The ghost algebra and the dilute ghost algebra

We introduce the ghost algebra, a two-boundary generalisation of the Temperley–Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang–Baxter equations, given existing solutions to the Yang–Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang–Baxter integrable.


Introduction
Lattice loop models are of interest in statistical mechanics, where they provide a tractable approach to modelling physical systems with non-local interactions.A key step towards solving such models is to construct a family of commuting operators called transfer tangles, dependent on one parameter u, often called the spectral parameter.There is a known method for constructing commuting transfer tangles for a two-dimensional square lattice using local operators called face operators satisfying certain relations.Indeed, if the face operators satisfy the Yang-Baxter equation (YBE), the boundary Yang-Baxter equations (BYBEs), and a local inversion relation, then they can be composed into a transfer tangle T (u) such that T (u)T (v) = T (v)T (u) for all u and v; see, for example, [1].The resulting model is then called Yang-Baxter integrable.
Lattice loop models are often encoded algebraically using diagram algebras.These algebras have a set of diagrams for a basis, on which multiplication is defined based on concatenation of diagrams.Perhaps the most famous is the Temperley-Lieb (TL) algebra [2,3].This algebra was first discussed by Temperley and Lieb in [2], where it was found to connect transfer matrices in ice-type models to certain graph colouring problems, and to the enumeration of weighted trees.It has since been applied to a range of physical models-from polymers and percolation [4] to quantum spin chains [5]-and has also been used in pure mathematics for knot theory [6,7].
In our loop model context, the TL algebra TL n (β) describes a fully-packed square lattice of non-crossing strings with closed boundary conditions.The one-and two-boundary TL algebras TL 1  n (β; α 1 , α 2 ) and TL 2 n (β; α 1 , α 2 , δ 1 , δ 2 ) generalise this to allow open boundary conditions on one or both sides of the lattice, respectively.As diagram algebras, each of these has a basis of rectangular string diagrams, considered up to connectivity, or more precisely, frame-preserving ambient isotopy.Some examples are given in Figure 1.Each diagram has n nodes on each of two opposing sides of the rectangle, between which non-crossing strings are drawn.In the one-and two-boundary algebras, one or both of the remaining sides become dotted boundary lines, to which strings may be connected at distinct points.Each node must have exactly one string endpoint attached to it, and each string must connect two distinct nodes, a node to a boundary, or two distinct boundaries.In the two-boundary TL algebra, however, the number of strings connected to each boundary must be even.This follows from the other rules in the one-boundary case, but must be separately imposed for the two-boundary case, as seen in Figure 1d.We will soon see that this evenness condition is essential for TL 2  n to be associative.The primary purpose of the ghost algebra constructed in this paper is to provide a twoboundary generalisation of the TL and one-boundary TL algebras that allows basis diagrams with odd numbers of strings connected to either boundary.In particular, we want this algebra to be associative, like TL n and TL 1  n , and to have subalgebras isomorphic to TL 1 n associated with each boundary, whose intersection is isomorphic to TL n .The specifics of multiplication in TL 1 n cause this to be nontrivial, however.The one-boundary TL algebra was introduced as the blob algebra b 1 n (β; α) by Martin and Saleur in [8].We use the slightly generalised three-parameter definition and diagrammatic presentation from Pearce, Rasmussen and Tipunin in [9], denoted TL 1 n (β; α 1 , α 2 ).The isomorphism TL 1  n (β; α 1 , α 2 ) ∼ = b 1 n (β; α 1 α 2 ) for α 2 = 0 is discussed in [9, App.A].The three parameters in TL 1  n (β; α 1 , α 2 ) arise during multiplication.First, the diagrams are concatenated.This may produce strings with both ends on the boundary, called boundary arcs, and loops.Each string endpoint on the boundary is assigned a parity by numbering them from left to right, starting from 1. Any boundary arcs are then removed and each replaced by a factor of a boundary parameter α 1 or α 2 , if the left endpoint is odd or even, respectively.Each loop is removed and replaced by a factor of the loop parameter β.For example, in TL 1  7 (β; α 1 , α 2 ), . (1.1) The two-boundary TL algebra has a similar diagrammatic multiplication.This algebra was introduced by Mitra, Nienhuis, de Gier and Batchelor in [10], with one parameter associated with each boundary, similar to the blob algebra.We use a slightly generalised definition with two parameters for each boundary, denoted TL 2 n (β; α 1 , α 2 , δ 1 , δ 2 ), for consistency with TL 1 n (β; α 1 , α 2 ).The parameters δ 1 and δ 2 are assigned to bottom boundary arcs with odd and even left endpoint, respectively.
Suppose we allow two-boundary diagrams with an odd number of strings attached to each boundary, and use the diagram multiplication rules from TL 2 n (β; α 1 , α 2 , δ 1 , δ 2 ).Then with n = 2, we have .
