On $tau^{(2)}$-model in Chiral Potts Model and Cyclic Representation of Quantum Group $U_q(sl_2)$

On τ ( 2 ) -mo del in C hiral P otts Mo del and Cyclic Representation of Quan tum Group U q ( sl 2 ) Shi-shyr Roan Institute of Mathematics A c ademia Sinic a T aip ei , T aiwan (email: mar o an@gate.sinic a.e du.tw ) Abstract W e iden tify the pr e c ise relationship b et ween the five-parameter τ (2) -family in the N -state chiral P otts mo del and XXZ chains with U q ( sl 2 )-cyclic r epresen tation. By studying the Y ang- Baxter relation of the six-vertex mo del, w e discov er an one-parameter family of L -op erators in terms of the q ua n tum group U q ( sl 2 ). When N is o dd, the N -state τ (2) -mo del can be regar de d as the XXZ chain of U q ( sl 2 ) cyclic representations with q N = 1. The s ymmet ry algebra of the τ (2) -mo del is describ ed by the quantum affine alg ebra U q ( b sl 2 ) via the canonical representation. In genera l for an arbitrar y N , w e show that the XXZ chain with a U q ( sl 2 )-cyclic representation for q 2 N = 1 is equiv alent to tw o copies of the same N -state τ (2) -mo del. 2008 P A CS: 05.50. +q, 02.20.Uw, 75.10Jm 2000 MSC: 17B37 , 17B80, 82B20 Key wor ds : τ (2) -mo del, XXZ c hain, Qu antum group U q ( sl 2 ) 1 In tro duction In the study of N -state chiral P otts mo del (CPM), Bazhano v and S t rogano v [12] disco v ered a fiv e- parameter family of L -op erators of τ (2) -mo del in the six-ve rtex mod e l with a particular field (see 1 [9] page 3), i.e. the solution of Y ang-Baxter (YB) equation for the asymmetric six-v ertex R -matrix, (see (2.1) and (2.2) in this pap er). The c h iral P otts transfer matrix can b e constructed as th e Bax- ter’s Q -op erato r of the τ (2) -mo del [11, 29]. By the functional relations b et ween the fu sion matrices of τ (2) -mo del and the c h iral P otts transfer matrix, one can compute the eigen v alue sp ectrum of the (homogeneous) sup erin tegrable C P M [1, 6, 7], where the τ (2) -degeneracy o ccurs with the sym met ry algebra d escrib ed by Onsager algebra [24]. F urthermore, one can discus s the eigen v alue sp ectrum and calculate the order paramete r in CPM thr o ugh functional-relation method [4, 5, 10, 22, 30]. Though muc h progress has b een made on issues related to the eigen v alues, w e still lac k enough information at presen t ab out the eigen v ectors in CP M. Consequen tly , some imp ortant problems whic h r equ ire the kn o wledge of eigen vect ors, su c h as the ca lculation of correlation functions, remain unsolv ed in the th e ory . On the other h a nd, the (”zero-field”) six-v ertex mo del ([8] Sect. 3) with the symmetric R -mat rix has b een a w ell-studied theory with pr ofoun d knowledge kno wn in literature ab out their eigen v alues and eigen vect ors. The un d erstanding of the structure w as fur ther extended to XXZ mo dels of h ig her spin [20], or more generally the mo del asso cia ted to a represen tation of the quantum