Integrability of planar-algebraic models

The quantum inverse scattering method is a scheme for solving integrable models in 1+1 dimensions, building on an R-matrix that satisfies the Yang–Baxter equation (YBE) and in terms of which one constructs a commuting family of transfer matrices. In the standard formulation, this R-matrix acts on a tensor product of vector spaces. Here, we relax this tensorial property and develop a framework for describing and analysing integrable models based on planar algebras, allowing non-separable R-operators satisfying generalised YBEs. We also re-evaluate the notion of integrals of motion and characterise when an (algebraic) transfer operator is polynomial in a single integral of motion. We refer to such models as polynomially integrable. In an eight-vertex model, we demonstrate that the corresponding transfer operator is polynomial in the natural Hamiltonian. In the Temperley–Lieb loop model with loop fugacity β∈C , we likewise find that, for all but finitely many β-values, the transfer operator is polynomial in the usual Hamiltonian element of the Temperley–Lieb algebra TLn(β) , at least for n⩽17 . Moreover, we find that this model admits a second canonical Hamiltonian, and that this Hamiltonian also acts as a polynomial integrability generator for small n and all but finitely many β-values.


Introduction
Based on Bethe's pioneering work [1], the Quantum Inverse Scattering Method [2,3,4,5] is a collection of techniques for solving integrable models in 1 + 1 dimensions. It is built on an R-matrix that acts on a tensor product of vector spaces, allowing the construction of a commuting family of transfer matrices whose spectral properties can be used to gain insight into the physical properties of the model. Although hugely successful, this approach does have its limitations. Using ideas of Jones, this was illustrated in the work [6] on a Temperley-Lieb loop model, where the basic building block replacing the R-matrix is a parameterised element of a planar algebra [7] whose representations do not naturally separate tensorially. This avenue was explored further in a series of papers by Pearce et al., including [8,9,10,11,12], where commutativity of the corresponding transfer operators is established in the algebraic setting and is therefore valid in any given representation. In the present work, we generalise and formalise this approach by developing a framework for describing and analysing two-dimensional integrable models whose algebraic structure is afforded by a planar algebra. In another recent study of integrable models, the underlying mathematical structure is a fusion category [13,14].
There is no consensus as to what constitutes quantum integrability, see [15] for a review. Here, we approach the problem from a statistical mechanical set-up based on a two-dimensional model described by an algebra A. We will view such a model as defined by a one-parameter family of transfer operators, T (u) ∈ A, and say that the model is integrable if [T (u), T (v)] = 0 for all u and v in some suitable domain. From T (u), one may extract a u-independent element of A and interpret it as the quantum hamiltonian of a one-dimensional system. To contrast, another common, and in some sense complementary, approach is to start with a hamiltonian of a one-dimensional quantum system, such as a quantum spin chain, and then construct a transfer operator as a generating function for the integrals of motion. For precision in our setting, we re-evaluate the notion of integrals of motion algebraically and distinguish between those arising from the model-defining transfer operator, which we generally refer to as the hamiltonians of the model, and those in the centraliser of the hamiltonians within the algebra A.
In a Yang-Baxter integrable model, the commutativity of the transfer matrices is a consequence of a set of local relations satisfied by the R-matrices, including the celebrated Yang-Baxter equations (YBEs) [16,17,18,19]. Depending on the model, these may be supplemented by Boundary YBEs [20] involving K-matrices encoding boundary conditions of the model. This extends to similar relations between the R-and K-operators in the Temperley-Lieb algebraic approach in [6]. Within the planaralgebraic framework developed here, we work with generalised YBEs which allow the auxiliary operators (the 'middle' operators) to be parameterised differently from the R-operators.
A key aim of this work is to examine the possibility of the transfer operator being polynomial in a spectral-parameter independent integral of motion, and we say that a model with that property is polynomially integrable. From this view, the ensuing integrability (in the sense of commuting transfer operators) is a consequence of a global property, in contrast to Yang-Baxter integrability which is local in origin. Despite the differing perspectives, we stress that they are often applicable simultaneously, as a model can be both Yang-Baxter and polynomially integrable. In fact, we find (under mild conditions on the underlying algebra A) that polynomial integrability follows from the diagonalisability of an integrable transfer operator -a ubiquitous property in a large class of (possibly Yang-Baxter) integrable models. The indicated shift in perspective will be explored in two concrete examples: a Temperley-Lieb loop model (Section 4) and an eight-vertex model (Appendix C).
The Temperley-Lieb loop model is formulated in the Temperley-Lieb planar algebra [7] incorporating the usual Temperley-Lieb algebra TL n (β) [21,22] and underlying the body of work initiated in [6]. We show that the model is freely Baxterisable in the sense that it is integrable for any choice of R-operator parameterisation. We then show that the model is also polynomially integrable for all β > 2 cos π n and u ∈ R. Using a new decomposition of the transfer operator T n (u, β), we classify the identity points (see below) for T n (u, β) and find that there are two canonical hamiltonians, one of which is the familiar one, h = −(e 1 + · · · + e n−1 ), while the other one does not appear to have been discussed in the literature. Using the cellularity [23] of TL n (β), we present a spectral analysis of the hamiltonians, with emphasis on their minimal polynomials. Armed with this, we show that T n (u, β) is polynomial in h for all but finitely many β ∈ C, at least for n ≤ 17. We also examine the finitely many β-values for which T n (u, β) fails to be polynomial in h and find that T n (u, β) is polynomial in the alternative hamiltonian for those values, at least for small n.
The paper is organised as follows. In Section 2, we recall basic properties of planar algebras (with some details deferred to Appendix A) and develop our planar-algebraic description of Yang-Baxter integrability. We thus introduce suitable R-and K-operators and use these to construct the transfer operators. A finite set of local relations, including generalised YBEs, provide sufficient conditions for integrability (Proposition 2.1). An identity point of a transfer operator is introduced as a value for the spectral parameter at which the transfer operator evaluates to a scalar multiple of the identity operator, and we use power-series expansions about these points to determine the corresponding principal hamiltonians. An algebraic characterisation of integrals of motion and the introduction of polynomial integrability conclude the section.
In Section 3, we consider polynomial integrability in a linear-algebraic setting and present necessary and sufficient conditions for parameter-dependent matrices to be expressible as polynomials in a parameter-independent matrix (Proposition 3.3). We then extend this result to elements of semisimple algebras (Proposition 3.6). We also prepare for our study of the Temperley-Lieb loop model and the aforementioned eight-vertex model, by introducing notions relevant to the employed spectral analysis and by reviewing basic properties of cellular algebras.
In Section 4, we turn to the Temperley-Lieb algebra where we find that the associated loop model is both freely Baxterisable (Proposition 4.9) and, for β generic, polynomially integrable (Proposition 4.10). The identity points of the model are then classified (Proposition 4.11), and two distinct hamiltonians are identified as candidates for an integral of motion in terms of which the transfer operator can be expressed as a polynomial. Spectral analysis confirms that both objects do the job for all but finitely many β ∈ C and all n ≤ 17. We then determine explicit polynomial expressions of the transfer operator in terms of these hamiltonians and find that, for all β ∈ C and n small, at least one of the polynomials is well-defined. Some technical details are deferred to Appendix B.
Section 5 contains concluding remarks, while in Appendix C, we recover the familiar quantum inverse scattering framework by specialising to tensor planar algebras. Our primary example of this type is an eight-vertex model, which we use to illustrate how the planar-algebraic framework simplifies in case the R-operators are tensorially separable. We show that the transfer matrix is diagonablisable and present an exact solution in the form of explicit expressions for all eigenvalues and corresponding eigenvectors of the transfer matrix. We then exploit similarities in the spectral properties of the transfer matrix and the canonical hamiltonian to establish that the transfer matrix is polynomial in the hamiltonian. Moreover, we decompose the transfer matrix into a linear combination of a complete set of orthogonal idempotents expressed in terms of the minimal polynomial of the hamiltonian. Throughout, we let N denote the set of positive integers, N 0 the set of nonnegative integers, F an algebraically closed field of characteristic zero, and R a commutative ring with identity.