(1.3) Associativity requires that these products are equal.They are scalar multiples of the same basis diagram, so their coefficients α 1 and α 2 must be equal.However, much of the structural richness of the TL 1 n (β; α 1 , α 2 ) subalgebra would be lost by considering only the cases where α 1 = α 2 , or, equivalently for α 2 = 0, b 1 n (β; 1).For example, as seen in [8,11], the representation theory of b 1 n (β; α) differs when α is specialised to different complex numbers.Hence we wish to allow α 1 = α 2 .
As seen above, naively including two-boundary diagrams with an odd number of strings connected to either boundary does not yield the desired associative algebra with potentially distinct α 1 and α 2 .Indeed, left-multiplication by any diagram with an odd number of strings on a boundary changes the parity of all the strings to its right on that boundary, because we count string parity starting from the left.Starting from the right gives the same problem with right-multiplication. Hence we seek an alternative method of keeping track of the parity of boundary connections that is robust to multiplication by such diagrams.
This motivates the introduction of ghosts: dots on the boundaries that allow us to keep track of the parity of strings.A ghost is left at each endpoint of each boundary arc removed during multiplication.When determining the parity of string endpoints on each boundary, we number the ghosts as well as the string endpoints.We also require the sum of the number of ghosts and the number of string endpoints on each boundary to be even.This allows us to have diagrams with odd numbers of strings connected to each boundary, and parity-dependent boundary parameters, while preserving associativity.
Similar problems with associativity occur when trying to produce generalisations of the dilute TL algebra with boundary, and these can also be resolved by introducing ghosts.The dilute TL algebra dTL n (β) describes dilute lattice configurations, where the strings need not be fully-packed.The basis diagrams for dTL n follow the same rules as the basis diagrams for TL n , except there may be empty nodes, with no attached strings, as seen in Figure 1e.Naturally, we would like one-and two-boundary generalisations of dTL n (β) to describe dilute systems with open boundary conditions on one or both boundaries, but existing versions lack some of our desired properties.In [12], Dubail, Jacobsen and Saleur use a dilute version of the blob algebra; after translating their diagrams from the blob picture to the boundary picture as in [9, App.A], this does effectively have distinct boundary parameters, but all diagrams have an even number of strings connected to the boundary.De Gier, Lee and Rasmussen study a dilute oneboundary lattice model in [13] that allows diagrams with an odd number of strings connected to the boundary, with a subalgebra isomorphic to TL 1 n (β; α 1 , α 2 ), but they found that nontrivial solutions to the BYBE exist only if all boundary parameters are equal.Alluqmani constructs a dilute blob algebra in [14] that also allows diagrams with an odd number of strings connected to the boundary, after converting to the boundary picture, but has only one boundary parameter.This algebra is actually based on the Motzkin algebra M n (β, η) (see e.g.[15]), but there is an isomorphism M n (β, η) ∼ = dTL n (β − 1) as long as η = 0, to be discussed in future work.
The dilute ghost algebra dGh 2 n constructed in this paper is an associative two-boundary generalisation of dTL n that allows diagrams with odd numbers of strings connected to the boundaries, has parity-dependent parameters at each boundary, and admits solutions to the BYBE for generic values of the parameters.It also has a one-boundary subalgebra dGh 1  n with these desired properties.
The layout of this paper is as follows.In Section 2, we define the ghost algebra.We first define the diagram basis in Section 2.1, and explain the multiplication in Section 2.2.In Section 3, we recall the definition of the dilute TL algebra, and then define the dilute ghost algebra diagrammatically.Section 4 discusses key subalgebras of the ghost algebra and the dilute ghost algebra, and how they fit together.In Section 5, we present lattice loop models associated with the ghost algebra and the dilute ghost algebra.We construct transfer tangles from bulk and boundary face operators satisfying the YBE, crossing symmetry, the local inversion relation, and the BYBEs.We use existing bulk face operators from the TL and dilute TL algebras, and give a classification of the new boundary face operators with ghosts that satisfy the BYBEs for generic parameter values.For the dilute case, this involves solving 110 functional equations for ten unknown functions, and takes up approximately a third of the paper.
The dimensions of the algebras introduced in this paper are given in Appendix A, and associativity is proven in Appendix B. In Appendix C, we show that the ghost algebra and its dilute and one-boundary counterparts are cellular, as defined by Graham and Lehrer in [16].We note that the TL and one-boundary TL algebras are cellular, and that this is closely tied to their diagrammatic presentations, using an anti-involution defined by reflection of basis diagrams.Accordingly, the ghost algebra and the dilute ghost algebra are constructed to be cellular with respect to this reflection anti-involution.However, relaxing this requirement leads to some natural generalisations of these algebras that are still associative and have paritydependent boundary parameters; the generalised algebras are discussed in Appendix D.

