Integrable models from singly generated planar algebras

Not all planar algebras can encode the algebraic structure of a Yang--Baxter integrable model described in terms of a so-called homogeneous transfer operator. In the family of subfactor planar algebras, we focus on the ones known as singly generated and find that the only such planar algebras underlying homogeneous Yang--Baxter integrable models are the so-called Yang--Baxter relation planar algebras. According to a result of Liu, there are three such planar algebras: the well-known Fuss--Catalan and Birman--Wenzl--Murakami planar algebras, in addition to one more which we refer to as the Liu planar algebra. The Fuss--Catalan and Birman--Wenzl--Murakami algebras are known to underlie Yang--Baxter integrable models, and we show that the Liu algebra likewise admits a Baxterisation. We also show that the homogeneous transfer operator describing a model underlied by a singly generated Yang--Baxter relation planar algebra is polynomialisable, meaning that it is polynomial in a spectral-parameter-independent element of the algebra.


Introduction
In [1], Jones proposed the "royal road" as a route to construct a conformal field theory from a socalled finite index subfactor, where one extracts, from the subfactor, Boltzmann weights of a critical two-dimensional lattice model, and uses these to construct a quantum field theory in the scaling limit. Despite the simplicity and promise of this proposal, Jones stressed himself that the available mathematical tools are likely insufficient in all but the simplest examples. The present work is an effort to expand our toolkit such that new and more scenic routes may one day be passable.
To set the stage, we recall that a subfactor planar algebra [2,3,4,5] is a shaded planar algebra, (A n,± ) n∈N 0 , that has a certain set of simple and physically well-motivated properties, including an inner product on each A n,± . A subfactor planar algebra is said to be singly generated [6,7,8] if it is generated by only one generator aside from the default Temperley-Lieb generators [9,10], and it is called a Yang-Baxter relation (YBR) planar algebra [11] if it satisfies certain algebraic relations generalising the familiar braid relations. According to Liu [11], there are three singly generated YBR planar algebras (as well as quotients thereof): the two-colour Fuss-Catalan (FC) [12] and the Birman-Wenzl-Murakami (BMW) [13,14] planar algebras, in addition to one more which we refer to as the Liu planar algebra.
Relations of the YBR form are typically needed to construct spectral-parameter dependent solutions to the Yang-Baxter equations [15,16,17,18], through a procedure called Baxterisation [19]. Indeed, the FC and BMW planar algebras are known to underlie Yang-Baxter integrable models [20,21], while, to the best of our knowledge, no integrability observation has been presented in the case of the Liu planar algebra. We remedy this by Baxterising the Liu algebra within the algebraic integrability framework put forward by the authors in [22].
Our main result is a classification of singly generated planar algebras that can encode the algebraic structure of a homogeneous Yang-Baxter integrable model. Here, up to technical details, a planar algebra is said to encode the homogeneous Yang-Baxter integrability if no proper planar subalgebra can take its place (the precise definition is given in Section 4.2), while a homogeneous Yang-Baxter integrable model is one that is described by a homogeneous transfer operator. Such a transfer operator is built in a particular 'uniform' way using a single R-operator and a pair of K-operators, each parameterised using a single parameter u ∈ C (see (4.4)) such that a set of local relations, including generalised Yang-Baxter equations (see Proposition 4.3), are satisfied. Moreover, a transfer operator is said to be polynomialisable [22] if it is polynomial in a spectral-parameter-independent algebra element. Our findings may now be summarised as follows. Theorem 1.1. Let A be a singly generated planar algebra. Then, there exists a homogeneous Yang-Baxter integrable model with A encoding the integrability if and only if A is a Yang-Baxter relation planar algebra.
Theorem 1.2. Let the homogeneous Yang-Baxter integrability of a model be encoded by a singly generated Yang-Baxter relation planar algebra. Then, there exist algebra-parameter choices and a suitable u-domain such that the corresponding transfer operator is polynomialisable.
As a unifying framework for describing the singly generated YBR planar algebras, inspired by the series of works [6,7,8,11], we find it convenient to introduce the unshaded proto-singly-generated (PSG) planar algebra (PS n ) n∈N 0 . Although PS n is infinite-dimensional for n ≥ 3 and does not encode a homogeneous Yang-Baxter integrable model, we demonstrate that the PSG planar algebra serves as an 'ambient' algebra admitting quotients isomorphic to the FC, BMW, and Liu planar algebras.
The paper is organised as follows. In Section 2, we recall the basics of shaded planar algebras, including the subfactor, YBR, and Temperley-Lieb variants. Certain technical details can be found in Appendix A and Appendix B.
In Section 3, we introduce the PSG planar algebra (PS n ) n∈N 0 . For each n ∈ N, we give a presentation of the corresponding algebra PS n .
In Section 4, we review the algebraic integrability framework developed recently in [22]. Using [23], we demonstrate that it suffices to consider unshaded planar algebras when addressing the integrability questions of our interest. We then show that the PSG planar algebra cannot encode Yang-Baxter integrability unless additional conditions are imposed on its generators such that the corresponding planar algebra is YBR.
In Section 5, we review the FC planar algebra and the corresponding FC algebras FC n . In the two-colour case, with the two colours having the same loop weight, the FC planar algebra is YBR, and we show that FC n is isomorphic to a quotient of PS n with parameters given in terms of the FC loop weight. We also recast the known Baxterisation [20] in our notation and find that it is the only nontrivial Baxterisation supported by our Yang-Baxter integrability framework.
In Section 6, we review the BMW planar algebra and the corresponding algebras BMW n . We show that BMW n is isomorphic to a quotient of PS n with parameters given in terms of the BMW parameters. We also recast the known Baxterisation [21] in our notation and find that it is the only nontrivial Baxterisation supported by our Yang-Baxter framework. A description of BMW n as a quotient of the braid-semigroup algebra on n strands is presented in Appendix C.
In Section 7, we review the planar algebra introduced by Liu in [11], here denoted by L. For each n ∈ N, we introduce the corresponding Liu algebra L n and show that it is isomorphic to a quotient of PS n with parameters given in terms of the loop weight appearing in L. We also present the Baxterisation of L within our integrability framework [22]. Some technical details are omitted but can be found in [24].
In Section 8, we recall the notion of polynomial integrability and show that the three singly generated YBR planar algebras (FC, BMW, and Liu) all encode Yang-Baxter integrable models described in terms of polynomialisable transfer operators. Section 9 contains some concluding remarks. Throughout, we let i denote the imaginary unit, N the set of positive integers, and N 0 the set of nonnegative integers.

Planar algebras
Here, we review the basics of planar algebras [2,3,5], including the definition of subfactor planar algebras and the subclass of Yang-Baxter relation planar algebras. We also present the Temperley-Lieb planar algebra [2] and the corresponding subfactor planar algebra, and recall its ubiquitous role. The usual Temperley-Lieb algebra [9,10], and how it arises from the corresponding planar algebra, is discussed in Appendix B.

