Solution M 1 The solution of t yp e M 1 is v alid only for the U q [ osp ( r | 2) (1) ] mo dels with r ≥ 1 and the K -matrix has the follo wing blo c k stru cture 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 )      , (27) where O a × b is a a × b n u ll matrix and K 1 ( x ) =  x − 1 2 β ( x − 1)( q 2 ζ − x )  I r × r . (28) Here an d in what follo ws I r × r denotes a r × r identit y matrix and the remainin g n on-n u ll en tries are giv en by k 1 , 1 ( x ) = x − 1 2 β ( x − 1)( q 2 ζ + 1) , k 1 ,N ( x ) = 1 2 β 1 ,N ( x 2 − 1) , k N , 1 ( x ) = − 1 2 β 2 β 1 ,N q 2 ζ ( x 2 − 1) , k N ,N ( x ) = x − x 2 β ( x − 1)( q 2 ζ + 1) . (29) where β = β 2 , 2 − β 1 , 1 and β 1 ,N are t wo free p arameters. W e remark her e that this solution for r = 2 consist of a particular case of the three p arameter solution giv en in the app end ix B for the U q [ osp (2 | 2) (1) ] v ertex m o del. 4.1.2 Solution M 2 The U q [ osp ( r | 4) (1) ] ve rtex m o dels admit the solution M 2 whose corresp onding K -matrix has the follo wing structure K − ( x ) =            k 1 , 1 ( x ) k 1 , 2 ( x ) k 1 ,N − 1 ( x ) k 1 ,N ( x ) k 2 , 1 ( x ) k 2 , 2 ( x ) O 2 × r 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 ) O 2 × r k N − 1 ,N − 1 ( x ) k N − 1 ,N ( x ) k N , 1 ( x ) k N , 2 ( x ) k N ,N − 1 ( x ) k N ,N ( x )            , (30) 8 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 ) = β 1 , 2 G ( x ) k 2 , 1 ( x ) = β 2 , 1 G ( x ) k 1 ,N − 1 ( x ) = β 1 ,N − 1 G ( x ) k N − 1 , 1 ( x ) = θ N − 1 q t N − 1 θ 2 q t 2 β 1 , 2 β 2 , 1 β 1 ,N − 1 G ( x ) k 2 ,N − 1 ( x ) = − β 2 , 1 β 1 ,N β 1 , 2 G ( x ) H f ( x ) k N − 1 , 2 ( x ) = q r − 2 β 1 , 2 β 2 , 1 β 1 ,N β 2 1 ,N − 1 G ( x ) H f ( x ) k 2 ,N ( x ) = − θ 2 q t 2 θ 1 q t 1 ǫ √ ζ β 1 ,N − 1 xG ( x ) k N , 2 ( x ) = − θ N q t N θ 2 q t 2 ǫ √ ζ β 2 , 1 β 1 , 2 β 1 ,N − 1 xG ( x ) k N − 1 ,N ( x ) = − θ N − 1 q t N − 1 θ 1 q t 1 ǫ √ ζ β 1 , 2 xG ( x ) k N ,N − 1 ( x ) = − θ N q t N θ 2 q t 2 ǫ √ ζ β 2 , 1 xG ( x ) k N , 1 ( x ) = θ N − 1 q t N − 1 θ 2 q t 2 β 2 2 , 1 β 1 ,N β 2 1 ,N − 1 G ( x ) H f ( x ) k 1 ,N ( x ) = β 1 ,N G ( x ) H f ( x ) , (31) With resp ect to the diagonal matrix elemen ts, w e h a ve the follo w ing expr essions k 1 , 1 ( x ) = [( β N ,N − β 3 , 3 ) x − ( β N ,N − β 1 , 1 − 2) xH f ( x ) + β 3 , 3 − β 1 , 1 ] G ( x ) x 2 − 1 +[1 + ǫq r − 1 p ζ ] ∆( x ) x 2 − 1 (32) for the recurrence relation k 2 , 2 ( x ) = k 1 , 1 ( x ) + ( β 2 , 2 − β 1 , 1 ) G ( x ) , k 3 , 3 ( x ) = k 1 , 1 ( x ) + ( β 3 , 3 − β 1 , 1 ) G ( x ) + ∆( x ) , k N − 1 ,N − 1 ( x ) = k 3 , 3 ( x ) + ( β N − 1 ,N − 1 − β 3 , 3 ) xG 2 ( x ) + ǫq r − 1 p ζ ∆ ( x ) , k N ,N ( x ) = x 2 k 1 , 1 ( x ) + ( β N ,N − β 1 , 1 − 2) xH f ( x ) G ( x ) . (33) where ∆( x ) = β 2 , 1 β 1 ,N β 1 ,N − 1  x − 1 1 − ǫq √ ζ  G ( x ) (34) The diagonal ent ries (33) dep end on the v ariables β α,α whic h are related to the free parameters β 1 , 2 , β 2 , 1 , β 1 ,N − 1 and β 1 ,N through the expressions β 2 , 2 = β 1 , 1 − 2 ǫ q 1 2 r − ǫ − β 2 , 1 β 1 ,N β 1 ,N − 1 (1 − ǫq p ζ ) , β 3 , 3 = β 2 , 2 + β 2 , 1 β 1 ,N β 1 ,N − 1 , β N − 1 ,N − 1 = β 1 , 1 − 2 ǫ q 1 2 r − ǫ + β 2 , 1 β 1 ,N β 1 ,N − 1 ( q 1 2 r + ǫ ) q p ζ , β N ,N = β 1 , 1 + 2 + β 2 , 1 β 1 ,N β 1 ,N − 1 ( q 2 + 1)( ǫq p ζ ) . (35) where β 2 , 1 = ǫ β 1 , 2 β 1 ,N − 1 β 1 ,N 1 q √ ζ − 2 ( q 1 2 r − ǫ )(1 − ǫq √ ζ ) ! β 1 ,N − 1 β 1 ,N (36) Therefore w e h a ve a solution with three fr ee parameters. 9 4.1.3 Solution M 3 This class of solution is v alid for all U q [ osp ( r | 2 m ) (1) ] vertex mo dels with m ≥ 3 and the corresp ond- ing K -matrix p ossess the follo wing general form K − ( x ) =                   k 1 , 1 ( x ) · · · k 1 ,m ( x ) k 1 ,r + m +1 ( x ) · · · k 1 ,N ( x ) . . . . . . . . . O m × r . . . . . . . . . k m, 1 ( x ) · · · k m,m ( 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 r + m +1 ,r + m +1 ( x ) · · · k r + m +1 ,N ( x ) . . . . . . . . . O m × r . . . . . . . . . k N , 1 ( x ) · · · k N ,m ( x ) k N ,r + m +1 ( x ) · · · k N ,N ( x )                   , (37) where K 3 ( x ) is a d iagonal matrix giv en b y K 3 ( x ) = k m +1 ,m +1 ( x ) I r × r . (38) With resp ect to the elemen ts of the last column, we h a ve th e follo wing expression k i,N ( x ) = − ǫ √ ζ θ i q t i θ 1 q t 1 β 1 ,i ′ xG ( x ) (39) i = 2 , . . . , m and i = r + m + 1 , . . . , N − 1 , In their turn the en tries of the fi rst column are mainly giv en b y k i, 1 ( x ) = θ i q t i θ 2 q t 2 β 2 , 1 β 1 ,i ′ β 1 ,N − 1 G ( x ) , (40) i = 3 , . . . , m and i = r + m + 1 , . . . , N − 1 . In the last r o w we h a ve k N ,j ( x ) = − ǫ √ ζ θ N q t N θ 2 q t 2 β 2 , 1 β 1 ,j β 1 ,N − 1 xG ( x ) (41) j = 2 , . . . , m and j = r + m + 1 , . . . , N − 1 , while the elements of th e fir st ro w are k 1 ,j ( x ) = β 1 ,j G ( x ) for j = 2 , . . . , m and j = r + m + 1 , . . . , N − 1. Concerning the elemen ts of the secondary diagonal, they are giv en b y k i,i ′ ( x ) = − q 2 θ 1 q t 1 θ i ′ q t i ′ (1 − ǫ q √ ζ ) 2 ( q + 1) 2 β 2 1 ,i ′ β 1 ,N G ( x ) H f ( x ) i = 2 , . . . , m , i 6 = i ′ and i = r + m + 1 , . . . , N − 1 , (42) 10 while the remaining en tries k 1 ,N ( x ) and k N , 1 ( x ) are determined by the follo wing expressions k 1 ,N ( x ) = β 1 ,N G ( x ) H f ( x ) k N , 1 ( x ) = θ N − 1 q t N − 1 θ 2 q t 2 β 1 ,N β 2 2 , 1 β 2 1 ,N − 1 G ( x ) H f ( x ) (43) The remaining matrix elemen ts k i,j ( x ) with i 6 = j are then k i,j ( x ) =    − ǫ √ ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q +1  β 1 ,i ′ β 1 ,j β 1 ,N G ( x ) , i < j ′ 2 ≤ i, j ≤ N − 1 1 ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q +1  β 1 ,i ′ β 1 ,j β 1 ,N xG ( x ) , i > j ′ 2 ≤ i, j ≤ N − 1 (44) and k 2 , 1 ( x ) = 2( − 1) m ( ǫq √ ζ − 1) q m − 2 (1 + ǫ √ ζ )( q 1 2 r − 1 + ( − 1) m ǫ )( q 1 2 r − ( − 1) m ǫ ) β 1 ,N − 1 β 1 ,N G ( x ) , k 1 ,m ( x ) = 2 ζ ( q + 1) 2 q m − 1 (1 − ǫq √ ζ )(1 + ǫ √ ζ )( q 1 2 r − 1 + ( − 1) m ǫ )( q 1 2 r − ( − 1) m ǫ ) β 1 ,N β 1 ,m ′ G ( x ) , (45) and the p arameters