Terminology and notation
Let A be an algebra.We say A is unital if it has a multiplicative identity.Let B be a vector subspace of A that is closed under multiplication.If B contains the multiplicative identity of A, then it is a unital subalgebra of A; if not, it is a non-unital subalgebra of A. Note that a nonunital subalgebra B of A may be a unital algebra in its own right; in this case, the multiplicative identity of B does not coincide with that of A, but is instead some other idempotent.
Let A 1 and A 2 be unital algebras with identities I 1 and I 2 respectively, and let ψ : A 1 → A 2 be a linear map that preserves multiplication.That is, ψ(xy) = ψ(x)ψ(y) for all x, y ∈ A 1 .If ψ(I 1 ) = I 2 , then ψ is a unital homomorphism; otherwise, it is a non-unital homomorphism.
All subalgebras and homomorphisms in this paper are unital unless otherwise specified.All algebras are over C, and all parameters are indeterminates, though they may be specialised to elements of C.
Each diagram algebra in this paper has a superscript indicating how many boundaries it has, or no superscript, if it has no boundary.

The ghost algebra
In this section, we define the ghost algebra using a diagrammatic presentation, with a basis given in Section 2.1 and multiplication given in Section 2.2.

Diagram basis
We define the ghost algebra Gh 2 n in terms of a diagram basis.A Gh 2 n -diagram is drawn on a rectangle where the top an bottom sides are dotted boundary lines, and n nodes are placed on each of the remaining sides.Non-crossing strings are drawn within the rectangle such that each node has exactly one string attached to it, and each string connects a node to another node, or to a boundary.Hence there are no loops, and no strings with each end connected to a boundary.There is no restriction on the number of strings attached to either boundary.
A finite number of filled black circles called ghosts may be drawn along the boundaries, except at the ends of any strings.At each boundary, we require that the sum of the number of ghosts and the number of strings attached to that boundary is even; this weaker evenness condition will ensure associativity of the algebra.
For example, some Gh A domain is the part of a boundary between a pair of adjacent strings, between a corner and its nearest string, or, if there are no strings connected to that boundary, the whole boundary.For example, the first Gh 2 6 -diagram above has four domains at the top boundary, and two at the bottom.
Two Gh 2 n -diagrams are considered equal if the same nodes are connected to each other or to the boundary, and in each domain, the number of ghosts is equivalent modulo 2. For example, the first and fourth diagrams in (2.1) are equal, while the first and sixth are not.It follows that each Gh 2 n -diagram can be drawn with at most one ghost in each of its domains.Note that strings connecting the top boundary to the bottom boundary are disallowed in Gh 2 n , while there may be arbitrarily many of these in TL 2 n -diagrams.This means Gh 2 n is finite-dimensional, like TL n and TL 1 n , while TL 2 n is infinite-dimensional.One can consider a quotient of TL 2 n obtained by removing top-to-bottom boundary arcs at the cost of an additional parameter, but to keep an even number of strings connected to each boundary, these must be removed in pairs.This is often called the symplectic blob algebra; see, for example, [17].
Formulas for the dimensions of the ghost algebra and all other algebras constructed in this paper are found in Appendix A, as well as a table of their dimensions for small n.

