Using quasi-triangularity to find the six-vertex R-matrix
Hand-in exercise 6 (for 09-01)
by J. Lamers

As you have proved last week, quasi-triangularity automatically leads to a solution of the Yang-Baxter equation. Since the quasi-triangularity relation R (Δ^op)= Δ R on V⊗V is linear in R, it is much easier to solve that the Yang-Baxter equation itself. In this exercise you are going to derive the six-vertex R-matrix by solving the quasi-triangularity condition. All equations refer to [GRS].

Define the coproduct and opposite coproduct

You may do this in any way you like, as long as the result is correct. A possible approach goes as follows:

Define the 2×2 matrices (Subscript[E, 0]),(Subscript[F, 0]),(Subscript[K, 0]),(Subscript[E, 1]),(Subscript[F, 1]),(Subscript[K, 1]) from equation (2.114) and check for typos by verifying the (Subscript[U, q])((Subscript[sl, 2]))-relations from (2.113) for (Subscript[E, 0]),(Subscript[F, 0]),(Subscript[K, 0]).

To define the coproduct, the following abbreviations may come in handy:

X_⊗Y_ := KroneckerProduct[X,Y]
Id = IdentityMatrix[2];

For quantum-affine (Subscript[sl, 2]) the coproduct Δ: V -> V⊗V gives 4×4 matrices that depend on the affine (= spectral) parameters u,v as well as the deformation parameter q. Here u is assigned to operators acting on the first factor of V⊗V, and v to operators acting on the second factor.

Define the coproduct Δ of the matrices from part 1 and check your code by verifying that Δ preserves the (Subscript[U, q])((Subscript[sl, 2]))-relations for (Subscript[E, 0]),(Subscript[F, 0]),(Subscript[K, 0]).

The opposite coproduct can be defined from Δ via the permutation operator

P={{1,0,0,0},{0,0,1,0},{0,1,0,0},{0,0,0,1}};

Although (Δ^op) is the composite of the braiding operator and Δ,the resulting operator equation involves conjugation by the permutation operator, (Δ^op)(X)=P Δ(X) (P^(-1)).

Using Sweedler's notation Δ(X)=Σ (Subscript[X, (1)])⊗(Subscript[X, (2)]) for a general operator X, check the last statement by hand by applying both sides of the equation for (Δ^op)(X) to a general vector (Subscript[v, 1])⊗(Subscript[v, 2]) ∈ V⊗V.

Define the opposite coproduct (Δ^op) of the matrices from part 1, and check the result with equation (2.118). (Hint: recall that the matrix product is written as X.Y in mathematica.)

Define the R-matrix

We also need a general 4×4 R-matrix on V⊗V:

R[u_,v_,q_]:=Array[(Subscript[R, ##])[u,v,q]&,{4,4}]

Implement quasi-triangularity

Find a way to implement the quasi-triangularity condition for the R-matrix, and check your code by checking that the six-vertex R-matrix defined by (2.50), with q=(e^(i γ)), satisfies the quasi-triangularity equation. (Of course we're cheating a little here: this R-matrix is the one that we are after! Hint: often it is convenient to define a matrix-valued function whose zeroes are equivalent to R (Δ^op)= Δ R.)

Solve the quasi-triangularity condition for (Subscript[U, q])((Subscript[sl, 2])) to find the six-vertex R-matrix

Since the quasi-triangularity equation is linear in the R-matrix, the entries of R are at best determined up to an overall normalization.
In case brute force does not work, you can take the following approach:

Since the coproducts Δ((Subscript[K, i])) are the simplest, use these to read off immediately that ten of the entries of the R-matrix vanish. Use this result to simplify the equations corresponding to (Subscript[E, i]),(Subscript[F, i]).

You should have 4 × 4 = 16 equations for six unknowns. For this system to be consistent, there should be many redundant equations. To find these we start by  looking for similarities between the equations. The first thing we note is that the equations come in pairs that are rather alike: this is related to the fact that (Subscript[E, 0]) and (Subscript[F, 1])have the same form, as do (Subscript[E, 1]) and (Subscript[F, 0]).

Use the similarity in the equations for the (Subscript[E, i]) to express the remaning nonzero offdiagonal elements of R in terms of each other. Use this result to simplify the remaining equations.

Solve the resulting equations by expressing the entries of R in terms of (Subscript[R, 1,1]).

Find a suitable normalization for (Subscript[R, 1,1])to recover the six-vertex R-matrix that we set out to find.

Bonus: checking the Yang-Baxter equation

Use Mathematica to check explicitly that your result satisfies the Yang-Baxter equation (Subscript[R, 12])(u-v)(Subscript[R, 13])(u)(Subscript[R, 23])(v)=(Subscript[R, 23])(v)(Subscript[R, 13])(u)(Subscript[R, 12])(u-v).