β 1 ,j are constrained by the relation β 1 ,j = − β 1 ,j +1 β 1 ,N − j β 1 ,N +1 − j j = 2 , . . . , m − 1 . (46) With regard to the diagonal matrix elemen ts, they are giv en by k i,i ( x ) =                      k 1 , 1 ( x ) + ( β i,i − β 1 , 1 ) G ( x ) , 2 ≤ i ≤ m k 1 , 1 ( x ) + ( β m +1 ,m +1 − β 1 , 1 ) G ( x ) + ∆( x ) , i = m + 1 k m +1 ,m +1 ( x ) + ( β r + m +1 ,r + m +1 − β m +1 ,m +1 ) xG ( x ) + ǫq r − 1 √ ζ ∆ ( x ) , i = r + m + 1 k i − 1 ,i − 1 ( x ) + ( β i,i − β i − 1 ,i − 1 ) xG ( x ) , r + m + 2 ≤ i ≤ N (47) The last term of the recurrence relation (47) is iden tified with k N ,N ( x ) = x 2 k 1 , 1 ( x ) + ( β N ,N − β 1 , 1 − 2) xG ( x ) H f ( x ) (48) to find k 1 , 1 ( x ) = [( β N ,N − β m +1 ,m +1 ) x − ( β N ,N − β 1 , 1 − 2) xH f ( x ) + β m +1 ,m +1 − β 1 , 1 ] G ( x ) x 2 − 1 +[1 + ǫq r − 1 p ζ ] ∆( x ) x 2 − 1 (49) where ∆( x ) = − 2( x − 1) G ( x ) (1 + ǫ √ ζ )( q 1 2 r − 1 + ( − 1) m ǫ )( q 1 2 r − ( − 1) m ǫ ) . (50) In their turn the diagonal parameters β i,i are fixed by the r elations β i,i =        β 1 , 1 + Λ m i − 2 P k =0 ( − 1 q ) k , 2 ≤ i ≤ m β r + m +1 ,r + m +1 − ( − 1) m ǫq 1 2 r Λ m i − r − m − 2 P k =0 ( − 1 q ) k , r + m + 2 ≤ i ≤ N (51) 11 and β m +1 ,m +1 = β 1 , 1 + Λ m  q q + 1 + ( − 1) m q 2 − m ( q + 1) 2 1 + ǫ √ ζ ǫ √ ζ  , (52) β r + m +1 ,r + m +1 = β m +1 ,m +1 + 2 q r − 2 ( ǫq √ ζ − 1) (1 + ǫ √ ζ )( q 1 2 r − 1 + ( − 1) m ǫ )( q 1 2 r − ( − 1) m ǫ ) , (53) with Λ m = − 2( − 1) m q m − 2 (1 + q ) 2 ǫ √ ζ (1 + ǫ √ ζ )( q 1 2 r − 1 + ( − 1) m ǫ )( q 1 2 r − ( − 1) m ǫ ) . (54) The class of solution M 3 has a total amoun t of m f ree p arameters namely β 1 ,r + m +1 , . . . , β 1 ,N . Here w e note that w e can take the limit m = 2 in the class M 3 to get a tw o parameter solution whic h is a p articular case of the th ree parameter solution of the class M 2 . 4.2 Bosonic Solutions Here we will tur n off the fermionic degree of fr eedom, i.e. k i,j ( x ) = 0 , for i 6 = j = { 1 , ..., m } and i 6 = j = { r + m + 1 , ..., N } in order to get K -matrix solutions w ith only b osonic degree of fr eedom. W e hav e found five classes of non-diagonal s olutions th at w e r efer in wh at follo ws as solutions of t yp e M 4 to t yp e M 8 : 4.2.1 Solution M 4 : This f amily of solutions is v alid only for the U q [ osp (2 | 2 m ) (1) ] v ertex mo del with m ≥ 1 and the corresp ondin g K -matrix has the follo wing blo c k diagonal structure K − ( x ) =         k 1 , 1 ( x ) I m × n 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 × n         (55) The non-null en tries 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 ) = β 2 x ( x 2 − 1)(1 − ζ ) x 2 − ζ , k m +2 ,m +1 ( x ) = 2 β x ( x 2 − 1) ζ (1 − ζ )( x 2 − ζ ) , k m +2 ,m +2 ( x ) = x 2 (1 − ζ ) x 2 − ζ . (56) where β = β m +1 ,m +2 is a f ree parameter. 12 4.2.2 Solution M 5 : The family M 5 is acceptable by the vertex m o del U q [ osp (3 | 2 m ) (1) ] and it is c haracterized by a K -matrix of th e 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            . (57) The non-diagonal matrix elemen ts are given b y the f ollo wing exp ressions k m +1 ,m +2 ( x ) = β m +1 ,m +2 G ( x ) k m +2 ,m +1 ( x ) = β m +2 ,m +1 G ( x ) k m +3 ,m +2 ( x ) = β m +3 ,m +2 xG ( x ) k m +2 ,m +3 ( x ) = β m +2 ,m +3 xG ( x ) (58) k m +1 ,m +3 ( x ) = β m +1 ,m +3 G ( x ) H b ( x ) k m +3 ,m +1 ( x ) = β m +3 ,m +1 G ( x ) H b ( x ) where H b ( x ) is g iv en by (26). In their turn the ab o ve parameters β i,j are constrained b y the relations β m +2 ,m +3 = ǫ √ q √ ζ β m +1 ,m +2 , β m +3 ,m +2 = ǫ √ q √ ζ β m +2 ,m +1 , β m +3 ,m +1 = β m +1 ,m +3  β m +2 ,m +1 β m +1 ,m +2  2 , β m +2 ,m +1 = − ǫ √ q √ ζ β m +1 ,m +2 β m +1 ,m +3 " √ q ( q + 1) β 2 m +1 ,m +2 β m +1 ,m +3 − 2 ζ ( q + ǫ √ ζ )( ǫ √ ζ − 1) # . (59) The diagonal en tries are then giv en by k 1 , 1 ( x ) =  2 x 2 − 1 − ǫ ( q x + ζ ) √ q ζ x ( x + 1) β m +1 ,m +3 β m +2 ,m +1 β m +1 ,m +2  G ( x ) H b ( x ) − ǫ √ q √ ζ β 2 m +1 ,m +2 β m +1 ,m +3 G ( x ) x + 1 k m +1 ,m +1 ( x ) =  2 x 2 − 1 − ǫ ( q − ζ ) √ q ζ ( x + 1) β m +1 ,m +3 β m +2 ,m +1 β m +1 ,m +2  G ( x ) H b ( x ) − ǫ √ q √ ζ β 2 m +1 ,m +2 β m +1 ,m +3 G ( x ) x + 1 k m +2 ,m +2 ( x ) = k m +1 ,m +1 ( x ) + ( β m +2 ,m +2 − β m +1 ,m +1 ) G ( x ) + ∆( x ) k m +3 ,m +3 ( x ) = k m +2 ,m +2 ( x ) + ( β m +3 ,m +3 − β m +2 ,m +2 ) xG ( x ) + ǫ p ζ ∆ ( x ) k N ,N ( x ) = x 2 k 1 , 1 ( x ) (60) where ∆( x ) = − √ q ( q + ǫ √ ζ ) β m +2 ,m +1 β m +1 ,m +3 β m +1 ,m +2 ( x − 1) G ( x ) , (61) 13 and the p arameters β m +1 ,m +1 , β m +2 ,m +2 and β m +3 ,m +3 are fixed by the r elations β m +1 ,m +1 = β 1 , 1 + ǫ √ ζ √ q β m +1 ,m +3 β m +2 ,m +1 β m +1 ,m +2 β m +2 ,m +2 = β 1 , 1 + 2 ǫ √ ζ ( q + ǫ √ ζ ) + ǫ √ ζ √ q β 2 m +1 ,m +2 β m +1 ,m +3 (62) β m +3 ,m +3 = β 1 , 1 + 2 + ǫ √ q √ ζ β m +1 ,m +3 β m +2 ,m +1 β m +1 ,m +2 . The solution M 5 p ossess t wo fr ee parameters n amely β m +1 ,m +2 and β m +1 ,m +3 . 4.2.3 Solution M 6 : The solution M 6 is adm itted for the U q [ osp (4 | 2 m ) (1) ] mo dels with the follo 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               . (63) The non-diagonal elemen ts are all group ed in the 4 × 4 central blo c k m atrix. With resp ect to this cen tral blo c k, the entries of th e secondary diagonal are giv en by k m +2 ,m +3 ( x ) = − β m +2 ,m +1 β m +1 ,m +2 k m +1 ,m +4 ( x ) k m +3 ,m +2 ( x ) = − q 2 ζ β m +1 ,m +2 Γ 2 m β m +2 ,m +1 β 2 m +1 ,m +4 k m +1 ,m +4 ( x ) (64) k m +4 ,m +1 ( x ) = q 2 ζ Γ 2 m β 2 m +1 ,m +4 k m +1 ,m +4 ( x ) , and the r emaining non-diagonal elemen ts can b e written as k m +1 ,m +2 ( x ) = β m +1 ,m +2 G 1 ( x ) k m +2 ,m +1 ( x ) = β m +2 ,m +1 G 1 ( x ) k m +1 ,m +3 ( x ) = β m +1 ,m +3 G 2 ( x ) k m +3 ,m +1 ( x ) = q 2 ζ β m +1 ,m +2 β m +1 ,m +3 Γ 2 m β m +2 ,m +1 β 2 m +1 ,m +4 G 2 ( x ) k m +2 ,m +4 ( x ) = − β m +2 ,m +1 β m +1 ,m +4 Γ m xG 1 ( x ) k m +4 ,m +2 ( x ) = − q 2 ζ β m +1 ,m +2 Γ m β m +1 ,m +4 xG 1 ( x ) k m +3 ,m +4 ( x ) = − q 2 ζ β m +1 ,m +2 β m +1 ,m +3 Γ m β m +1 ,m +4 β m +2 ,m +1 xG 2 ( x ) k m +4 ,m +3 ( x ) = − q 2 ζ β m +1 ,m +3 Γ m β m +1 ,m +4 xG 2 ( x ) , (65) 14 where G 1 ( x ) =  ζ − q 2 x q 2 ( x − 1) + β m +1 ,m +3 Γ m β m +2 ,m +1 β m +1 ,m +4  q 2 ( x − 1) ( ζ − q 2 x 2 ) k m +1 ,m +4 ( x ) β m +1 ,m +4 , G 2 ( x ) =  ζ − q 2 x x − 1 + ζ β m +2 ,m +1 β m +1 ,m +4 β m +1 ,m +3 Γ m  ( x − 1) ( ζ − q 2 x 2 ) k m +1 ,m +4 ( x ) β m +1 ,m +4 (66) and Γ m = β m +1 ,m +2 β m +1 ,m +3 β m +1 ,m +4 + 2 ζ q 2 − ζ . (67) In their turn the diagonal en tries are give n by the follo wing exp ressions k 1 , 1 ( x ) =  ζ − q 2 x ( x + 1)( ζ − q 2 x 2 )  β m +1 ,m +2 β m +2 ,m +1 Γ m + β m +1 ,m +2 β m +1 ,m +3 ζ β m +1 ,m +4  q 2 β m +1 ,m +3 Γ m β m +1 ,m +4 β m +2 ,m +1 + ( ζ + q 2 x 2 ) x  + 2  ζ − q 2 x 2  x ( x 2 − 1) ( ζ − q 2 ) ) k m +1 ,m +4 ( x ) β m +1 ,m +4 (68) with the recurrence relation k m +1 ,m +1 ( x ) = k 1 , 1 ( x ) + ( β m +1 ,m +1 − β 1 , 1 ) G 1 ( x ) + ∆ 1 ( x ) , k m +2 ,m +2 ( x ) = k m +1 ,m +1 ( x ) + ( β m +2 ,m +2 − β m +1 ,m +1 ) G 1 ( x ) , 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 ) + ( β m +4 ,m +4 − β m +3 ,m +3 ) xG 2 ( x ) , k N ,N ( x ) = x 2 k 1 , 1 ( x ) , (69) where ∆ 1 ( x ) = Γ m  Γ m β m +1 ,m +3 q 2 x + ζ β m +1 ,m +4 β m +2 ,m +1  β 2 m +1 ,m +4 β m +2 ,m +1 ( x − 1) k m +1 ,m +4 ( x ) x ( ζ − q 2 x 2 ) ∆ 2 ( x ) = β m +1 ,m +2  ζ β 2 m +2 ,m +1 β 2 m +1 ,m +4 − q 2 β 2 m +1 ,m +3 Γ 2 m  Γ m β 3 m +1 ,m +4 β m +2 ,m +1 x ( ζ − q 2 ) k m +1 ,m +4 ( x ) ζ ( ζ − q 2 x 2 ) . (70) The v ariables β i,j are giv en in terms of the fr ee p arameters β m +1 ,m +2 , β m +2 ,m +1 , β m +1 ,m +3 and β m +1 ,m +4 through th e relations here β m +2 ,m +2 = β 1 , 1 + 2 ζ ζ − q 2 − β m +1 ,m +2 β m +2 ,m +1 Γ m β m +3 ,m +3 = β 1 , 1 + 2 ζ ζ − q 2 − q 2 ζ Γ m β m +1 ,m +2 β 2 m +1 ,m +3 β m +2 ,m +1 β 2 m +1 ,m +4 β m +4 ,m +4 = β 1 , 1 + 2 ζ ζ − q 2 − q 2 ζ β m +1 ,m +2 β m +1 ,m +3 β m +1 ,m +4 . ( 71) W e remark h ere that the f orm of this class of solutio n differs fr om general form (25), which is reco v ered by the condition G 1 ( x ) = G 2 ( x ), but r ed ucing th e n um b er of free parameters. 15 4.2.4 Solution M 7 : The vertex mo del U q [ osp (2 n | 2 m ) (1) ] adm its th e solution M 7 for n ≥ 3, wh ose K -matrix has the follo wing blo 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            (72) The central blo ck matrix cluster all non-diagonal elements d ifferent f rom zero. Concerning that cen tral blo c k, w e h a ve the follo win g expressions determin in g entries of th e b orders, k i, 2 n + m ( x ) = ǫ √ ζ q t i − t m +1 β m +1 ,i ′ xG ( x ) , i = m + 2 , . . . , 2 n + m − 1 k 2 n + m,j ( x ) = ǫ √ ζ q t 2 n + m − t m +2 β m +2 ,m +1 β m +1 ,j β m +1 , 2 n + m − 1 xG ( x ) , j = m + 2 , . . . , 2 n + m − 1 k i,m +1 ( x ) = q t i − t m +2 β m +2 ,m +1 β m +1 ,i ′ β m +1 , 2 n + m − 1 G ( x ) , i = m + 3 , . . . , 2 n + m − 1 k m +1 ,j ( x ) = β m +1 ,j G ( x ) . j = m + 2 , . . . , 2 n + m − 1 (73) The en tries of the secondary diagonal are giv en by k i,i ′ ( x ) =                β m +1 , 2 n + m G ( x ) H b ( x ) , i = m + 1 q 2 m q t m +1 − t i ′  q + ǫ √ ζ q + 1  2 β 2 m +1 ,i ′ β m +1 , 2 n + m G ( x ) H b ( x ) , i = m + 2 , ..., 2 n + m − 1 q t 2 n + m − 1 − t m +2 β 2 m +2 ,m +1 β m +1 , 2 n + m β 2 m +1 , 2 n + m − 1 G ( x ) H b ( x ) , i = 2 n + m (74) and the r emaining non-diagonal elemen ts are determined by the expression k i,j ( x ) =              ǫ √ ζ q t i − t m +1  q + ǫ √ ζ q + 1  β m +1 ,i ′ β m +1 ,j β m +1 , 2 n + m G ( x ) , i < j ′ , m + 1 < i, j < 2 n + m 1 ζ q t i − t m +1  q + ǫ √ ζ q + 1  β m +1 ,i ′ β m +1 ,j β m +1 , 2 n + m xG ( x ) , i > j ′ , m + 1 < i, j < 2 n + m . (75) and k m +1 ,m + n ( x ) = 2 ǫ √ ζ (1 − ǫ √ ζ )( q + ǫ √ ζ ) ( − 1) n (1 + q ) 2 ζ ( q n − ( − 1) n ǫ √ ζ )( q n − 1 − ( − 1) n ǫ √ ζ ) β m +1 , 2 n + m β m +1 ,n + m +1 G ( x ) , k m +2 ,m +1 ( x ) = − 2 ǫ √ ζ (1 − ǫ √ ζ ) q ( q + ǫ √ ζ ) ( q n − ( − 1) n ǫ √ ζ )( q n − 1 − ( − 1) n ǫ √ ζ ) β m +1 , 2 n + m − 1 β m +1 , 2 n + m G ( x ) . ( 76) In their turn the diagonal en tries k i,i ( x ) are giv en by k 1 , 1 ( x ) =  x − ǫ √ ζ 1 − ǫ √ ζ   xq n − ( − 1) n ǫ √ ζ q n − ( − 1) n ǫ √ ζ   xq n − 1 − ( − 1) n ǫ √ ζ q n − 1 − ( − 1) n ǫ √ ζ  2 G ( x ) x ( x 2 − 1) (77) 16 k i,i ( x ) =                      k 1 , 1 ( x ) + Γ n ( x ) , i = m + 1 k m +1 ,m +1 ( x ) + ( β i,i − β m +1 ,m +1 ) G ( x ) , i = m + 2 , ..., m + n k n + m,n + m ( x ) , i = n + m + 1 k n + m,n + m ( x ) + ( β i,i − β n + m,n + m ) xG ( x ) , i = n + m + 2 , ..., 2 n + m x 2 k 1 , 1 ( x ) , i = N (78) where Γ n ( x ) = − 2 ζ ( q x + ǫ √ ζ ) (1 − ǫ √ ζ )( q n − ( − 1) n ǫ √ ζ )( q n − 1 − ( − 1) n ǫ √ ζ ) G ( x ) x . (79) The diagonal parameters β i,i are then fixed by the r elations β i,i =                          β 1 , 1 − q + ǫ √ ζ (1 + q ) 2 ∆ n , i = m + 1 β m +1 ,m +1 + ∆ n i − m − 2 X k =0 ( − q ) k , i = m + 2 , ..., n + m β n + m,n + m , i = n + m + 1 β n + m,n + m − ǫ ( − 1) n q 2 ∆ n i − n − m − 2 X k =0 ( − q ) k , i = m + 2 , ..., n + m (80) and the au x iliary p arameter ∆ n is giv en b y ∆ n = 2 ζ (1 + q ) 2 (1 − ǫ √ ζ )( q n − ( − 1) n ǫ √ ζ )( q n − 1 − ( − 1) n ǫ √ ζ ) . (81) Besides the ab ov e r elations the follo wing constraints sh ould also holds β m +1 ,m + j = − β m +1 ,j + m +1 β m +1 , 2 n + m − j β m +1 , 2 n + m +1 − j j = 2 , . . . , n − 1 , (82) and β m +1 ,m + n +1 , . . . , β m +1 , 2 n + m are regarded as the n free parameters. Th e case n = 2 is a particular solution of the three parameter class M 6 . 