Multiplication
Let Gh 2 n be the complex vector space with the set of all Gh 2 n -diagrams as its basis.To turn this into an algebra, we define multiplication on Gh 2 n -diagrams, and extend this bilinearly to the whole space.This algebra is dependent on nine parameters, β, α 1 , α 2 , α 3 , γ 12 , γ 3 , δ 1 , δ 2 and δ 3 .We take these to be indeterminates, but they may be specialised to fixed values in C.
To multiply two Gh 2 n -diagrams, we first concatenate them.Each loop that appears is removed and replaced by a factor of β.The ghosts and strings are then numbered along each boundary, starting from 1 on the left.Any string that is attached to a boundary at each end is called a boundary arc.Each boundary arc is removed, leaving a ghost at each of its endpoints, and replaced by a parameter according to which boundary each end was attached to, and the parity of the connection points.This is summarised in Table 1 With this multiplication, Gh 2 n is an associative unital algebra.Associativity is proven in Appendix B. The identity is the diagram with n horizontal strings linking the nodes on either side of the rectangle; this is the same as for TL n , TL 1  n and TL 2 n .We note that Gh 2 n has subalgebras isomorphic to TL n (β), TL 1 n (β; α 1 , α 2 ) and TL 1 n (β; δ 1 , δ 2 ); these are spanned by the diagrams with no boundary connections, no ghosts and no bottom boundary connections, and no ghosts and no top boundary connections, respectively.There is also a subalgebra spanned by the diagrams that have no bottom boundary connections, but may have ghosts; we call this the one-boundary ghost algebra Gh 1 n (β; α 1 , α 2 , α 3 ).The diagrams with no top boundary connections span a subalgebra isomorphic to Gh 1 n (β; δ 1 , δ 2 , δ 3 ).We also note that some boundary arcs of different parities are assigned the same parameters; see the arcs listed for the parameters α 3 , δ 3 , γ 12 and γ 3 in Table 1.This is done for another point of consistency with the TL n and TL 1 n subalgebras.The algebras TL n and TL 1 n are cellular algebras, in the sense of Graham and Lehrer [16].The definition of cellularity requires an antiinvolution on the algebra, and for TL n and TL 1 n , the anti-involution is given by reflecting basis diagrams about a vertical line.Assigning the parameters α 3 , δ 3 , γ 12 and γ 3 to certain pairs of boundary arcs of different parities means the ghost algebra is also cellular with respect to the reflection anti-involution; this is proven in Appendix C.
If these boundary arcs were assigned distinct parameters, the resulting algebra is still associative and unital, and has the desired TL n and TL 1 n subalgebras.We call this the generalised ghost algebra Gh 2 n , and it is discussed in Appendix D, alongside its dilute counterpart, the generalised dilute ghost algebra d Gh 2 n .

The dilute ghost algebra
Recall from Section 1 that each basis diagram in the two-boundary TL algebra TL 2 n is required to have an even number of strings connected to each boundary.In fact, each basis diagram of the one-boundary TL algebra TL 1  n also has an even number of strings attached to the boundary.Indeed, each string has two ends, and each of the 2n nodes must have exactly one end of a string attached to it, so there are an even number of string endpoints remaining, and these are all attached to the boundary.This is why the TL 2 n -diagrams with no strings connected to the bottom boundary are precisely the basis diagrams of TL 1  n , and thus TL 2 n (β; α 1 , α 2 , δ 1 , δ 2 ) has a subalgebra isomorphic to TL 1 n (β; α 1 , α 2 ) spanned by these diagrams.However, this relies on the assumption that every node has exactly one string endpoint attached to it.Weakening this requirement to allow empty nodes, with no attached strings, we enter the realm of dilute algebras.

