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Using the most elementary methods and considerations, the solution of the star-triangle condition

In two-dimensional lattice vertex models in which the state of a lattice point is specified by the states of the four links to its neighboring points, the matrix

in which

have only vertical indices (for example

The matrix coproduct is a mapping Δ :

which in matrix index language would be

Because the ⊗_{•} product is fundamentally a matrix product (which is associative), Δ is coassociative, so Δ and a compatible counit map

If

For a lattice vertex model with

The transfer matrix for a lattice model with such a matrix

by summing over the last set of horizontal indices

This is the trace of the

Define the complementary product in which the tensor is on the horizontal indices, and matrix product on the vertical (note that the horizontal space sub-matrices depend on different parameter sets, carefully compare with Equation 3)

The matrices

For commutativity of the transfer matrix [

since if we write out the horizontal components

(explicitly using invertibility of

A sufficient condition that makes Equation 10 true is [

Equation 13 becomes the Yang–Baxter equation (with a spectral parameter) if

If Equation 13 is written out in detail for the six-vertex model, with

Suppose that one begins with a matrix

The purpose of this paper is to show that Equation 14 is a necessary condition for the matrix coalgebra of the spectrum-generating operators

We will prove that Equation 14 is necessary for commutativity of the transfer matrices for the six-vertex model by constructing a complete closed set of quadratic operator products that annihilate the entire vector space basis of the physical states. The operator products are found by exploiting the recursive nature of the coalgebra, and the requirement that the coproduct be an algebra homomorphism. If these products annihilate the lowest dimensional (single-site) state space, the recursions guarantee that they annihilate the state space for the model with arbitrarily long rows. In other words these operators are identically zero in any physical representation. We refer to these products as quadratic “zero-operators” [^{0}

and from there building a collection of states

that are eigenvectors of the transfer matrix

The set of parameters {_{1}, _{2}, …, λ_{r}

For the eight-vertex model the

The plan of the article is to first establish those quadratic zero-operators that are compatible with the matrix coalgebra structure, and to show that this will require Equation 14 for closure, making it a necessary condition. The next step is to show that [

The motivation behind this work is the desire to have a way of attacking lattice models that may not satisfy the Yang–Baxter equation. Perhaps

We seek out binary product relations of matrix representations (2^{n}^{n}_{n}_{n}_{n}_{n}_{1} = _{1} = _{1} = _{1} =

We will use notation

and so forth.

The coproducts (Equation 6)

give us another (the next higher dimensional) representation of the same coalgebra, the vector space of the representation is 2^{n}^{+1}-dimensional, and we can decompose a product of operators as

Let v be any basis vector of the vector space of the 2^{n}_{n}_{n}_{n}_{n}_{n}^{n}^{+1}-dimensional representation space Ф_{n}_{+1} upon which _{n}_{+1}, _{n}_{+1}, _{n}_{+1}, _{n}_{+1} act is the set of all

Apply the operator products Equation 21 to the basis of the vector space {v⊗ ↓, v⊗ ↑} of states for the model with _{n}

We can see that these binary products close in the sense that exactly the same set of binary products appear on both sides of each equation, a powerful constraint on the form of algebraic relations compatible with the coproduct.

The relations of Equations 25 and 26 can be written as

if the operator products _{1}, …, _{4} are given by

The operators given in Equation 28 annihilate the entire vector space of states for the _{1}) and (_{2}) annihilate the entire

In the next section we will show that Equations 23 and 24 can also be written with right-hand sides expressed entirely in terms of (_{i}_{3}) and (_{4}) also have recursion relations of the same form, making a complete set of recursions for the entire

the action of (_{3})_{n}_{+1} on a basis of the _{i}_{n}_{4}). This is the core of the method; in the matrix coalgebra structure, sets of operator products (here the

Equation 29 follow directly from Equation 28 by the fact that the coproduct is an algebra homomorphism in the bialgebra (for instance Δ(_{n}_{1}), …, (_{4}) for

We will also show that commutativity of the transfer matrix is one of the algebraic relations (a binary product that identically annihilates the entire vector space of states) if and only if this same constraint is imposed. The constraint is the star-triangle relation.