4.2.5 Solution M 8 : F or n ≥ 2 the vertex mo del U q [ osp (2 n + 1 | 2 m ) (1) ] admits the f amily of solutions M 8 whose K -matrix is of the 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 +1+ m ( x ) . . . . . . . . . k 2 n +1+ m,m +1 ( x ) · · · k 2 n + m, 2 n + m ( x ) O 2 n +1 × m O m × m O m × 2 n +1 k N ,N ( x ) I m × m            (83) In th e cen tral blo c k matrix we find all n on-diagonal elements different from zero similarly to the structure of the solution M 7 . The b orders of the cen tral blo ck are then determined by the follo wing 17 expressions k i, 2 n + m +1 ( x ) = ǫ √ ζ q t i − t m +1 β m +1 ,i ′ xG ( x ) i = m + 2 , . . . , 2 n + m k 2 n + m +1 ,j ( x ) = ǫ √ ζ q t 2 n + m +1 − t m +2 β m +2 ,m +1 β m +1 ,j β m +1 , 2 n + m xG ( x ) j = m + 2 , . . . , 2 n + m k i,m +1 ( x ) = q t i − t m +2 β m +2 ,m +1 β m +1 ,i ′ β m +1 , 2 n + m G ( x ) i = m + 3 , . . . , 2 n + m k m +1 ,j ( x ) = β m +1 ,j G ( x ) j = m + 2 , . . . , 2 n + m (84) The secondary diagonal elemen ts are give n b y k i,i ′ ( x ) =              β m +1 , 2 n + m +1 G ( x ) H b ( x ) , i = m + 1 q 2 m q t m +1 − t i ′  q + ǫ √ ζ q + 1  2 β 2 m +1 ,i ′ β m +1 , 2 n + m +1 G ( x ) H b ( x ) , i = m + 2 , ..., 2 n + m q t 2 n + m − t m +2 β ´ 2 m +2 ,m +1 β m +1 , 2 n + m +1 β 2 m +1 , 2 n + m G ( x ) H b ( x ) , i = 2 n + m + 1 (85) The remaining non-diagonal en tries are determined by k i,j ( x ) =              ǫ √ ζ q t i − t m +1  q + ǫ √ ζ q + 1  β m +1 ,i ′ β m +1 ,j β m +1 , 2 n + m +1 G ( x ) , i < j ′ , m + 2 < i, j < 2 n + m 1 ζ q t i − t m +1  q + ǫ √ ζ q + 1  β m +1 ,i ′ β m +1 ,j β m +1 , 2 n + m +1 xG ( x ) , i > j ′ , m + 2 < i, j < 2 n + m . (86) and k m +2 ,m +1 ( x ) = − ( − 1) n q ζ  q + ǫ √ ζ q + 1  2 β m +1 ,n + m β m +1 ,n + m +2 β m +1 , 2 n + m β 2 m +1 , 2 n + m +1 G ( x ) , k m +1 ,m + n ( x ) = ǫ ( q + 1)( − 1) n ( q m + 1 2 − ( − 1) n ǫ ) 2 " q m β 2 m +1 ,n + m +1 β m +1 ,n + m +2 + 2( q + 1) √ ζ (1 − ǫ √ ζ )( q + ǫ √ ζ ) β m +1 , 2 n + m +1 β m +1 ,n + m +2 # G ( x ) , (87) while the diagonal matrix elemen ts are giv en by the r elations k 1 , 1 ( x ) =  x − ǫ √ ζ 1 − ǫ √ ζ  xq m + 1 2 − ( − 1) n ǫ q m + 1 2 − ( − 1) n ǫ ! 2 2 G ( x ) x ( x 2 − 1) + q m ( xq 2 m +1 − 1) ( q + 1) √ ζ ( x − ǫ √ ζ )( q + ǫ √ ζ ) ( q m + 1 2 − ( − 1) n ǫ ) 2 β 2 m +1 ,m + n +1 β m +1 , 2 n + m +1 G ( x ) x ( x + 1) (88) k i,i ( x ) =                                  k 1 , 1 ( x ) + Γ( x ) , i = m + 1 k m +1 ,m +1 ( x ) + ( β i,i − β m +1 ,m +1 ) G ( x ) , i = m + 2 , ..., n + m k m +1 ,m +1 ( x ) + ( β n + m +1 ,n + m +1 − β m +1 ,m +1 ) G ( x ) + ∆( x ) , i = n + m + 1 k n + m +1 ,n + m +1 ( x ) + ( β n + m +2 ,n + m +2 − β n + m +1 ,n + m +1 ) xG ( x ) + ǫ √ ζ ∆( x ) , i = n + m + 2 k i − 1 ,i − 1 ( x ) + ( β i,i − β i − 1 ,i − 1 ) xG ( x ) , i = n + m + 3 , ..., 2 n + m + 1 x 2 k 1 , 1 ( x ) , i = N (89) 18 The auxiliary functions ∆( x ) and Γ( x ) are ∆( x ) = − q n ( q + ǫ √ ζ ) ζ ( q + 1) 2 β m +1 ,n + m β m +1 ,n + m +2 β m +1 , 2 n + m +1 ( x − 1) G ( x ) Γ( x ) = ( − 1) n +1 ( xq + ǫ √ ζ )( q + ǫ √ ζ ) ǫ √ ζ x ( q + 1) 2 β m +1 ,n + m β m +1 ,n + m +2 β m +1 , 2 n + m +1 G ( x ) , (90) and the p arameters β m +1 ,m + j are constrained b y the r ecurrence form u la β m +1 ,j + m = − β m +1 ,j + m +1 β m +1 , 2 n + m +1 − j β m +1 , 2 n + m +2 − j j = 2 , . . . , n − 1 . (91 ) In their turn the diagonal parameters β i,i are fixed by β i,i =                  β m +1 ,m +1 + Q n,m i − 2 − m X k =0 ( − q ) k , i = m + 2 , ..., n + m β n + m +2 ,n + m +2 − ( − 1) n q 2 m − n +1 ǫ p ζ Q n,m i − 3 − n − m X k =0 ( − q ) k , i = n + m + 3 , ..., 2 n + m + 1 (92) and β m +1 ,m +1 = β 1 , 1 − ǫ ( − 1) n q m − n + 1 2  q + ǫ √ ζ q + 1  2 β m +1 ,n + m β m +1 ,n + m +2 β m +1 , 2 n + m +1 β n + m +1 ,n + m +1 = β 1 , 1 − 2 ǫ ( − 1) n q m + 1 2 − ( − 1) n ǫ + q m (1 + ǫq m − n + 3 2 )( q n − ( − 1) n ) ( q + 1)( q m + 1 2 − ( − 1) n ǫ ) β 2 m +1 ,n + m +1 β m +1 , 2 n + m +1 β n + m +2 ,n + m +2 = β n + m +1 ,n + m +1 + q 2 m +1 − n ( q + ǫ √ ζ ) 2 ( q + 1) 2 β m +1 ,n + m β m +1 ,n + m +2 β m +1 , 2 n + m +1 − q 2 m − n + 1 2 ( q + ǫ √ ζ ) ( q + 1) β 2 m +1 ,n + m +1 β m +1 , 2 n + m +1 (93) where Q n,m = − 2( q + 1) 2 ( ǫ √ ζ − 1)( q m + 1 2 − ( − 1) n ǫ ) 2 + q m ( q + 1)( q + ǫ √ ζ ) √ ζ ( q m + 1 2 − ( − 1) n ǫ ) 2 β 2 m +1 ,n + m +1 β m +1 , 2 n + m +1 . ( 94) This solution has alt ogether n + 1 fr ee parameters corresp onding to the s et of v ariables β m +1 ,n + m +1 , . . . , β m +1 , 2 n + m +1 . 4.3 Complete K-matrices The complete K -matrices are solutions w ith all entries different from zero. This kind of solution will b e present only in four class: the mo d els with one or tw o b osonic degree of freedom, U q [ osp (1 | 2 m ) (1) ] and U q [ osp (2 | 2 m ) (1) ] resp ectiv ely and those mo dels w ith only tw o fermionic degree of fr eedom, U q [ osp (2 n | 2) (1) ] and U q [ osp (2 n + 1 | 2) (1) ] . The sp ecial cases U q [ osp (1 | 2) (1) ] and U q [ osp (2 | 2) (1) ] will b e presente d in app endices. 19 4.3.1 Solution M 9 : The family M 9 consist of a solution of th e reflection equation where all entries of the K -matrix are n on -null. This solution is admitted only by the U q [ osp (1 | 2 m ) (1) ] vertex mo d el. The asso ciated K -matrix is of the general form (13) and th e matrix elements of the b orders are mainly giv en b y k i,N ( x ) = − ǫ √ ζ θ i q t i θ 1 q t 1 β 1 ,i ′ xG ( x ) i = 2 , . . . , N − 1 k i, 1 ( x ) = θ i q t i θ 2 q t 2 β 2 , 1 β 1 ,i ′ β 1 ,N − 1 G ( x ) i = 3 , . . . , N − 1 k N ,j ( x ) = − ǫ √ ζ θ N q t N θ 2 q t 2 β 2 , 1 β 1 ,j β 1 ,N − 1 xG ( x ) j = 2 , . . . , N − 1 k 1 ,j ( x ) = β 1 ,j G ( x ) j = 2 , . . . , N − 1 . (95) The secondary diagonal is c h aracterized b y entries of the form k i,i ′ ( x ) =                β 1 ,N G ( x ) H f ( x ) , i = 1 − q 2 θ 1 q t 1 θ i ′ q t i ′  1 − ǫq √ ζ q + 1  2 β 2 1 ,i ′ β 1 ,N G ( x ) H f ( x ) , i 6 = { 1 , m + 1 , N } θ N − 1 q t N − 1 θ 2 q t 2 β 1 ,N β 2 2 , 1 β 2 1 ,N − 1 G ( x ) H f ( x ) , i = N . (96) and the r emaining non-diagonal elemen ts are giv en by k i,j ( x ) =        − ǫ √ ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q + 1  β 1 ,i ′ β 1 ,j β 1 ,N G ( x ) , i < j ′ , 2 < i, j < N − 1 1 ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q + 1  β 1 ,i ′ β 1 ,j β 1 ,N xG ( x ) , i > j ′ , 2 < i, j < N − 1 . (97) and k 1 ,m ( x ) = i √ q q − 1 β 2 1 ,m +1 β 1 ,m +2 G ( x ) , k 2 , 1 ( x ) = ( − 1) m +1 q 2 m  ǫq √ ζ − 1 q + 1  2 β 1 ,m β 1 ,m +2 β 1 , 2 m β 2 1 ,N G ( x ) , k 1 ,N ( x ) = i 2 q ( q 3 2 + ( − 1) m ǫ ) √ q + ( − 1) m ǫ (1 + ǫ √ ζ )( ǫq √ ζ − 1) √ ζ ( q + 1) 2 β 2 1 ,m +1 G ( x ) . (98) In their turn the diagonal en tries k α,α ( x ) are give n by the f ollo wing expression k 1 , 1 ( x ) = [( β N ,N − β m +1 ,m +1 ) x − ( β N ,N − β 1 , 1 − 2) xH f ( x ) + β m +1 ,m +1 − β 1 , 1 ] G ( x ) x 2 − 1 +  1 + ǫ √ ζ x 2 − 1  ∆( x ) , with k i,i ( x ) =    k 1 , 1 ( x ) + ( β i,i − β 1 , 1 ) G ( x ) , i = 2 , ..., m k i − 1 ,i − 1 ( x ) + ( β i,i − β i − 1 ,i − 1 ) xG ( x ) , i = m + 3 , ..., N (99) 20 and k m +1 ,m +1 ( x ) = k 1 , 1 ( x ) + ( β im +1 ,m +1 − β 1 , 1 ) G ( x ) + Γ( x ) , k m +2 ,m +2 ( x ) = k m +1 ,m +1 ( x ) + ( β im +2 ,m +2 − β m +1 ,m +1 ) xG ( x ) + ǫ p ζ Γ( x ) , where the au x iliary f unction Γ( x ) is giv en b y Γ( x ) = − q m +2 ( ǫq √ ζ − 1) ( q + 1) 2 β 1 ,m β 1 ,m +2 β 1 ,N ( x − 1) G ( x ) . (100) The parameters β 1 ,j are constrained b y the r ecurrence relation β 1 ,j = − β 1 ,j +1 β 1 ,N − j β 1 ,N +1 − j , j = 2 , . . . , m − 1 ( 101) while β i,i are fixed by β i,i =                        β 1 , 1 + Q m i − 2 X k =0 ( − 1 q ) k , i = 2 , ..., m β m,m + ∆ 1 , i = m + 1 β m,m + ∆ 2 , i = m + 2 β m +2 ,m +2 − ( − 1) m ǫ √ q Q m i − m − 3 X k =0 ( − 1 q ) k , i = m + 3 , ..., N (102) where Q m = 2( − 1) m q m − 1 ( q + 1) 2 ǫ √ ζ ( √ q − ( − 1) m ǫ )( q 3 2 + ( − 1) m ǫ )(1 + ǫ √ ζ ) , ∆ 1 = − 2[ q (1 + ǫ √ ζ ) + ǫ √ ζ ( q + 1)] ( √ q − ( − 1) m ǫ )( q 3 2 + ( − 1) m ǫ )(1 + ǫ √ ζ ) , ∆ 2 = 2( q + 1)  q − ǫ √ ζ  ( √ q − ( − 1) m ǫ )( q 3 2 + ( − 1) m ǫ )(1 + ǫ √ ζ ) . (103) In this solution the ha ve a total amount of m free parameters, n amely β 1 ,m +1 , . . . , β 1 , 2 m . 4.3.2 Solution M 10 : The series of s olutions M 10 is v alid for the U q [ osp (2 | 2 m ) (1) ] mo del and the corr esp onding K -matrix also p ossess all entries differen t from zero. In the first and last columns, th e matrix elemen ts are mainly giv en b y k i, 1 ( x ) = θ i q t i θ 2 q t 2 β 2 , 1 β 1 ,i ′ β 1 ,N − 1 G ( x ) i = 3 , . . . , N − 1 k i,N ( x ) = − ǫ √ ζ θ i q t i θ 1 q tt 1 β 1 ,i ′ xG ( x ) i = 2 , . . . , N − 1 (104) 21 while the ones in the first and last ro w s are r esp ectiv ely k 1 ,j ( x ) = β 1 ,j G ( x ) j = 2 , . . . , N − 1 k N ,j ( x ) = − ǫ √ ζ θ N q t N θ 2 q t 2 β 2 , 1 β 1 ,j β 1 ,N − 1 xG ( x ) j = 2 , . . . , N − 1 (105) In the secondary diagonal w e ha v e the follo w ing expression determining the m atrix elemen ts k i,i ′ ( x ) =                        β 1 ,N G ( x ) H f ( x ) , i = 1 − q 2 θ 1 q t 1 θ i ′ q t i ′  1 − ǫq √ ζ q + 1  2 β 2 1 ,i ′ β 1 ,N G ( x ) H f ( x ) , i 6 = { 1 , m + 1 , m + 2 , N } − q 2 θ 1 q t 1 θ i ′ q t i ′ ( q + ǫ √ ζ )(1 − ǫq √ ζ ) q 2 − 1 β 2 1 ,i ′ β 1 ,N G ( x ) H b ( x ) , i = { m + 1 , m + 2 } θ N − 1 q t N − 1 θ 2 q t 2 β 1 ,N β 2 2 , 1 β 2 1 ,N − 1 G ( x ) H f ( x ) , i = N . (106) recalling that H b ( x ) = q x + ǫ √ ζ q + ǫ √ ζ H f ( x ) = x − ǫq √ ζ 1 − ǫq √ ζ . (107) In their turn the other non-diagonal en tries satisfy the relation k i,j ( x ) =        − ǫ √ ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q + 1  β 1 ,i ′ β 1 ,j β 1 ,N G ( x ) , i < j ′ , 2 < i, j < N − 1 1 ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q + 1  β 1 ,i ′ β 1 ,j β 1 ,N xG ( x ) , i > j ′ , 2 < i, j < N − 1 , (108) and k 1 ,m ( x ) = i ( q + 1) q − 1 β 1 ,m +1 β 1 ,m +2 β 1 ,m +3 G ( x ) , k 1 ,m +1 ( x ) = 2 iǫ √ ζ ( q + 1) q ( √ ζ − ( − 1) m )( q √ ζ + ( − 1) m )(( − 1) m − ǫ ) β 1 ,N β 1 ,m +2 G ( x ) , k 2 , 1 ( x ) = i ( − 1) m +1 q ζ  q √ ζ − ǫ q √ ζ + ǫ   q 2 ζ − 1 q 2 − 1  β 1 ,m +1 β 1 ,m +2 β 1 ,N − 1 β 2 1 ,N G ( x ) , (109) and the p arameters β 1 ,j are required to satisfy the recurrence relation β 1 ,j = − β 1 ,j +1 β 1 ,N − j β 1 ,N − j +1 j = 2 , . . . , m − 1 . (110) Considering no w th e diagonal en tries, they are given b y k 1 , 1 ( x ) = [ ( β N ,N − β m +1 ,m +1 ) x − ( β N ,N − β 1 , 1 − 2) xH f ( x ) + β m +1 ,m +1 − β 1 , 1 ] G ( x ) x 2 − 1 , k i,i ( x ) =          k 1 , 1 ( x ) + ( β i,i − β 1 , 1 ) G ( x ) , i = 2 , ..., m + 1 k m +1 ,m +1 ( x ) , i = m + 2 k m +1 ,m +1 ( x ) + ( β i,i − β m +1 ,m +1 ) xG ( x ) , i = m + 3 , ..., N , (111) 22 where the p arameters β α,α are determined by the expressions β i,i =                  β 1 , 1 + ∆ m i − 2 X k =0 ( − 1 q ) k , i = 2 , ..., m β m,m + 4 √ ζ (1 − ǫ √ ζ ) ( q − 1)( √ ζ − ( − 1) m )( q √ ζ + ( − 1) m )( ǫ − ( − 1) m ) , i = m + 1 , m + 2 β m +1 ,m +1 + [ q j − m − 2 − q j − m − 3 − ( − 1) j − m ( q + 1)] q N − j +1 ( q − 1)( q m − ( − 1) m ) , i = m + 3 , ..., N (112) with ∆ m = 2( − 1) m ( q + 1) 2 q 2 ( q − 1)(1 − ( − 1) m √ ζ )( ǫ − ( − 1) m ) . (113) This solution has altogether m + 1 free parameters, namely β 1 ,m +2 , . . . , β 1 ,N . 4.3.3 Solution M 11 : The class of solutions M 11 is v alid for the verte x mo del U q [ osp (2 n | 2) (1) ] and the corresp ond ing K -matrix con tains only non-null entries. The b order elemen ts are mainly giv en by the follo w ing expressions k i,N ( x ) = − ǫ √ ζ θ i q t i θ 1 q t 1 β 1 ,i ′ xG ( x ) i = 2 , . . . , N − 1 k i, 1 ( x ) = θ i q t i θ 2 q t 2 β 2 , 1 β 1 ,i ′ β 1 ,N − 1 G ( x ) i = 3 , . . . , N − 1 k N ,j ( x ) = − ǫ √ ζ θ N q t N θ 2 q t 2 β 2 , 1 β 1 ,j β 1 ,N − 1 xG ( x ) j = 2 , . . . , N − 1 k 1 ,j ( x ) = β 1 ,j G ( x ) j = 2 , . . . , N − 1 . (114) The secondary diagonal is constituted by elemen ts k i,i ′ ( x ) giv en b y k i,i ′ ( x ) =                β 1 ,N G ( x ) H f ( x ) , i = 1 − q 2 θ 1 q t 1 θ i ′ q t i ′  1 − ǫq √ ζ q − 1   q + ǫ √ ζ q + 1  β 2 1 ,i ′ β 1 ,N G ( x ) H b ( x ) , i 6 = { 1 , N } θ N − 1 q t N − 1 θ 2 q t 2 β 1 ,N β 2 2 , 1 β 2 1 ,N − 1 G ( x ) H f ( x ) , i = N . (115) where the f unctions H b ( x ) and H f ( x ) w ere already giv en in (107). The remaining non-diagonal en tries are determined b y the exp ression k i,j ( x ) =        ǫ √ ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q − 1  β 1 ,i ′ β 1 ,j β 1 ,N G ( x ) , i < j ′ , 2 < i, j < N − 1 1 ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q − 1  β 1 ,i ′ β 1 ,j β 1 ,N xG ( x ) , i > j ′ , 2 < i, j < N − 1 , (116) and k 1 ,n +1 ( x ) = i √ ζ ( q − 1) q ( q n − 1 + ( − 1) n )( q n − 2 + ( − 1) n ) β 1 ,N β 1 ,n +2 G ( x ) 23 k 2 , 1 ( x ) = i ( − 1) n ( q + 1) q n − 1  q n − 1 + ( − 1) n q n − 2 + ( − 1) n  β 1 ,N − 1 β 1 ,N G ( x ) . and the p arameters β 1 ,j are required to satisfy β 1 ,j = − β 1 ,j +1 β 1 ,N − j β 1 ,N +1 − j j = 2 , . . . , n. (11 7) With resp ect to the diagonal ent ries, they are giv en b y k 1 , 1 ( x ) = [( β N ,N − β n +1 ,n +1 ) x − ( β N ,N − β 1 , 1 − 2) xH f ( x ) + β n +1 ,n +1 − β 1 , 1 ] G ( x ) x 2 − 1 , k i,i ( x ) =          k 1 , 1 ( x ) + ( β i,i − β 1 , 1 ) G ( x ) , i = 2 , ..., n + 1 k n +1 ,n +1 ( x ) , i = n + 2 k n +1 ,n +1 ( x ) + ( β i,i − β n +1 ,n +1 ) xG ( x ) , i = n + 3 , ..., N , (118) where the p arameters β α,α are determined by the expressions β i,i =                      β 1 , 1 − ( − 1) n + i q ( q + 1)( q n − 2 + ( − 1) n ) [ q i − 1 + q i − 2 − ( − 1) i ( q − 1)] , i = 2 , ..., n + 1 β n +1 ,n +1 , i = n + 2 β n +1 ,n +1 + ( − 1) n ( q + 1) ( q n − 2 + ( − 1) n ) i − n − 3 X k =0 ( − q ) k , i = n + 3 , ..., N − 1 β 1 , 1 + 2 − q 2 +1 q ( q +1)  q n − 1 +( − 1) n q n − 2 +( − 1) n  . i = N (119) The v ariables β 1 ,n +2 , . . . , β 1 ,N giv e u s a total amoun t of n + 1 f ree parameters. 4.3.4 Solution M 12 : The solution M 12 also d o es not con tain null en tries and it is v alid for the U q [ osp (2 n + 1 | 2) (1) ] v er tex mo del. C onsidering fi rst the non-diagonal en tr ies, we h a ve the f ollo wing expression determining b order elemen ts, k i,N ( x ) = − ǫ √ ζ θ i q t i θ 1 q t 1 β 1 ,i ′ xG ( x ) i = 2 , . . . , N − 1 k i, 1 ( x ) = θ i q t i θ 2 q t 2 β 2 , 1 β 1 ,i ′ β 1 ,N − 1 G ( x ) i = 3 , . . . , N − 1 k N ,j ( x ) = − ǫ √ ζ θ N q t N θ 2 q t 2 β 2 , 1 β 1 ,j β 1 ,N − 1 xG ( x ) j = 2 , . . . , N − 1 k 1 ,j ( x ) = β 1 ,j G ( x ) j = 2 , . . . , N − 1 , (120) and the f ollo wing on e for th e en tries of the s econdary d iagonal k i,i ′ ( x ) =                β 1 ,N G ( x ) H f ( x ) , i = 1 − q 2 θ 1 q t 1 θ i ′ q t i ′  1 − ǫq √ ζ q − 1   q + ǫ √ ζ q + 1  β 2 1 ,i ′ β 1 ,N G ( x ) H b ( x ) , i 6 = { 1 , i ′ , N } θ N − 1 q t N − 1 θ 2 q t 2 β 1 ,N β 2 2 , 1 β 2 1 ,N − 1 G ( x ) H f ( x ) , i = N . (121) 24 where the f unctions H b ( x ) and H f ( x ) w ere already giv en in (107). The remaining non-diagonal en tries are determined b y the exp ression k i,j ( x ) =        ǫ √ ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q − 1  β 1 ,i ′ β 1 ,j β 1 ,N G ( x ) , i < j ′ , 2 < i, j < N − 1 1 ζ θ i q t i θ 1 q t 1  1 − ǫq √ ζ q − 1  β 1 ,i ′ β 1 ,j β 1 ,N xG ( x ) , i > j ′ , 2 < i, j < N − 1 , (122) and k 2 , 1 ( x ) = ( − 1) n ζ ( ǫq √ ζ − 1) 2 q 2 − 1 β 1 ,n +1 β 1 ,n +3 β 1 ,N − 1 β 2 1 ,N G ( x ) , k 1 ,n +1 ( x ) = ǫ ( − 1) n ( q + 1) ( √ q − ( − 1) n ǫ ) 2 " β 2 1 ,n +2 β 1 ,n +3 − 2 i ( q − 1) √ ζ q ( q √ ζ − ǫ )( √ ζ − ǫ ) β 1 ,N β 1 ,n +2 # G ( x ) , (123) and the p arameters β 1 ,j are required to satisfy β 1 ,j = − β 1 ,j +1 β 1 ,N − j β 1 ,N +1 − j j = 2 , . . . , n. (12 4) With resp ect to the diagonal ent ries, they are giv en b y k 1 , 1 ( x ) = [( β N ,N − β n +2 ,n +2 ) x − ( β N ,N − β 1 , 1 − 2) xH f ( x ) + β n +2 ,n +2 − β 1 , 1 ] G ( x ) x 2 − 1 +  1 + ǫ √ ζ x 2 − 1  ∆( x ) (1 25) k i,i ( x ) =                k 1 , 1 ( x ) + ( β i,i − β 1 , 1 ) G ( x ) , i = 2 , ..., n + 1 k n +1 ,n +1 ( x ) + ( β n +2 ,n +2 − β n +1 ,n +1 ) G ( x ) + ∆( x ) , i = n + 2 k n +2 ,n +2 ( x ) + ( β n +3 ,n +3 − β n +2 ,n +2 ) xG ( x ) + ǫ √ ζ ∆ ( x ) , i = n + 3 k n +3 ,n +3 ( x ) + ( β i,i − β n +3 ,n +3 ) xG ( x ) , i = n + 4 , ..., N where ∆( x ) = i ( ǫq √ ζ − 1) q n − 3 ( q 2 − 1) β 1 ,n +1 β 1 ,n +3 β 1 ,N ( x − 1) G ( x ) (126) The parameters β α,α are d etermin ed by the expr essions β i,i =                                    β 1 , 1 + ( − 1) n Q n [( − q ) i − 2 ( q + 1) + (1 − q )] , i = 2 , ..., n + 1 β n +1 ,n +1 − iq ( q √ ζ − ǫ ) q − 1 " β 2 1 ,n +2 β 1 ,N − ( ǫq + √ ζ ) q n − 1 ( q + 1) β 1 ,n +1 β 1 ,n +3 β 1 ,N # , i = n + 2 β n +2 ,n +2 + iq ( q √ ζ − ǫ ) ( q − 1) " ǫβ 2 1 ,n +2 √ ζ β 1 ,N − ( ǫq + √ ζ ) q n − 2 ( q + 1) β 1 ,n +1 β 1 ,n +3 β 1 ,N # , i = n + 3 β n +3 ,n +3 + ( q + 1) 2 q n − 3 ǫ p ζ Q n i − n − 4 X k =0 ( − q ) k , i = n + 4 , ..., N − 1 β 1 , 1 + 2 − iǫ ( − 1) n √ ζ  q 2 + 1 q 2 − 1  ( ǫq p ζ − 1) 2 β 1 ,n +1 β 1 ,n +3 β 1 ,N , i = N (127) and the au x iliary p arameter Q n is giv en b y Q n = iǫ ( ǫq √ ζ − 1) √ ζ ( q 2 − 1) β 1 ,n +1 β 1 ,n +3 β 1 ,N . (128) This solution p ossess n + 2 fr ee parameters, namely β 1 ,n +2 , . . . , β 1 ,N . 25 5 Concluding Remarks In this work we ha ve presente d the general set of r egular solutions of th e graded reflection equation for th e U q [ osp ( r | 2 m ) (1) ] vertex mo del. Our findings can b e summarized in to four classes of diagonal solutions and t w elve classes of n on-diagonal ones. Although the R matrix of the U q [ sl ( r | 2 m ) (2) ] v ertex model is s imilar to the R mat rix of th e U q [ osp ( r | 2 m ) (1) ] v ertex mo d el, their K -matrices solutions are different b y the num b er of free p arameters in eac h solution (see [16]). It s eems to o muc h the num b er of the K -matrix solutions. W e h a ve tried to use a particular gauge tr ansformation in order to relate them b ut without success. Th e main d ifficult is the difference of the n u m b er of free p arameters in eac h class of solution M i . Here we remark that w e kno w from the osp (1 | 2) (1) and sl (2 | 1) (2) mo dels that solutions of the graded Y ang-B axter equation can b e mapp ed to (the A (2) 2 and B (1) 1 mo dels, resp ectiv ely) solutions of the non-graded equations b y graded p ermutatio n s and gauge tr an s formations [26, 27]. Although our complete solutions w ere deriv ed from solutions based on sup eralgebras, we could use th e non-graded f ormalism