The dilute Temperley-Lieb algebra
The dilute Temperley-Lieb algebra dTL n (β) is another diagram algebra, with a basis of dTL ndiagrams drawn on the familiar rectangle of 2n nodes.Non-crossing strings are drawn within the rectangle, and each must connect a node to a different node.Each node must have at most one attached string, but may have none; empty nodes are drawn as unfilled circles.Strings must not connect nodes to the boundaries, however, so we remove the boundary lines to indicate this.Some example dTL 5 -diagrams are , , , .
Two dTL n -diagrams are equal if the strings connect the same nodes in each.
Multiplication is similarly defined on pairs of diagrams by concatenation, and extended bilinearly to the whole space.Any loops produced are removed and replaced by a factor of β, as for TL n , but now if a string meets an empty node during concatenation, the product of those diagrams is set to zero.For example, in dTL 6 , = 0, = β .
This algebra is associative and unital.To state the identity, let diagrams with k dashed strings represent the sum of the 2 k dTL n -diagrams obtained by drawing or not drawing each dashed string.For example, in dTL 5 , The identity in dTL n is then Observe that the set of all TL n -diagrams is a subset of the set of all dTL n -diagrams, and since the TL n -diagrams have no empty nodes, their products are the same in dTL n as in TL n .The span of the TL n -diagrams is closed under multiplication, but it does not contain the identity of dTL n , since this is a linear combination of dTL n -diagrams, including some with empty nodes.Hence the TL n -diagrams in dTL n (β) span a non-unital subalgebra isomorphic to TL n (β).
There is a unital subalgebra of dTL n (β) isomorphic to TL n (β+1), however.This is generated by the dTL n identity, and diagrams of the form . . .

Diagram basis and multiplication
The dilute ghost algebra dGh 2 n is defined in terms of a basis of dGh 2 n -diagrams.A dGh 2 n -diagram consists of non-crossing strings drawn within a two-boundary rectangle with 2n nodes.Each string must connect a node to another node, or to a boundary, and each node must have at most one string attached to it.Ghosts may be drawn in the domains of the boundaries, and we require that the number of ghosts plus the number of strings on each boundary is even.
Two dGh 2 n -diagrams are equal if the strings connect the same nodes to each other, or to the same boundary, and the numbers of ghosts in corresponding domains are equivalent modulo 2.
These dGh 2 n -diagrams are multiplied by concatenation, where each loop is removed and replaced by a factor of β, and each boundary arc is removed and replaced by a factor of the corresponding parameter from Table 1, leaving a ghost at each endpoint.The parity of boundary arc endpoints is again determined by counting the strings and ghosts along each boundary, starting from 1 on the left.If, during concatenation, a string meets an empty node, the product is set to zero.If not, we remove the extra vertical line in the middle to produce a scalar multiple of a dGh 2 n -diagram, tightening the strings and removing extra pairs of ghosts from each domain, if desired.
For example, in dGh 2  5 , we have = 0, ( and This algebra is associative, as proven in Appendix B, and unital, with the same identity (3.4) as dTL n , except with the two dotted boundaries drawn in.
The dGh 2 n -diagrams without any strings attached to the bottom boundary span a subalgebra that we call the one-boundary dilute ghost algebra dGh 1 n (β; α 1 , α 2 , α 3 ).The dGh 2 ndiagrams without any strings connected to the top boundary span a subalgebra isomorphic to dGh 1 n (β; δ 1 , δ 2 , δ 3 ), and the intersection of these two subalgebras is isomorphic to dTL n (β).

