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Baxter’s Q-operators for supersymmetric spin chains 9 0 0 Vladimir V. Bazhanov1 and Zengo Tsuboi2 2 n 1 Department of Theoretical Physics, a Research School of Physical Sciences and Engineering, J Australian National University, Canberra, ACT 0200, Australia. 3 2 2 Okayama Institute for Quantum Physics, ] h Kyoyama 1-9-1, Okayama 700-0015, Japan. t - p e h Journal reference: Nucl. Phys. B 805 [FS] (2008) 451-516 [ DOI: 10.1016/j.nuclphysb.2008.06.025 3 Report number: OIQP-07-13 v 4 7 2 4 Abstract . 5 We develop Yang-Baxter integrability structures connected with the quantum affine su- 0 peralgebra U (sl(21)). Baxter’s Q-operators are explicitly constructed as supertraces of 8 q | 0 certain monodromy matrices associated with (q-deformed) bosonic and fermionic oscillator : algebras. Therebare six different Q-operatorsin this case, obeying a few fundamental fusion v relations,which imply allfunctional relationsbetweenvarious commuting transfermatrices. i X The results are universal in the sense that they do not depend on the quantum space of r states and apply both to lattice models and to continuous quantum field theory models as a well. 1email: [email protected] 2email: [email protected], URL: http://www.pref.okayama.jp/kikaku/kouryoushi/english/kouryoushi.htm 1 Contents 1 Introduction 3 2 Yang-Baxter equation, transfer matrices and Q-operators 6 2.1 The universal R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 T-operators (transfer matrices) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Q-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Lattice R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Functional relations 17 3.1 Wronskian-type relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 T-Q relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Fusion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Applications in continuous quantum field theory. 23 4.1 T- and Q-operators in conformal field theory . . . . . . . . . . . . . . . . . . . . 23 4.2 Connections with the spectral theory of differential equations. . . . . . . . . . . . 27 5 Algebraic proof of the functional relations 30 5.1 Symmetry transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Additional notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2.1 Shifted modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2.2 Modified versions of the maps ρ and ρ . . . . . . . . . . . . . . . . . . . 32 i i 5.2.3 Fock spaces for oscillator algebras . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Wronskian-type relation (3.2) for Q-operators . . . . . . . . . . . . . . . . . . . . 34 5.4 Factorization formula (3.10) for a typical representation . . . . . . . . . . . . . . 38 5.5 Wronskian type T-Q relation (3.9) . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.5.1 Factorization formula (5.57) for infinite-dimensional representations . . . 42 6 Concluding remarks 44 Appendices 45 A. Highest weight representations of U (gl(21)) . . . . . . . . . . . . . . . . . . . . . . 45 q | B. Representations of the Borel subalgebra (U (sl(21))) . . . . . . . . . . . . . . . 50 + q B | C. Quantum affine superalgebra analogue of the first and second Weyl formulae . . . . 56 D. Expansion of the universal R-matrix . . . . . .b. . . . . . . . . . . . . . . . . . . . 