Shaded planar algebras
Informally, a complex shaded planar algebra [2,3,5] is a collection of complex vector spaces (A n,± ) n∈N 0 where vectors can be 'multiplied planarly' to form vectors, and where (A n,± ) n∈N 0 is a common abbreviation for (A n,ε ) ε∈{+,−} n∈N 0 . A basis for A n,± consists of disks with 2n nodes (connection points) on their boundary, whereby a boundary is composed of nodes and boundary intervals, and the boundary intervals are labelled alternatingly by + or −. 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 shaded planar tangles are the diagrammatic objects, defined up to ambient isotopy, that facilitate such combinations. The shading of a planar tangle forms a 'checkerboard' pattern that 'matches' the +/− labelling of the boundary intervals.
In general, a planar tangle has the following features: • An exterior disk called the output disk.
• A finite set of interior disks called input disks.
• A finite set of non-intersecting strings connecting the nodes of disks.
• For each disk, a marked boundary interval.
• A checkerboard shading of the region outside the input disks.
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) and is necessarily even, while the shading of D, denoted by ζ(D), is +, respectively −, if the (exterior) marked boundary interval borders an unshaded, respectively shaded, region. The marked boundary intervals on the disks 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 A η(D)/2, ζ(D) , in such a way that the nodes match up, the marked intervals are aligned, and the shading is consistent. The details of the map P T specify how one should remove the input disks in the picture and identify the image as a vector in A η(D T 0 )/2, ζ(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 ∈ A 1,+ , v 2 ∈ A 2,− and v 3 ∈ A 3,+ . Note that we have not specified any details about (A 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 (A n,± ) n∈N 0 and the action of planar tangles as multilinear maps (2.2) such that naturality is satisfied, one arrives at a shaded planar algebra. For each n ∈ N 0 , the identity tangles are introduced as id n,+ := . . . , (2.5) with corresponding linear maps P r n,1,± : A n,± → A n,∓ and P r n,−1,± : A n,± → A n,∓ . If A n,± has no null vectors, then P id n,± acts as the identity operator. Here, we say that a nonzero v ∈ A n,± is a null vector if P T (v) = 0 for every planar tangle T for which P T has domain A n,± . Moreover, if neither A n,+ nor A n,− has any null vectors, then P r n,−ℓ,∓ • P r n,ℓ,± = P id n,± for each ℓ ∈ {−1, 1}. Details are presented in Appendix A.2.

Remark.
A zero planar algebra [5], where the vector spaces (A n,± ) n∈N 0 are arbitrary and all planar tangles act as the zero map, exclusively contains null vectors. We note (i) that each vector space of a planar algebra can be extended to include arbitrarily many null vectors, and (ii) for each planar algebra with null vectors, except for a zero planar algebra, there exists a corresponding planar algebra without null vectors (obtained by omitting them).
A shaded 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 tangles , P M n,± : A n,± × A n,± → A n,± , (2.6) induce a multiplication on A n,+ and A n,− , respectively, and we write vw = P M n,± (v, w) ∈ A n,± for v, w ∈ A n,± , where v, respectively w, is replacing the lower, respectively upper, disk in M n,± .
Remark. If the shading of a region is unspecified, as it may depend on the parity of n, we use the 'banded' pattern illustrated in (2.6). For simplicity, the region containing the 'dots' is coloured white.
The ensuing algebra, A n,± , is associative and, if it has no null vectors, unital, with unit whose dependence on n may be ignored by writing 1 ± for 1 n,± . For details, see Appendix A.2.

Subfactor planar algebras
Special classes of planar algebras follow when additional structure is imposed on the vector spaces A n,± or on the multilinear maps P T . By doing so, we will encounter subfactor planar algebras in the following.
In general, there are no constraints on the dimensions of the vector spaces A n,± , but the planar algebra is called evaluable if, for each shading +/−, dim(A 0,± ) = 1 and dim(A n,± ) < ∞ for all n ∈ N. In that case, the evaluation map e : A 0,± → C, (2.8) mapping the 'empty disk' to the scalar 1, provides an isomorphism, A 0,± ∼ = C. induce notions of left and right traces, respectively: A planar algebra is said to be spherical if for all n ∈ N. We note that sphericality requires A 0,+ ∼ = A 0,− . Let · † denote the operator that acts by reflecting a planar tangle about a line perpendicular to the marked exterior boundary interval, and let * : A n,± → A n,± , n ∈ N 0 , denote a conjugate linear involution. Analogous to naturality, compatibility between the two maps manifests itself in a simple relation, which must hold for all planar tangles T and all (v 1 , . . . , v |D T | ) ∈ × D∈D T A η(D)/2, ζ(D) . A planar algebra (A n,± ) n∈N 0 endowed with the maps · † and · * satisfying (2.12) is known as involutive. In that case, P Id † n,± () = P Id n,± () * , 1 n,± = 1 * n,± , (2.13) and for p ∈ A n,± , we have with the indicated multiplication induced by M n,± . An involutive planar algebra (A n,± ) n∈N 0 admits the sesquilinear maps An involutive planar algebra inherits the qualifier positive (semi-)definite if both trace forms enjoy it for all n ∈ N 0 . If the involutive planar algebra is spherical, then the two trace maps (and hence trace forms) are identical.
An evaluable, spherical and positive-definite planar algebra is called a subfactor planar algebra. It is implied that a positive-definite planar algebra is involutive, so a subfactor planar algebra (A n,± ) n∈N 0 is involutive, with each A n,± having the corresponding unique trace form (2.16) as inner product. In that case, we introduce P ′ tr (c) n,± := e • P tr (c) n,± (2.17) and refer to the map as the trace norm. Moreover, the multiplication on A n,± is induced by the corresponding planar tangle in (2.6), and since evaluability implies dim A n,± < ∞, it follows that A n,± is a finite-dimensional semisimple algebra, see e.g. [3]. Consequently, A n,± is isomorphic to a direct sum of matrix algebras, facilitating the use of linear-algebraic techniques in the analysis of subfactor planar algebras. As a simple but ubiquitous example of a subfactor planar algebra, we recall the Temperley-Lieb subfactor planar algebra [2] in Section 2.4 below.
Remark. The positive-definiteness of subfactor planar algebras implies that they do not have any null vectors. Given the relevance of subfactor planar algebras to the classification in Theorem 1.1, we will henceforth assume that P id n,± is the identity operator and that the algebra with multiplication induced by M n,± is unital.

Yang-Baxter relation planar algebras
Here, we recall a variant of subfactor planar algebras that will play an important role in our discussion of Yang-Baxter integrability. Let (A n,± ) n∈N 0 be a subfactor planar algebra, with B 2,± denoting a basis for A 2,± . Following [11], a triple (x, y, z) ∈ A 2,− × A 2,+ × A 2,− , respectively (x, y, z) ∈ A 2,+ × A 2,− × A 2,+ , is said to satisfy a Yang-Baxter relation (YBR) if respectively, for some C a,b,c x,y,z , D a,b,c x,y,z ∈ C. A subfactor planar algebra is called a YBR planar algebra if every triple of vectors in A 2,− × A 2,+ × A 2,− and A 2,+ × A 2,− × A 2,+ satisfies a YBR.
Remark. Although a YBR planar algebra is a subfactor planar algebra, we are suppressing that qualifier, in line with the convention in [11].
While we adopt the form (2.19) of the YBRs introduced in [11], the invertibility of the elementary rotation tangles (see Corollary A.2 in Appendix A) ensures that the characterisation of a planar algebra as a YBR planar algebra does not depend on the particular choices of input disk markings (and consequently shadings) in (2.19), on either side of any of the two YBRs. However, we do need a YBR for each shading of the output disk, as in (2.19).
In Section 4.1, we describe how one can associate a homogeneous Yang-Baxter integrable model to any planar algebra satisfying a particular set of sufficient conditions, including YBRs. It is thus natural to expect that YBR planar algebras play an important role in the classification of Yang-Baxter integrable planar-algebraic models. Indeed, we find (Proposition 4.5 in Section 4.4) that a so-called singly generated planar algebra that is not a YBR planar algebra does not encode the structure of a homogeneous Yang-Baxter integrable model. Singly generated planar algebras are discussed in Section 3.

