Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra $U_q(sl(2))$ leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.


Introduction
The "Yang-Baxter maps" [1] are invertible maps of a Cartesian product of two identical sets X , R : XˆX Þ Ñ XˆX (1.1) satisfying the "functional" or "set-theoretic" Yang-Baxter equation [2], This equation states an equality of two different composition of the three maps R 12 , R 13 and R 23 , acting on different factors in a product of three sets XˆXˆX (for instance, R 12 coincides with the map (1.1) in the first and second factors and acts as an identity map in the third one; the maps R 13 and R 23 are defined similarly). The interest to the Yang-Baxter maps is motivated mostly by their connection to discrete integrable evolution equations. Important examples of the Yang-Baxter maps as well as some classification results were obtained in [3][4][5][6][7][8][9][10][11]. Mention also a related consistency around a cube condition [12], which is in many cases can be associated with some set-theoretic Yang-Baxter equation. One common limitation of the existing methods is that Hamiltonian structures of the arising maps generally remain unclear. As a result there are no regular procedures for quantization of the Yang-Baxter maps or their applications to Hamiltonian evolution systems.
In this paper we address these problems and present a new approach to the Yang-Baxter maps, which is based on the theory of quantum groups [13,14] and naturally connected to Baxter's commuting transfer matrices [15] and the Quantum Inverse Problem Method [16]. The key role in our approach will be played by the universal R-matrix [13]. This notion is associated with the so-called, quasitriangular Hopf algebras, which include a very important class of the q-deformed (affine) Lie algebras and super-algebras (more precisely, their universal enveloping algebras). Let A be a quasitriangular Hopf algebra, then there exists an invertible element R P A b A, belonging to the tensor product of two algebras A, called the universal R-matrix, which by construction satisfies the quantum Yang-Baxter equation Similarly to Eq.(1.2) the indices here indicate how the universal R-matrix is embedded into the triple product A b A b A (for example, R 12 P A b A b 1, and similarly for R 13 and R 23 ). Interestingly, the universal R-matrix allows one to construct a map from the tensor square of the algebra A to itself [17,18], acting as R : px b yq Þ Ñ px 1 b y 1 q " R px b yq R´1 , x, y P A . (1.5) It will automatically satisfy Eq.(1.2) in virtue of (1.3) and commutation relations of the quantum algebra A. Thus, for any quasitriangular algebra A the map (1.5) is a solution the set-theoretic Yang-Baxter equation (1.2), where the algebra A serves as a set X . At this point it should be noted that a quasi-triangular algebra A is, of course, not just an abstract structureless set (as it is implicitly assumed in the setting of Eq.(1.2)), but has algebraic relations between its elements, which must be taken into account in order for the Eq.(1.2) to hold. Obviously, this is a slight generalization of the meaning of Eq.(1.2), but this is a well needed generalization, since the most important applications of the Yang-Baxter maps to dynamical systems naturally require to equip the set X with an additional structure for purposes of quantization or considerations of Hamiltonian systems. Indeed, we show that the map (1.5) defines Hamiltonian evolution equations for a discrete integrable quantum system in 2D (with discrete space and time) with an algebra of observables formed by a tensor power of the quantum algebra A. The functional Yang-Baxter equation (1.2) then implies that these evolution equation obey the quantum version of the consistency-around-a-cube condition [12]. Further, the map (1.5) possesses a discrete analog of the zero curvature representation. This allows one to construct a full set of mutually commuting integrals of motion using the standard approach of commuting transfer matrices [15] and the Quantum Inverse Problem Method [16]. The unitary Heisenberg evolution operator for the system are expressed through the matrix elements of the universal R-matrix.
Remarkably, all steps of our new approach to the Yang-Baxter maps admit a meaningful quasiclassical limit. As a result for any quasitriangular algebra A one obtains a classical Yang-Baxter map which automatically possesses properties of a Hamiltonian map, since it preserves the tensor product structure of Poisson algebras arising in the quasiclassical limit of the quantum algebra A. The action functional of the corresponding classical discrete integrable system is determined by the quasi-classical asymptotics of the universal R-matrix.
In this paper we illustrate the above ideas for the case of the algebra U q pslp2qq, which is the simplest example of a quasitriangular Hopf algebra. The quantum case is considered in Sect. 2 and the classical one in Sect. 3. The considerations are completely parallel. For the reader's convenience we preserve the same numeration of the corresponding subsections in both cases. Note the discrete integrable equations arising here for the algebra U q pslp2qq are related to the discrete Liouville equation, various variants of which were previously considered in [19][20][21][22].
2 Quantum Yang-Baxter map for the algebra U q pslp2qq In this subsection we briefly summarise the basic properties of the algebra U q pslp2qq. A more detailed description of this algebra well suited to our purposes can be found in [23]. The universal enveloping algebra A " U q pslp2qq is generated by elements H, E, F , satisfying the relations, rH, Es " 2 E , rH, F s "´2 F , rE, F s " pq´q´1q pq H´q´H q . (2.1) Here we use a normalisation, where the elements E and F are multiplied by the factor pq´q´1q with respect to the usual choice. Below, it will be convenient to also use the element K " q H , for which the commutation relations (2.2) become The quadratic Casimir operator, which commutes with all other elements of the algebra, has the form The algebra (2.2) is a Hopf algebra with the co-multiplication ∆, the co-unit ǫ, and the antipode S. The co-multiplication is a map from the algebra A to its tensor square ∆ : The co-unit ǫ is a map from A to complex numbers. It is defined as The co-multiplication and co-unit define algebra homomorphisms, i.e, ∆pabq " ∆paq ∆pbq and ǫpabq " ǫpaqǫpbq. The antipode S is defined as 1 Correspondingly, for the element H one has Note that the antipode is an algebra anti-homomorphism, i.e., Spabq " Spbq Spaq. It satisfies an important properties where ∆ 1 is another co-multiplication obtained from ∆ by interchanging factors in the tensor product, ∆ 1 " σ˝∆, σpx b yq " py b xq. (2.10)