in ord er to construct them. Thus, the K matrices w ith b oth b osonic and fermionic entries are indicating a sy m metry breaking at the b oundaries but do not ha ve an y con tradiction with the corresp onding b ound ary in tegrabilit y . These results pa ve the wa y to construct, solv e and stud y p h ysical prop erties of the un derlying quan tum s p in c hains with op en b oundaries, generalizing the previous efforts made for the case of p erio dic b oundary conditions [18, 19 ]. Although w e exp ect that the Alg ebraic Bethe Ansatz solution of the mo dels co nstructed from the diagonal solutions pr esen ted here can b e obtained by adapting the results of [22], the algebraic-functional metho d p resen ted in [23] ma y b e a p ossibilit y to treat the non-diagonal cases. F or further r esearc h, an in teresting p ossibilit y would b e th e inv estigatio n of soliton non- preserving b oundary conditions [14, 24] for qu an tum s pin c h ains based on q -deformed Lie algebras and sup eralgebras, whic h can also b e p erformed by adapting the method described in [9]. W e exp ect th e r esults pr esented here to motiv ate f urther develo pmen ts on the sub ject of integrable op en b ound aries for vertex mo d els based on q -d eform ed Lie sup er algebras. In particular, the classification of the solutions of the graded reflection equ ation for others q -deformed Lie sup eralgebras, w hic h w e h op e to rep ort on a fu ture w ork. 6 Ac kno wledgmen ts W e thank to W . Galle as for his v aluable discussions. Th is w ork is partially supp orted b y the Brazilian researc h councils CNPq and F APESP . 26 References [1] R.J. Baxter, E xactly solve d mo dels in statistic al me chanics , Academic Press, New Y ork, (1982). [2] L.D. T akh ta jan an d L.D. F add eev, Russian Math. Surveys 34, (1979) 11. [3] V.E. Korep in, G. Izergin and N.M. Bogoliub o v, Quantum Inverse Sc attering M e tho d and Cor- r elation F unctions , C am br idge Univ. Press, Cam br idge, (1993). [4] J.L. Cardy , N ucl. P hys. B 275, (1986) 200. [5] E.K. Sklya n in, J. Phys. A: Math. Gen. 21, (1988) 2375. [6] I.V. Chered n ik, The or. Math. Phys. 61, (1984) 977. [7] V.V . Bazhano v, Phys. L ett. B 159, (1985) 321. [8] M. Jimbo, Comm. Math. Phys. 102, (1986 ) 247. [9] R. Malara and A. Lima-San tos, J. Stat. M e ch.: The or. E xp., (2006) P09013. [10] G.L. Li, R.H. Y ue and B.Y. Hou, N ucl. Phys. B 586, (2000) 711 ; A. Gonzalez-Ruiz, N ucl. Phys. B 424, (1994) 468 . [11] A.J. Brac k en, X.Y. Ge, Y.Z. Zh ang and H.Q. Zh ou, N ucl. Phys. B 516, (1998) 588 . [12] M.J. Martins and X.W. Guan, Nucl. Phys. B 562, (1999) 433. [13] D. Arnaudon, J. Av an, N. Cramp e, A. Doik ou, L. F rappat and E. Ragoucy , Nucl. Phys. B 668, (2003) 469 ; G.L. Li, K.J. Shi and R.H. Y ue, Nucl. Phys. B 687, (2004) 220. [14] D. Arnaudon, J. Av an, N. Cr amp e, A. Doik ou, L. F rappat and E. Ragoucy , J. Stat. Me ch.: The or. E xp., (2004) P08005. [15] W. Galleas, Nu cl. P hys. B 777, (2007) 352. [16] A. Lima-San tos and W.Gallea s, R efle ction matric es for the U q [ sl ( r | 2 m ) (2) ] vertex mo del , [ArXiv: nlin.SI /08063659]. [17] V.V. Bazhano v and A.G. Sh adrik o v, The or. Math. Phys. 73, (1987) 1302. [18] W. Galleas and M.J. Martins, Nucl. Phys. B 699, (2004) 455. [19] W. Galleas and M.J. Martins, Nucl. Phys. B 732, (2006) 444. 27 [20] W. Galleas and M.J. Martins, Nucl. Phys. B 768, (2007) 219. [21] L . Mezincescu and R.I. Nep omechie, J. Phys. A: Math. Gen. 24, (1991) L17 ; L. Mezincescu and R.I. Nep omec hie, Int. J. M o d. Phys. A6, (1991) 5231 . [22] G.L. Li and K .J. S hi, J. Stat. Me ch.: The or. Exp., (2007) P01018. [23] W. Galleas, Nu cl. P hys. B 790, (2008) 524. [24] A. Doik ou, J. Phys. A: Math. Gen. 33, (2000) 8797. [25] A. Lima-San tos, Nuc l. Phys. B 558, (1999) 637. [26] M. J. Martins, Nucl. Phys. B 450, (1995) 768. [27] E . C . Fireman, A. Lima-San tos and W. Utiel, N ucl. P hys. B 626, (2002) 435. App endix A: Th e U q [ osp (1 | 2 ) (1) ] case The reflection equ ation asso ciated with the U q [ osp (1 | 2) (1) ] v ertex mo del admits more general solu- tions than the corresp onding ones obtained from th e general series presented in the section 3. In this case the reflection matrices were previously studied in [25] and we h a ve obtained th e follo wing solutions K − ( x ) = Diag (1 , q 3 2 + ǫx q 3 2 + ǫ , x 2 ) (A.1) and K − ( x ) =        x − 1 2 β ( q + 1) q ( x − 1) 0 1 2 β 13 ( x 2 − 1) 0 x + 1 2 β q ( xq − 1)( x − 1) 0 − 1 2 β 2 q β 13 ( x 2 − 1) 0 x + 1 2 β ( q + 1) q x ( x − 1)        (A.2) where w e ha ve t w o free parameters β = β 22 − β 11 and β 13 . In addition to the solutions (A.1) and (A.2) w e also ha ve a solution in the general form K − ( x ) =        k 11 ( x ) β 12 G ( x ) β 13 x √ q − ǫ √ q − ǫ G ( x ) β 21 G ( x ) k 22 ( x ) − iǫq β 12 xG ( x ) β 13  β 21 β 12  2 x √ q − ǫ √ q − ǫ G ( x ) − iǫq β 21 xG ( x ) k 33 ( x )        . (A.3) where β 21 = ǫq 3 2 ( q − 1) " β 2 12 β 13 − 2 i ( √ q + ǫ ) √ q ( q 3 2 − ǫ ) # β 12 β 13 (A.4) 28 The diagonal en tries are then giv en by k 11 ( x ) = ( x √ q − ǫ ) ( √ q − ǫ ) 2 " x (1 + q 2 ) − ǫ √ q ( q + 1) q 3 2 − ǫ # 2 G ( x ) x 2 − 1 +  x √ q (1 + q 2 ) − √ q ( q − 1) − ǫ ( q + 1) ( √ q − ǫ )( q − 1)  i √ q β 2 12 β 13 G ( x ) x + 1 k 22 ( x ) = k 11 ( x ) + ( β 22 − β 11 ) G ( x ) + ∆( x ) k 33 ( x ) = x 2 k 11 ( x ) + ( β 33 − β 11 − 2)  x √ q − ǫ √ q − ǫ  xG ( x ) , ( A.5) where ∆( x ) = − ǫq 2 ( √ q − ǫ ) " i √ q ( q − 1) β 2 12 β 13 + 2 ( √ q − ǫ )( q 3 2 − ǫ ) # ( x − 1) G ( x ) . (A.6) This solution has altogether t wo fr ee parameters β 1 , 2 and β 1 , 3 and the remaining v ariables β i,j are giv en by β 22 = β 11 − 2 ǫq 3 2 ( √ q − ǫ )( q 3 2 − ǫ ) −  1 q − 1 + ǫ √ q  i √ q β 2 12 β 13 , β 33 = β 11 − 2 ǫ √ q ( q + 1) ( √ q − ǫ )( q 3 2 − ǫ ) −  1 + q 2 q − 1  i √ q β 2 12 β 13 . (A.7) App endix B: The U q [ osp (2 | 2 ) (1) ] case The set of K -matrices asso ciated with the U q [ osp (2 | 2) (1) ] v ertex mo del includes b oth diagonal and