Temperley-Lieb planar algebra
Let T n,± denote the complex vector space spanned by disks with (i) 2n nodes where each node is connected to another node via a set of non-intersecting strings (defined up to ambient isotopy), and (ii) a ± checkerboard shading (see Section 2.1). Examples of such Temperley-Lieb disks are The dimension of T n,± is given by a Catalan number, as dim T n,± = 1 n + 1 2n n . (2.21) The Temperley-Lieb (TL) planar algebra TL(δ) is the graded vector space (T n,± ) n∈N 0 , together with the natural diagrammatic action of shaded planar tangles, with each loop replaced by a factor of the parameter δ ∈ C. To illustrate, we present the example From [2], we know that TL(δ) is spherical and involutive, with the involution · * defined as the conjugatelinear map that acts by reflecting every disk about a line perpendicular to the marked boundary interval. Let T denote the set of all δ-values such that the planar algebra TL(δ) is positive semi-definite. For each δ ∈ T , the TL subfactor planar algebra TL(δ) is then defined as the quotient of TL(δ) by the kernel of the trace norm (2.18). According to [10], For δ ∈ 2 cos π m+2 | m ∈ N , the kernel of the trace norm is nontrivial, so TL(δ) and TL(δ) are nonisomorphic, while for δ > 2, TL(δ) is positive-definite, in which case TL(δ) ∼ = TL(δ).
Remark. Here and throughout, the sans-serif font distinguishes a subfactor planar algebra, such as TL(δ), from the corresponding (not necessarily subfactor) planar algebra, here TL(δ).
In a so-called singly generated (subfactor) planar algebra, dim A 1,± = 1 and dim A 2,± = 3, so A 2,± has a basis consisting of the two Temperley-Lieb disks (2.24) and one additional disk, hence the terminology. Moreover, the vector spaces A n,± for n > 2 are generated by the action of planar tangles on disks in A 2,± . Section 3 below is devoted to the study of (unshaded) singly generated planar algebras.
Remark. Although a singly generated planar algebra is a subfactor planar algebra, we are suppressing that qualifier, in line with the convention in [6].
3 Proto-singly-generated algebra Subfactor planar algebras, including singly generated planar algebras, are shaded by default. The shading is necessary, in general, to relate a subfactor planar algebra to the so-called standard invariant of the corresponding subfactor ; this story is outlined in [25]. However, the shading of a planar algebra (A n,± ) n∈N 0 need not carry any nontrivial information. In that case, the shading can be ignored, giving rise to the corresponding unshaded planar algebra (A n ) n∈N 0 , see e.g. [23].
Remark. Omitting the subscript indicating the shading of a given shaded planar tangle, we are referring to the corresponding unshaded version of the planar tangle. A similar convention is adopted for the vector spaces (A n ) n∈N 0 spanned by unshaded disks.
As will be clear after Section 4.3, the singly generated planar algebras relevant to the integrability questions of our interest necessarily admit an unshaded description. Accordingly, we will henceforth restrict to the class of unshaded singly generated planar algebras. About these, we have the following key result involving the proto-singly-generated planar algebra PS (ϵ) (α, δ) constructed in Section 3.1 and Section 3.2 below. This algebra was conceived in [6].
Indeed, PS (ϵ) (α, δ) is defined in terms of relations that, for some ϵ, α, δ, are satisfied by any given unshaded singly generated planar algebra. Here, δ is the loop weight arising in is the unshaded Temperley-Lieb generator. We note that e * = e.

Planar algebra
We now introduce a planar algebra whose vector spaces are spanned by planar tangles with labelled disks, and where planar tangles act on these vector spaces in the natural way. For each n ∈ N 0 , let S n denote a set whose elements label disks with 2n nodes. With S := n∈N 0 S n , an S-labelled tangle is thus a planar tangle whose input disks each have been labelled by an element of S. For such a label set S, the unshaded universal planar algebra consists of the vector spaces (A n (S)) n∈N 0 where, for each n, A n (S) is spanned by all S-labelled tangles with 2n nodes on their output disk, together with the planar-tangle action colloquially named 'what you see is what you get', illustrated in (3.29), see also [2,26]. We note that the elements of S have no additional structure. Accordingly, the list of cardinalities, |S 0 |, |S 1 |, |S 2 |, . . ., characterises a universal planar algebra. A universal planar algebra is infinite-dimensional. Indeed, even if S k = ∅ for all k ∈ N 0 , then each A n (S) is spanned by the corresponding set of (unshaded) Temperley-Lieb disks, together with those same disks but with all possible combinations of loops. To tame the dimensionality of a universal planar algebra, relations are imposed on (A n (S)) n∈N 0 . For any set C of vectors in (A n (S)) n∈N 0 , we thus let I(C) denote the planar ideal generated by C, and (A n (S, C)) n∈N 0 the corresponding quotient planar algebra.

Remark.
A PSG planar algebra is not evaluable (since dim A n (S, C) = ∞ for n > 2) nor necessarily having a positive-definite trace form for each n. It follows that additional structure must be imposed on A n (S, C) for n > 2 to obtain a singly generated planar algebra.
where P ′ tr 1 is defined as in (2.17). With that, the positive-definiteness of A 2 (S, C) similarly implies that from which it then follows that δ > 1.
The evaluations in (3.4) and (3.5) involve the Jones-Wenzl idempotent w n (B.6) for n = 1 and n = 2, respectively: w 1 = 1 1 and w 2 = 1 2 − 1 δ e. For general n ∈ N, we have w * n = w n and where U n is the n th Chebyshev polynomial of the second kind. We note that the first equality in (3.7) is a special case of Lemma 3.2 below.
Remark. If (A n (S, C)) n∈N 0 is positive semi-definite for a given value of δ, then one obtains a well-defined subfactor planar algebra by quotienting out the ideal generated by all the vectors a ∈ A n (S, C) for which P ′ trn (a * a) = 0 for all n ∈ N 0 . In the degenerate case δ = 1, for example, the Temperley-Lieb planar subalgebra is trivialised by quotienting out the ideal generated by The conditions C are determined in Section 3.2 below, where we find that the PSG planar algebra is unique, up to specification of parameters, including the loop weight δ. From here onward, we accordingly opt for the abridged notation PS n ≡ A n (S, C), n ∈ N 0 , with S and C as above.