group U q ( sl 2 ). When q is a N th ro ot of unit y , it wa s sho wn in [15, 16, 17, 23, 25, 27] that the degeneracy of XXZ m odels o c curs with the extra sl 2 -lo op -algebra s y m met ry induced from the structure of the quantum affine algebra U q ( b sl 2 ). F or o dd N , as a sp ecial mem b er of a one- parameter U q ( sl 2 )-cyclic rep resen tation, the spin-( N − 1) / 2 XXZ c hain can b e identified with the sup erin tegrable N -state τ (2) -mo del [28]. Hence the sup erint egrable τ (2) -mo del also carries the sl 2 - lo o p-algebra s y m met ry for the degeneracy (of certain sectors), compatible with the Onsager-alg ebra symmetry inherited from the c hiral Potts transfer matrix [28]. Th e in terrelation of these t w o sym- metries can lead to s ome useful information ab out eigen ve ctors of the sup erint egrable mo del (see, e.g. [2, 3]). The aim of this pap er is to extend the equiv alen t relationship b et w een the N -state τ (2) - mo del and XXZ mo del with U q ( sl 2 )-cyclic represen tation to the whole fi v e-parameter τ (2) -family found in [12] for an y N . Through the YB relat ion in the six-ve rtex mod el , w e find a c haracterization of quant um group U q ( sl 2 ) through a one-parameter L -op erators of the YB solution (see (2.5), (2.6) in the pap er). When q is a ro ot of unit y , the alge bra U q ( sl 2 ) p ossesses a three-parameter family of cyclic represen tation. T ogether with the rescaling of the sp ectral p a rameter, XXZ c hains w ith U q ( sl 2 )-cyclic representa tion also carry a fi ve-paramete r L -operators. By extending the argument in [28], w e sho w the identifica tion of τ (2) -mo dels and the XXZ c hains with U q ( sl 2 )-cyclic represen- tation. Note that suc h a pr ec ise connection b et wee n τ (2) -family in C PM an d th a t XXZ-family w it h symmetric R -matrix seems not to ha v e app eared in the literature b efore, to the b e st of the author’s kno wledge, even in o dd N case except a one-parameter family of τ (2) -mo dels in [28]. W e hop e an iden tification of these mo dels will h e lp to pr o vide some usefu l clues to und e rstand the eigen v ectors of C PM in certain sp ecial cases. This pap er is organized as follo w s . In section 2, w e recall the definition of τ (2) -mo del, qu a nt um group U q ( sl 2 ), and the quan tum affine alge bra U q ( b sl 2 ). In sectio n 3, w e fi rst describ e the three- parameter family of cyclic r e presenta tions of U q ( sl 2 ) for a ro ot of unity q . The case of q N = 1 f o r o dd N is discussed in subsection 3.1 where the τ (2) -mo del is iden tified with the XXZ c hain with cyclic r e presenta tion of U q ( sl 2 ), h e nce the quantum space carries a canonical r e presentat ion of th e 2 quan tum affine algebra U q ( b sl 2 ). I n subsection 3.2, we stu dy the N -state τ (2) -mo del for an arbitrary N . W e show that a XXZ c hain of U q ( sl 2 )-cyclic representa tion with q 2 N = 1 is equiv alent to tw o copies of the same N -state