62 References 64 2 1 Introduction The method of the Q-operator, introduced by Baxter in his seminal paper [1] on the exact solution of the eight-vertex model, finds many applications in the theory of integrable quantum systems. Itsrelationshiptothealgebraicstructureofquantumgroups[2,3]wasunraveledin[4,5]. This method does not require the existence of the “bare” vacuum state and therefore has wider applicability than the traditional approaches to integrable systems such the coordinate [6] or algebraic Bethe Ansatz [7]. Here we consider integrable models of statistical mechanics and quantum field theory associated with the quantum affine superalgebra U (sl(21)). The fundamental R-matrix serving q | these models was found by Perk and Schultz [8], b q−e11 xqe11−21 (q−1 q)q−12xe21 (q−1 q)q−21xe31 − − − R(x) = q (q−1 q)e12 q−e22 xqe22−21 (q−1 q)q−12xe32 , (1.1) − − −  (q−1 q)e13 (q−1 q)e23 qe33 xq−e33−21   − − −    wheree isa3 3matrixwhose(k,l)elementisδ δ . Itdefinesan“interaction-round-a-vertex” ij ik jl × model on the square lattice with three different states, s = 1,2,3, for each lattice edge. The states “1” and “2” will be referred to as bosonic (even) states and the state “3” as a fermionic (odd) state. We will call this model as the 3-state gl(21)-Perk-Schultz model or just as the | “3-state model”. It is worth noting that the paper [8] contains more general R-matrices with an arbitrary number of states per edge associated with the U (sl(m n)) superalgebras. We also q | remark that closely related R-matrices for the non-graded case of U (sl(m)) were previously q given by Cherednik [9]. b The edge configurations of the whole lattice in the 3-state gl(21)-Pebrk-Schultz model obey | simplekinematicconstraints, analogous tothe“arrowconservationlaw”intheordinary6-vertex model. Here we consider the periodic boundary conditions in the horizontal direction. Then for every allowed edge configuration the numbers m ,m ,m , counting the edges of the types “1”, 1 2 3 “2” and “3” in a horizontal row are the same for all rows of the lattice. The row-to-row transfer matrix of the model reduces to a block-diagonal form, where the blocks are labeled by these conserved numbers. Note that, m +m +m = L, where L is the horizontal size of the lattice. 1 2 3 The spin chain Hamiltonian connected with the gl(21)-Perk-Schultz model describes the | trigonometric generalization [8,10,11] of the supersymmetric t-J model [12]. Both models (rational and trigonometric, and also their multicomponent analogs) were studied by many authors (see for example, [13]- [48]). Owing to the edge-type conservation properties discussed above these models can be solved via the “nested” Bethe Ansatz [12,13]. The problem of the diagonalization oftheHamiltonian isthen reducedto thesolution of certain algebraic equations, called the Bethe Ansatz equations, where the number of unknowns depends edge occupation numbers m ,m ,m . It is important to note that the integrability of the model is not affected 1 2 3 byanintroductionoftwoarbitrary“horizontalfields”(or“boundarytwists”)whichrequiresonly simplemodificationstothetransfermatrixandtotheHamiltonian. Wefoundthisgeneralization to be extremely useful. In the following we will always consider the non-zero field case. Itwasremarkedmanytimes[23,24,34,39,44,45,49–51]thatthereareequivalent,butdifferent, formsoftheBetheAnsatzinthemodel. Infact, itiseasy toarguethatthereareprecisely3! = 6 differentBetheAnsa¨tzeinthiscase. Theyarerelatedbypermutationsoftheoccupationnumbers m ,m ,m . Indeed, there are three ways choose the bare vacuum state and then two ways to 1 2 3 proceed on the second “nested” stage of the Bethe Ansatz. Of course, the super-symmetry does 3 not play any special role in this respect. Exactly the same counting also takes place for all models related with U (sl(3)) algebra, see [52,53]. It is important to realize, however, that the q above arguments fully apply only to a generic non-zero field case. If the fields are vanishing (or