Universal R-matrix
The algebra U q pslp2qq is a quasi-triangular Hopf algebra. This means that there exists an element R P A b A, called the universal R-matrix, which satisfies the properties where R 12 " R b 1, R 23 " 1 b R and R 13 " pσ b 1q R 23 . Together with (2.7) and (2.9) this definition implies the quantum Yang-Baxter equation and two simpler relations Note, that the universal R-matrix is not unique as an element of AbA. However, if one assumes, that R P B´b B`, where B`and B´are Borel subalgebras of A, generated by the elements pH, F q and pH, Eq respectively, then the universal R-matrix is uniquely defined by the following formal series [13] R " q (2.14)

Quantum Yang-Baxter map
Let X " tK, E, F u denotes the set of generating elements of the algebra A and X 1,2 denote these sets in the corresponding components of the tensor product A b A, Let us now explicitly calculate the map (1.5). Evidently, it is completely determined by its action on the generating elements of the tensor product, Using (2.14) and (2.2) one reproduces the result of [18] $ ' ' ' & "set-theoretic multiplication" δ which acts on two sets of generating elements δ : pX 1 , X 2 q Þ Ñ X 1 , (2.19) and write it as Explicitly, it is defined as  Consider the tensor product A b A b A of three algebras A and let pX 1 , X 2 , X 3 q denote three sets of generators in the corresponding components of the product. Define a functional operator R 12 R 12 pX 1 , X 2 , X 3 q :" pX 1 1 , X 1 2 , X 3 q " pRpX 1 , X 2 q, X 3 q (2.23a) which acts as (2.16) on the first two sets and does not affect the third one. Similarly, define operators At this point it is worth commenting on a relationship between the operator notations used in Sect.2.2, and the set-theoretic substitutions R ij defined here. Note, that successive similarity transformations acting on a list of generator sets pX 1 , X 2 , X 3 . . .q with quantum operators R ij is equivalent to a successive application of the corresponding set-theoretic map R ij taken in the reverse order. For instance, one can easily verify R 12 R 13 pX 1 , X 2 , X 3 q R´1 13 R´1 12 " R 13 pR 12 pX 1 , X 2 , X 3 qq , R 12 R 13 R 23 pX 1 , X 2 , X 3 q R´1 23 R´1 13 R´1 12 " R 23 pR 13 pR 12 pX 1 , X 2 , X 3 qqq . (2.24) Eq.(2.12) then immediately implies that the set-theoretic maps R ij satisfy the functional Yang-Baxter equation R 23 pR 13 pR 12 pX 1 , X 2 , X 3 qqq " R 12 pR 13 pR 23 pX 1 , X 2 , X 3 qqq . (2.25) and, more generally, To reformulate the remaining properties of the universal R-matrix introduce additional notations, δ 12 pX 1 , X 2 , X 3 q :" pδpX 1 , X 2 q, X 3 q , δ 23 pX 1 , X 2 , X 3 q :" pX 1 , δpX 2 , X 3 qq .