Relationships between new and existing algebras
In this section, we describe some notable unital and non-unital subalgebras of the ghost and dilute ghost algebras, culminating in the commutative diagrams in Figures 2a and 2b.
The two-boundary TL algebra TL n -diagram obtained by replacing any top-to-bottom boundary arcs with factors of γ 12 or γ 3 and leaving ghosts at their endpoints, as is done in Gh 2 n multiplication.This is not injective, since TL 2 n is infinite-dimensional, while Gh 2 n is finite-dimensional.From Section 3, dGh 2  n has one-boundary subalgebras isomorphic to dGh 1 n (β; α 1 , α 2 , α 3 ) and dGh 1 n (β; δ 1 , δ 2 , δ 3 ), spanned by the dGh 2 n -diagrams with no strings attached to the bottom and top boundaries, respectively.Within these subalgebras, the diagrams without any boundary TL n (β)  connections span a subalgebra isomorphic to dTL n (β), and the diagrams without empty nodes span subalgebras isomorphic to Gh 1 n (β; α 1 , α 2 ) and Gh 1 n (β; δ 1 , δ 2 ), respectively.Recall that dTL n has a non-unital subalgebra isomorphic to TL n (β), spanned by the diagrams without empty nodes.The dilute ghost algebra similarly has a non-unital subalgebra isomorphic to Gh 2 n , spanned by the dGh 2 n -diagrams without empty nodes.These relationships are summarised in a commutative diagram in Figure 2a.Since unital subalgebras are often more useful than non-unital subalgebras, it is fortunate that we also have similar unital subalgebras of these dilute algebras.Recall that a dashed string in a dilute diagram is the sum of the diagrams with a solid string and no string in that position.When a dashed string is connected to a dashed string in multiplication, the result is a dashed string.A dashed loop is then the sum of a loop and no loop, so the parameter associated with a dashed loop is effectively the loop parameter, plus one.Similarly, the parameter associated with a dashed boundary arc is effectively the parameter associated with the corresponding solid boundary arc, plus one.It follows that we have an injective unital homomorphism from Gh , obtained by turning the solid strings in each Gh 1 n -diagram into dashed strings.There are analogous injective unital homomorphisms between the ghost and dilute ghost algebras, and between TL n (β) and dTL n (β − 1).This is summarised in Figure 2b.
We note that the zero-and one-boundary algebras TL n , dTL n , TL 1 n , Gh 1 n and dGh 1 n also have injective unital homomorphisms into the corresponding algebras with the same parameters and subscript n + 1.This is given by adding a string linking the (n + 1)th nodes on each side of the diagram, or a dashed string in the dilute cases.This process does not work for the twoboundary algebras, since the added string would intersect any strings connected to the bottom boundary.

Loop models
In Sections 5.1 and 5.2, we present lattice loop models associated with the ghost algebra and the dilute ghost algebra, respectively.For each algebra, the goal is to produce a one-parameter family of commuting transfer tangles T (u).These are built from the bulk face operators discussed in Sections 5.1.1 and 5.2.1, and the boundary face operators discussed in Sections 5.1.2and 5.2.2.We follow a standard construction of the transfer tangles such that, if the face operators satisfy the Yang-Baxter equation (YBE), the local inversion relation and the boundary Yang-Baxter equations (BYBEs), then the resulting transfer tangles commute, as proven in [1, §2.4].Since the bulk of these lattices behaves identically to the TL or dilute TL lattices, we use known bulk face operators from each that satisfy the YBE and local inversion relation.Any new boundary behaviour due to the ghosts is captured by the boundary face operators; much of Sections 5.1.2and 5.2.2 is spent classifying boundary face operators that satisfy the BYBEs.The final transfer tangles are given in Sections 5.1.3and 5.2.3.

Ghost algebra
The ghost algebra can be used to describe a fully-packed lattice loop model with two boundaries.An example of such a lattice is given in Figure 3.These lattices are constructed from two possible bulk squares Figure 3: Fully-packed lattice that can be described using the ghost algebra Gh 2 3 .
and five possible boundary triangles for each boundary,

Bulk face operator
The bulk squares of the ghost lattice model are the same as those in the Temperley-Lieb lattice model, so we can use the well-known Temperley-Lieb bulk face operator, where β = 2 cos(λ).This satisfies the Yang-Baxter equation (YBE) We also have crossing symmetry and the local inversion relation (5.6)