Planar algebras
Here, we develop a general planar-algebraic description of Yang-Baxter integrability. Much of the material would be known to experts, but the details and generality of the framework does not seem to have been outlined before in the literature. This includes the formulation and treatment of the transfer operators in Section 2.2, the generalised Yang-Baxter equations in Section 2.4, the definition of hamiltonians and integrals of motion in Sections 2.6 and 2.7, and the introduction of polynomial integrability in Section 2.8.
To set the stage and fix our notation, we first recall some basic properties of planar algebras [7,24,25]. In the following, focus will be on so-called unshaded planar algebras, while some technical details are deferred to Appendix A.

Algebraic structure
Informally, a planar algebra [7,24,25] is a collection of vector spaces (P n ) n∈N 0 where vectors can be 'multiplied planarly' to form vectors. A basis for P n consists of disks with n nodes (connection points) on their boundary, whereby a boundary is composed of nodes and boundary intervals. The specification of the internal structure of the basis disks is a key part of the definition of any given planar algebra. When disks are combined ('multiplied'), every node is connected to a single other node (possibly on the same disk) via non-intersecting strings, and planar tangles are the diagrammatic objects, defined up to ambient isotopy, that facilitate such combinations.
Planar tangles have the following general features: (i) an exterior disk, called the output disk, (ii) a finite set of interior disks, called input disks, (iii) a finite number of non-intersecting strings connecting nodes of disks pair-wise or forming loops not touching any of the disks, and (iv) a marked boundary interval on each disk. An example of a planar tangle is given by input disks output disk loop strings marked intervals We denote the output disk of the planar tangle T by D T 0 and the set of input disks by D T . The number of nodes on the (exterior) boundary of a disk D is denoted by η(D). The boundary marks on each disk disambiguate the alignment of the input and output disks and are indicated by red rectangles, see (2.1).
Planar 'multiplication' is induced by the action of the planar tangles as multilinear maps. For a planar tangle T , this is denoted by Pictorially, P T acts by filling in each of the interior disks D ∈ D T with an element of the corresponding vector space P η(D) , in such a way that the nodes match up and the marked intervals are aligned. The details of the map P T specify how one should remove the internal disks in the picture and identify the image as a vector in P η(D T 0 ) . If D T = ∅, we simply write P T () for the image under P T , and stress that P T () is distinct from T . Taking T as in (2.1), we present the example where v 1 ∈ P 2 , v 2 ∈ P 4 and v 3 ∈ P 6 . Note that we have not specified any details about (P n ) n∈N 0 or about the action of the planar tangles.
Remark. Unlike in the picture of T in (2.3), disks in D T are not labelled; however, to apply the orderedlist notation for the vectors in P T (v 1 , v 2 , v 3 ), it is convenient to label the disks accordingly. Once drawn as in the second picture in (2.3), no labelling is needed.
Planar tangles can be 'composed', and consistency between this composition and the associated multilinear maps is often referred to as naturality, see Appendix A.1 for details. By specifying the vector spaces (P n ) n∈N 0 and the action of planar tangles as multilinear maps (2.2) such that naturality is satisfied, one arrives at a planar algebra. As an example, for each n ∈ N 0 , let P 2n denote the vector space spanned by all planar tangles T with no input disks and η(D T 0 ) = 2n, while P 2n+1 = {0}, and let the multilinear maps act by the composition of tangles. By construction, this data satisfies naturality. Revisiting (2.3) for this particular planar algebra, we have (2.4) In this example, each vector space P 2n is infinite-dimensional, as a set of disks with identical internal connections but different numbers of loops is considered linearly independent. A natural condition on a planar algebra is dim P 0 = 1. In this case, the empty disk and that with any number of loops are linearly dependent. Accordingly, there exists a linear map e : P 0 → F, here referred to as the evaluation map, that maps the 'empty tangle' to 1, and we may identify P 0 with F. Naturality implies that each loop formed in the image under P T can be removed and replaced by a common scalar weight. Another natural condition on a planar algebra is to forbid null vectors. Here, a nonzero v ∈ P n is called a null vector if P T (v) = 0 for every planar tangle T for which D T = {D} with η(D) = n.
In the remainder of this work, all planar algebras will have dim P 0 = 1 and P 2n+1 = {0}, and P 2n will have no nonzero null vectors for all n ∈ N 0 . To distinguish such a planar algebra from the general discussion above, we will use the notation (A n ) n∈N 0 , where A n ≡ P 2n is a vector space spanned by disks with 2n marked boundary points.
A planar algebra (A n ) n∈N 0 is not an algebra in the usual sense; however, it contains a countable number of standard algebras. Indeed, for each n ∈ N 0 , the planar tangle induces a multiplication on A n , and we write vw = P Mn (v, w) ∈ A n , where v, respectively w, is replacing the lower, respectively upper, disk in M n . Naturality implies that the ensuing standard algebra A n is associative and unital, where

Transfer operators
Let (A n ) n∈N 0 be a planar algebra, and, for each n, let B n denote a basis for A n . Without loss of generality, we may assume that 1 n ∈ B n , and it is then convenient to introduce We also introduce the parameterised algebra elements where k a , r a , k a : Ω → F, with Ω ⊆ F a suitable domain, and the transfer tangle We are now in a position to define the corresponding transfer operator as T n (u) := P Tn K(u), R(u), . . . , R(u), K(u) , (2.10) where K(u) and K(u), respectively, are inserted into the left-and rightmost disk spaces in (2.9). It follows that T n (u) is an element of A n . Using the diagrammatic representations and we refer to R(u) as the corresponding R-operator, and to K(u) and K(u) as the corresponding K-operators.
Remark. Different internal colours in (2.11) are used to indicate that the parameterisations are independent. In fact, it is for later convenience that we use the same A 2 -element R(u) in every available position in (2.10) and hence in (2.12), and that we let the coefficients in (2.8) be functions of a single variable only. Generalisations are possible and in some cases straightforward but will not be considered here.
Within this planar-algebraic framework, the transfer operator can be recast as a partial-trace expression reminiscent of more traditional descriptions. To this end, we introduce the trace and partial trace   where j = 1, . . . , n and i = 1, . . . , n − 1, respectively. Similar traces and partial traces can be defined by 'closing' on the left; however, for our purposes, we only need the right-closing (partial) traces. Using the last diagram in (2.12), we now obtain

Crossing symmetry
For each n ∈ N, we introduce the rotation tangles , (2.17) where the numbers of spokes are such that η(D r n,±1 0 ) = η(D r n,0 0 ) = n, and let r n,±k , k ∈ N, denote the composition of r n,±1 with itself k times. Accordingly, and we note that r n,l = r n,l mod n for all l ∈ Z. Naturality ensures that P r 2n,k : A n → A n is invertible for all n, k, see Appendix A.2 for details. We say that the K-and R-operators are crossing symmetric if P r 2,1 (K(u)) =c K (u)K(c K (u)), P r 4,1 (R(u)) =c R (u)R(c R (u)), P r 2,1 (K(u)) =c K (u)K(c K (u)), (2.19) for some scalar functionsc K , c K ,c R , c R ,c K , c K : Ω → F such that P r 2,2 (K(u)) = K(u), P r 4,4 (R(u)) = R(u) and P r 2,2 (K(u)) = K(u). If the relation for R only holds with r 4,1 replaced by r 4,2 , the R-operator is considered partially crossing symmetric. We say u iso ∈ F is an isotropic point if P r 2,1 (K(u iso )) = K(u iso ), P r 4,1 (R(u iso )) = R(u iso ), P r 2,1 (K(u iso )) = K(u iso ). (2.20) Remark. Rigid planar algebras [26] use rigid planar isotopy in place of the (full) planar isotopy inherent in the planar algebras in Section 2.1, essentially discarding the rotation tangles r n,±1 in (2.17). Rigid planar algebras thus provide a natural algebraic framework for describing models without rotation symmetry, such as the A 2 models discussed in [27].