28)
Rpδ 12 pX 1 , X 2 , X 3 qq " δ 12 pR 23 pR 13 pX 1 , X 2 , X 3 qqq , (2.29) Further, from (2.13) it follows that where the co-unit ǫ and the antipode S are defined in (2.7). We would like to stress that the above relations (2.28)-(2.32) are not specific to the algebra U q pslp2qq and must hold for any quasi-triangular Hopf algebra. In the case of U q pslp2qq they could be verified using the explicit form of the map R and its inverse given in (2.17) and (2.18). The equations (2.28) and (2.30) are illustrated in Fig. 2 2.5 R-matrix form of the U q pslp2qq defining relations As remarked before the universal R-matrix is not uniquely defined by the relations (2.11). One can easily to check, that if R 12 P B´b B`, given by (2.14), satisfies (2.11), then so does the element which does not coincide with R 12 (this is why the algebra U q pslp2qq is called a quasi-triangular Hopf algebra). In particular, the new element R˚satisfies the Yang-Baxter equation which is a simple corollary of (2.12) and (2.33). Similarly, one can derive "mixed" relations Let π 1 2 denotes the 2-dimensional (spin 1 2 ) representations of the algebra (2.1), Define the L-operators by evaluating the universal R-matrix in the 2-dimensional representation in the first space (to be called an auxiliary space). Using (2.14) and (2.33), one gets which are operator-valued matrices, whose elements belong to U q pslp2qq and act in the "quantum space". Choosing the two-dimensional representation for the latter one gets the block R-matrices (2.39) where the internal blocks act in the second space of the product C 2 b C 2 . It is convenient to define the parameter-dependent L-operator The latter, of course, coincides with the R-matrix of the six-vertex model [24]. It possesses an important property where P is the permutation matrix, interchanging the factors of the product Let us now specialize the Yang-Baxter equations (2.12), and (2.34) by choosing the 2-dimensional representations (2.35) in the first and second spaces and then using (2.36) and (2.38). The matrix elements of the L-operators will then act in the third space, which now becomes the quantum space. As a result one obtains the "R-matrix form" [25] of the commutation relations (2.1) (2.44) The symbol 9 b here denotes the tensor product of two-by-two matrices L˘, acting in different factors of C 2 b C 2 , and operator product for their matrix elements. With an account of (2.42) and the definitions (2.40), (2.41), the above relations can be combined into a compact form of the parameter-dependent Yang-Baxter equation which is well-known in the theory of the six-vertex model (see [26,27] for recent advances in this area). Finally, note that the co-multiplication ∆, defined in (2.5), can be written as where the symbol b denotes the matrix product of the two-by-two matrices (2.37) and the tensor product for their operator-valued elements, belonging to different copies of the algebra U q pslp2qq. There are two equations in (2.46), where the superscripts are either all pluses or all minuses, simultaneously.

Heisenberg-Weyl realisation
The algebra U q pslp2qq admits an important homomorphism into the Heisenberg-Weyl algebra, W q , generated by two elements u and v, Namely, if one sets where the element z commutes with u and v, then all commutation relations (2.2) will be satisfied in virtue of (2.47). The commuting element z parametrises the Casimir operator (2.3), Let us now consider the representation (2.48) as a change of variable for the set of generators tK, E, F u to the set u, v, z ( and rewrite the map (2.17) in the new variables. Assume the same meaning to the subscripts 1, 2 as in (2.17) (they distinguish components of the tensor product A b A), for instance, and similarly for v 1,2 and z 1,2 . From (2.17) it follows then (2.51) where we have used the notation Note that the commuting elements z 1 and z 2 , obviously, remain invariant. They enter the map (2.51) as (spectral) parameters. Similarly for the inverse map, one obtains Figure

Discrete quantum evolution system
Here we introduce a p1`1q-dimensional integrable quantum evolution system with discrete space and time. In doing this we follow the scheme suggested in [28,29] and further developed in [30][31][32]. Remind that the quantum Yang-Baxter map (2.16) acts on a tensor square of the algebra A, It can be represented graphically as in Fig.3(a). We use this map to define a discrete quantum evolution system, where the algebra of observables is formed by a tensor power of the algebra A (all factors here are just independent copies of the algebra A; the indices indicate positions of the corresponding factors in the product). Let denotes a composition of the map (2.55), acting on the i-th and j-th factors of the product (2.56), followed by the permutation operator σ ij , swapping these factors with each other. With this notation we can define the map where the operator S cyclically shifts the factors in the product (2.56), The map (2.58) defines an evolution for one step of the discrete time. By construction it preserves the tensor product structure of the algebra of observables where each algebra A 1 n is isomorphic to the algebra A. So the evolution map is an automorphism of the algebra of observables, which preserves its tensor product structure (often referred to as an "ultra-local" structure). For a geometric interpretation consider the square lattice drawn diagonally as in Fig. 4, with 2N sites per row and periodic boundary conditions in the horizontal (spatial) direction. Now assign the factors of the product (2.56) to the set of connected edges forming a horizontal "saw", as shown in Fig. 4. In the same way assign the algebras A 1 n in the product of (2.60) to edges of the saw, shifted by one time step above from the initial saw. Note Figure 4: Assignment of dynamical variables to edges of the square lattice. Due to periodic boundary conditions the points P 1 , P 2 , P 3 on the left side should be identified with their images P 1 1 , P 1 2 , P 1 3 on the right side of the lattice.
that these conventions are completely consistent with the graphical representation of the map (2.55), corresponding to one quadrilateral, shown in Fig.3(a). The elements of the algebra (2.56) constitute the set of dynamical variables of the system at any fixed value of time. With the Heisenberg-Weyl realization (2.48) the generating elements of this algebra are expressed through the elements tu i , v i , z i u, where the index i " 1, 2, . . . 2N is considered as the spatial coordinate, numerating the factors in (2.56). Combining (2.51) with (2.58) and (2.60), it is easy to see that U pz 2n`1 q " z 2n´1 , U pz 2n q " z 2n , n " 1, 2, . . . , N , where z 2N`1 " z 1 . For simplicity, consider the homogeneous case, when with arbitrary z 1,2 . Then all z's stay unchanged under the time evolution, while the remaining variables transforms as where n " 1, 2, . . . N , (2.64) and the cyclic boundary conditions u 2N`1 " u 1 and v 2N`1 " v 1 are implied. With this realization the algebra of observables at any fixed moment of the discrete time reduces to the (spatially localized) tensor product of the Heisenberg-Weyl algebras (2.47), exactly as one would expect for the equal-time commutation relations of the canonical variables in a discrete quantum system.

