Three Blocks Solvable Lattice Models and Birman--Murakami--Wenzl Algebra

Birman--Murakami--Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the $SU(2)$ $2\times 2$ fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.


. Introduction
Solvable lattice models in two dimensions play a major role as models of phase transitions. For a review see Baxter's book [1]. They also have a beautiful role in mathematics. For example, they allow for the definition of knot invariants, which are crucial in the classification of knots. For a review see [2], and ref. therein. This paper is concerned with Interaction Round the Face (IRF) solvable lattice models. The first examples are the Ising and the hard hexagon models. These have been vastly generalized, see, e.g. [2]. In particular, in ref. [3], the construction of such models from conformal field theory was described. The conformal field theories in two dimensions were first suggested by BPZ [4], constructing the so called minimal models. They were subsequently extended to WZW affine conformal field theories [5,6,7]. The main examples of solvable IRF models, as discussed here, are related to the WZW theories.
The algebraic structure of IRF models plays a significant role in their solution.
It is also important in applications to knot theory. The face transfer matrix obeys some polynomial equation, the order of which we call the number of blocks. This is further elucidated in the following discussion. For two blocks it is known that the algebra is Temperly-Lieb algebra [8], yielding the, so called, graph-state IRF models.
Our aim here is to treat the three block case. We prove that three block IRF models obey the Birman-Murakami-Wenzl (BMW) algebra, which was suggested in connection to knot theory [9,10]. This is our main result. We exemplify by showing that the fused 2 × 2 SU (2) models obey the BMW algebra.
These results are important to the understanding of IRF models, the construction of new ones, and is mathematically interesting, particularly in knot theory. We hope to further the understanding the algebraic structure of IRF models, in general.

.2. Interaction round the face lattice models
Fusion IRF models were constructed in ref. [3]. We consider a square lattice.
Fix a rational conformal field theory O. We denote the primary fields by [p] where the unit field is denoted by [1]. On the sites of the lattice models we have some primary fields. The interaction is around the faces of the model. The partition function of the model is given by where the sum is over all primary fields sitting on the sites and is some Boltzmann weight to be defined. Here u is some parameter called the spectral parameter. The model is defined by the admissibility condition that the Boltzmann weights vanishes unless the fields around the face obey [3], Accordingly, we denote this model as IRF(O, h, v). We find it convenient to define the face transfer matrix, through its matrix elements, In this language, the conditions for the solvability of the model, which is the celebrated Yang Baxter equation becomes, The Yang Baxter equation implies that the transfer matrices for all values of u commute with one another.
At the limit u → i∞ the Yang Baxter equation (YBE) becomes, This is the braid group relation. n is the number of blocks. The exponent pre-factor is necessary to make the limit finite as is discussed below.
We have a natural candidate for the braid group which obeys the admissibility condition eq. (3). This is the braiding matrix of the conformal field theory O, which expresses the braiding of the conformal blocks [11]. This matrix is denoted by C and it obeys the Hexagon relation [11], which for h = v is equivalent to the braid group relations, eq. (7-8). We define With this definition the Boltzmann weights ω obey the admissibility condition for the lattice model IRF(O, h, v), and the face matrix, X i , obeys the braid group relations, eqs. (7)(8).
Denote by ∆ p the dimension of the primary field [p]. Assume also that h = v where the product is in terms of the fusion rules, and ψ i are some primary fields.
We denote by ∆ i the dimension of the primary field ψ i , and n is the number of primary fields in the product. We call this the n block case. The face operator X i , which is equal to the braiding matrix, has then the eigenvalues, where ǫ i = ±1, according to whether the product is symmetric or anti-symmetric.
The matrix X i then is seen to obey an nth order polynomial equation, The fact that X i obeys an nth order equation allows us to define projection operators. We define, The projection operators obey the relations, We shall assume that the theory is real.
where ∆ i is the dimension of the field ψ i , and ∆ 0 = 0. The trigonometric solution for the Yang Baxter equation is then where the functions f a (u) are defined by . (17) In many cases it was verified explicitly that the conjectured Boltzmann weights, eqs. (16)(17), obey the YBE equation. For a review see, e.g., ref. [2].
In what follows we shall assume that for our conjectured Boltzmann weights, the YBE indeed holds. It appears that all solvable IRF models can be derived in this way from some conformal field theory, provided that they have a second order phase transition.
The Boltzmann weights, X i (u) obey several properties [3]. The first one is crossing, where S a,b is the modular matrix, and λ = ζ 0 is the crossing parameter.
Define the element of the algebra, Since X i (0) = 1 we find from the crossing relation an expression for E i , From this expression it easy to verify that E i obeys the Temperly-Lieb algebra, For the square of E i we find, In the case of two blocks n = 2 this solves completely for X i (u) showing that the solution is a graph state IRF [3]. We conclude that for two blocks the algebra is Temperly-Lieb. It is noteworthy that for any number of blocks, E i obeys the Temperly-Lieb algebra.
Another relation which is evident is the unitarity, where .