Baxterisation and integrability
A model described by the transfer operator T n (u) is integrable on Ω if with Ω ⊆ F a suitable domain, and we say that the model is Yang-Baxter integrable if the R-and K-operators satisfy a set of (local) relations that imply (2.21). Following [28], the associated parameterisations (2.8) are then called a Baxterisation. As given, however, the functions in (2.8) are unspecified, so a natural goal is to constrain them to ensure Yang-Baxter integrability. As we allow for different parameterisations of the auxiliary operators in each of the Yang-Baxter equations below, our considerations are more general than what is usually done in the literature.
The following proposition is formulated in the diagrammatic representation used in (2.11) and (2.12) but is readily reformulated in planar-algebraic terms. Importantly, each of the relations in (2.24)-(2.26) is local in the sense that there exists an 'ambient planar tangle' with a suitable marking, relative to which it holds. In fact, the invertibility of the linear maps P r n,±k associated with the rotation tangles in Section 2.3, implies that the specific marking of the ambient planar tangle is immaterial. To illustrate, we have Proof. Using the following familiar manipulations [29], (2.28) we arrive at the result. In (2.28), YBE4 and Inv4 refer respectively to YBE1 and Inv1 with u and v interchanged and 1 replaced by 4, c.f. (2.27).
Remark. Proposition 2.1 provides sufficient conditions for integrability; they are in general not necessary. The elements in (2.23) depend on u and v, but their parameterisations may be different from the ones of R. We therefore say that the YBEs (2.25) are generalised. While the work [30] on dimers also relies on more than one YBE to establish commutativity of transfer operators, the auxiliary operators playing the role of (2.23) are all used as building blocks in the construction of the dimer transfer operator. We stress that the auxiliary operators in the generalised YBEs (2.25) do not necessarily appear in the transfer operator; they are in many ways a means to an end in establishing commutativity.
Remark. Relations like the BYBEs (2.26) are often referred to as reflection equations.

Sklyanin's formulation
The partial trace appearing in (2.15) is diagrammatic in origin. In fact, for a given planar algebra, there need not be a vector space over which the trace may be interpreted as being performed. However, if the algebra and the corresponding R-operator satisfy certain conditions, we can identify such an auxiliary vector space. For each m, n ∈ N 0 , the quadratic tangle , induces a tensor product between A m and A n within A m+n . For ease of notation, for a ∈ A m and b ∈ A n , we write a ⊗ b = P Km,n (a, b) ∈ A m+n . We say that R(u) ∈ A 2 is separable if it can be written as an element of A 1 ⊗ A 1 . Diagrammatically, this amounts to a decomposition of the form The auxiliary vector space of the corresponding transfer operator is then the rightmost channel, here coloured blue: We also say that a planar algebra is braided, respectively symmetric, if each vector space A n admits a representation of the n-strand braid, respectively symmetric, group algebra. In the symmetric case, applying the permutation element of A 2 to the separable R-operator (2.30), we obtaiň

Hamiltonian limits
We say that u * ∈ F is an identity point of T n (u) if T n (u * ) is proportional to 1 n . Around each identity point for which the proportionality constant is nonzero, we will use a power-series expansion of the transfer operator to define associated hamiltonian operators.
In preparation, for Ω ⊆ F a suitable domain, we define with the image written in our usual diagrammatic representation. Note that Composing g with the evaluation map, we obtain the scalar function The following proposition provides sufficient conditions for the existence of identity points.
Proposition 2.2. Let u * ∈ Ω and suppose there exist l u * , r u * ∈ F such that Then, u * is an identity point, with T n (u * ) = l n u * ĝ (u * )1 n or T n (u * ) = r n u * ĝ (u * )1 n , respectively.
Proof. If the left relation in (2.38) holds, then Similar arguments apply if the right relation holds.
Remark. If both relations in (2.38) hold andĝ(u * ) = 0, then l n−k u * r k u * = l n−k ′ u * r k ′ u * for all k, k ′ ∈ {0, 1, . . . , n}, hence l u * = r u * . Now, let u * be an identity point, and suppose Ω contains an open subset of F containing u * . If T n (u) is not just proportional to 1 n , then there exist unique k ∈ N and unique H n,u * / ∈ F1 n such that . We refer to H n,u * as the hamiltonian associated with the given identity point (and given transfer operator), and note that wherep k is a polynomial of degree at most k. Up to a possible rescaling, we refer to h n,u * as the principal hamiltonian associated with the identity point u * . If confusion is unlikely to occur, we may write h u * or h n instead of h n,u * . Different identity points associated with T n (u) may give rise to different principal hamiltonians. Hamiltonians arising from different Baxterisations are also likely to correspond to different elements of A n .
Remark. If u * is such that (2.38) holds and T n (u * ) = 0, then one may renormalise the original transfer operator such that the limit u → u * yields a well-defined nonzero operator, see e.g. [8]. This will be illustrated in Section 4.6.
If T n (u) describes an integrable model on Ω, then we have the familiar relations

Hamiltonians and integrals of motion
Let A be an associative algebra with basis B, and let T (u) ∈ A denote a transfer operator of some model. Then, where t a : Ω → F for each a ∈ A, with Ω ⊆ F a suitable domain. As a space of scalar functions, we have and let B T denote a basis for F. Relative to this, we can decompose the transfer operator as where a f ∈ A for each f ∈ B T . We also introduce and note that dim H T ≤ dim F. Related to this, we have the A-subalgebra Any nonzero element h ∈ H T (such that h / ∈ F1 for A unital) could conceivably be regarded as the hamiltonian of the model, although the physics would likely guide the selection. The hamiltonians H n,u * in (2.40) are examples of such algebra elements. Here, we refer to the elements of H T generally as hamiltonians and to certain preferred choices, h ∈ H T , as the principal hamiltonians. In general distinct from this notion of hamiltonians, we view the centraliser of h in A, here denoted by C A (h), as the space (in fact, subalgebra) of h-conserved quantities.
If the model is integrable in the sense that then A T is commutative, and given a choice of h, every hamiltonian is an h-conserved quantity. The converse need not be true; that is, an h-conserved quantity need not be a hamiltonian of the model, as we generally only have the inclusion H T ⊆ C A (h). Also, noting that C A (H T ) is a subalgebra of A, we view its centre, Z(C A (H T )), as the subalgebra of integrals of motion of the model. In summary, these various algebras of the integrable model are related as

Polynomial integrability
Let A be an associative algebra and {T (u) | u ∈ Ω} ⊆ A a one-parameter family of commuting elements, with Ω ⊆ F. If there exists b ∈ A such that, for all u ∈ Ω, then [T (u), T (v)] = 0 is trivially satisfied. If the commuting family corresponds to the transfer operators of some model, e.g. {T n (u) | u ∈ Ω} ⊆ A n in the planar-algebraic setting above, and satisfies (2.53) for some b n ∈ A n , then we say that the model is polynomially integrable. A polynomially integrable model may thus be seen as trivially integrable. It may seem surprising at first that a nontrivial physical model could be polynomially integrable, but that is exactly what we find. In fact, a hamiltonian operator may play the role of b in (2.53), in which case the transfer operators are polynomial in the hamiltonian. Examples of this will be provided in the following.