Defining relations
Here, we determine the relations satisfied by the vectors in a distinguished PS 2 -basis of the form {1 2 , e, s}. Taking inspiration from the classification approach in [11], selecting s as conveniently as possible is key in the following. Lemma 3.2. Let (A n,± ) n∈N 0 be a subfactor planar algebra and {p 1 , . . . , p k } a basis for A n,± for some n and shading +/−, and suppose {p 1 , . . . , p k } is a complete set of mutually orthogonal idempotents, with multiplication induced by M n,± . Then, P ′ tr n,± (p i ) > 0 and p * i = p i for all i = 1, . . . , k.
Proof. By construction, for some scalars c ij , where, by (2.14), while the positive-definiteness implies that Now, let PS 2 be endowed with the multiplication induced by the unshaded companion to (2.6). Since dim PS 0 = 1, the idempotent P 0 := 1 δ e satisfies dim(P 0 PS 2 P 0 ) = 1 and is therefore primitive. By assumption, PS 2 is positive-definite, hence semisimple, and because {1 2 , P 0 } ⊂ PS 2 and dim PS 1 = 1, it follows that PS 2 is commutative, see e.g. [5]. The set {P 0 } ⊂ PS 2 can thus be extended to a PS 2 -basis, {P 0 , P 1 , P 2 }, consisting of a complete set of mutually orthogonal (and primitive) idempotents, where we note that We now introduce where p 1 , p 2 ∈ C, and for {1 2 , e, s} to be a basis for PS 2 , it must hold that p 1 ̸ = p 2 . It follows that hence P 2 r 2,1 (s)e = 0, and that P tr 2 (s) = p 1 P tr 2 (P 1 ) + p 2 P tr 2 (P 2 ). (3.17) For convenience, we choose p 1 , p 2 such that P tr 2 (s) = 0, (3.18) noting that (3.14) then ensures that p 1 ̸ = p 2 (as required) and implies that p 1 , p 2 ̸ = 0. Without loss of generality, we may further choose the normalisation of s such that p 1 p 2 = −1, thereby obtaining where α := p 1 + p 2 , noting that α can take any value in C \ {−2i, 2i}. By construction, P r 2,1 (s) = ϵ 1 1 2 + ϵ e e + ϵs (3.20) for some ϵ 1 , ϵ e , ϵ ∈ C, while (3.18) implies that (since δ ̸ = 0) and, by sphericality, that P 3 r 2,1 (s)e = 0. Using P 4 r 2,1 = id and δ 2 ̸ = 1, it follows that Subsequently, recalling that P r 2,2 = P 2 r 2,1 , the relations P r 2,2 (1 2 ) = 1 2 and P r 2,2 (e) = e imply that This requirement may be implemented by setting Under the conjugate linear involution · * , we have wherep denotes the complex conjugate to the scalar p. Using the analysis above implies that we thus have a PSG planar algebra (PS n ) n∈N 0 , where a basis for PS 2 is given diagrammatically by To illustrate the corresponding action of planar tangles, we have We note that the conditions P ℓ r 2,1 (s)e = 0, where P 0 r 2,1 is the identity map on PS 2 , correspond to the following diagrammatic relations in PS 1 : In fact, this property of s was a motivating factor behind (3.18). Moreover, the tracelessness of s allows us to represent the trace-form inner product relative to the ordered basis {1 2 , e, s} as confirming the positive-definiteness of PS 2 for δ > 1, c.f. (3.6).
To summarise, the PSG planar algebra PS (ϵ) (α, δ) is the quotient planar algebra (A n (S, and With the parameters as above, PS (ϵ) (α, δ) is the unique planar algebra satisfying the conditions outlined in the paragraph containing (3.3). It follows that any unshaded singly generated planar algebra can be obtained by possibly specialising the parameters ϵ, α, δ and by taking a quotient in such a way that each PS n , n ∈ {3, 4, . . .}, is spherical, involutive and positive-definite. These observations conclude the proof of Proposition 3.1.

Presentation
We proceed to describe the algebra that arises when endowing the vector space PS n with the multiplication induced by the unshaded companion to (2.6). For each n ∈ N, δ > 1, and each pair (α, ϵ) ∈ C\{−2i, 2i} × n (α, δ) is thus defined as the unital associative algebra ⟨e i , s i | i = 1, . . . , n − 1⟩ subject to the relations with 1 denoting the unit. Following from (3.35), we also have the relations and     with the marked boundary interval linking the two horizontal edges via an invisible arc on the left. Multiplication is then implemented by vertical concatenation, where the diagram representing the product ab is obtained by placing the diagram representing b atop the one representing a.
Remark. The PSG planar algebra 'includes' the PSG algebras but not the other way around. A planar algebra (A n ) n∈N 0 offers a consistent way to define operations that are not accessible to the individual algebras A n themselves, such as (unshaded versions of) the rotations (2.5) and traces (2.9).
We stress that there are no nontrivial relations involving s i s i±1 s i without also involving terms with four or more s-factors. Accordingly, for n = 3, there are infinitely many linearly independent vectors of the form (s 1 s 2 ) k , hence dim PS 3 = ∞, manifesting the non-evaluability of the planar algebra.

Yang-Baxter integrable models
We now review the integrability framework developed in [22], and introduce homogeneous transfer operators, generalised Yang-Baxter equations, and homogeneous Yang-Baxter integrability. We also show that a singly generated planar algebra encoding a homogeneous Yang-Baxter integrable model must (i) admit an unshaded description, and (ii) be YBR.

Transfer operators
Here, we recall the planar-algebraic Yang-Baxter integrability framework developed recently in [22]. Although we will be using terminology usually reserved for shaded planar algebras, including subfactor and YBR planar algebras, we will accordingly and henceforth work with unshaded versions of these planar algebras.
For each n ∈ N, we follow [22] and introduce the transfer tangle  where B n denotes a basis for A n , while k a , r a , k a : Ω → C, with Ω ⊆ C a suitable domain. We refer to u parameterising the operators in (4.2), as the corresponding spectral parameter.
Remark. The set Ω indicates a domain over which R(u), K(u), and K(u) are well-defined. Typically, Ω contains an open set in C, allowing power-series expansions of R(u), K(u), and K(u).
The associated homogeneous transfer operator is defined as where K(u) and K(u), respectively, are inserted into the left-and rightmost disk spaces in (4.3), while a copy of R(u) is inserted into each of the remaining 2n disks, and we note that We refer to R(u) as the corresponding R-operator, and to K(u) and K(u) as the corresponding Koperators. For convenience, here we have adopted a slightly different convention compared to the definition of the transfer tangle in [22]. As in [22], the homogeneous transfer operator (4.3) may be thought of as a double-row transfer operator on the strip, that is, a particular generalisation of a Sklyanin-type transfer matrix [27].
Remark. More general transfer operators may be constructed, for example by including 'inhomogeneities' at the level of the R-operator. Spectral inhomogeneities are thus introduced by varying the spectral parameter of the R-operator depending on its position within the transfer tangle, while algebraic inhomogeneities are introduced by varying the parameterisation in the construction of the R-operator (as an element of A 2 ) depending on its position within the transfer tangle, thereby introducing more than one R-operator. We refer to transfer operators with any of these features as inhomogeneous. However, as we will exclusively consider homogeneous transfer operators, we often omit the qualifier "homogeneous".
Proposition 4.1. If the Rand K-operators are self-adjoint with respect to the involution · * , then so is the transfer operator T n (u) for each n ∈ N.
Proof. Using (2.12), we have where the second equality follows from (4.4) and the self-adjointness of the Rand K-operators.
We denote the regular representation of A n by Corollary 4.2. Let (A n ) n∈N 0 be a subfactor planar algebra, and suppose the Rand K-operators are self-adjoint with respect to the trace form. Then, ρ n (T n (u)) is diagonalisable for all n ∈ N.
Proof. Proposition 4.1 and the self-adjointness of the Rand K-operators imply that T n (u) is self-adjoint. By the spectral theorem, ρ n (T n (u)) is therefore diagonalisable.
With reference to the unshaded versions of the elementary rotation tangles in (2.5), we say that the Rand K-operators (4.2) are crossing symmetric if for some scalar functionsc K , c K ,c R , c R ,c K , c K : Ω → C.