Birman Murakami and Wenzl algebra
Our purpose is to generalize the two blocks result to three blocks. Our main result is that every three block theory obeys the Birman-Murakami-Wenzl algebra (BMW) [9,10]. First, for this purpose, we list the relationships of the algebra. There are two generators of the algebra, G i and E i . The relations are, These relations imply the additional relations, Here, l and m are the two parameters of the algebra. The BMW algebra is defined here according to Kauffmann's 'Dubrovnik' version of the algebra [12], which is used in the defenition of Kauffmann's polynomial.
The BMW algebra is known to have a canonical Baxterization [9,10]. This is given by two parameters λ and µ. The parameters are related to m and l, as will be described below. The face operator is defined by, Here 1 stands for the unit matrix. With the definition of the BMW algebra, eqs.
(25-26), U i (u) obeys the Yang Baxter equation, We shall concentrate now on the three block case, n = 3. In this case, the solution for the face transfer matrix, eqs. (16)(17), becomes, We also define According to the unitarity condition, eqs. (23-24), X i X t i is proportional to one. We wish to scale them, so that they are the inverse of one another. The scale constant can be read from eqs. (23-24), the unitarity condition, where w = 4 sin(ζ 0 ) sin(ζ 1 ), is the necessary scale constant.
Our purpose now is to connect our solution to the YBE X i (u), eqs. (19), with algebraic solution U i (u), eq. (27), To do this, we identify ζ 0 with λ (as before) and ζ 1 with µ. We also identify the generators G i and G −1 i to be proportional to X i and X t i respectively, The phase is arbitrary, and we fixed it to be compatible with the BMW algebra, Our purpose is to show that with this definition the BMW algebra is indeed obeyed.
We substitute the scale definition, eq. (34), into the expression for the Baxterized BMW algebra, U i (u), eq. (33). We find after some algebra that X i (u) and U i (u) are identical, for any P 0 i , P 1 i and P 2 i , provided only that they obey P 0 i + P 1 i + P 2 i = 1, which is true.
We conjectured that X i (u) obeys the Yang Baxter equation, eqs. (5)(6). This was proved for many cases. It then follows that U i (u) obeys the YBE also. By expanding the face weight, as in eq. (33), this, in turn, implies that G i and We get now to determining the parameters l and m. We have the equation, eq. (25), where E i = U i (λ). This is the Skein relation. In particular E i obeys the Temperly-Lieb algebra, eqs. (21-22), which is part of the BMW algebra, eqs.
We know from the BMW algebra that we should have, Substituting l and m we find that this relation is indeed obeyed. Let us move now to a concrete example. This is the model IRF(SU (2) k , [2], [2]).
Namely, the conformal field theory SU (2) k , which is a WZW affine theory, at level k, with the admissibility fields h = v = [2]. We denote by l the weights of SU (2) k representations, which are twice the isospin, l = 0, 1, 2, . . . , k. So [2] is the field with isospin one.
The fields appearing in the product hv are The dimensions of the fields in the theory are given by So we find for ζ 0 and ζ 1 the values, according to eq. (15), and Thus the face transfer matrix, X i (u), is given by eq. (29), with the substitution of the above values of ζ 0 and ζ 1 . In fact, the exact same weights were given in a paper by Pasquier [13]. In this paper it is proved that X i (u) obeys the Yang Baxter equation, with the appropriate choice of P 0 i and P 1 i . It thus follows, according from the discussion above, that G i and E i obeys the BMW algebra, with λ = π/(k + 2) and µ = 2λ. The values of l and m for this model are, Explicit expressions for all the Boltzmann weights of the model are listed in the appendix. We checked numerically that all the relations of the BMW algebra, eqs. (25-26), are obeyed for this model. We defined G i and G −1 i as in eq. (34).
We checked for levels k = 8, 9, 10, 12 and we find a complete agreement with the BMW algebra, with all the relations fulfilled. We conclude that this model obeys the BMW algebra. .

Discussion
In this paper we proved that any three block IRF model obeys the BMW algebra, along with its Baxterization. We treated explicitly the fused model IRF(SU (2) k , [2], [2]), and showed that it obeys the BMW algebra, also by direct computations.
Other important models are the BCD IRF models constructed by Jimbo et. al. [14]. These models can be described as Thus, our analysis in this paper is applicable to these models, proving that they obey the BMW algebra.
Other three block theories were described in ref. [15], at the level of the conformal data, as theories with low number of primary fields. It will be interesting to study the IRF models associated with these models, as well. In particular, from our analysis we expect them to obey the BMW algebra.
An important question, left to the future, is to find the algebras corresponding to IRF models with more than three blocks. We believe that our methods can be useful to study these models, as well. Some of the relations of the general models are known [3], already. Finding these algebras will be of considerable importance to knot theory.
The weights are taken from ref. [16]. The weights were originally computed in ref. [17], by the fusion procedure. We shifted the primary field a → a + 1 so the weights range over a = 1, 2, . . . , k + 1, giving the dimension of the representation.