τ (2) -mo del. Finally we close in section 4 with a brief concluding remark. Notatio n: W e use standard notations. F or a p ositiv e integ er N greate r than one, C N denotes the ve ctor space of N -cyclic v ectors w i th the canonical base | n i , n ∈ Z N (:= Z / N Z ). W e fix the N th ro ot of unity ω = e 2 π i N , and X , Z , the W eyl C N -op erato rs : X | n i = | n + 1 i , Z | n i = ω n | n i ( n ∈ Z N ) , whic h satisfy X N = Z N = 1 and the W eyl relation: X Z = ω − 1 Z X . 2 N -state τ (2) -mo del and quan tum group U q ( sl 2 ) The N -state τ (2) -mo del [11 , 12, 30] (also called the Baxter-Bazhano v-S t rogano v mo del [19]) is the fiv e-parameter family of L -op e rators of C 2 -auxiliary , C N -quan tum space with en tries expressed b y W eyl op erators X , Z : L ( t ) = 1 − t c b ′ b X ( 1 b − ω ac b ′ b X ) Z − t ( 1 b ′ − a ′ c b ′ b X ) Z − 1 − t 1 b ′ b + ω a ′ ac b ′ b X ! , (2.1) where t is th e sp ect ral v ariable, and a , b , a ′ , b ′ , c are non-zero complex parameters. It is kn o wn that the L - op erator (2.1) satisfies the YB equation R ( t/t ′ )( L ( t ) O aux 1)(1 O aux L ( t ′ )) = (1 O aux L ( t ′ ))( L ( t ) O aux 1) R ( t/t ′ ) for the asymm etry six-v ertex R -matrix R ( t ) =        tω − 1 0 0 0 0 t − 1 ω − 1 0 0 t ( ω − 1) ( t − 1) ω 0 0 0 0 tω − 1        . ( 2.2) The mon o dromy matrix, N L ℓ =1 L ℓ ( t ) where L ℓ ( t ) = L ( t ) at site ℓ , a gain satisfies the ab o v e YB relation, w hose ω -t wisted trace defines the τ (2) -matrix: τ (2) ( t ) = tr C 2 L O ℓ =1 L ℓ ( ω t ) , (2.3) comm uting with the spin-shift op erator X (:= Q ℓ X ℓ ). The quantum group U q ( sl 2 ) is the asso cia ted C -algebra generated by K ± 1 2 , e ± with the relations K 1 2 K − 1 2 = K − 1 2 K 1 2 = 1 and K 1 2 e ± K − 1 2 = q ± 1 e ± , [ e + , e − ] = K − K − 1 q − q − 1 . (2.4) 3 Then the t w o-b y -tw o matrix with U q ( sl 2 )-en tr i es defin e s a one-parameter family of L -op erators: L ( s ) = ρ − 1 sK − 1 2 − s − 1 K 1 2 ( q − q − 1 ) e − ( q − q − 1 ) e + sK 1 2 − ρs − 1 K − 1 2 ! , ρ 6 = 0 ∈ C , (2.5) whic h satisfy the YB equation R 6v ( s/s ′ )( L ( s ) O aux 1)(1 O aux L ( s ′ )) = (1 O aux L ( s ′ ))( L ( s ) O aux 1) R 6v ( s/s ′ ) . (2.6) for the six-vertex (symmetric) R -matrix [18, 21]: R 6v ( s ) =        s − 1 q − sq − 1 0 0 0 0 s − 1 − s q − q − 1 0 0 q − q − 1 s − 1 − s 0 0 0 0 s − 1 q − sq − 1        . Indeed, the YB relatio n (2.6) f o r L in (2. 5) is the necessary and sufficien t condition of the constr aint (2.4) in the d e finition of quantum group U q ( sl 2 ). Usin g th e lo cal L -op erator (2.5), one constru c ts the m o no drom y matrix N L ℓ =1 L ℓ ( s ) with en tries in ( L N U q ( sl 2 ))( s ), L O ℓ =1 L ℓ ( s ) = A L ( s ) B L ( s ) C L ( s ) D L ( s ) ! again satisfying (2.6). The structure of th e quantum affine algebra U q ( b sl 2 ) is describ ed b y the leading and lo west terms of the ab o v e entries in th e mono drom y matrix, A + = lim s →∞ ( ρ − 1 s ) − L A L ( s ) , A − = lim s → 0 ( − s ) L A L ( s ) , B ± = lim s ± 1 →∞ ( ± s ) ∓ ( L − 1) B L ( s ) q − q − 1 , C ± = lim s ± 1 →∞ ( ± s ) ∓ ( L − 1) C L ( s ) q − q − 1 , D + = lim s →∞ s − L D L ( s ) , D − = lim s → 0 ( − ρ − 1 s ) L D L ( s ) . One has A + = A − 1 − , A ∓ = D ± . Denote T − = B + , S − = B − and S + = C + , T + = C − . The op erators k − 1 0 = k 1 = A 2 − = D 2 + , e 1 = S + , f 1 = S − , e 0 = T − , f 0 = T + generate the quan tum affine algebra U q ( b sl 2 ) with the Hopf-algebra structure △ ( k i ) = k i ⊗ k i , i = 0 , 1 , △ ( e 1 ) = k 1 ⊗ e 1 + ρ − 1 e 1 ⊗ k 0 , △ ( f 1 ) = k 1 ⊗ f 1 + ρf 1 ⊗ k 0 , △ ( e 0 ) = ρ − 1 k 0 ⊗ e 0 + e 0 ⊗ k 1 , △ ( f 0 ) = ρk 0 ⊗ f 0 + f 0 ⊗ k 1 , Indeed, th e explicit expr e ssion of generators of U q ( b sl 2 ) is A − = K 1 2 ⊗ · · · ⊗ K 1 2 , A + = K − 1 2 ⊗ · · · ⊗ K − 1 2 , S ± = P L i =1 K 1 2 ⊗ · · · ⊗ K 1 2 | {z } i − 1 ⊗ e ± ⊗ K − 1 2 ⊗ · · · ⊗ K − 1 2 | {z } L − i ρ ∓ ( L − i ) , T ± = P L i =1 ρ ± ( i − 1) K − 1 2 ⊗ · · · ⊗ K − 1 2 | {z } i − 1 ⊗ e ± ⊗ K 1 2 ⊗ · · · ⊗ K 1 2 | {z } L − i . (2.7) 4 Note that (2.6) is still v alid w hen c h an ging the v ariable s by αs for a non-zero complex α . Giv en a fi nite -dimensional r ep resen tation σ : U q ( sl 2 ) − → En d ( C d ), the L -op erator of C 2 -auxiliary and C d -quan tum space, L ( s ) = σ ( L ( αs )), satisfies the YB relation (2.6). In particular, by setting the parameter ρ = 1 in (2.5), with α = q d − 2 2 and σ the s p in- d − 1 2 (highest-w eigh t) represen tation of U q ( sl 2 ) on C d = ⊕ d − 1 k =0 Ce k : K 1 2 ( e k ) = q d − 1 − 2 k 2 e k , e + ( e k ) = [ k ] e k − 1 , e − ( e k ) = [ d − 1 − k ] e k +1 , where [ n ] = q n − q − n q − q − 1 and e + ( e 0 ) = e − ( e d − 1 ) = 0, on e obtains the w ell-kno wn L -op erator of XXZ c h ai n of spin - d − 1 2 (see, e.g. [20, 26, 27] and references therein). 3 The equiv alence of τ (2) -mo dels and XXZ chai ns with cyclic rep- resen tation of U q ( sl 2 ) In this section, we consider the case when q is a ro o t of un it y . F or a N th ro ot of unit y q , th e re exists a three-parameter family of cyclic repr esentati on σ φ,φ ′ ,ε of U q ( sl 2 ), lab eled by non-zero complex n umbers φ, φ ′ and ε , whic h acts on cyclic C N -v ectors b y K 1 2 | n i = q − n + φ ′ − φ 2 | n i , e + | n i = q ε q φ + n − q − φ − n q − q − 1 | n − 1 i , e − | n i = q − ε q φ ′ − n − q − φ ′ + n q − q − 1 | n + 1 i , (3.1) (see, e.g. [13, 14]). Th e L -op erato r, L ( s ) = σ φ,φ ′ ,ε L ( s ) (3.2) giv es rise to the transfer matrix of the XXZ c h a in with the U q ( sl 2 )-cyclic represen tation σ φ,φ ′ ,ε , T ( s ) = ( ⊗ σ φ,φ ′ ,ε )(tr C 2 L O ℓ =1 L ℓ ( s )) , (3.3) whic h comm utes with K 1 2 (:= ⊗ ℓ K 1 2 ℓ , th e pro duct of lo cal K 1 2 -op erato rs ). 