Yang-Baxter integrability
We say that a model described by the transfer operator with Ω ⊆ C a suitable domain. The following result is established in [22] using standard diagrammatic manipulations [28]. Remark. As they appear in (4.11), we label the equations YBE 1 , YBE 2 , and YBE 3 .
We denote the 'auxiliary' operators in (4.9) by and will refer to them as Y -operators, as short for 'YBE operators'. We refer to Y i (u, v) as the spatial inverse of Y i (u, v) if the pair satisfy (4.10), and stress that pairs for different i = 1, 2, 3 are, in general, independent of one another. Since the Y -operators need not be expressible in terms of the R-operator itself, we say that the inversion identities (4.10), the YBEs (4.11), and the boundary YBEs (4.12) are generalised. We refer to YBE 1 under the specialisation Y 1 (u, v) = R(uv) as the standard YBE.
Traditionally, a model is referred to as Yang-Baxter integrable if the Rand K-operators satisfy a set of local relations, including YBEs, that imply (4.8). The corresponding Rand K-operators are then said to provide a Baxterisation [19]. Based on our homogeneous transfer operator (4.3), a prototypical set of such local relations is presented in Proposition 4.3. We accordingly refer to the ensuing integrability as homogeneous Yang-Baxter integrability, and say that the corresponding Rand K-operators provide a homogeneous Baxterisation. We view such a Baxterisation as specious if where a 1 , a 1 ∈ A 1 , a 2 ∈ A 2 and k, r, k : Ω → C, because, in that case, we have T n (u) = k(u)k(u)r 2n (u)a, a = P Tn a 1 , a 2 , . . . , a 2 , a 1 ∈ A n , (4. 16) from which (4.8) trivially follows. In the following, we will disregard specious Baxterisations.
Remark. As indicated in the Remark following (4.4), a model may be described by a transfer operator with inhomogeneities. Adopting the approach in Proposition 4.3, one would then seek to use a modified set of sufficient conditions to establish the Yang-Baxter integrability of the model. Accordingly, such models are absent in the 'homogeneous specialisation' discussed here.
We say that a planar algebra (A n ) n∈N 0 encodes the homogeneous Yang-Baxter integrability if {K(u), K(u) | u ∈ Ω} and {R(u) | u ∈ Ω} together with the action of planar tangles generate the full vector spaces A 1 and A 2 , respectively, not just subspaces. This notion will play a key role in subsequent sections.
As indicated, our focus will be on singly generated planar algebras, where dim A 1 = |B 1 | = 1. We may accordingly normalise the K-operators in (4.2) so that they equal the unit element, We have opted to keep the exposition above general, but will be assuming that (4.17) holds in the following. In particular, the boundary YBEs (4.12) now reduce to solve YBE 2 and YBE 3 , respectively.
Proof. By Corollary A.2 in Appendix A, P r n,k is invertible for all n, k ∈ N, so applying one of these maps to both sides of an equation yields an equivalent relation. If Y 1 (u, v) = R(uv), then applying P r 3,−1 and P r 3,1 to YBE 2 and YBE 3 , respectively, yields conditions equivalent to Remark. As we will see in sections 5-7, the singly generated planar algebras admit Baxterisations, where the Y -operators are of the form presented in Proposition 4.4. While the present work thus does not exploit fully the generality of the YBEs (4.11), we stress that, in the general setting, the generalised YBEs are inequivalent to the standard YBE.

Unshaded planar algebras
A key observation for us is that the singly generated planar algebras that do not admit an unshaded description cannot encode a homogeneous Yang-Baxter integrable model within the algebraic integrability framework outlined in Section 4.1 and Section 4.2.
To see this, let (A n,± ) n∈N 0 denote a singly generated planar algebra encoding a homogeneous Yang-Baxter integrable model, and consider the following linear maps that reverse the shading on the vectors in A n,+ and A n,− : ι n,+ : A n,+ → A n,− , → ; ι n,− : A n,− → A n,+ , → , (4.20) here illustrated for n = 2. Following [23], there exists an unshaded planar algebra (A n ) n∈N 0 corresponding to (A n,± ) n∈N 0 if and only if the map ι n,∓ • ι n,± acts as the identity on A n,± for all n ∈ N 0 . Now, consider a shading consistent with the transfer operator (4.4), where are shaded R-operators which, by homogeneity, satisfy and the K-operators are shaded units, c.f. (4.17). As the planar algebra encodes the integrability of the model, {R ± (u) | u ∈ Ω} together with the action of planar tangles, generates the vector space A 2,± . Using (4.23), it follows that ι 2,∓ • ι 2,± acts as the identity on A 2,± , so the corresponding singly generated planar algebra (A n,± ) n∈N 0 admits an unshaded description. For our homogeneous Yang-Baxter integrability purposes, it thus suffices to consider unshaded planar algebras only. We stress that a shaded planar algebra not admitting an unshaded description could, in fact, encode the structure of an integrable model; however, the corresponding transfer operator would necessarily be inhomogenous, see the Remark following (4.4) and the comments made in Section 9.