Zero curvature representation
Consider again the Yang-Baxter map (2.55). Let L1 and L2 denote L-operators (2.37), with elements belonging respectively to the first and second algebra of the tensor product A 1 b A 2 , appearing in (2.55). Next, recall the Yang-Baxter equation for the universal R-matrix, given in (2.12). Cyclically shifting its indices p1, 2, 3q Ñ p3, 1, 2q and then choosing the 2-dimensional representation (2.35) in the (new) space 3, one obtains This equation involves a matrix product of L's (as two-by-two matrices) and a tensor product for their operator-valued matrix elements. Without the use of the indices 1, 2, the same equation can be written as For instance, the first equation on the last line reads This equation, as well as the other two matrix equations in (2.67) are simple corollaries of the explicit expression of the Yang-Baxter map, given in (2.17). Conversely, the equations (2.67) can be used as an alternative definition of the Yang-Baxter map independent of the notion of the universal R-matrix. Evidently, Eqs.(2.67) provide a "Zero Curvature Representation" (ZCR) for the map (2.17) and for the local equations of motion of the discrete quantum evolution system considered in the previous subsection. Indeed, as illustrated in Fig.3(b), the two-by-two matrices L can be viewed as edge transition matrices for an auxiliary discrete linear problem. Eqs.(2.67) then imply that products of transition matrices along the paths Q 0 Q 1 Q 2 and Q 0 Q 1 1 Q 2 are exactly the same, i.e., independent of the choice of path between Q 0 and Q 2 .
Using the notation (2.40) it is convenient to rewrite the ZCR (2.67) in the form where λ is arbitrary.