takesomespecialvalues)bthenonlyafewoftheseBetheAnsätzearewelldefined,whiletheother suffer from the “beyond the equator” problem, first encountered in [54] for the XXX-model. The three-state gl(21)-Perk-Schultz R-matrix is just one representative of an infinite set | of R-matrices associated with the U (sl(21)) algebra. These R-matrices can be constructed q | as different specializations of the universal R-matrix. In particular, there are the so-called fusion [55] R-matrices related to the mbatrix representations of the finite-dimensional algebra U (gl(21)). Other important R-matrices connected with the (q-deformed) oscillator algebras q | and continuous quantum field theory. There are two ways this variety could be used. First, one canconsidermodelswithdifferentquantumspacesofstates. Second,forthesamequantumspace there are “higher” or “fusion” transfer matrices corresponding to different finite-dimensional representation of U (gl(21)) in the “auxiliary” space. All these transfer-matrices are operator- q | valued functions of a (multiplicative) spectral variable “x”. We will call them the T-operators. TheT-operators with differentvalues of xforma commutingfamily of operators. Theysatisfy a number of important functional relations, called the fusion relations (see eqs.(3.12) below). For the case of U (sl(21)) related models a complete set of these relations was proposed in [33–35] q | (see also, [28,30,45,46,56,57]). However, a direct algebraic proof of these relations in full generality, i.e., bfor anarbitrary quantumspaceandageneric valueof thedeformation parameter q, was not hitherto known. Here we fill this gap. The precise form of the functional relations, obviously, depends on the normalization of the T-operators. Here we use a distinguished normalization determined by the universal R-matrix (see Eqs. (2.19) and (2.39) below). The functional relations then take a universal form, which do not contain various model-dependent scalar factor. Such factors are usually present in the transfer matrix relations in lattice theory. To restore these factors in our approach one needs to explicitly calculate the specializations of the universal R-matrix for particular models. Here we compute these factors for the 3-state lattice model and in the case of the continuous conformal field theory, arising in quantization of the AKNS soliton hierarchy [58]. An important part in the theory of integrable quantum systems is played by the so-called Q-operators, introduced by Baxter in his pioneering work on the eight-vertex model of lattice statistics [1]. The Q-operators belong to the same commuting family of operators, as the T- operators. In this paper we present a complete algebraic theory of the Q-operators for the models related with U (sl(21)) algebra. There are six different Q-operators in this case. We q | denote them as Q (x) and Q (x), i = 1,2,3. They are single-valued functions in the whole i i complex plane of the varbiable of x, except the origin x = 0, where they have simple algebraic branching points, Q (e2πix) = e2πiSkQ (x), Q (e2πix) = e−2πiSkQ (x), k = 1,2,3, (1.2) k k k k Here S , S and S , such that S +S +S = 0, are constant operators (acting in the quantum 1 2 3 1 2 3 space) given by certain linear combinations of the Cartan generators of U (sl(21)) and external q | field parameters. They commute among themselves and with all the other operators in the commuting family. Their eigenvalues S are conserved quantum numbers, bwhich in the case of i the 3-state lattice model reduce to the edge occupation m , m and m , mentioned above (see 1 2 3 Eq.