Singly generated planar algebras
With reference to the notion of homogeneous Yang-Baxter integrability outlined above, including the definition (4.3) of homogeneous transfer operators and non-specious Baxterisations, we have the following result.
Proof. Since singly generated planar algebras that do not admit an unshaded description cannot encode a homogeneous Yang-Baxter integrable model within our algebraic integrability framework, Proposition 3.1 allows us to focus on the PSG planar algebra and its quotients. It thus suffices to show that if one does not impose conditions turning the PSG planar algebra into a YBR planar algebra, then there exist no Rand Y -operators such that (i) the YBE is satisfied, and (ii) the Y -operator has a spacial inverse.
Remark. If r s is the zero function, then the attempted Baxterisation is, in fact, encoded by the Temperley-Lieb planar algebra, and not by a singly generated planar algebra, hence not of relevance here.
From (4.24), we get (4.28) First, suppose (s 1 s 2 s 1 − s 2 s 1 s 2 ) is linearly independent of the other algebra elements in (4.27). Then, for the corresponding term to vanish, r s or y s must be zero, with the observation following (4.26) subsequently implying that the function y s must be zero. As r s ̸ = 0 and y s = 0, the vanishing of the (s 2 e 1 − e 2 s 1 )-term in (4.27) implies that r e (u)r s (v) − ϵ r s (u)r 1 (v) y e (u, v) = 0. Since y s = 0, the spacial invertibility of the Y -operator implies that y e ̸ = 0, so amounting to a u, v-independent constant. It follows that the R-operator is of the form in (4.15), so the Baxterisation is specious. Finally, if (s 1 s 2 s 1 − s 2 s 1 s 2 ) can be expressed as a linear combination of other terms in (4.27), then we are in a quotient of the PSG planar algebra that is YBR.
In sections 5-7, we present concrete examples of YBR planar algebras encoding homogeneous Yang-Baxter integrability, thereby establishing the existence of planar algebras satisfying the assumptions in Proposition 4.5. Indeed, we use Liu's classification in Theorem 4.6 below to show that, for every singly generated YBR planar algebra, there exists a corresponding Yang-Baxter integrable model. Theorem 4.6 (Liu [11]). A singly generated YBR planar algebra is isomorphic to a quotient of an FC, BMW, or Liu planar algebra.
The planar algebras listed in Theorem 4.6 are recalled in Section 5, Section 6, and Section 7, respectively. In Section 8, we show that these models are not only Yang-Baxter integrable but also polynomially integrable. The paragraph including Theorem 1.1 and Theorem 1.2 in Section 1 summarises these key findings.
Remark. If the FC, BMW, or Liu planar algebra in Theorem 4.6 is positive semi-definite, then its quotient by the kernel of the trace norm (that is, the quotient by the ideal generated by all nonzero v for which P trn (v * v) = 0 for some n) is isomorphic to the corresponding YBR planar algebra. However, to keep dim A 2 (S, C) = 3 in (3.3), we will only apply this quotient operation for n > 2, c.f. the Remark following (3.7). and (iii) every node is connected to another node with the same colour, using non-intersecting strings defined up to ambient isotopy. Examples of such Fuss-Catalan disks are The vector-space dimensions are given by Fuss-Catalan numbers, as dim F The k-coloured Fuss-Catalan planar algebra FC (k) (γ 1 , . . . , γ k ) is the vector-space collection (F (k) n,± ) n∈N 0 together with the following action of shaded planar tangles [5]: (i) for each string within a shaded planar tangle, draw k − 1 parallel strings in the adjacent unshaded region and assign each a label c k . . . , c 1 starting from the original string (for k > 1, the tangle shading is thus encoded in the string labels and can thereafter be omitted), (ii) if a loop is formed with the colour c l , then it is removed and replaced by the scalar weight γ l , and (iii) the output vector is given by the output disk with the given colour labels and ensuing string connections. To illustrate, we have where c 1 and c 2 correspond to the colours cyan and black, respectively. If γ 1 = · · · = γ m , then the colours of the strings are immaterial and the corresponding planar algebra admits an unshaded description.
As we are concerned with unshaded singly generated planar algebras, we denote by FC(γ) the unshaded planar algebra corresponding to FC (2) (γ, γ), and refer to it simply as Fuss-Catalan (FC). From [12], we know that FC(γ) is spherical and involutive, with the involution · * defined as the conjugate linear map that acts by reflecting every disk about a line perpendicular to its marked boundary interval.
Let F denote the set of all γ such that FC(γ) is positive semi-definite. For each γ ∈ F, the FC subfactor planar algebra FC(γ) is then defined as the quotient of FC(γ) by the kernel of the trace norm. Details of the set F are presented in [12], including 2 cos π m+2 | m ∈ N ⊂ F. For γ in that discrete subset, FC(γ) is positive semi-definite, while for γ > 2, FC(γ) is positive-definite, in which case FC(γ) ∼ = FC(γ).

Presentation
For each n ∈ N, the FC algebra FC n (γ) is now defined by endowing the (unshaded) vector space F (2) n with the multiplication induced by the unshaded planar tangle M n following from (2.6). We note that the FC algebra is both unital (with unit denoted by 1) and associative, and that it is a * -algebra with involution inherited from the FC planar algebra. Using a diagrammatic representation similar to the one in (3.39), FC n (γ) is generated by the following algebra elements:  where each label below a diagram labels a pair of string endpoints. For γ ̸ = 0, the FC algebra admits [12] a presentation, with relations Following from (5.8), we also have the relations and For γ = 0, the relations (5.8), (5.9), and (5.10) still hold, but the relations in (5.10) do not all follow from the relations in (5.8), and should be imposed separately.
We let F n denote the set of all γ such that the trace form (2.16) is positive semi-definite on F (2) n , noting that F ⊆ F n for all n ∈ N 0 . For each γ ∈ F n , FC n (γ) is then defined as the quotient of FC n (γ) by the kernel of the trace norm.
Remark. By renormalising the P i generators, introducingP i := P i /γ, the relations (5.8)-(5.10) only depend on the loop weight through δ = γ 2 , as we then have

(5.19)
Remark. Although ⟨E 1 , . . . , E n−1 ⟩ and ⟨P 1 , . . . , P n−1 ⟩ are subalgebras of FC n (γ), the P -generators do not form a planar subalgebra of FC(γ). It follows from the Remark after (4.26) that only r P is required to be nonzero when exploring homogeneous integrability encoded by FC n (γ).
It is known [20] that FC n (γ) admits a Baxterisation. In the notation (5.18), r 1 is nonzero, so R may be normalised such that r 1 (u) = 1. UsingP = P/γ and the homogeneous Baxterisation then reads and Using the crossing symmetry the Y -operators can be expressed explicitly in terms of R-operators, and the conditions (4.10) and (4.11) reduce to the following single inversion identity and single (standard ) YBE: Relative to the involution · * on the FC algebra, the R-operator (5.21) is self-adjoint for all u ∈ R \ {δ − 1}.
Remark. We have verified that, up to a normalising factor, the generalised Yang-Baxter framework of Proposition 4.3 does not admit any other non-specious solution of the form (5.18), with r P nonzero, than the one presented above. 6 Birman-Wenzl-Murakami algebra

Planar algebra
Let W n denote the vector space spanned by disks with 2n nodes such that, within each disk, (i) the nodes are connected pairwise by strings, defined up to regular isotopy, (ii) strings may intersect but not self-intersect, (iii) two strings cannot intersect one another more than once, and (iv) strings cannot form loops. To illustrate, we present the following examples and non-examples: , , and , , . (6.1) The dimension of W n is given by dim W n = (2n − 1)!!.
For each pair of scalars τ ̸ = 0 and q / ∈ {−1, 0, 1}, the Birman-Wenzl-Murakami (BMW) planar algebra BMW(τ, q) is the collection of vector spaces (W n ) n∈N 0 together with the natural diagrammatic action of planar tangles, defined up to regular isotopy and subject to the relations Remark. Self-intersecting or loop-forming strings may arise as the result of a planar tangle acting on vectors in (W n ) n∈N 0 , hence the relevance of relations like (6.3).
The planar algebra BMW(τ, q) is spherical and, for |τ | = |q| = 1 or τ, q ∈ R, involutive [8,29]. In these cases, the involution · * is defined as the conjugate linear map that acts by 'reflecting' respectively 'flipping' every disk about a line perpendicular to its marked boundary interval, as indicated by * recalling that q ̸ = ±1. We let B denote the set of all (τ, q) such that BMW(τ, q) is positive semi-definite. For each (τ, q) ∈ B, the BMW subfactor planar algebra BMW(τ, q) is then defined as the quotient of BMW(τ, q) by the kernel of the trace norm. Details of the set B are presented in [8], including the observation that B ̸ = ∅.

Presentation
For each n ∈ N, the BMW algebra BMW n (τ, q) is defined by endowing the vector space W n with the multiplication induced by the unshaded planar tangle M n following from (2.6). We note that the BMW algebra is both unital (with unit denoted by 1) and associative, and that, for |τ | = |q| = 1 or τ, q ∈ R, it is a * -algebra with involution inherited from the BMW planar algebra. As is well-known [30,31,32], the generators of BMW n (τ, q) can be represented diagrammatically as     The algebra BMW n (τ, q) admits [29] a presentation, with relations It follows from these relations that e 2 i = δe i , e i e i±1 e i = e i , g i e j = e j g i , e i e j = e j e i , |i − j| > 1, (6.10) with δ as in (6.5), and that g i e i±1 e i = g −1 i±1 e i , e i e i±1 g i = e i g −1 i±1 , e i g i±1 e i = τ e i . (6.11) We note that it suffices to list one of the two relations g i e i = τ −1 e i and e i g i = τ −1 e i in (6.9). We let B n denote the set of all (τ, q) such that the trace form (2.16) is positive semi-definite on W n , noting that B ⊆ B n for all n ∈ N 0 . For each (τ, q) ∈ B n , BMW n (τ, q) is then defined as the quotient of BMW n (τ, q) by the kernel of the trace norm.