We prove that (_{3}) and (_{4}) have recursion relations of the form Equation 27, so that the operator products (_{i}_{n}_{+1} for _{i}_{n}

Let us explicitly calculate the action of (_{3})_{n}_{+1} on the basis of the _{n}

Force closure of the set of relations {_{1}, _{2}, _{3}, _{4}}; is there an

This requires that there be a solution to the equations

(and of course there might not be a solution) the first and third of these giving

Note that the proposed zero-operators Equation 27 are unchanged by (

which can be factored into

or

the familiar solution of the star-triangle relation for the six-vertex model. The (

The second of Equation 29 results in

which are identical to Equation under interchange of

Following the same set of steps, (which we do not repeat here) we discover that exactly the same condition gives us the desired recursion relation for the action of (_{4})_{n}_{+1} on a basis of the _{1})_{n}_{2})_{n}_{3})_{n}_{4})_{n}

The star-triangle relations are a necessary condition for the operators Equation 28 to annihilate the entire vector space of physical states, a full set of recursions for the four zero-operators Equation 28 can be constructed in this way.

Combining _{3} and _{4} we obtain one of two important spectrum-generating relations used in the algebraic Bethe ansatz

The other portion of the full bialgebra needed by the algebraic Bethe ansatz is the

Let v be any basis of the

and finally

The first two sets of these identities suggest examining the combinations

all of which annihilate the _{i}

The second product law needed to complete the algebraic Bethe ansatz is made by combining (_{7}) with (_{8})

Relation Equation 14 imposed in the standard way by re-parameterization

simplifies the coefficients of Equations 39 and 46

to the standard forms seen elsewhere

The analogous integrability condition for the five-vertex model found by constructing the zero-operators is much simpler [

which is not a re-parameterization of the six-vertex model; the matrix of Boltzmann weights is singular;

and so no invertible

By building the binary product relations for each subalgebra that exhibits this closure inherent in the matrix coalgebra structure, we can complete the entire set of algebraic relations that make the operator coalgebra into a bialgebra. This is tedious, but each step is simply a repetition of what we have done for the

From the relations

we find that (doing away with the subscripts; we know what these matrices act upon)

Calling

(_{9}) and (_{10}) could be zero-operators if for some

which requires that

or

which is identical to Equation 33 with _{10} = 0) to hold, it is necessary for (_{9}) and (_{10}) annihilate the other half of the basis, namely v⊗ ↑, and a short calculation verifies that they do. The same conclusion is drawn for the five-vertex model, (_{10}) is a zero-operator and transfer matrices with different spectral parameters commute [

Despite the lack of elegance that other constructions of bialgebras utilizing a solution _{+}) with generators

uniquely produces

Starting with a two-dimensional representation of the Yangian Hopf algebra

with matrix coproduct, analytical construction of the zero-operators leads to the well-known product rules [

and finally when applied to the five-vertex or hexagonal lattice dimer model with spectral parameters

closely related to the Yangian, but which has no _{+}) and the five-vertex model do not, yet the methods of this article lead uniquely to their bialgebra structure equations.

Once the spectrum-generating relations Equation 49 have been established and a pseudo-vacuum state

The vector

satisfies the requirements of the Bethe ground state (which we show inductively)

This state is an eigenstate of the transfer matrix

Products of the

The eigenvalues are

provided the set {_{1}, _{2}, _{r}

which eliminates the unwanted terms in the expansion of

The conditions for elimination of the unwanted terms in the five-vertex model (hexagonal lattice dimer model) is far simpler [

with a similar relation involving

one is led to Equation 50, and this easily extends to the higher excitations [

The method of constructing zero-operators, quadratic operator products that annihilate the entire state-space which are preserved by the coproduct, can be used to deduce the conditions (on the model parameters) under which a lattice model has commuting transfer matrices. Coassociativity of the coproduct operation, which is the operation by which the transfer matrix is built up from local Boltzmann weights, is used to obtain recursion relations for a set of operators that in their lowest dimensional representation annihilate the state space. The recursions guarantee that they will annihilate all higher-dimensional state spaces. This promotes the operator coalgebra to a bialgebra. If the bialgebra can be shown to possess an antipode it may actually be a Hopf algebra, and the existence of an

In this article we have shown that a necessary condition for the zero-operator recursions to exist (for the six vertex model) is that the Boltzmann weights are constrained by

the star-triangle relations. One of these zero-operators is in fact [

The zero-operator method does not make use of the Yang–Baxter equation or require the existence of an

The zero-operators themselves are linear combinations of products of the

The methods used here are particularly amenable to the use of computer algebra systems. All of the calculations here were facilitated, and verified, by the use of REDUCE [