Endomorphism algebras
Having outlined our integrability framework for planar algebras in Section 2, we digress with (i) an examination of parameterised families of linear operators, introducing the notion of spurious spectral degeneracies, of which there can only be finitely many, (ii) a discussion of parameter-dependent matrices (respectively algebra generators) expressible as polynomials in a parameter-independent matrix (respectively algebra generator), and (iii) a brief review of some basic properties of cellular algebras. A key result of this section is Proposition 3.6 which gives necessary and sufficient conditions for a transfer operator to be polynomial in a single integral of motion, c.f. (2.53). Other results include Proposition 3.1 and Corollary 3.7, which are used in Section 4.8 and Section 4.5, respectively. In the following, we specialise F to C.

Spectral degeneracies
Let A(x) be an x-dependent linear map C n → C n that admits a matrix representation where every matrix element is polynomial in x, and denote the associated characteristic polynomial by By construction, c(x, λ) is polynomial in x and λ, and the eigenvalues λ will in general depend on x. If the spectrum of A(x) for x an indeterminate possesses fewer than n distinct eigenvalues, we say that the corresponding spectral degeneracies are permanent. Let l denote the number of distinct eigenvalues for x an indeterminate. If A(x 0 ) has fewer than l distinct eigenvalues for some value x 0 ∈ C, we refer to the additional, non-permanent spectral degeneracies as spurious. As Proposition 3.1 below asserts, there are only finitely many values x 0 for which spurious degeneracies arise. Here and elsewhere, we allow "finitely many" to mean zero. First, recall that a polynomial f (x, y) is said to be irreducible if it cannot be expressed as the product of two non-constant polynomials. For such a polynomial, the equation f (x, y) = 0 determines an algebraic function y(x), and we recall that an algebraic function has a finite number of branches and at most algebraic singularities.
Also, two polynomials f (x, y) and g(x, y) are relatively prime if they do not share a common nonconstant polynomial factor. In this case, there are only finitely many values x 0 for which the y-polynomials f (x 0 , y) and g(x 0 , y) have a common root (see for example Theorem 3 on page 300 of [31]). We can now establish the following result which is used in Section 4.8.
Proposition 3.1. The spectrum of A(x) is non-spurious for all but finitely many x ∈ C.
Proof. The characteristic polynomial of A(x) admits a factorisation of the form and c j (x, λ) are relatively prime or scalar multiples of each other. In the latter case, the pairing only contributes permanent degeneracies to the spectrum of A(x). Because each pair of relatively prime factors (of which there are finitely many) will contribute finitely many spurious degeneracies, and since each of the t irreducible factors individually can contribute finitely many spurious degeneracies, the result follows.

Block Toeplitz
We let M n (R) denote the set of n×n matrices with elements from R. For each z ∈ M n (C), we denote the centraliser of z in M n (C) by C(z) and let c z and m z denote the characteristic and minimal polynomial, respectively. Recall that z is non-derogatory if c z = m z . Equivalently, it is non-derogatory if and only if every eigenspace of z is at most one-dimensional; and if and only if z generates its own centraliser, that is, C(z) = C[z], see e.g. [32]. Note that if z ∈ M n (C) is similar to a non-derogatory matrix, then z is non-derogatory. We say that a linear operator on C n is non-derogatory if there exists a basis relative to which its matrix representation is non-derogatory.
For f an analytic function, we then have An upper-triangular Toeplitz matrix in M r (R) is characterised by a tuple in R ×r , a r = (a 1 , . . . , a r ), as We refer to a matrix of the form diag T (a [1] as block-diagonal upper-triangular Toeplitz (BT) and say it has block-partitioning r 1 , . . . , r s . Note that a block-partitioning need not be unique. Indeed, the 2 × 2 identity matrix admits the block-partitionings 2 and 1, 1.
for some r 1 , . . . , r s ∈ N and λ 1 , . . . , λ s ∈ C. By (3.4), and since each p x (J r i (λ i )) is an upper-triangular Toeplitz matrix, it follows that B(x) is BT. As to "⇐", let where r 1 , . . . , r s is a block-partitioning of B(x) and where λ 1 , . . . , λ s ∈ C are all distinct. Since b is nonderogatory, it generates its own centraliser. Because of the shared block-partitioning, we have Remark. The proof of Lemma 3.2 holds for any block-partitioning of B(x). Indeed, the key properties of the matrix b constructed in (3.9) are that it is non-derogatory and that [b, B(x)] = 0.
Proof. Because the matrices C(x) are diagonalisable and in involution, there exists a basis for C n consisting of x-independent common eigenvectors, see e.g. [33]. Let S ∈ M n (C) be constructed by concatenating the basis vectors. Then, for every x ∈ Ω, S −1 C(x)S is diagonal, hence BT, and the result follows from Proposition 3.3.
Remark. Corollary 3.4 implies that any model described by a family of commuting and diagonalisable transfer matrices is polynomially integrable.
As discussed in Section 3.3 below, Proposition 3.3 and Corollary 3.4 elevate from matrices to elements of finite-dimensional unital associative semisimple algebras.

Semisimple algebras
Let A be a finite-dimensional unital associative algebra, and let rad(A) denote its radical. Motivated by the discussion of square matrices in Section 3.2, we say that b ∈ A is non-derogatory if it generates its own centraliser, that is, if admits a unique expression of the form where c 0 , . . . , c d−1 are scalars.
Recall that A has finitely many non-isomorphic irreducible modules, L 1 , . . . , L r , each of which is finite-dimensional, and (3.12) Letting ρ i denote the representation corresponding to L i for each i = 1, . . . , r, the homomorphism is indeed surjective with kernel given by rad(A). We also recall that A is semisimple if rad(A) = {0}. Equivalently, A is semisimple if and only if and if and only if any A-representation, including the regular representation, is completely reducible. In the following, we let Lemma 3.5. Let A be a finite-dimensional unital associative semisimple algebra, and let b ∈ A. Then, b is non-derogatory if and only if ρ(b) is non-derogatory.
Proof. As to "⇒", let b be non-derogatory and ψ ∈ End so ρ(b) is non-derogatory. As to "⇐", let ρ(b) be non-derogatory and c ∈ C A (b). Since bc = cb and ρ is a homomorphism, we have We now have the following algebraic version of Proposition 3.3.
Proposition 3.6. Let A be a finite-dimensional unital associative semisimple algebra over C, and let if and only if there exists an x-independent basis for L such that the matrix representation of ρ(U (x)) is BT.
Since ρ is an homomorphism, we have and since ρ(b) is x-independent, there exists an x-independent L-basis relative to which the matrix representation of ρ(b) is in JCF. By Lemma 3.2, the corresponding matrix representation of ρ(U (x)) is BT. As to "⇐", let B be an x-independent L-basis relative to which the matrix representation, M , of ρ(U (x)) is BT. Construct ψ ∈ End C (L) such that its matrix representation, P , relative to B is in JCF with block-partitioning matching that of M , and with all the Jordan blocks having distinct eigenvalues. This matrix representation is thus non-derogatory, so by Lemma 3.5, b := ρ −1 (ψ) is non-derogatory. By the construction of ψ, we have P M = M P , hence ψρ(U (x)) = ρ(U (x))ψ, and since ρ is an isomorphism, The following algebraic counterpart to Corollary 3.4 is used in Section 4.5.
Corollary 3.7. Let A be as in Proposition 3. 6, Ω ⊆ C a suitable domain, and {C(x) ∈ A | x ∈ Ω} a commutative family of operators such that there exists a basis relative to which the matrix representation Remark. Subfactor planar algebras [7,25] are a class of planar algebras that present themselves in the context of polynomial integrability. They are semisimple and possess a natural inner product, so establishing diagonalisability of the transfer operator in a given representation amounts to showing that the operator is self-adjoint with respect to the inner product. In fact, the global property of polynomial integrability is reduced to satisfying the local YBEs, BYBEs, inversion identities, and self-adjointness of the R-and K-operators.