3.1 The iden t ifi cation of the N -state τ (2) -mo d el and the XXZ ch ain of U q ( sl 2 ) - cyclic represen tation wit h q N = 1 for o d d N In the subsection, w e consider the case N o dd, and write N = 2 M + 1. Let q b e the p rimitiv e N th ro ot- of-unit y with q − 2 = ω . Th en one can express th e cyclic represen tation (3.1) in terms of the W eyl op erato rs X , Z : K 1 2 = q φ ′ − φ 2 Z 1 2 , e + = q ε ( q φ +1 Z − 1 2 − q − φ − 1 Z 1 2 ) X − 1 q − q − 1 , e − = q − ε ( q φ ′ +1 Z 1 2 − q − φ ′ − 1 Z − 1 2 ) X q − q − 1 , hence follo w the expression: K − 1 = q φ − φ ′ Z − 1 , K − 1 2 e + = − q − φ − φ ′ 2 + ε − 1 (1 − q 2 φ +2 Z − 1 ) X − 1 q − q − 1 , K − 1 2 e − = q φ + φ ′ 2 − ε +1 (1 − q − 2 φ ′ − 2 Z − 1 ) X q − q − 1 . (3.4) 5 Using the F ourier basis c | n i = 1 N P j ∈ Z N ω − nj | j i , one ma y con v ert the W eyl operator ( Z − 1 , X ) to ( X, Z ):  Z − 1 X c | 0 i , · · · , Z − 1 X d | N − 1 i  =  c | 0 i , · · · , d | N − 1 i  X Z , (3.5) hence represen t K − 1 , K − 1 2 e ± in (3.4) b y K − 1 = q φ − φ ′ X, K − 1 2 e + = − q − φ − φ ′ 2 + ε − 1 (1 − q 2 φ +2 X ) Z − 1 q − q − 1 , K − 1 2 e − = q φ + φ ′ 2 − ε +1 (1 − q − 2 φ ′ − 2 X ) Z q − q − 1 . By the ab o ve exp ressio n and introdu c ing the sp ectral v ariable t = λ − 1 s 2 for a non-zero complex λ , the m odified L -op erator of (3.2), − s K − 1 2 L ( s ), is gauge equ iv alen t to 1 − tλρ − 1 q φ − φ ′ X q − ε (1 − q − 2 φ ′ − 2 X ) Z − tλ q ε (1 − q 2 φ +2 X ) Z − 1 − tλ + ρ q φ − φ ′ X ! , whic h is the same as the L -op erator (2.1) of τ (2) -mo del b y the follo wing id entificatio n of parameters: a = λ − 1 ρ q − φ − φ ′ − ε , a ′ = ρ q φ + φ ′ + ε +2 , b = q ε , b ′ = λ − 1 q − ε , c = ρ − 1 q φ − φ ′ , equiv alen tly q ε = b , q 2 φ = ω a ′ c b , q 2 φ ′ = b ′ ac , ρ 2 = ω aa ′ bb ′ , λ = 1 bb ′ . (3.6) This im p lie s the transfer matrix (3.3) of XXZ c hain is equiv alent to the τ (2) -transfer matrix (2.3). Note th a t the p rodu c t of lo cal op erator K − 1 ’s of XXZ c hain is no w corresp onding to a scalar m ultiple of the spin-shift op erator X , whic h commutes with the τ (2) -matrix. T he represen tations with ε = 0 , φ = φ ′ in (3.1) form the one-parameter cyclic repr ese nt ation of U q ( sl 2 ) discuss e d in [28] section 4. In p a rticular, the case φ = φ ′ = M is the s p in- N − 1 2 highest-w eigh t representat ion of U q ( sl 2 ). It is kn o wn that the d e generacy of τ (2) -mo del in CPM o ccurs in the alternating sup erin tegrable case (see, [30] section 4.3), where the τ (2) -mo del is c haracterized b y the L -op erator (2.1) with the parameter a = ω m b ′ , a ′ = ω m ′ b , c = ω n , (m , m ′ , n ∈ Z ) , or equiv alen tly , w ith the repr e sen tation parameters φ, φ ′ , ε and ρ, λ in (3.6) giv en by φ = − (m ′ + n + 1) , φ ′ = m + n , q ε = b , ρ = q − (1+m+m ′ ) , λ = 1 bb ′ . (3.7) In this case, th e re exist the normalized N th p o wer of S ± , T ± in (2.7), S ± ( N ) = S ± N [ N ]! , T ± ( N ) = S ± N [ N ]! . As φ in (3.7) is an intege r, b y the algebraic-Bethe-ansatz tec hn ique, one can show the degeneracy of the alternating sup erin tegrable τ (2) -mo del (for certain sectors) p o ssesses the symmetry algebra gen- erated by S ± ( N ) , T ± ( N ) , whic h can b e iden tified with the sl 2 -lo op algebra under certain constraint s b et wee n the intege rs m , m ′ and n, as in the discussion of the homogeneous sup erin tegrable case in [15, 23, 27 ]. Ho we v er, the precise structure ab out th e symmetry algebra for arbitrary m , m ′ , n remains unknown to b e identified. 