Quotient description
Proposition 6.1. For δ > 1, with δ parameterised as in (6.5), and for each µ ∈ {−1, 1}, we have Proof. The proposed algebra isomorphism uses the same notation, e i , for the Temperley-Lieb generators, and sets 15) or equivalently, With this, one verifies that the relations (6.9) imply the relations (3.35) and the vanishing of (6.14). Likewise, the relations (3.35) together with the vanishing of (6.14) are seen to imply the relations (6.9).
Remark. For ϵ = 1, s ∈ W 2 is invariant under P r 2,1 , as becomes evident when rewriting (6.15) as since P r 2,1 (1 2 ) = e, P r 2,1 (e) = 1 2 , Remark. The algebra BMW n (τ, q) is occasionally referred to as Kauffman's Dubrovnik version [32]. It differs from the one in [14], which is based on and consequently admits a description as a quotient of PS

Baxterisation
Relative to the canonical W 2 -basis {1 2 , e, g}, we introduce the parameterised R-operator as R(u) = r 1 (u)1 2 + r e (u)e + r g (u)g, u = r 1 (u) + r e (u) + r g (u) , (6.20) with r 1 , r e , r g : Ω → C. We note that Remark. Since e i is quadratic in g i , the function r g is required to be nonzero when exploring homogeneous integrability encoded by BMW n (τ, q), see Remark following (4.26).
It is known [21], see also [33], that BMW n (τ, q) admits a Baxterisation. For each ω ∈ {−τ q, τ q −1 }, using the homogeneous Baxterisation then reads and Using the crossing symmetry the Y -operators can be expressed explicitly in terms of R-operators, and the conditions (4.10) and (4.11) reduce to the following single inversion identity and single (standard ) YBE: Relative to the involution · * on the BMW algebra with τ, q ∈ R, the R-operator (6.23) is self-adjoint for all u ∈ R \ {q 2 , ω}.
Remark. We have verified that, up to a normalising factor, the generalised Yang-Baxter framework of Proposition 4.3 does not admit any other non-specious solution of the form (6.20), with r g nonzero, than the one presented above.

Planar algebra
The Liu planar algebra is constructed as a quotient planar algebra much akin to the PSG planar algebra in Section 3. With C (ϵ) (α, δ) as in (3.34), we thus introduce where δ ̸ = 0 and ϵ ∈ {−i, i}. Following [11], the Liu planar algebra L (ϵ) (δ) is then defined as the quotient planar algebra (A n (S, C (ϵ) L (δ))) n∈N 0 , where S is as in (3.33). Remark. In the PSG planar algebra in Section 3, for PS 1 and PS 2 to be positive-definite, we have δ > 1, see (3.6). Here, in our definition of the Liu planar algebra, we relax this condition to δ ̸ = 0.
From here onward, we opt for the abridged notation L n ≡ A n (S, C The Liu planar algebra L (ϵ) (δ) is spherical and involutive [11], with the involution · * defined as the conjugate linear map that acts by reflecting every disk about a line perpendicular to its marked boundary interval, as * recalling that s * = s for α = 0, see comment following (3.25). We let L (ϵ) denote the set of all δ such that L (ϵ) (δ) is positive semi-definite. For each δ ∈ L (ϵ) and ϵ ∈ {−i, i}, the Liu subfactor planar algebra L (ϵ) (δ) is then defined as the quotient of L (ϵ) (δ) by the kernel of the trace norm. Details of the set L (ϵ) are presented in [11], including i qm+q −1

Presentation and quotient description
For each n ∈ N, ϵ ∈ {−i, i} and δ ̸ = 0, the Liu algebra L (ϵ) n (δ) is defined by endowing the vector space L n with the multiplication induced by the unshaded planar tangle M n following from (2.6), subject to the relations (7.1). We note that the Liu algebra is both unital (with unit denoted by 1) and associative, and that it is a * -algebra with involution inherited from the Liu planar algebra. The generators can be represented diagrammatically as in (3.39), and the algebra admits a presentation ⟨e i , s i | i = 1, . . . , n − 1⟩ subject to the relations and and e i s i±1 e i = 0, s i e i+1 s i = s i+1 e i s i+1 . (7.8) We note that it suffices to list one of the two relations e i s i = 0 and s i e i = 0 in (7.4).
Remark. It follows from the presentation above that the minimal vanishing polynomial in s i is s 3 i − s i , so s i is not invertible.
The Liu algebra differs from the FC and BMW algebras in that there is no known basis for L (ϵ) n (δ) in terms of which the relations (7.4)-(7.8) admit a natural diagrammatic representation. In Section 7.4, we consider a basis which includes a braid [11], and where (7.5) can be interpreted as a type-III Reidemeister move. However, in this basis, some of the other relations fail to have a natural diagrammatic interpretation.
By comparing the presentation above with the one of PS
Using this result together with Proposition 4.3 and Proposition 4.4, as well as (7.13) and (7.14), we obtain the homogeneous Baxterisation of the Liu algebra in Proposition 7.3 below. To describe it, we find it useful to introduce the function ϕ : 7.17) and the parameter provides a homogeneous Baxterisation of L (ϵ) (7.20) Remark. As expressed in (7.19), the R-operator appears ill-defined at u = 1. However, following a simple rewriting, it may be evaluated at u = 1, yielding R(1) = 1 2 . The Baxterisation is thus well-defined for all u ∈ C for which ϕ(u) ̸ = δ, that is, for all u ∈ C \ {−∆}.
Unlike the R-operators in the FC and BMW algebras, the R-operator (7.19) is not crossing symmetric, that is, there do not exist scalar functionsc R and c R such that P r 2,1 [R(u)] =c R (u)R(c R (u)). (7.21) This explains why the Y -operators are not expressible in terms of the R-operator itself, as is the situation in the FC and BMW cases, c.f. (5.22)-(5.24) and (6.24)-(6.26), respectively. In the Liu case, the conditions (4.10) and (4.11) reduce to the inversion identities Relative to the involution · * on the Liu algebra with δ ∈ R, the R-operator (7.19) is self-adjoint for all u ∈ C such that |u| = 1 and u ̸ = −∆.
Remark. We have verified that, up to a normalising factor, the generalised Yang-Baxter framework of Proposition 4.3 does not admit any other non-specious solution of the form (7.11), with r s nonzero, than the one presented in Proposition 7.3.

Braid limits
For each µ ∈ {−1, 1}, the R-operator (7.19) yields well-defined L 2 -elements under the specialisation u = 0 and in the limit |u| → ∞, and we collect the ensuing four elements in the set so, for δ ̸ = ±i, b is invertible, with inverse given by Although not obtained as limits of our R-operator, the elements of H also feature in [11], where it is shown that for each b ∈ H, the generators {b 1 , . . . , b n−1 } ⊂ L n satisfy and under a specialisation of δ, this set generates an Iwahori-Hecke algebra. Here, b i ∈ L n denotes the element 'acting' as b on the i th and (i + 1) th nodes and as the identity elsewhere. This justifies referring to (7.24) as braid limits, noting that the particular limits and pre-factors in (7.24) are a consequence of the normalisation choice in (7.19).