Cellular algebras
Here, we recall the definition and some basic properties of cellular algebras to be used in Section 4.2 and Appendix C.1. Proofs will not be reproduced but may be found in [23]. The following definition is taken from [23]. is an injective map with image an R-basis for A.
• If λ ∈ Λ and S, T ∈ M (λ), then for any element a ∈ A we have
Let the left, respectively right, A-modules W (λ) and W (λ) * be defined as the free R-modules with basis {C S | S ∈ M (λ)} and A-actions We have the natural R-module isomorphism 27) and the bilinear form Letting u, v, w ∈ W (λ) and a ∈ A({λ}), we note that the form is symmetric, The radical of φ λ is defined as Let {C S 1 , . . . , C S d λ } be an ordered basis for W (λ), where S 1 , . . . , S d λ ∈ M (λ). Then, the symmetric matrix is referred to as the corresponding Gram matrix. It holds that and if these relations are satisfied, the form φ λ is said to be non-degenerate. It follows that the represen- is faithful if det G λ = 0 for all λ ∈ Λ.

Temperley-Lieb algebras
Here, we work over C and let the so-called loop fugacity, β, be an indeterminate unless otherwise stated. Most of the introductory material in Sections 4.1 and 4.2 can be found in standard references on Temperley-Lieb algebras, such as [34,35,36,37,7,23,38]. New results include the decompositions in Proposition 4.4 and Corollary 4.6 in Section 4.3, the free Baxterisation in Section 4.4, the classification of identity points and the determination of the corresponding principal hamiltonians in Section 4.6, the analysis of minimal hamiltonian polynomials in Section 4.7, and the discussion of polynomial integrability in Section 4.8.

Definition
For each n ∈ N 0 , let B n be the set of 2n-tangles without input disks and loops. An element of B n is called a connectivity diagram and is thus a disk with 2n nodes connected by non-intersecting strings, defined up to regular isotopy. Examples are Denote by T n the complex vector space spanned by the elements of B n . The dimension of T n is given by a Catalan number, The Temperley-Lieb planar algebra TL(β) is the graded vector space (T n ) n∈N 0 , together with the natural diagrammatic action of planar tangles, with each loop replaced by a factor of a parameter β. For each n ∈ N, the Temperley-Lieb algebra TL n (β) is obtained by endowing the vector space T n with the multiplication induced by the planar tangle M n in (2.5). In accordance with (2.6), this is a unital algebra, and with reference to (2.7), it holds that It is common to represent the connectivity diagrams in B n by rectangular diagrams, here drawn such that the marked boundary interval corresponds to the leftmost vertical side, as illustrated by

←→
. subject to e j e i e j = e j , |i − j| = 1, (4.7) e i e j = e j e i , |i − j| > 1. Remark. In physical applications, the parameter β typically takes on a real (or complex) value and is often referred to as the loop fugacity.
For each n ∈ N and β an indeterminate, there exists a unique w n ∈ TL n (β) such that w 2 n = w n , e i w n = w n e i = 0, i = 1, . . . , n − 1. (4.9) We will have more to say about this Jones-Wenzl idempotent [40] in Section 4.7.

Standard modules and cellularity
For each n ∈ N, let D n := {n − 2k | k = 0, . . . , ⌊ n 2 ⌋}, (4.10) which is a naturally ordered set with min(D n ) = 1 2 (1 − (−1) n ), and introduce the map † : where x † is constructed by reflecting x about the horizontal. Extended linearly to T n , this yields an anti-automorphism on TL n (β). For each d ∈ D n , let B n,d ⊆ B n denote the set of n-diagrams with exactly d nodes on the lower edge connected to nodes on the upper edge. The d strings connecting these 2d nodes are referred to as through-lines. We now let L n,d denote the set of all n-diagrams with d through-lines, whose upper edge has been discarded along with any string having both endpoints on the upper edge. The elements of L n,d are referred to as (n, d)-link states and the elements of as n-link states. To illustrate, give rise to the same (6, 2)-link state. In fact, the (n, d)-link states may be viewed as equivalence classes of n-diagrams. From that perspective, the two 6-diagrams in (4.13) are seen as representatives of the same (6, 2)-link state. The vector space becomes a TL n (β)-module by defining an action of the algebra generators on the link states such that a 2 (a 1 v) = (a 2 a 1 )v for all a 1 , a 2 ∈ TL n (β) and v ∈ V n,d . The action defining the familiar standard module, V n,d , is first given diagrammatically for n-diagrams acting on (n, d)-link states in the 'natural way', see e.g. [38], and then extended linearly to all of TL n (β) and all of V n,d .
For each pair x, y ∈ L n,d , let x, y n,d be constructed by reflecting the link state x about the horizontal, placing it below the link state y, connecting the strands in the natural way, and replacing any loop by a factor of β, see e.g. [38] for details. This extends to a bilinear map, Relative to the (n, d)-link state basis L n,d , the nonzero elements of the corresponding Gram matrix G n,d are all monomials in β. The Gram determinant is thus polynomial in β, and following [37], it can be expressed as where U k (x) is the k th Chebyshev polynomial of the section kind.
For each pair x, y ∈ L n,d , let |x y| n,d be constructed by reflecting the link state y about the horizontal, placing it above the link state x and connecting the d defects non-intersectingly. This extends to a bilinear map, |· ·| n,d : V n,d × V n,d → TL n,d (β), (4.17) where TL n,d (β) is the subset of TL n (β) whose elements have exactly d through-lines. It follows that |x y| n,d z = y, z n,d x, ∀ x, y, z ∈ V n,d . When clear, the subscripts of ·, · n,d and |· ·| n,d will be suppressed, writing ·, · and |· ·|, respectively. It now follows [23] that TL n (β) is cellular with cell datum (D n , L n , |· ·|, †), where the dagger provides an adjoint operation relative to the bilinear form ·, · on V n,d : x, ay = a † x, y , a ∈ TL n (β), x, y ∈ V n,d . In preparation for the discussion in Section 4.7, let V n := span C (L n ), (4.20) and note that, as vector spaces, We also let ρ n,d denote the representation corresponding to the standard module V n,d , and let so relative to an ordered V n -basis of the form L n,sn ∪ L n,sn+2 ∪ · · · ∪ L n,n , s n := 1 2 (1 − (−1) n ), (4.24) the matrix representation of ρ n is block-diagonal. Moreover, from Section 3.4, we have that ρ n is faithful for all β ∈ C for which d∈Dn det G n,d = 0. In particular, ρ n is faithful for β an indeterminate and fails to be faithful for at most finitely many values β ∈ C.