Commuting Integrals of Motion
The phase space of the quantum evolution system (2.60) possesses 4N degrees of freedom (2N coordinates and 2N momenta). Let us show that the system has exactly 2N linearly independent mutually commuting integrals of motion (IM). Introduce two transfer matrices and Tpλq " Tr´L1 It follows from (2.44) and (2.45) that these transfer matrices form a commuting family " Further, using the zero curvature representation (2.69) one can easily show that the transfer matrices Tpλq and Tpλq are, in fact, time-independent, i.e., they are unchanged under the discrete evolution map (2.58), Tpλq " U`Tpλq˘. (2.72) Let pH k , E k , F k q P A k be the set of generators belonging to the k-th algebra in the tensor product (2.56). Define the operators which obey the following relations To within simple factors the transfer matrices (2.70) are N -th degree polynomials in the variable λ 2 (or λ´2), Tpλq " λ´N N ÿ n"0 λ 2n G n (2.77) whose coefficients are commuting integrals of motion.
From the definition (2.70) it is easy to see that there are only 2N independent coefficients, since two pairs of them coincide The remaining coefficients can be expressed as "ordered sums" of products of the operators (2.73), which are analogous to the ordered integrals that appear in the context of the continuous conformal field theory [33] (the so-called non-local integrals of motion). For instance, the simplest non-trivial coefficients read Finally note that the transfer matrices (2.70) can be viewed as modified versions of the transfer matrix of the six-vertex model [24].
(2.88) Further, let |s 1 , s 2 y " |s 1 , α 1 , β 1 y b |s 2 , α 2 , β 2 y (2.89) denotes basic vectors in the tensor product π α 1 ,β 1 b π α 2 ,β 2 of two such representations, where for brevity the dependence on the parameters α 1,2 and β 1,2 in the LHS of (2.89) is not explicitly shown. It is useful to introduce additional notations Equations (2.50) and (2.51) provide recurrence relations for the matrix elements of the Rmatrix. For instance, consider the second equation in the left column of (2.51), namely If this equation is "sandwiched" in between the vectors xs 1 , s 2 | and |s 1 1 , s 1 2 y it gives In this way four similar equations in (2.51) and (2.53) lead to the following four recurrence relations (2.93) We assume that the matrix elements xs 1 , s 2 | R 12 |s 1 1 , s 1 2 y are analytic in the the strip´b ď Ims i , Ims 1 i ď b. Then the above recurrence relations uniquely define these matrix elements up to an inessential normalization factor, xs 1 , s 2 | R 12 |s 1 1 , s 1 2 y " V β 1´α2 ps 2´s1 qV α 1´β2 ps 1 2´s 1 1 qV α 1´α2 ps 1 2´s1 qV β 1´β2 ps 2´s where V α psq " e iπ{8´iπs 2 ϕpiα´sq , The function ϕpsq is the "non-compact" quantum dilogarithm [36] ϕpsq " expˆ1 4 (2.96) and the parameter η is defined in (2.82). Note that essentially the same expression (2.94) (but in a slightly modified form) was also obtained in ref. [37]. The non-compact quantum dilogarithm satisfies the quantum pentagon identity [22] ϕppqϕpxq " ϕpxqϕpp`xqϕpxq , where x, p are the elements of Heisenberg algebra (2.83). The functions s V α psq and V α psq are neither positively defined nor symmetric with respect to the substitution s Ñ´s. However, they are Fourier-dual to each other ż R e 2πisx V α psq ds " V α pxq . (2.98) The factorized form for the kernel of the R-matrix corresponds to a factorized operator expression acting in the product of two Heisenberg-Weyl algebra (2.47) (but not the full algebra U q pslp2qq), where P 12 is the permutation operator, P 12 |s 1 , α 1 , β 1 y b |s 2 , α 2 , β 2 y " |s 2 , α 1 , β 1 y b |s 1 , α 2 , β 2 y, (2.100) which swaps the coordinates s 1 and s 2 , but keep the parameters α i and β i unchanged 4 . This R-matrix coincides with that for the Dehn twist in quantum Teichmüller theory [39]. Explicit correspondence will be established in Appendix. The elements of the matrix inverse to (2.94) are given by xs 1 1 , s 1 2 |R´1|s 1 , s 2 y " Vα 1´α2 ps 1 2´s1 qVβ 1´β2 ps 2´s where Vαpsq " e iπ{8´iπs 2 ϕpiη`iα´sq . where α is purely imaginary and a real positive ǫ defines circumventions of the poles. Note, that in the case when η is real and the spectral parameter α is purely imaginary, the superscript '˚' can be viewed as the complex conjugation. which reduces the (discrete time) evolution to an internal automorphism of the algebra of observables. To write this operator explicitly, first remind that we have assumed that the elements z k take only two values, as in (2.62). Next, consider the representation of algebra (2.56) π O " π α 1 ,β 1 b π α 2 ,β 2 b π α 1 ,β 1¨¨¨b π α 2 ,β 2 (2.106) given by the product of the representations (2.88) with alternating values of the parameters α and β. Further, let | s y " |s 1 , α 1 , β 1 y b |s 2 , α 2 , β 2 y b |s 3 , α 1 , β 1 y b¨¨¨b |s 2N , α 2 , β 2 y (2.107) denotes the corresponding basis vectors. Then the matrix elements of U are given by 6 x s | U | s 1 y " N ź n"1 xs 2n´1 , s 2n | R |s 1 2n`1 , s 1 2n y (2.108) The physical regime corresponds to imaginary values of α i , β i . In this case R-matrix is unitary and therefore the Heisenberg evolution operator is unitary as well.
describes the initial and final states of the system. Then

2.110)
3 Quasi-classical limit and classical Yang-Baxter map

Poisson algebra
Consider the quasi-classical limit where the parameter b is the same as in (2.82). In this limit the algebra (2.2) becomes a Poisson algebra (i.e., a commutative algebra with an associative Poisson bracket t , u, obeying the Leibniz rule). Making a substitution and replacing commutators in (2.2) by Poisson brackets, one obtains the following Poisson algebra, The central element (2.3) takes the form The structure of the Hopf algebra in this limit remains intact. In particular, one can define the co-multiplication, co-unit and antipode, which, obey the same properties as for the quantum algebra U q pslp2qq. We will denote them by the symbols ∆, ǫ and S, respectively. Define the co-multiplication ∆ as a map from the Poisson algebra P to its tensor square ∆ : P Þ Ñ P b P, (3.6) acting as Similarly to (2.6) and (2.7) define the co-unit ǫ, ǫ pkq " 1 , ǫ peq " ǫ pf q " 0, (3.8) and the antipode S, S pkq " k´1, S peq "´e k´1, S pf q "´k f . (3.9) Note that co-multiplication and co-unit are homomorphisms of the Poisson algebra, while the antipode is an antihomomorphisms. In particular, this means that The relations (2.9) remain exactly the same as in the quantum case, where ∆ 1 " σ˝∆ is another co-multiplication obtained from (3.7) by interchanging factors in the tensor product.