(1.9) below). From an algebraic point of view the Q-operators are very similar to the transfer matrices. They are constructed as traces of certain (in general, infinite-dimensional) monodromy matrices, arising as specializations of the universal R-matrix to the representations of the fermionic 4 and bosonic q-oscillator algebras. Using some special decomposition properties of products of these infinite-dimensional representations we show that the Q-operators satisfies a few fundamental functional relations. There are four independent Wronskian-type relations between the Q-operators, c12 = c13Q1(q+21x)Q1(q−21x) c23Q2(q+12x)Q2(q−12x) − c12 = c13Q1(q−21x)Q1(q+21x) c23Q2(q−12x)Q2(q+12x), − (1.3) c Q (x) = Q (qx)Q (q−1x) Q (q−1x)Q (qx), 21 3 1 2 1 2 − c Q (x) = Q (qx)Q (q−1x) Q (q−1x)Q (qx), 12 3 1 2 1 2 − where cij = (zi zj)/√zizj, with z1 = q2S1, z2 = q2S2 and z3 = z1z2 = q−2S3. − The fusion transfer matrices are expressed as polynomial combinations of the Q-operators. In particular, the fundamental transfer matrix (corresponding to the three-dimensional representation in the auxiliary space) is given by c12T(x)= c13Q1(q23x)Q1(q−23x) c23Q2(q32x)Q2(q−23x) (1.4) − With an account of (1.3) the last formula can be transformed to any of the six equivalent forms T(x)= p Qi(q−pi−21x) +p Qi(qpi+2pj−21x)Qk(qpi−pj−12x) +p Qk(qpi+pj+2pk−21x), (1.5) i Qi(qpi−12x) j Qi(qpi−12x) Qk(qpi+pj−12x) k Qk(qpi+pj−21x) where p = p = p = 1 and (i,j,k) is any permutation of (1,2,3), which are the standard 1 2 3 − Bethe Ansatz type expressions for the transfer matrix. All the above functional relations are written inthenormalization of theuniversalR-matrix (used forbothT andQoperators). They can be easily adjusted for the traditional normalization in the lattice theory, where the corresponding eigenvalues T(x) and Q (x) and Q (x) become finite polynomials of x. For instance, i i for the 3-state lattice model T(x) = pif(q2pi−21x)QQii((qq−ppi−i−1212xx)) +pjf(q−12x) QiQ(qi(pqi+pi2−pj12−x21)x)QQkk((qqppii+−ppjj−−1221xx)) +pkf(q−12x) QkQ(qkp(iq+ppi+j+pj2−pk21−x12)x) (1.6) where f(x) = (1 x)L and − mi L−mi Q (x) = xSi (1 x/x(i)), Q (x) = x−Si (1 x/x(i)). (1.7) i − ℓ i − ℓ ℓ=1 ℓ=1 Y Y (i) (i) The zeroes x , ℓ = 1,2,...,m and x , ℓ = 1,2,...,L m , i = 1,2,3, satisfy the Bethe { ℓ } i { ℓ } − i Ansatz equations, p f(q+pix(i)) Q (q+2pjx(i))Q (q−pjx(i)) i ℓ = i ℓ k ℓ , ℓ = 1,2,...,L m i − pj f(q−pixℓ(i)) Qi(q−2pix(ℓi))Qk(q+pjx(ℓi)) − p Q (q−pjx(k))Q (q+2pkx(k)) j = i ℓ k ℓ , ℓ =1,2,...,m (1.8) k −pk Qi(q+pjx(ℓk))Qk(q−2pjx(ℓk)) 5 where, as before, (i,j,k) denotes an arbitrary permutation of (1,2,3). Further, the eigenvalues of S , S and S , entering the exponents in (1.7), are given by 1 2 3 2S = L m +b , 2S = L m +b 2S = L m b b , (1.9) 1 1 1 2 2 2 3 3 1 2 − − − − − − where m and m and m = L m m are the edge occupation numbers, and b and b are 1 2 3 1 2 1 2 − − arbitrary field parameters. Eqs.(1.8) provide six self-contained sets of the Bethe Ansatz equations only involving zeros, belonging to a pair of the eigenvalues (A (x),A (x)), i = j. Once any such pair is determined, i j 6 the remaining zeroes can be found from the functional equations The organization of the paper is as follows. The algebraic definitions of the R-matrices, transfer matrices and Q-operators are given in Section 2. This section also contains neces- sary information about the representation theory of U (sl(21)). The functional relations are q | presented in Section 3. Their applications in continuous quantum field theory and their connections to the spectral theory of ordinary differential eqbuations are considered in Section 4. A direct algebraic proof of the functional relations is given in Section 5. Technical details of calculations are removed to four