6 3.2 XXZ c hain of U q ( sl 2 ) -cyclic represen tation with q 2 N = 1 as tw o copies of t he N -state τ (2) -mo d el for an arbitrary N By extendin g the argumen t of the previous subsection, we now identify the N -state τ (2) -mo del with a XXZ c hain for an arbitrary N . Let q b e a prim itive (2 N )-th ro ot of unity with q − 2 = ω , hence q N = − 1. Consider the thr e e p a rameter f amily of U q ( sl 2 )-cyclic rep resen tations σ φ,φ ′ ,ε in (3.1) on the vecto r space V of 2 N -cyclic v ectors: V := M { C | n i ′ | n ∈ Z 2 N } . Denote th e v ectors in V : | n i := | n i ′ + | n + N i ′ , ( n ∈ Z 2 N ) , | n i − := | n i ′ − | n + N i ′ , (0 ≤ n ≤ N − 1) . Then | n i = | n + N i . W e define | n i − for n ∈ Z by the N -p erio dic condition: | n + N i − = | n i − . Hence w e can ident ify {| n i} or {| n i − } with the canonical basis of th e ve ctor space of N -cyclic v ectors, and one has the decomp ositio n of V : V = V + ⊕ V − , V + := X n ∈ Z N C | n i , V − := X n ∈ Z N C | n i − . The spin -sh ift op erator | n i ′ 7→ | n + 1 i ′ of V is d ecomp osed as the sum of X on V + and e X on V − , where X | n i = | n + 1 i ( n ∈ Z N ) , e X | n i − = ( | n + 1 i − n 6≡ N − 1 (mo d N ) , −| n + 1 i − n ≡ N − 1 (mo d N ) . T ogether with the op erat or Z : Z | n i = ω n | n i , Z | n i − = ω n | n i − , one finds ( X , Z ) , ( e X , Z ) are W eyl op erato rs of V + or V − resp ectiv ely with X N = Z N = 1 , e X N = − 1. Under the cyclic rep resen tation σ φ,φ ′ ,ε (3.1) of U q ( sl 2 ), the subspaces V + , V − are interc hanged und e r the ge nerators K 1 2 , e ± , w it h the exp ression: K 1 2 | n i = q − n + φ ′ − φ 2 | n i − , K 1 2 | n i − = q − n + φ ′ − φ 2 | n i (0 ≤ n ≤ N − 1) , e + | n i = q ε q φ + n − q − φ − n q − q − 1 | n − 1 i − (1 ≤ n ≤ N ) , e − | n i = q − ε q φ ′ − n − q − φ ′ + n q − q − 1 | n + 1 i − ( − 1 ≤ n ≤ N − 2) , e + | n i − = q ε q φ + n − q − φ − n q − q − 1 | n − 1 i , e − | n i − = q − ε q φ ′ − n − q − φ ′ + n q − q − 1 | n + 1 i (0 ≤ n ≤ N − 1) . (3.8) Using the ab o ve form ulas, one fin d s that K − 1 and K − 1 2 e ± are op erators of V + and V − , whic h can b e exp ressed by X, Z , and e X , Z r e sp ectiv ely suc h that the relation (3.4) (usin g q ) holds for V + , and also v alid for V − when replacing X by e X . By usin g the V − -basis, | n ii := q n | n i − for 0 ≤ n ≤ N − 1, the W eyl pair ( e X , Z ) is con v erted to ( q − 1 X, Z ). One can express the op erators K − 1 2 e ± b y K − 1 2 e + | n ii = − q − φ − φ ′ 2 + ε (1 − q 2 φ +2 Z − 1 ) X − 1 q − q − 1 | n ii , K − 1 2 e − | n ii = q φ + φ ′ 2 − ε (1 − q − 2 φ ′ − 2 Z − 1 ) X q − q − 1 | n ii . 