Quasi-classical expansion of the universal R-matrix
In the limit (3.1) the universal R-matrix (2.14) becomes singular. It is not difficult to show that where k, e, f are the generators of the Poisson algebra (3.4) and Li 2 pxq is the Euler dilogarithm (3.13)

Classical Yang-Baxter map
Even though the quasi-classical limit of the universal R-matrix becomes singular (see (3.12) above) its adjoint action on the elements of the tensor product of two algebras (2.2), appearing in (2.16), is well-defined [17]. Therefore the q Ñ 1 limit of the quantum Yang-Baxter map (2.16) is well defined. We will denote it by a special symbol to distinguish it from its quantum counterpart. Let x " pk, e, f q denotes the set of generators of Poisson algebra P and x 1,2 denotes the corresponding sets in the two components of the tensor product P b P, Similarly to (2.22) the classical map R is completely defined by is action to two sets of generators in the tensor square P b P, The formulae (2.17) lead to We would like to stress that this map preserves the tensor product structure of two Poisson algebras R : P b P Ñ P b P, (3.18) which means that the "primed" quantities tk 1 i , e 1 i , f 1 i u, i " 1, 2, have exactly the same Poisson brackets (3.4) as the elements tk i , e i , f i u. The inverse map reads Similarly to the quantum case (see (2.32)) the direct and inverse maps are related by the antipode (3.9) R˝pS b Sq " pS b Sq˝R´1 . (3.20)

Properties of the classical Yang-Baxter map
The quasi-triangular structure of the quantum algebra is also inherited on the classical level. Therefore, most of the relations of Sect. 2.4 remains literally unchanged in the classical case. For completeness we summarize them below. Let be sets of generating elements of different copies of the Poisson algebra (3.4). Introduce the set-theoretic multiplication δ which acts on a two sets generators δ : and write it as Explicitly, one has which is essentially the same formulae as (3.7). As in quantum case, the map (3.22) is only defined on two sets of generating element and the direction of the arrow in (3.22) is reversed with respect to (3.6). Consider the tensor product P b P b P of three Poisson algebras (3.4) and let px 1 , x 2 , x 3 q denote three sets of generators in the corresponding components of the product. Define a functional operator R 12 R 12 px 1 , x 2 , x 3 q :" pRpx 1 , x 2 q, x 3 q (3.25a) which acts as (3.16) on the first two sets and does not affect the third one. Similarly, define operators R 13 " σ 23˝R12˝σ23 , In the q Ñ 1 limit Eq.(2.26) simply reduces to the set-theoretic Yang-Baxter equation for the classical maps R ij R 23˝R13˝R12 " R 12˝R13˝R23 . (3.26) To formulate the remaining properties of the classical map R introduce additional notations, Then Eqs.(2.11) lead to

28)
Rpδ 12 px 1 , x 2 , x 3 qq " δ 12 pR 23 pR 13 px 1 , x 2 , x 3 qqq , (3.29) where the permutation operator σ is defined in (2.10). Further, from (2.13) it follows that where the co-unit ǫ and the antipode S are defined in (3.9). We would like to stress that the above relations (3.26), (3.28)-(3.32) are not specific to the Poisson algebra (3.4) and must hold for any (quadratic) Poisson algebra which arise in the quasi-classical limit of a quasi-triangular Hopf algebra. In the case under consideration these relations could be verified using the explicit form of the map R and its inverse given in (2.17) and (2.18). Note that, contrary to the quantum case, the above relations only involve ordinary substitutions for a set of commuting variables, since the Poisson algebra P is a commutative algebra. The fact, that the map (3.18) is a "canonical transformation" (in the sense of Hamiltonian dynamics), preserving the Poisson brackets of the generating elements, does not play any role for the validity of (3.26)-(3.32).

r-matrix form of the Poisson algebra
As noted before, the universal R-matrix (3.12) as well as the Yang-Baxter equations (2.12) and (2.34) becomes singular in the quasiclassical limit (3.1), (3.2). We will return to these singular equations in more details in Sect.3.10.

Heisenberg-Weyl realisation (canonical variables)
The three-dimensional Poisson algebra (3.4) has one central element (3.5). Therefore it must essentially reduce to the classical Heisenberg-Weyl algebra, defined by the bracket The required coordinate transformation (cf. (2.48)) is The variable z parametrises the central element (3.5), c " z`z´1 , (3.42) and has vanishing Poisson brackets with u and v. In the new variables the map (3.17) reduces to the classical analog of (2.51) Note that the parameters z 1,2 remain unchanged under the map. Similarly for the inverse map one obtains (3.46)