Appendices. Some of our results in Sect. 3 partially overlap with those in [44] devoted to some models in the rational case q = 1. Our approach to the Q-operators is different from that of [44]; in particular, it is applicable for an arbitrary quantum space and to generic values of q. 2 Yang-Baxter equation, transfer matrices and Q-operators 2.1 The universal R-matrix The quantum affine algebra = U (sl(21)) [59] (see also [60]) is generated by the elements q A | h ,h ,h , e ,e ,e and f ,f ,f , which are of two types: “fermionic” and “bosonic”. The 0 1 2 0 1 2 0 1 2 elements e ,e ,f ,f are fermionic, whbile all the other generating elements are bosonic. It is 0 2 0 2 convenient to assign the parity 1, X = e ,e ,f ,f , p(X) = 0 2 0 2 (2.1) 0, X = e ,f ,h ,h ,h , 1 1 0 1 2 (cid:26) such that p(XY)= p(X)+p(Y) (mod 2), X,Y , (2.2) ∈ A and introduce the generalized commutator [X,Y] = XY ( 1)p(X)p(Y)qYX. (2.3) q − − Note, in particular, that [X,Y] [X,Y] is reduced to the ordinary commutator when at least 1 ≡ one of the elements X,Y is even and to the anti-commutator when both of them are odd. The algebra U (sl(21)) is defined by the following commutation relations q | b qhi q−hi [h ,h ] = 0, [h ,e ]= a e , [h ,f ] = a f , [e ,f ]= δ − , (2.4) i j i j ij j i j − ij j i j ij q q−1 − where i,j = 0,1,2, the Serre relations e2 = f2 = [e ,[e ,e ] ] = [f ,[f ,f ] ] = 0, j = 0,2, (2.5) j j 1 1 j q−1 q 1 1 j q−1 q 6 and the extra Serre relations [e , [e , [e , [e , e ] ]]] = [e , [e , [e , [e , e ] ]]] , (2.6) 0 2 0 2 1 q−1 q 2 0 2 0 1 q−1 q [f , [f , [f , [f , f ] ]]] = [f , [f , [f , [f , f ] ]]] . (2.7) 0 2 0 2 1 q−1 q 2 0 2 0 1 q−1 q As usual, (a ) denotes the Cartan matrix ij 0 1 1 − (a ) = 1 2 1 , (2.8) ij 0≤i,j≤2  − −  1 1 0 −   Note, that the sum k = h +h +h , (2.9) 0 1 2 isacentralelement, commutingwithallotherelements ofthealgebra. Inthispaper,weconsider the case k = 0. The algebra =U (sl(21)) is a Hopf algebra with the co-multiplication q A | b ∆ : s (2.10) A −→ A⊗ A defined as ∆(h ) = h 1+1 h , i i s s i ⊗ ⊗ ∆(e ) = e 1+qhi e , (2.11) i i s s i ⊗ ⊗ ∆(f ) = f q−hi +1 f , i i s s i ⊗ ⊗ where i = 0,1,2 and denotes the graded tensor product, such that s ⊗ (A B)(C D)= ( 1)p(B)p(C)AC BD (2.12) s s s ⊗ ⊗ − ⊗ Thereis another co-multiplication ∆′ obtained from (2.11) by interchanging factors of thedirect products, ∆′ = σ ∆, σ (X Y) = ( 1)p(X)p(Y)Y X, X,Y . (2.13) s s ◦ ◦ ⊗ − ⊗ ∈ A The Borel subalgebras and are generated by h ,h ,h ,e ,e ,e and + − 0 1 2 0 1 2 B ⊂ A B ⊂ A h ,h ,h ,f ,f ,f , respectively. There exists a unique element [61,62] 0 1 2 0 1 2 , (2.14) + − R∈ B ⊗B satisfying the following relations ∆′(a) = ∆(a) ( a ), R R ∀ ∈ A (∆ 1) = 13 23, (2.15) s ⊗ R R R (1 ∆) = 13 12 s ⊗ R R R where 12, 13, 23 and 12 = 1, 23 = 1 , 13 = (σ 1) 23. The s s R R R ∈ A⊗ A⊗ A R R⊗ R ⊗R R ⊗ R element is called the universal R-matrix. It satisfies the Yang-Baxter equation R 12 13 23 = 23 13 12 , (2.16) R R R R R R 7 which is a simple corollary of the definitions (2.15). The universal R-matrix is understood as a formal series in generators in . Its dependence on the Cartan elements can be isolated + − B ⊗B as a simple factor, = qK, = h h h h , (2.17) 0 2 2 0 R R K − ⊗ − ⊗ where the “reduced” universal R-matrix = series in (e 1) and (1 f ) , (2.18) j j R ⊗ ⊗ is a series in (e 1) 1 and (1 f ) 1 , j = 0,1,2, and does not contain Cartan j + j − ⊗ ∈ B ⊗ ⊗ ∈ ⊗B elements. Remind that we assume k = h +h +h = 0. A few first terms in (2.18) can be 0 1 2 readily calculated directly from the definitions (2.14) and (2.15), 2 (q q−1)2 = 1 (q q−1) ( 1)p(j)e f + − (e )2 (f )2 R − − − j ⊗s j q2+1 1 ⊗s 1 j=0 X +(q q−1) e e f f ( 1)p(i)p(j)q−aije e f f + , (2.19) i