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 M of local Boltzmann weights can be used to construct the row-to-row transfer matrix by a matrix coproduct operation. In the six-vertex model a link between two lattice points has only two states, call them 0, 1 or ↑, ↓ subject to the “ice-rule” (two arrows point into a vertex, two point out). The matrix M of local Boltzmann weights for the six-vertex model has two types of indices, “vertical” and “horizontal” [1],

in which

have only vertical indices (for example ). All four matrices A, B, C, D typically depend on a set of parameters λ. The matrix C has a non-zero kernel which is not annihilated by B, and A and D are invertible. This will be a principal ingredient of the solution of the model by the Bethe ansatz.

The matrix coproduct is a mapping Δ : → × (a tensor product on vertical indices, a dot or matrix product on horizontal)

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 ε (see for example [2,3]) define a coalgebra structure on the set .

If is an algebra and the coproduct and counit of the coalgebra structure on are compatible with the product and unit of the algebra (Δ and ε are algebra homomorphisms), then is a bialgebra (simultaneous coalgebra and algebra). The compatibility with the coalgebra structure is a very strong constraint on the algebraic product rules. The existence of an antipode map extends the bialgebra to a Hopf algebra [3].

For a lattice vertex model with n sites per row, we construct the L-matrix of Boltzmann weights whose horizontal components are higher dimensional representations of the elements (λ), (λ), (λ), … of

The transfer matrix for a lattice model with such a matrix M of Boltzmann weights is obtained from an iterated coproduct,

by summing over the last set of horizontal indices

This is the trace of the L-matrix over its horizontal components

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 A, B, C, D and , , and are functions of the parameter-sets λ (un-primed depend on λ, primed are functions of λ'), which for the six-vertex model is a collection of three energies upon which a, b, c depend.

For commutativity of the transfer matrix [T(λ), T(λ')] = 0, which makes the model integrable if a sufficiently large set of mutually commuting matrices can be found, it is sufficient that there exists an invertible matrix R such that [1]

since if we write out the horizontal components

(explicitly using invertibility of R) and take traces we obtain the desired commutativity

A sufficient condition that makes Equation 10 true is [1]

Equation 13 becomes the Yang–Baxter equation (with a spectral parameter) if R = M(μ) for some choice of parameter μ, and this is not possible if M is not invertible.

If Equation 13 is written out in detail for the six-vertex model, with λ = (a, b, c), λ' = (a', b', c') and μ= (a'', b'', c'') and μ is eliminated from the resulting system of cubic equations, one obtains a much simpler constraint under which Equation 12 is true, which is the solution of the star-triangle relation

Suppose that one begins with a matrix M of local Boltzmann weights (possibly not invertible), which we think of as a low-dimensional representation of some operator coalgebra, and uses it to construct a transfer matrix for a lattice model by the matrix coproduct construction. Under what conditions can the coalgebra be extended to a bialgebra, or perhaps even a Hopf algebra?

The purpose of this paper is to show that Equation 14 is a necessary condition for the matrix coalgebra of the spectrum-generating operators , , , with lowest dimensional representation given in Equation 2 (the six-vertex model) to extend to a bialgebra with matrix coproduct. A bialgebra need not be a Hopf algebra, and a Hopf algebra need not have an R-matrix. We show that, at least in the case of the six-vertex model, commuting transfer matrices are one of the bialgebra product relations. The star-triangle relation is therefore necessary for integrability. Whether or not the star-triangle equation is necessary for [T(λ), T(λ')] = 0 for any given model is still an open problem (see for example P. 418 of [4]), one which seems to have been abundantly acknowledged but largely unaddressed in the literature.

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” [5], since they evaluate to zero on the basis of physical states. These relations establish the unique bialgebra structure compatible with the matrix coalgebra, and the given lowest dimensional representation of the coalgebra, namely the M-matrix, and are the familiar algebraic relations used in the algebraic Bethe ansatz. The algebraic Bethe ansatz begins with establishing a “vacuum” state Ф⁰

and from there building a collection of states

that are eigenvectors of the transfer matrix

The set of parameters {λ₁, λ₂, …, λ_r} for which this can be done is determined by eliminating “unwanted” vectors using the zero-operator product rules.

For the eight-vertex model the operator is non-singular, and one must perform local gauge transformations {, …, } → {', …, '} so that ' has a zero-eigenvector in order to finish the program of quadrature by the algebraic Bethe ansatz [6]. For the simple dimer model, one can find two vacuum states (both and have kernels [7]) and modify the Bethe ansatz accordingly. It may be very difficult or impossible to select parameters that eliminate unwanted terms, but that is a separate issue.

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 [ + ,' + '] is also a quadratic zero-operator if Equation 14 holds true, so that it is a necessary condition for commutativity of the transfer matrix.

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 M is not invertible, or a suitable R matrix cannot be found. It may still be possible to apply the algebraic Bethe ansatz if the operator coalgebra can be extended to o bialgebra by the means described here.