7 Using th e F ourier transform (3.5) of the b a sis {| n i} and {| n ii} , b y the s ame argument in subsection 3.1, one can sho w that − sK − 1 2 L ( s ) for the L -op erator (3.2) is gauge equiv alen t to t w o copies of the same L -op erator (2.1) of τ (2) -mo del with the parameter giv en b y (3.6) (replacing q b y q ). T herefore the transfer matrix (3.3) of XXZ c hain is equ iv alen t to the sum of t w o copies of τ (2) -transfer matrix (2.3). Note that neither one of the t w o copies of τ (2) -mo del inh e rits the U q ( sl 2 ) stru ctur e of XXZ c h ai n, in wh ic h one copy is sen t to another by (3.8). Th e symmetry structure of τ (2) -mo del related to U q ( sl 2 ) is different from that in s u bsectio n 3.1 for the o dd N case. Neverthele ss, the τ (2) -mo del with certain constrain ts on the parameter again p ossesses the sl 2 -lo op -algebra symm etry that arises from the U q ( b sl 2 )-structure of the corresp onding XXZ-mo del. In particular, in the homogeneous sup erin tegrable case, the Onsager-algebra symmetry of the d e generate τ (2) -eigenspace is extended to the symmetry of s l 2 -lo op algebra (in certain sectors). The relationship of these t w o symmetries will hop efully , though n o t immediately apparent, lead to a solution of the eigen vec tor problem in the s up er integrable CPM (for an arbitrary N ) along the line in [2, 3]. 4 Concluding Rema rks By stu dying the general solution of YB equation of the six-v ertex mo del, we fi nd a one-parameter L -op erato rs asso ciated to the quan tum group U q ( sl 2 ), whic h carries a three-parameter family of cyclic representat ion when q is a ro ot of unity . W e ha ve show ed that the XXZ c hain of U q ( sl 2 )-cyclic represent ation with q 2 N = 1 is equiv alen t to the sum of t w o copies of a N -state τ (2) -mo del. In particular, the Onsager-algebra sym m e try of the homogeneous s u perintegrable CPM is enlarged to the sl 2 -lo op -algebra symmetry induced f rom the corresp onding XXZ-mo del. Wh en N is o dd and q is a primitiv e N th r oot of u n it y , the N -state τ (2) -mo del can also b e identified with the XXZ chain of a cyclic represent ation of U q ( sl 2 ), hence the qu antum space carr ies the stru ctur e of quan tum affine algebra U q ( b sl 2 ). As a s pecial ca se, the homogeneous sup erin tegrable τ (2) -mo del in CPM is equiv alen t to the s p in-( N − 1) / 2 XXZ c hain with the canonical U q ( b sl 2 )-structure [28], by whic h the sl 2 -lo op -algebra symmetry of the sup erint egrable τ (2) -mo del was deriv ed in [23, 27]. In the recen t study of eigenv alues of the c hiral P otts mo del with alternating rapid ities [30], the τ (2) -degeneracy is also fou n d in the alternating sup erin tegrable case. By ident ifying these τ (2) -mo dels with the XXZ c hains via (3.7 ), we find the sl 2 -lo op -algebra s ymmetry f o r the degenerate τ (2) -eigenspace in certain cases. 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