Polynomial integrability
Following [22], let A be a finite-dimensional unital associative semisimple algebra, and suppose a model is described by a transfer operator T (u) ∈ A, where u ∈ Ω with Ω ⊆ C a suitable domain. We then say that the model is polynomially integrable (on Ω) if there exists b ∈ A such that in which case T (u) is said to be polynomialisable. Indeed, the integrability of such a model readily follows from (8.1). For the existence of b, we have the following corollary to results in [22], wherein an element of A is said to be diagonalisable if the regular representation of the element is.
Corollary 8.1. Let A be a finite-dimensional unital associative semisimple algebra, and suppose that {C(u) ∈ A | u ∈ Ω} is a one-parameter family of commuting and diagonalisable operators, with Ω ⊆ C a suitable domain. Then, there exists b ∈ A such that C(u) ∈ C[b] for all u ∈ Ω.
Remark. The algebraic structure underlying the polynomial integrability of a model is, in a sense, as simple as possible, owing to its characterisation as a polynomial ring in a single (parameter-independent) variable. We stress that this result applies in the familiar setting where A is a matrix algebra. In this case, polynomial integrability arises if the transfer matrix belongs to a commuting family of diagonalisable matrices -a common situation for such models.
Corollary 8.1 indicates that YBR planar algebras offer prototypical examples of algebras underlying polynomially integrable models, as they (i) have a natural Yang-Baxter integrable structure, (ii) have an in-built inner product (the positive-definite trace form), and (iii) are semisimple. For such models, the polynomial integrability thus follows from the local properties (4.10)-(4.12) of Yang-Baxter integrability, provided the Rand K-operators (4.2) are self-adjoint, c.f. Corollary 4.2. Proposition 8.2 below is a consequence of this.

Singly generated algebras
For each of the singly generated planar algebras FC, BMW, and Liu, we refer to the homogeneous transfer operator built using the R-operator parameterised in (5.21), (6.23), and (7.19), respectively, as the canonical transfer operator on Ω, with Ω ⊆ C a suitable domain (that depends on the underlying algebra). In each case, this transfer operator is the unique (up to renormalisations and reparameterisations) algebra element encoding homogeneous Yang-Baxter integrability.
Using Liu's Theorem 4.6, the comments following Corollary 8.1, and results obtained in the previous three sections, we can now account for the polynomialisability of the transfer operator in Theorem 1.2. With notation as in Section 5, Section 6, and Section 7, the following result thus gives conditions on the various algebra-defining and spectral parameters, ensuring that the respective canonical transfer operator is polynomialisable.
(BMW) : Let n ∈ N and (τ, q) ∈ B n ∩ R 2 , and suppose T n (u) ∈ BMW n (τ, q) is the corresponding canonical transfer operator, with u ∈ R \ {q 2 , ω}. Then, T n (u) is polynomialisable. Remark. As the TL subfactor planar algebra is a planar subalgebra of every subfactor planar algebra, we have δ ∈ T ⊂ R for any subfactor planar algebra, see (2.23). In the Liu case, in particular, it thus holds that L (ϵ) n for all n ∈ N and ϵ ∈ {−i, i}.
We are yet to determine a polynomial integrability generator (i.e. b in (8.1)) for any of the FC, BMW, and Liu models. In [22], we do it for the Temperley-Lieb model (see e.g. [34]), for small n, and find that the transfer operator is polynomialisable outside the domain specified by the corresponding version of Proposition 8.2. We anticipate similar results for the FC, BMW, and Liu models, that is, we expect that the domains of polynomial integrability stated in Proposition 8.2 are not maximal.

Discussion
Based on Liu's classification [11] and the algebraic integrability framework developed in [22], we have shown that the only singly generated (subfactor) planar algebras that can encode homogeneous Yang-Baxter integrability are the YBR planar algebras. In the process, we have provided a Baxterisation of the Liu planar algebra, and characterised the FC, BMW, and Liu algebras as quotients of the proto-singlygenerated algebra developed in [6,7,8,11].
In a forthcoming paper, we show how the present work offers a natural framework for defining and describing a Yang-Baxter integrable model with an underlying Iwahori-Hecke algebraic structure, see e.g. [35] for an exposition of the Iwahori-Hecke algebra. Continuing [33], we also intend to make precise how the (2 × 2)-fused Temperley-Lieb algebra [36,37,38,39] arises as a quotient of the BMW algebra.
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 of the affine category of a given planar algebra [40,41], see also [42,43,44] on the so-called periodic Temperley-Lieb algebra. We hope to return elsewhere with a discussion of how our findings extend to the annular setting; preliminary results can be found in [24].
A natural continuation of the present work would be to examine doubly generated planar algebras, for example as singly generated extensions of the FC, BMW, or Liu algebras. We would also find it interesting to study models with inhomogeneous Yang-Baxter integrability, with or without shading, and to explore the physical properties of the various models, including integrals of motion and the continuum scaling limit. see [5,24] for more details.
Lemma A.1. If A n,± has no null vectors, then P id n,± is the identity operator.
Proof. Let v ∈ A n,± and T be a planar tangle for which P T has domain A n,± . By naturality, we then have P T • D id n,± (v) = P T (P id n,± (v)), (A.3) hence P T (v − P id n,± (v)) = 0, (A.4) so v−P id n,± (v) ∈ ker(P T ). Since A n,± has no null vectors, it follows that P id n,± (v) = v for all v ∈ A n,± .
Together with naturality, Lemma A.1 implies the following result.
Proposition A.3. Let A n,± be endowed with the multiplication induced by M n,± , and suppose A n,± has no null vectors. Then, A n,± is unital, with unit 1 n,± .

B Temperley-Lieb algebra
The Temperley-Lieb planar algebra (T n,± ) n∈N 0 discussed in Section 2.4 admits an unshaded version: the (unshaded) TL planar algebra (T n ) n∈N 0 , where the shading of planar tangles and disks in (T n,± ) n∈N 0 is omitted. For each n ∈ N, the corresponding TL algebra TL n (δ) is defined by endowing the vector space T n with the multiplication induced by the unshaded companion to (2.6). We note that TL n (δ) is both unital (with unit denoted by 1) and associative, and that it is a * -algebra with involution inherited from (T n ) n∈N 0 . As is well-known [30], the generators of TL n (δ) can be expressed diagrammatically as

C Braid-semigroup algebra quotient
For each n ∈ N, let the braid-semigroup algebra BS n be the associative unital algebra ⟨b 1 , . . . , b n−1 ⟩ subject to with unit denoted by 1. Since {b k 1 | k ∈ N} is linearly independent, BS n is infinite-dimensional for n > 1. This may be remedied by quotienting out the ideal generated by an appropriate set of polynomials. In the following, we demonstrate how the BMW algebra BMW n (τ, q) can be obtained as such a quotient. Although this is likely known to experts, we have not been able to find it in the literature and include it for the interested reader.
We note that the BMW algebra can similarly be expressed as a quotient of the braid-group algebra but find the semigroup description above more appealing. In particular, the invertibility of the basic generators in BMW n follows from (C.3), while it is 'built-in' if starting with the braid-group algebra.