j s j i i j s i j − ⊗ − − ⊗ ··· Xi6=j n o The symbol p(j) denotes the parity of the corresponding element e , namely j p(0) = p(2) = 1, p(1) = 0. (2.20) The higher terms in (2.19) soon become very complicated and their general form is unknown. This complexity should not be surprising, since the universal R-matrix contains infinitely many nontrivial solutions of the Yang-Baxter equation associated with U sl(21) . Fortunately, for q | applicationsoneonlyneedscertainspecializationsofuniversalR-matrix,whichcanbecalculated (cid:0) (cid:1) explicitly . Almost all these specializations are associated with the evabluation homomorphisms from the infinite-dimensional algebra U sl(21) (and from its Borel subalgebras) into finite- q | dimensional algebras. The most important case is the evaluation map to the finite-dimensional (cid:0) (cid:1) quantum algebra U gl(21) . This algebrabis generated by the elements E , i = 0,1,2 and E , q ii ij | (i,j) = (1,2),(2,1),(2,3),(3,2), for which we also use the notations (cid:0) (cid:1) E = E , E = E , F = E , F = E , (2.21) α 12 β 23 α 21 β 32 and H = E E , H = E +E , H = E +E . (2.22) α 11 22 β 22 33 α+β 11 33 − Theelements E andF areodd,p(E )= p(F ) = 1, allother generators areeven. Theysatisfy β β β β the following relations (written with the generalized commutator (2.3)) qHαi q−Hαi [E ,E ]= 0, [E ,E ]= (δ δ )E , [E ,F ] = δ − , ii jj ii kl ik − il kl αi αj αi,αj q q−1 − (2.23) E2 = F2 = [E ,[E ,E ] ] = [F ,[F ,F ] ] = 0, β β α α β q−1 q α α β q−1 q where the Greek indices α and α take two values α or β. Introduce the following elements i j E = qE22+2E33[E ,E ] , E = [E ,E ] q−E22−2E33 (2.24) 13 12 23 q 31 32 21 q−1 E = q−E22−2E33[E ,E ] , E = [E ,E ] qE22+2E33 (2.25) 13 12 23 q−1 31 32 21 q 8 Let x be a complex (spectral) parameter. Define the evaluation map Ev : U (sl(21)) U (gl(21)) x q q | −→ | as follows, b Ev (h )= E E , Ev (h )= E E , Ev (h ) =E +E , x 0 11 33 x 1 11 22 x 2 22 33 − − − Ev (e )= xE , Ev (e )= E , Ev (e ) = E , (2.26) x 0 31 x 1 12 x 2 23 − Ev (f ) = x−1E , Ev (f )= E , Ev (f ) = E . x 0 13 x 1 21 x 2 32 Onecan check thatthismapisan algebrahomomorphismas allthedefiningrelations (2.4)-(2.7) becomes corollaries of (2.23)1. Abriefintroduction into therepresentation theory ofU (gl(21)) isgiven in theAppendixA. q | We alsosummarizesomeimportantfacts here. Letπ , withµ = (µ ,µ ,µ ), suchthatµ µ µ 1 2 3 1 2 − ∈ Z , denotes theirreduciblefinite-dimensionalrepresentation oftheU (gl(21)) withthehighest ≥0 q | weight µ and the highest weight vector 0 > defined as | E 0 >= E 0 >=0, E 0 >= µ 0 >, i = 1,2,3. (2.27) 12 23 ii i | | | | Any such representation is realized by linear transformations End(V) of some graded vector space V = V V , where p(V ) = 0 and p(V ) = 1. The latter is always a subspace of the 0 1 0 1 ⊕ vector space generated by a free action of the elements E and E on the highest weight vector 21 32 (note that the action of E changes the parity, while the action of E leaves it unchanged). 32 21 There are bases of V, called homogeneous, where all basis vectors v have definite parities, i { } i.e., v V or v V for any v . Let A be an arbitrary matrix A End(V), and A denote i 0 i 1 i ij ∈ ∈ ∈ its matrix elements in a homogeneous basis Av = v A . The supertrace of A over V is k j j jk defined as Str A= ( 1)p(vj)A . V j − jj P Further, let π (x) be the representation of U (sl(21)) obtained by the composition of π µ P q | µ with the evaluation map (2.26), π (x) = π Ebv . (2.28) µ µ x ◦ Then the finite-dimensional R-matrices, obtained from the universal R-matrix, R (x /x ) = (π (x ) π (x ))[ ] (2.29) µ1µ2 1 2 µ1 1 ⊗ µ2 2 R satisfy appropriate specializations of the