Transfer operators
The connectivity bases for T 2 and T 1 are given by B 2 = {1 2 , e} and B 1 = {1 1 }, respectively. Imposing that the coefficients to the unit elements are nonzero in (2.8), the R-and K-elements can be normalised such that where u ∈ C. Diagrammatically, this R-element is given by To emphasise its dependence on β, we occasionally write T n (u, β) instead.
Proof. The results follow from repeated application of (4.28) with v = u, to (4.27). Moreover, there exist a 0 , . . . , a 2n ∈ TL n (β) such that and using (4.36), it follows that a 2n−i = a i , i = 0, . . . , n − 1, sõ where T To shine further light on the structure of T n (u, β), we introduce the following parameterised elements of TL n (β): For n ∈ N and each pair j, k ∈ N 0 such that j + k ≤ n − 2, let Proof. The result (4.42) follows by induction on n, decomposing the two rightmost R-operators as in (4.28) and using that both sides of (4.42) reduce to 1 1 for n = 1. The result (4.43) follows similarly.
For k ∈ N 0 and x ∈ C, we let [k] x := 1 + x + · · · + x k−1 (k > 1), [1] x := 1, [0] x := 0. The result now follows by induction on n, applying (4.42) to the first term on the right in (4.46) and the induction hypothesis to the second term. Corollary 4.6. The transfer operator decomposes uniquely as where Γ (n) a (u, β) is polynomial in u, β for every a ∈ B ′ n .
Proof. With the parameterisation (4.25), the decomposition of T n (u, β) into connectivity diagrams (elements of B n ) involves only coefficients that are polynomial in u, β, and since B n is a linearly independent set, the decomposition is unique. The restriction to a summation over B ′ n is permitted (and required for uniqueness) because S Remark. The expression (4.27) for T n (u, β) may be formally extended to n = 0, yielding T 0 (u, β) = β1 0 . With 1 0 ≡ 1, this becomes T 0 (u, β) = β. where p a (u, v, β) is polynomial in u, v, β for every a ∈ B ′ n .
Proof. The result follows from Corollary 4.6, noting that the restriction to a summation over B ′ n is permitted because of (4.3).
Let (indeterminate or nonzero scalar) q be such that β = q + q −1 , (4.50) and define F n := q −n T n (−q), F n := q n T n (−q −1 ). (4.51) Although these braid elements are known, see e.g. [41], to be central and satisfy F n = F n , we now present a proof of this in our notation, for completeness.
Proposition 4.8. For every n ∈ N, we have Proof. Property (i) follows from Corollary 4.5. As to (ii), we have It follows that hence F n ∈ Z TL n (β) .

Yang-Baxter integrability
The parameterisations (4.25) provide a Baxterisation for all β, as where and solve the inversion relations, YBEs and BYBEs in Proposition 2.1. It follows that T n (v) commutes with T n (u) (that is, T n (v) ∈ C TL n (β) (T n (u))) for all v for which the functions y (i) (u, v) and y (i) (u, v), i = 1, 2, 3, are well-defined. The next result extends this commutativity to all u, v, β ∈ C. Proposition 4.9. For every β ∈ C, we have Proof. For each β, Proposition 2.1 implies that the commutator is zero for all u, v ∈ C except possibly along the algebraic curves defined by setting the denominators in (4.57) and (4.58) to zero. That is, for every (u, v) ∈ C 2 not on any of the curves, the polynomials in Proposition 4.7 satisfy p a (u, v, β) = 0. It follows that, for all a ∈ B ′ n , p a (u, v, β) = 0 for all (u, v) ∈ C 2 , hence [T n (u, β), T n (v, β)] = 0 for all u, v ∈ C.
We note that the auxiliary operators (4.56) can be expressed in terms of the R-operator as (4.61) Accordingly, for the Temperley-Lieb planar algebra, the full generality of the sufficient conditions (2.24)-(2.26) is not exploited. To contrast, we refer to [42] for an example of an integrable planar-algebraic model relying nontrivially on the generality of the sufficient conditions. We stress that the integrability (4.59) is present without constraining the parameterisation of the Roperator in (4.25). We accordingly refer to the Baxterisation as free and the Temperley-Lieb loop model as freely Baxterisable. The usual (here renormalised) trigonometric Baxterisation [6], is not free, as y 2 (u) + βy(u) + 1 = sin 2 λ sin 2 u . (4.63) It nevertheless provides a solution to the relations in Proposition 2.1, as we can normalise the auxiliary operators such that In this case, the YBEs (2.25) assume the standard additive form. Crossing symmetry of the R-operator and the (trigonometric) transfer operator amounts to

Polynomial integrability
Having established the (Yang-Baxter) integrability of the Temperley-Lieb loop model in Proposition 4.9, we now turn to the integrals of motion of the model. It is known, see for example [43], that the bilinear form ·, · n,d , with the corresponding scalar field restricted to R, is an inner product for large enough β ∈ R. In fact, it follows from (4.16) that, for β > 2 cos π n , the form is an inner product and TL n (β) is semisimple. Since the transfer operator is self-adjoint, T † n (u, β) = T n (u, β), with respect to the form, it is thus diagonalisable for β > 2 cos π n and u ∈ R. Using this, the following result follows from Corollary 3.7.
Remark. For β > 2, the Temperley-Lieb planar algebra is, in fact, a subfactor planar algebra. Proposition 4.10 implies that there exists a u-independent integral of motion b n ∈ TL n (β) such that T n (u, β) ∈ R[b n ], and following Section 3.3, ρ n (b n ) and ρ n (T n (u, β)) will have closely related Jordan decompositions. We note that the usual hamiltonian (see (4.69) below) has this property when T n (u, β) is expanded to lowest nontrivial order in u. In the following, we derive the principal hamiltonians associated with T n (u, β), see (4.69) and (4.74), and use spectral analysis to argue that both of these TL n (β)-elements can indeed play the role of b n , at least for small n. We supplement this result by determining an explicit polynomial expression for T n (u, β) in terms of each of the hamiltonian elements, and find that they are well-defined for all u ∈ C and all but finitely many β-values in C. The restrictions on u and β in Proposition 4.10 can thus be relaxed accordingly, at least for small n. Moreover, we find that, for small n, T n (u, β) is polynomial in at least one of the two principal hamiltonians for all β, u ∈ C, see the discussion following (4.102).
Remark. In addition to establishing polynomial integrability, our focus in the following is on probing the naturally arising principal hamiltonians as candidates for the integral of motion b n . In fact, one could also explore whether any given specialisation of T n (u), where u is fixed to some value, could play the role of b n . We have indeed examined a number of such candidates, including the braid-limit element F n in (4.51), but the standard hamiltonian (4.69) has so far had the fewest number of exceptional points, see the discussion in Section 4.8.  Proof. It follows from (4.41) that the connectivity diagram corresponding to e 1 · · · e n−1 only appears in S (n) j,k for j = k = 0, and with coefficient u n−2 . By Proposition 4.4, the element thus appears in T n (u, β) with coefficient u n−1 (β + 2u)(2 + βu). This expression vanishes exactly for the indicated values for u.

Hamiltonian limits
To determine the hamiltonian associated with the identity point u * = 0, we use Proposition 4.4 to compute For n ≥ 2 and all β, we may thus choose the familiar (see e.g. [44,6,8]) as the principal hamiltonian associated with u * = 0.
Remark. There is also a 'hidden' identity point at infinity, see the Remark, following (4.27), that addresses the extension of the domain for u from C to the Riemann sphere. The corresponding principal hamiltonian is proportional to h 0 .
Hamiltonians associated with the identity points u * = − β 2 = 0 and u * = − 2 β do not seem to have been discussed before in the literature. To determine the corresponding principal hamiltonians, hβ 2 and h -2 β , we expand as and and h n,0 are not. We also note that