Discrete classical evolution system
The quantum evolution system considered in Section 2.7 admits a straightforward classical limit. Remind, that the classical Yang-Baxter map (3.16) acts on the tensor square of the Poisson algebra (3.4), Correspondingly, the algebra of observables (2.56) becomes a tensor power Poisson algebra (3.4), Similarly to the quantum case the factors of this product are assigned to edges of one horizontal saw, corresponding to a fixed moment of the discrete time, as shown in Fig. 4 (one just need to replace all A's there by P's to get the classical picture). The classical evolution map is defined as where q R ij " σ ij˝Rij is the map (3.47), acting on the i-th and j-th factors of the product (3.48), followed by the permutation operator σ ij . The operator S cyclically shifts (to the right) the factors in the product (3.48), similarly to (2.59). The one-step time evolution is defined as For the realisation (3.41) each edge will be assigned a triple of dynamical variables tu i , v i , z i u, i " 1, 2, . . . , 2N . As in the quantum case we assume the homogeneous arrangement (2.62). Then all z's stay unchanged under the time evolution. The remaining variables possess the canonical Poisson brackets t log u i , log v j u " δ ij , i, j " 1, 2, . . . , 2N . (3.51) which are preserved by the time evolution. Taking classical limit in (2.63) and introducing discrete time variable t, one immediately obtains the Hamiltonian equations of motion where g n,t " 1´v 2n´1,t pz 1´u2n´1,t qpu 2n,t´1 {z 2 q u 2n´1,t v 2n,t . (3.53)

Zero curvature representation
The quantum zero curvature representation (2.67) admits a straightforward quasi-classical limit. Let ℓ1 and ℓ2 denote the classical ℓ-operators (3.33) with elements belonging respectively to the first and second Poisson algebra of the tensor product P 1 b P 2 . Next, let r ℓ1 and r ℓ2 are defined in a similar way, but with elements from the "transformed" algebras P 1 1 and P 1 2 , arising from the map (3.47). Then Eqs.(2.67) lead to the relations ℓ1 ℓ2 " r ℓ2 r ℓ1 , ℓ1 ℓ2 " r ℓ2 r ℓ1 , ℓ1 ℓ2 " r ℓ2 r ℓ1 , (3.54) which can be easily verified using explicit formulae for the classical map (3.17). Conversely, the equations (3.54) can be used as an alternative definition of the classical Yang-Baxter map independent of the notion of the universal R-matrix for the quantum algebra. The geometric interpretation of the relations (3.54) as the zero curvature representation remains essentially the same as in the quantum case, see Fig.3(b) and the relevant discussion at the end of Sect.2.8. Finally, note that Eqs.(3.54) can be conveniently rewritten as where λ is arbitrary, similarly to (2.69).

Involutive Integrals of Motion
The following considerations essentially repeat those for the quantum case in Sect. It follows from (3.39) that these quantities form an involutive family for arbitrary values of λ and λ 1 . Further, using the zero curvature representation (2.69) one can easily show that tpλq and tpλq are, indeed, time-independent, i.e., they are unchanged under the discrete evolution map (3.49), tpλq " U cl`t pλq˘, tpλq " U cl`t pλq˘.

The Lagrangian equation of motion and the action
In view of the quasi-classical correspondence, considered in the section, the Lagrangian function and the action of the classical evolution system (3.52) is determined quasi-classical asymptotics of the universal R-matrix (3.12). More specifically, consider the expression for its matrix elements given by Eq.(2.94). It involves the spectral parameters α 1,2 , β 1,2 and the spin variables s 1,2 and s 1 1,2 . For a consistent quasi-classical limit these variables should become infinite, provided that the quantities where the Euler dilogarithm Li 2 pxq defined in (3.13). It is useful to note that Bλ a pσq Bσ "´logpe σ`e´a q, Bλ a pσq Bσ " logpe σ´e´a q . (3.61) With these notations one can show that This formula easily follows from the asymptotics of the functions (2.95) for b Ñ 0, (3.64) Eqs (3.52) are the (first order) Hamiltonian equations of motion for our dynamical system. To obtain the corresponding (second order) Lagrangian equations of motion one may consider the quasi-classical asymptotics of the matrix elements of the evolution operator (2.110). Substituting (3.62) therein and calculating the integral by the saddle point method one gets where A`tσu pclq˘i s the classical action of the system L`σ 2n´1,t , σ 2n,t , σ 2n`1,t`1 , σ 2n,t`1˘( 3.66) calculated on the stationary spin configuration tσu pclq defined as BA`tσuB σ k,tˇt σu"tσu pclq " 0 , k " 1, . . . , 2N, t " 1, . . . , T´1 . (3.67) The boundary spins tσ k,0 u 2N k"1 and tσ k,T u 2N k"1 are kept fixed.
The same result can be obtained by the standard method of Hamiltonian mechanics. Remind that the classical map (3.43) involves the set of six independent variables u 1,2 , v 1,2 , z 1,2 . They are simply related to the variables in (3.59), Next, the map (3.43) is a canonical transformation, preserving the Poisson bracket Its generation function L is defined by the equations Rewrite (3.43) in the form Integrating (3.70) and using the variables (3.68), one obtains the same Lagrangian (3.63).