Yang-Baxter equation (2.16), R (x /x )R (x /x )R (x /x )= R (x /x )R (x /x )R (x /x ) (2.30) µ1µ2 1 2 µ1µ3 1 3 µ2µ3 2 3 µ2µ3 2 3 µ1µ3 1 3 µ1µ2 1 2 2.2 T-operators (transfer matrices) First, let us review standard definitions of the transfer matrices in lattice models. TheR-matrix R (x), defined in (2.29), acts in the graded product of two representation spaces V V . It µν µ s ν ⊗ is convenient to consider the first of these spaces as an “auxiliary space” and the second one as a “quantum space”. The transfer matrix for a homogeneous periodic chain of the length L is defined as follows, T (xν,y,L) = Str D R (x/y) R (x/y) R (x/y) , (2.31) µ | πµ µν ⊗s µν ⊗s···⊗s µν (cid:16) L−times (cid:17) 1Notethereisanother(non-equivalent)ev|aluationmapobtainedf{rzom(2.26),ifE andE} arereplacedwith 31 13 E and E . 31 13 9 where the tensor product is taken with respect to the quantum spaces V = π (y), while the ν ν matrix product and the supertrace is taken with respect to the auxiliary space V = π (x). µ µ The boundary operator D reads D= qb1E11+b2E22+(b1+b2)E33 = v q−b1h0+b2h2 , (2.32) x E where E ,E ,E are defined in (2.23) and b ,b deno(cid:2)tes two arbit(cid:3)rary horizontal field pa- 11 22 33 1 2 rameters. The transfer matrix (2.31) acts in a Hilbert space T (xν,y,L) : (ν) (ν), (ν) = V V V , (2.33) µ ν s ν s s ν | H → H H ⊗ ⊗ ···⊗ L−times which is the graded product of L copies of the space V =|π (y). {z } ν ν The symbols µ and ν in the notation for the transfer matrix T (xν,y,L), obviously, refer µ | to the auxiliary and quantum spaces respectively. For the same quantum space (ν) there H is an infinite number of different transfer matrices, corresponding to different choices of the representation µ in the auxiliary space. The Yang-Baxter equation (2.30) implies that these matrices form a commuting family [T (x ν,y,L),T (x ν,y,L)] = 0, for all µ ,µ ,x ,x . (2.34) µ1 1| µ2 2| 1 2 1 2 Note that due to the invariance property of the R-matrix (which trivially follows from (2.11) and the first relation in (2.15)) (D D )R (x) = R (x)(D D ), D = π D , (2.35) µ s ν µν µν µ s ν µ µ ⊗ ⊗ (cid:2) (cid:3) the commutativity (2.34) is not affected by the presence of non-zero fields in definition (2.31). Below we will derive algebraic relation between different transfer matrices, using decomposition properties of products of representations of the quantum affine superalgebra U (sl(21)) q | in the auxiliary space. We would like to stress that our results are independent on the quantum space of the model. To facilitate these considerations it is useful to make a model-indepbendent definition of the transfer-matrices, T (x) = Str 1 (q−b1h0+b2h2 1) (2.36) µ πµ(x)⊗ ⊗ R h i (cid:0) (cid:1) where istheuniversalR-matrix, andb andb aretheexternalfieldparameters. Thisformula 1 2 R defines a “universal” T-operator which is an element of the Borel subalgebra associated with − B the quantum space. To specialize it for a particular model one needs to choose an appropriate representation of . For example, choosing the latter to be the product π (y) π (y) − ν s ν s B ⊗ ⊗ π (y), where π (y) is defined by (2.28), one immediately obtains2 the lattice transfer s ν ν ··· ⊗ matrix (2.31) T (xν,y,L) = π (y) π (y) π (y) T (x) . (2.37) µ ν s ν s s ν µ | ⊗ ⊗ ···⊗ (cid:0) L−times (cid:1)h i Another important example of the |specialization{ozf (2.36), relat}ed with the continuous super- conformal field theory associated with U (sl(21)) algebra, is considered in Section 4. q | It is convenient to define new operators b z = q2S1 = qh2+b1, z = q2S2 = q−h0+b2, z = q−2S3 = z z , (2.38) 1 2 3 1 2 2To evaluatethe RHSof (2.37) one needs torepeatedly use thethird equation in (2.15). 10