Minimal hamiltonian polynomials
Since TL n (β) is finite-dimensional, corresponding to each a ∈ TL n (β), there exists a unique monic polynomial, of least positive degree, that annihilates a -the so-called minimal polynomial of a. Let m Proof. For ρ n faithful, the minimal polynomial of ρ n (h u * ) is the same as that of h u * , irrespective of β being an indeterminate or taking on a complex value. Specialising β to a complex value may introduce spurious (see Section 3.1 and the remark following (4.82)) degeneracies in the spectrum of ρ n (h u * ), and such degeneracies could reduce the degree of the minimal polynomial of ρ n (h u * ). This explains the first inequality. The second inequality follows from the existence of an c n -dimensional representation, ρ n , that is faithful for β an indeterminate.
Remark. To appreciate the inequality l (n) u * ,β ≤ c n directly, note that ρ n is a c n -dimensional representation that is faithful for all but finitely many β-values. The degree of the minimal polynomial for β complex and generic is thus bounded by c n , and, possibly rescaled to remain well-defined, the corresponding minimal polynomial will remain annihilating when specialising β to one of these values. Such a rescaling can be chosen such that the rescaled polynomial is nonzero when specialising β, and the degree of this rescaled polynomial may decrease upon specialisation (this happens if and only if the rescaling multiplies the leading monomial by a factor that is zero when specialised) but cannot increase. Proof. For ρ n faithful, the minimal polynomial of h u * is the same as that of ρ n (h u * ), and h u * is nonderogatory if and only if ρ n (h u * ) is non-derogatory. The latter is also equivalent to m ρn(hu * ) = c ρn(hu * ) , hence to deg(m ρn(hu * ) ) = c n . Since ρ n is faithful for β an indeterminate, the result follows.
Through direct computation, we have found that the spectrum of ρ n (h 0 ) for β = −2 is non-degenerate for n = 2, . . . , 17. It follows that l We likewise find that the spectrum of ρ n (h n,-2 β ) for β = π + π −1 is non-degenerate for n = 2, . . . , 6, hence l (n) -2 β ,π+π −1 = c n , n = 2, . . . , 6. (4.81) The specific β-values in these computations are immaterial, as long as they are 'sufficiently generic'. This conjecture implies that, for every n ∈ Z ≥2 and u * ∈ {0, − 2 β }, we have Remark. Necessary conditions for strict inequalities in (4.79) are the existence of spurious respectively permanent degeneracies in the spectrum of ρ n (h u * ). However, these are not sufficient conditions as the Jordan-block structure may 'prevent' a corresponding reduction in the degree of the minimal polynomials.

Transfer-operator hamiltonian polynomials
The above spectral analysis of the principal hamiltonians h n,u * for small n, together with Proposition 4.18 below, indicates that the hamiltonians are viable candidates for an integral of motion in terms of which the Temperley-Lieb transfer operator T n (u, β) is polynomial. We proceed by presenting explicit polynomial expressions of T n (u, β) in terms of the hamiltonians for small n, and by offering conjectures about the form of such polynomials for general n.
Proof. Let the spectrum of ψ(h u * ) be non-degenerate. Then, the characteristic and minimal polynomials of ψ(h u * ) agree, so ψ(h u * ) is non-derogatory. Since ψ is faithful, it follows that h u * is non-derogatory, and since T n (u, β) commutes with h u * , we have T n (u, β) ∈ h u * TL n (β) .
Remark. In the following, we will use that ρ n is faithful for β an indeterminate and for all but finitely many β ∈ C.
Because the matrix elements of ρ n (T n (u, β)) are polynomial in β, Proposition 3.1 implies that there are at most finitely many values β ∈ C for which the spectrum of ρ n (T n (u, β)) possesses spurious degeneracies. Combined with the non-degeneracy observations implying (4.80) and (4.81), it follows from Proposition 4.18 that for every n = 2, . . . , 17, T n (u, β) is polynomial in h 0 for all but finitely many β-values, and that for every n = 2, . . . , 6, T n (u, β) is polynomial in h -2 β for all but finitely many β-values. For n = 2, for example, we have h -2 β = −h 0 for β = 0, and valid for all β ∈ C. In the following, we will argue that T n (u, β) ∈ C[u][h u * ], u * ∈ {0, − 2 β }, for every n ∈ Z ≥2 and all but finitely many β-values. We refer to these values as h u * -exceptional and note that the number and values of them will depend on n and u * .
If Conjecture 4.19 holds, then every h u * -exceptional β-value will be a root of f n,u * (β). The converse need not be true since T n (u, β) could be polynomial in h u * even if β is a root of f n,u * (β), see below. Letting E n,u * denote the set of h u * -exceptional β-values and Z n,u * the set of roots (or zeros) of f n,u * (β), we thus have E n,u * ⊆ Z n,u * .
For β r ∈ E n,u * , T n (u, β r ) is not expressible as a polynomial in h u * . However, observing that we conclude that E n,0 ∩ E n,-2 β = ∅, n = 3, 4, 5. It follows that, for n = 3, 4, 5 and every β ∈ C, T n (u, β) is polynomial in at least one of the two hamiltonians: h 0 and h -2 β .

Discussion
We have developed a general framework for describing integrable models based on planar algebras, and we have revisited the notion of integrals of motion from an algebraic perspective, introducing polynomial integrability as a fundamental characteristic. We applied the framework to the Temperley-Lieb loop model and to an eight-vertex model, and discussed their polynomial integrability. It would be interesting to explore further what polynomial integrability can teach us about an integrable model -in general but also in specific models of particular physical relevance. For example, we would like to understand how polynomial integrability might recast T -systems [45,46], Y -systems [47,48] and functional relations more generally [49], including the inversion relations for critical dense polymers described by TL n (β = 0) [8,12], and the similar higher-order relations in [11].
In lattice-model language, our transfer operators are constructed on the strip. When extending to the cylinder or annulus, transfer operators may be constructed from affine tangles, in which case the operators are morphisms in the affine category of a given planar algebra [50,51]. We expect that most of our polynomial integrability considerations carry over to the periodic case.
On both the strip and cylinder, the polynomial integrability of the Temperley-Lieb loop model seems to have intricate implications for our understanding and analysis of the continuum scaling limit. If indeed the hamiltonian generates its own centraliser for all finite n, it would require reconciliation with established insight [52] into how the conformal integrals of motion [53] inherit behaviour displayed by their finite-size counterparts.
Preliminary results indicate that properties and results similar to the ones reported here for the Temperley-Lieb loop model, and for the special eight-vertex model in Appendix C, apply for the standard six-and eight-vertex models [54,55,56,18,19], RSOS models [57,58], the dilute loop models related to the O(n) models [59,60], fused [61] Temperley-Lieb loop models [62,63,10,11], and models built on Brauer [64] and Birman-Wenzl-Murakami algebras [65,66]. We hope to return elsewhere with more on this.
can be glued inside the tangle T : Consistency between the composition of planar tangles and the action of the tangles as multilinear maps (2.2) is often referred to as naturality and corresponds to see [25,67] for more details.