non-diagonal solutions. The three solutions intrinsical ly diagonal tw o conta in only one free parameter β an d they are given b y K − ( x ) = Diag (1 , x β ( x − 1) + 2 β ( x − 1) − 2 x , x β ( q 4 − x ) + 2 x β ( xq 4 − 1) + 2 , x 2 ) , K − ( x ) = Diag (1 , 1 , x β ( x − 1) + 2 β ( x − 1) − 2 x , x β ( x − 1) + 2 β ( x − 1) − 2 x ) (B.1) and one w ithout free parameters K − ( x ) = Diag (1 , x q 2 + ǫx xq 2 + ǫ , x q 2 + ǫx xq 2 + ǫ , x 2 ) . (B.2) W e ha v e also found th e follo wing n on-diagonal solutions K − ( x ) =             x + α − β 2 2 β ( x − 1) 0 0 β 14 2 ( x 2 − 1) 0 x + α + xβ 2 2 β ( x − 1) 0 0 0 0 x + α + xβ 2 2 β ( x − 1) 0 α 2 β 14 ( x 2 − 1) 0 0 x − α − β 2 2 β x ( x − 1)             (B.3) 29 con taining three f r ee parameters α, β and β 14 , and one free parameter solution in the form K − ( x ) =              q 2 x 2 − 1 q 2 − 1  1 x 0 0 0 0 x 1 2 β 23 ( x 2 − 1) 0 0 2 β 23  q q 2 − 1  2 ( x 2 − 1) x 0 0 0 0  q 2 x 2 − 1 q 2 − 1  x             . (B.4) Concerning the complete solution, we could not find a complete solution for this mo del. 30 App endix C: T he main reflecti on equations In th is a pp end ix w e p resen t the main sequence of the step nece ssary to get a general solution of the reflection equation. First w e consider the (i,j) comp onent of the matrix equation (12). By differen tiating it with resp ect to v and b y taking v = 0, we obtain N 4 algebraic equations E [ i, j ] = 0 in volving the single v ariable u an d N 2 parameters β i,j for the matrix elemen ts k i,j ( u ). Although there are man y equations ( N = r + 2 m, r ≥ 1 , m ≥ 1 ) a few of th em are actually indep endent. Analyzing these E [ i, j ] = 0 equati ons, the simplest are those inv olving o nly t wo matrix elemen ts of th e t yp e k i,i ′ ( u ) (secondary diagonal) k i,i ′ ( u ) =          β i,i ′ β 1 ,N k 1 ,N ( u ) , i = 1 , 2 , ..., m β i,i ′ β m +1 ,N − m k m +1 ,N − m ( u ) , i = m + 1 , ..., m + r β i,i ′ β 1 ,N k 1 ,N ( u ) , i = m + r + 1 , ..., N . (C.1) Moreo v er, we can use the follo w ing pairs of equations E [ i + iN − N , N 2 − i N + j ] = 0 an d E [ N 2 + 1 − ( N 2 − i N + j ) , N 2 + 1 − ( i + iN − N )] = 0 (C.2) for eac h i = { 1 , 2 , · · · , m } with j = i + 1 , · · · , N + 1 − i , in order to get the matrix elemen ts k i,j ( u ) and k N +1 − j,N +1 − i ( u ) in terms of k i,i ′ ( u ). This p r o cedure allo ws to find all non-diagonal entries k i,j ( u ) with fermionic d egree of fr eedom ( i, j = 1 , 2 , ..., m and m + r + 1 , ..., N ) in terms of k 1 ,N ( u ) k i,j ( u ) = F ( u )[ β i,j c 1 ( u ) d 1 , 1 ( u ) + β j ′ ,i ′ b ( u ) d i,j ′ ( u )] k 1 ,N ( u ) β 1 ,N ( i < j ′ ) (C.3) and k i,j ( u ) = F ( u )[ β i,j c 2 ( u ) d 1 , 1 ( u ) + β j ′ ,i ′ b ( u ) d i,j ′ ( u )] k 1 ,N ( u ) β 1 ,N ( i > j ′ ) (C.4) where F ( u ) = b 2 ( u ) − a 1 ( u ) d 1 , 1 ( u ) b 2 ( u ) d 1 , 2 ( u ) d 2 , 1 ( u ) − c 1 ( u ) c 2 ( u ) d 1 , 1 ( u ) 2 (C.5) and all non d iagonal en tries k i,j ( u ) with b osonic d egree of freedom ( i, j = m + 1 , m + 2 , ..., m + r ) in terms of k m +1 ,N − m ( u ) k i,j ( u ) = B ( u )[ − β i,j c 1 ( u ) d m +1 ,m +1 ( u ) + β j ′ ,i ′ b ( u ) d i,j ′ ( u )] k 1 ,N ( u ) β 1 ,N ( i < j ′ ) (C.6) and k i,j ( u ) = B ( u )[ − β i,j c 2 ( u ) d m +1 ,m +1 ( u ) + β j ′ ,i ′ b ( u ) d i,j ′ ( u )] k m +1 ,N − m ( u ) β m +1 ,N − m ( i > j ′ ) ( C.7) where B ( u ) = b 2 ( u ) − a m +1 ( u ) d m +1 ,m +1 ( u ) b 2 ( u ) d 1 , 2 ( u ) d 2 , 1 ( u ) − c 1 ( u ) c 2 ( u ) d m +1 ,m +1 ( u ) 2 . (C.8) 31 Note in these expressions that we h a ve u sed the id en tity d i,j ( u ) d j,i ( u ) = d 1 , 2 ( u ) d 2 , 1 ( u ) , i 6 = j, j ′ . (C.9) Sim ultaneously , w e can lo ok at the equations of th e typ e E [ i, i ] = 0, with aid of the p rop erty d i,j ( u ) = d j ′ ,i ′ ( u ), in order to get the sy m metric relations k i,j ( u ) k j,i ( u ) = β i,j β j,i = ⇒ β i,j = β j,i β i ′ ,j ′ β j ′ ,i ′ , i > j and i 6 = j, j ′ . (C.10) Next, the equatio ns E [ ´ ı, N + 1 − i ] = 0 give us the quadratic relat ions ( ǫ = ± 1) b et w een the parameters β i,j and β j ′ ,i ′ : β i,j = − ǫ √ ζ θ i q t i θ j ′ q t j ′ β j ′ ,i ′ (C.11) Substituting these relations int o (12), the s implest equation are n o w th ose inv olving the matrix elemen ts k 1 ,N ( u ) and k m +1 ,N − m ( u ): β m +1 ,N − m H b ( u ) k 1 ,N ( u ) = β 1 ,N H f ( u ) k m +1 ,N − m ( u ) (C.12) where the H { f ,b } ( u ) functions are giv en by (26) and w e get the general structure (25) for the non-diagonal K -matrix elemen ts. Solving eac h pair of equations { E [1 , j ] = 0 , E [1 , ( j − 1) N + 1] = 0 } w e find the diagonal matrix elements { k 1 , 1 ( u ) , k j,j ( u ) } , for j = 2 , ..., N − 1. The pair { k N − 1 ,N − 1 ( u ) , k N ,N ( u ) } can b e obtained solving the equation { E [ N 2 , N 2 − 1] = 0 , E [ N 2 , N 2 − N ] = 0 } . Again, substituting these relations into (12) we can see that several equations of the t yp e E [ m + 1 , j ] = 0 giv e us the constraints b etw een parameters w ith b osonic and fermionic en tr ies β m +1 ,b β 1 ,f = 0 , m + 1 < b ≤ N − m , m + r < f ≤ N (C.13) It means, in general, that w e can g et solutions with only a t yp e of degree of freedom. F or a particular choice β m +1 ,b = 0 or β 1 ,f = 0, we will fi nd the solutions M 1 to M 8 present ed in the section 4 of this pap er. F r om t he reflecti on e quations, we can see for the U q [ osp (1 | 2 m ) (1) ] c ase that there is no relations of the t yp e (C.1) with b osonic degree of freedom and for th e U q [ osp (2 | 2 m ) (1) ] case th at there is only one b osonic relation k N − m,m +1 ( u ) = β N − m,m +1 β m +1 ,N − m k m +1 ,N − m ( u ) (C.14) In addition, for the U q [ osp ( r | 2)] case wee ha v e only one fermionic relation k N , 1 ( u ) = β N , 1 β 1 ,N k N , 1 ( u ) (C.15) Therefore, f or these cases we can solve th e coupled equations (C.12) without making u se of the constrain t equations of the t yp e (C.13). Thus, we found the four complete solutions M 9 to M 12 . 32