Discrete Liouville Equations
It is instructive to write down the Lagrangian equations of motion (3.67) explicitly. The corresponding Hamiltonian equations (3.52) are written in terms of the variables u k,t , v k,t , k " 1, . . . , 2N , where the variables v k,t (playing the role of coordinates) are related to σ k,t in (3.66) as follows v 2n´1,t " e´b 1`σ2n´1,t , v 2n,t " e´b 2`σ2n,t , n " 1, . . . , N, t " 0, . . . , T . Below we present a general solution of both the Lagrangian and Hamiltonian equations of motion. It will be convenient to introduce the "light cone" coordinates: x " pn, t´nq, e 1 " p1, 0q, e 2 " p0, 1q . (3.77) Moreover, we will use the variables u 1,2 pxq and v 1,2 pxq instead of u k,t and v k,t , The Hamiltonian equations of motion in these notations are easily obtained from (3.72) by substituting the variables tv 1 , v 2 , v 1 1 , v 1 2 u therein with tv 1 pxq, v 2 pxq, v 1 px`e 1 q, v 2 px`e 1 qu and similarly for tu 1 , u 2 , u 1 1 , u 1 2 u. Their arrangement on lattice is shown in Fig. 7 (cf. Fig. 5).
The resulting equations imply a simple relation u 1 pxqu 2 pxq " u 1 px`e 1 qu 2 px`e 2 q .
(3.80) which ensures the existence of the τ -function, defined by a system of two first order difference equation With this substitution the system (3.74) can be rewritten as x`e 1`e2 z 2 τ x´τx`e 1 , v 1 pxq v 2 pxq " z 2 τ x`e 1 τ x`e 2´τ x τ x`e 1`e2 pz 1 τ x´τx`e 2 qpz 2 τ x´τx`e 1 q . (3.82) Here we have only three equations, since (3.80) is satisfied automatically by virtue of (3.81).
Integrating the first two equations, one obtains where x " px 1 , x 2 q are the light cone coordinates, x " x 1 e 1`x2 e 2 , x 1 " n , x 2 " t´n , (3.84) and α x 1 , β x 2 are arbitrary constants of integration. The third equation in (3.82) reduces to a single equation for the τ -function, This is the inhomogeneous discrete Liouville equation, involving two arbitrary functions α x 1 and β x 2 . The corresponding geometric arrangement of the τ -functions is shown in Fig.7. This is a generalization of the homogeneous discrete Liouville equation (with α x 1 " 1, β x 2 " 1), previously studied in [19][20][21][22]. It appears that the inhomogeneity does not bring notable complications. Indeed, a general solution to Eq.(3.85) is given by the formula where φ x 1 , γ x 2 are arbitrary functions, while f x 1 and g x 2 are determined by the following first order difference equations The solution contains 4N arbitrary constants α x 1 , β x 2 , φ x 1 , γ x 2 (x 1 , x 2 " 1, . . . , N ), whose number precisely coincides with the dimension of the phase space of our classical discrete evolution system (3.50).

Conclusion
In this paper we have shown that the theory of integrable maps (and more specifically the socalled Yang-Baxter maps) can be naturally included as a part of the modern theory of integrable system, based on the theory of quantum groups, the Quantum Inverse Problem Method and their classical counterparts. The considerations apply to both quantum and classical theories (the latter arise as the quasiclassical limit of the quantum case). One of the main advantages of our algebraic approach is that the resulting discrete integrable evolution systems automatically possess meaningful Hamiltonian structures. In addition, the problem of quantization of the Yang-Baxter maps appears to be solved from the very beginning. We have illustrated the entire scheme on the example on the quantized Lie algebra U q pslp2qq (and its Poisson algebra limit for the classical case). Completely parallel presentations of the quantum and classical cases are given in Sect. 2 and Sect. 3 respectively. In particular, a general solution the classical discrete Liouville equations arising in this context (both in the Lagrangian and Hamiltonian forms) is given in Sect. 3.11.
Although the above considerations were mainly restricted to the case of U q pslp2qq, our approach is rather general and can be applied to any quantized Lie (super) algebra as well as their affine extensions. We hope to consider some of these problems in future publications. The Boltzmann weights for the Faddeev-Volkov model (which is related to the quantized affine algebra U q p p sl 2 q) are defined as [38], W α psq " 1 F α e 2παs ϕps`iαq ϕps´iαq , W α psq " W η´α psq , (A. 6) where F α " e iπα 2`i πp1´8η 2 q{24 Φp2iαq .

Appendix B. Relation to R-matrix for quantum Teichmüller theory
The R-matrix (2.99) coincides with the inverse of R-matrix (44) from [39]. Details are the following. Moving quadratic exponents in (2.99) to the right, one could obtain the left hand side of the following identity: ie´i πpx 1´x2 q 2 e´i πpp 2 1`p 2 2 q e´i πpx 1´x2 q 2 P 12 " e´2 πip 1 p 2 .

(B.1)
This identity can be easily proven by consideration of a kernel or of a normal symbol of the left and right hand sides and performing the Gauss integration.
This form of R 12 literally coincides with R´1, formula (44) from [39], after indentifying corresponding variables. The authors are grateful to R. M. Kashaev for this observation.