A.2 Unitality
Let (P n ) n∈N 0 be a planar algebra.
Lemma A.1. If P n has no nonzero null vectors, then P r n,0 is the identity operator.
Proof. Let v ∈ P n and T be a planar tangle for which D T = {D} with η(D) = n. By naturality, we then have so v − P r n,0 (v) ∈ ker(P T ). Since P n has no nonzero null vectors, it follows that P r n,0 (v) = v for all v ∈ P n .
Together with naturality, Lemma A.1 implies the following result.
Corollary A.2. If P n has no nonzero null vectors, then P r n,k is invertible for every k ∈ Z.
Now, let (A n ) n∈N 0 be a planar algebra of the type used in the paragraph containing (2.5), and recall 1 n := P Idn () (2.6).
Proposition A.3. Let A n be endowed with the multiplication induced by M n , and suppose A n has no nonzero null vectors. Then, A n is unital, with unit 1 n .
Proof. Let D l and D u denote the lower respectively upper disk in the planar tangle M n , and let v ∈ A n . Then, naturality implies that B TL n (β) polynomials B.1 Principal hamiltonian h 0 For n = 2, . . . , 7 and β an indeterminate, the minimal polynomial for h 0 is given by In a matrix representation of ρ n (h 0 ), the off-diagonal elements are independent of β, whereas the diagonal elements are of the form −iβ, i ∈ {0, . . . , ⌊ n 2 ⌋}. Since the number of elements equal to −iβ is ⌊ n it follows that m We also note that the degree of the monic β-polynomial multiplying h i in m (n) 0 (h) is given by c n − i, and that this β-polynomial is even (respectively odd) if the degree is even (respectively odd).
For β = 0 and n ≥ 2, there are spurious degeneracies in the spectrum of ρ n (h 0 ), so l and for β an indeterminate, its minimal polynomial is given by We note that the numerators of the fractions multiplying the even monic β-polynomials in these minimal polynomials are the same as the coefficients to the similar terms in (B.1)-(B.4). We also note that the degree of the monic β-polynomial multiplying h i in m where f n,0 (β) is as in Conjecture 4.19 andã n,0 i,k (β) are polynomials such that no root of f n,0 (β) is a root ofã n,0 i,k (β) for all i, k. The form of the contribution x n−3 (β − x)h 0 + h 2 0 follows from continuing the expansion (4.67) to third order in ǫ: and thatã n,0 i,k (β) is even (respectively odd) if its degree is even (respectively odd). This is seen to correspond to the parity of n + i + k + 1.
Conjecture B.4. Let n ∈ Z ≥3 and β an indeterminate. Then,T n (x) admits a unique decomposition of the formT i,k (β) is even (respectively odd) if its degree is even (respectively odd). This is seen to correspond to the parity of n + i + k.

C Tensor planar algebras
Here, we specialise to tensor planar algebras and thereby recover the familiar quantum inverse scattering method framework, in which case the R-operators are tensorially separable, and outline how the planaralgebraic framework simplifies. To illustrate, we consider a specialisation of the zero-field eight-vertex model [55,56] that satisfies the free-fermion condition, and whose principle hamiltonian corresponds to the Ising model hamiltonian. The general model was solved by Baxter in [18], see also [19], while our presentation highlights the underlying polynomial integrable structure of a particular specialisation, by analysing the spectral properties of the transfer operator and the associated hamiltonian.
We thus show that the transfer matrix of this specialised eight-vertex model is diagonablisable and present an exact solution. Although the model is Yang-Baxter integrable, its simplicity allows us to use standard techniques to obtain explicit expressions for all eigenvalues and corresponding eigenvectors of the transfer matrix. We then exploit similarities in the spectral properties of the transfer matrix and the canonical hamiltonian to establish that the transfer matrix is polynomial in the hamiltonian. Moreover, we decompose the transfer operator into an explicit linear combination of a complete set of orthogonal idempotents expressed in terms of the minimal polynomial of the hamiltonian.

C.1 Definition and cellularity
For each n ∈ N 0 and ℓ ∈ N, we let V n denote the ℓ 2n -dimensional vector space spanned by disks with 2n labelled boundary points where each label is taken from the set {1, . . . , ℓ}. As the disks do not come equipped with any further (interior or otherwise) structure, we have V n ∼ = V ⊗2n , where V is an ℓ-dimensional vector space.
The tensor planar algebra is the graded vector space (V n ) n∈N 0 , together with the following action of planar tangles: If a string in the planar tangle connects a pair of boundary points with different labels, then the output is the zero vector, and if not, then the labels of the output vector are given by the labels at the opposite string endpoints. If both endpoints of a string are on the exterior boundary of the planar tangle, then the output vector is a sum obtained by varying the common label of the two endpoints.
Following from compatibility with the gluing of planar tangles, and using the evaluation map e, a loop is accordingly replaced by the scalar ℓ. To illustrate, with T as in (2.3) and a, b ∈ F, we have We now equip each V n with the multiplication induced by the corresponding multiplication tangle, M n in (2.5), and identify the first n labels clockwise from the marked boundary interval as characterising an incoming vector, with the remaining n labels characterising an outgoing vector. The vector space V n thus has the structure of an endomorphism algebra, and is consequently cellular.

C.2 Transfer operator
Let bases for V 1 and V 2 be given by respectively, such that, viewed as matrices relative to the natural basis orderings, e k j are ℓ × ℓ matrix units with 1 in position (j, k) and zeros elsewhere. By construction, so every element of V 2 is separable. In particular, the V 2 basis vectors decompose as e k m j l = e k j ⊗ e m l , j, k, l, m ∈ {1, . . . , ℓ}. (C.5) As parameterised elements of V 1 and V 2 , respectively, the K-and R-operators (2.8) are here written as where K k j , R k m j l , K in terms of which we construct the Sklyanin transfer operatorŤ (u) ∈ V n , as in (2.33). Accordingly,Ť (u) can be expressed familiarly as a vector-space trace over an auxiliary copy of End F (V ) in V n+1 . In the following, the auxiliary space is the (n + 1) th copy, and the corresponding trace is denoted by Tr n+1 . For each i = 1, . . . , n, we first introduce where the indices i and n + 1 denote the copies of V on which the operators act nontrivially, as well as The transfer operator can then be expressed in terms of the 'pre-trace' transfer operator L n+1 (u) := R n,n+1 (u) · · · R 1,n+1 (u)K 1 (u)R n+1,1 (u) · · · R n+1,n (u)K n+1 (u) (C.12) asŤ n (u) = Tr n+1 (L n+1 (u)). (C.13)

C.3 Eight-vertex model
In the remainder of this appendix, we let dim(V ) = 2 (C.14) and fix the parameterisation to an eight-vertex model, characterised by 1, j = m, l = k, u, j = l, k = m, j = k, u, j = k, l = m, j = l, 0, otherwise, Working in the natural matrix representation where the e k j are matrix units, and where 1 denotes the 2 × 2 identity matrix, the R-and K-operators can be expressed in terms of Pauli matrices as It follows that and using the standard notation σ α m,i := 1 i−1 ⊗ σ α ⊗ 1 m−i , α ∈ {x, y, z}, i ∈ {1, . . . , m}, m ∈ N, (C. 19) we then have the following result.
It follows that n (u) ∈ V n , and consequently thať The next result allows us to determine the polynomial structures of L n+1 (u) andŤ n (u).
We let |± denote eigenvectors of σ x , Proof. The result follows from the multiplicative expression (C.29).
The following result readily follows from Proposition C.4.
Remark. The eigenvalues ofŤ n (u) and h n are related as λ s (ǫ) = 2 1 + 2ǫµ s + O(ǫ 2 ) , ∀ s ∈ {±} n . (C.43) The form of the minimal polynomial of h n follows from Proposition C.6. To fix our notation, we define the polynomials The next result provides details of this polynomial. In preparation, let λ k (u) and µ k denote the eigenvalues corresponding to the (joint) eigenspaces V n,k , k = 0, 1, . . . , n − 1, ofŤ n (u) and h n , respectively. For ease of reference, we recall their expressions, λ k (u) = 2(1 + u 2 )(1 − u) 2k (1 + u) 2(n−1−k) , µ k = n − 1 − 2k, (C. 46) and introduce the n × n matrix Since this is a Vandermonde matrix with µ i = µ j for all i = j, it is invertible, and the inverse can be evaluated explicitly. For every n ∈ N, we let m 0 (h n ) ≡ h 0 n ≡ 1 n .
Proposition C.8. For every n ∈ N and all u ∈ F, we havě   In terms of these polynomials, we now define Finally, for each k = 0, 1, . . . , n − 1, we have for some polynomial q k (x). It follows that We have verified Conjecture C.12 for n = 1, . . . , 180.