We seek out binary product relations of matrix representations (2ⁿ × 2ⁿ matrices) of the operators , , , , which we denote by _n, _n, _n, _n of the coalgebra that are invariant under the coproduct operation of Equation 6 (Δ is an algebra homomorphism) by which higher-dimensional representations are constructed from the lower. We work in the vector space of physical states. For the lowest dimensional representation, the 2 × 2, these four matrices are ₁ = A, ₁ = B, ₁ = C, and ₁ = D, and the basis for the vector space of physical states is

We will use notation A = A(λ), A' = A(λ') with λ, λ' being different parameter sets (a, b, c), (a', b', c'). The action of simple associative matrix products of these operators on the basis of physical states is

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 2ⁿ⁺¹-dimensional, and we can decompose a product of operators as

Let v be any basis vector of the vector space of the 2ⁿ-dimensional representation Ф_n, upon which _n, _n, _n, _n act. Then the basis of the 2ⁿ⁺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 + 1 sites per row, in which v ∈ Ф_n is any n-site vector

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 OP₁, …, OP₄ are given by

The operators given in Equation 28 annihilate the entire vector space of states for the n = 1 representation, {↑, ↓}. This is easy to verify using Equation 19. Therefore Equation 27 have all zeros on the right-hand side, and the operator products (OP₁) and (OP₂) annihilate the entire n = 2 vector space of states.

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 (OP_i) for i = 1, 2, 3, 4, and (OP₃) and (OP₄) also have recursion relations of the same form, making a complete set of recursions for the entire AB-subalgebra;

the action of (OP₃)_n+1 on a basis of the n + 1-vector space can be written entirely in terms of the actions of (OP_i)_n, i = 1, 2, 3, 4 on the basis of the n-vector space provided a, b, c and a', b', c' obey the star triangle equation, and the same for (OP₄). This is the core of the method; in the matrix coalgebra structure, sets of operator products (here the AA', AB', BA', BB') have recursion relations for their actions on basis vectors that close within the set. A complete set of such recursion relations mean that these binary operator products annihilate the entire vector space for any n, since the recursions are single-step, and they annihilate the n = 1 vector space. Matrices that annihilate the entire basis of a vector space are identically zero. These operator combinations are the zero matrix on the spaces of physical states, and we have constructed those binary product laws (algebraic relations between , preserved by the coproduct). From these we establish the product relations for the subalgebra spanned by and compatible with the coalgebra structure.

Equation 29 follow directly from Equation 28 by the fact that the coproduct is an algebra homomorphism in the bialgebra (for instance Δ(') = Δ()Δ('), and the only question is closure—is the coproduct of (OP_n) a linear combination of only (OP₁), …, (OP₄) for n = 1, 2, 3, 4. If so, then if each operator annihilates the physical basis, so does its coproduct, and the operator is zero on the basis of all physical representations. The coproduct is an algebra homomorphism requirement of compatibility of coalgebra and algebra structure in the bialgebra; it is what guarantees the existence of our basis-annihilating operator products.

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.

Once the spectrum-generating relations Equation 49 have been established and a pseudo-vacuum state with = 0 has been found, the eigenvalues of the transfer matrix can be found by the algebraic Bethe ansatz.

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 operators applied to the vacuum state will produce vectors that are eigenvalues of the transfer matrix if the unwanted vectors in the expansion of T(θ)Ф have zero coefficients

The eigenvalues are

provided the set {θ₁, θ₂, …, θ_r} is chosen such that

which eliminates the unwanted terms in the expansion of T(θ)Ф [6].

The conditions for elimination of the unwanted terms in the five-vertex model (hexagonal lattice dimer model) is far simpler [5,8], from

with a similar relation involving and , together with [, '] = 0.

one is led to Equation 50, and this easily extends to the higher excitations [5,8].

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 R-matrix would make it quasi-triangular, but these steps are not needed for a complete quadrature. Only two sub-algebras, the AB and DB, are required to construct Bethe vectors and perform the algebraic Bethe ansatz, assuming that the resulting equations for the elimination of unwanted terms can be solved. Simple possession of these subalgebras is no guarantee that this can be done.

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 [T, T'], T = + , and therefore a necessary condition for commutativity of transfer matrices with different Boltzmann weights is the star-triangle condition. This condition is well-known to be sufficient [1].

The zero-operator method does not make use of the Yang–Baxter equation or require the existence of an R-matrix. It is a direct determination of the spectrum-generating bialgebra of a lattice model, requiring only the lowest dimensional non-trivial representation of the bialgebra (the M matrix) and has been used to perform the algebraic Bethe ansatz for models with a singular M-matrix [5,8,7], in which the bialgebra is closest in structure to a Yangian quantum group, but possesses no antipode. This approach for constructing a bialgebra is complementary to the RTT method [3,10] which begins with a solution to the Yang–Baxter equation (an invertible R-matrix constructed from M) and produces a bialgebra with matrix coalgebra.

The zero-operators themselves are linear combinations of products of the , , and operators for pairs of spectral parameters. The commutator of the transfer matrices is just such an object, and so the method is particularly well-suited to the problem of establishing necessary spectral conditions under which the transfer matrices will commute.

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 [11].