Boundary face operators
Unlike the YBE and the local inversion relation, the BYBEs involve the boundary face operators.The boundary face operators in the TL lattice models use only the first triangle for each boundary from (5.2), while the one-and two-boundary TL lattice models also use the second triangle for the top or both boundaries, respectively.Thus, for our lattice model to have any new boundary behaviour due to the ghosts, we cannot simply use the existing BYBE solutions from the zero-, one-and two-boundary TL lattice models.Hence in this section, we classify solutions to the BYBEs involving our new boundary triangles with ghosts.
The BYBE for the top boundary of the ghost lattice model is and the BYBE for the bottom boundary is Given the bulk face operator (5.3), we seek solutions for the boundary face operators satisfying (5.7) and (5.8).Since the bottom boundary equation is just a reflection of the top boundary equation about a horizontal line, it suffices to solve the top boundary equation, and reflect the solution for the bottom boundary, as long as we also swap the parameters α i ↔ δ i .Note also that the BYBEs are homogeneous, so our solutions for the boundary face operator with parameter u will have an overall scaling factor, that may depend on u.Note that the top boundary BYBE involves no bottom boundary triangles, and has two places on each side that strings could be connected to.This means the top boundary BYBE is contained within the one-boundary ghost algebra on two nodes, Gh 1  2 .Hence we will solve this BYBE within Gh 1  2 , so our diagrams will have no bottom boundary.Since the top boundary BYBE is exactly the same for the one-boundary ghost algebra Gh 1  n , and the bottom triangle with no boundary connections solves the bottom boundary BYBE, this means the one-boundary ghost lattice model is also Yang-Baxter integrable, with the transfer tangle given in Section 5.1.3,using that bottom triangle alone as the bottom boundary face operator. Let (5.9) and let (5.10) Theorem 5.1.Let the face operator for Gh 2 n be given by (5.3).Then the BYBE (5.7) admits two solutions for the boundary face operator (5.9).Solution I is ) ) ) Proof.We consider the expression as a linear combination of the basis diagrams, and note that the BYBE (5.7) holds if and only if the coefficient of each of these diagrams is zero.Expanding (5.20), there are twenty-four diagrams whose coefficients are not trivially zero.We first consider the diagrams with four strings connected to the top boundary.Their coefficients are of the form with i, j ∈ {1, 2, 3, 4}.For these to be zero for all u, v, the expression in the larger brackets must be zero.If b i = 0 for some i ∈ {1, 2, 3, 4}, then for each j, we have Next, we consider the diagrams , , , . (5.24) Setting the coefficient of the first diagram equal to zero gives If c 1 = 0, this can be rearranged to find and therefore for some κ ∈ C. Then (5.28) Substituting this into the coefficients of the remaining three diagrams from (5.24), and setting them equal to zero, we find (5.31) Hence we must have or for some ρ ∈ C.
If c 1 = 0, but c i = 0 for some i ∈ {2, 3, 4}, then setting the coefficients of the diagrams (5.24) to zero analogously implies that (5.32) or (5.33) must hold.If c i = 0 for all i, both are true.Applying either of (5.32) and (5.33) is sufficient for the coefficients of all diagrams to be zero, and after some rescaling, this yields the two solutions from the theorem.

Transfer tangle
The transfer tangle T (u) is constructed from the bulk and boundary face operators as

Dilute ghost algebra
The dilute ghost algebra can be used to describe a dilute lattice loop model with two boundaries.
An example of such a lattice is given in Figure 4.The dilute lattice is constructed from nine possible bulk squares and ten possible boundary triangles for each boundary, , Figure 4: Dilute lattice that can be described using the dilute ghost algebra dGh 2 3 .

Bulk face operators
The bulk squares of the dilute ghost lattice model are the same as those in the dilute Temperley-Lieb lattice model, so we can use an existing face operator from that model.We use a face operator from Batchelor and Yung [18], given by where w 1 (u) = sin(2φ) sin(3φ) + sin(u) sin(3φ − u), (5.38) w 3 (u) = sin(2φ) sin(u), (5.40) and β = −2 cos(4φ).These bulk face operators satisfy the YBE where we note that this has dashed strings instead of the solid strings used for the ghost algebra lattice model.We also have crossing symmetry (5.46)

Boundary face operators
In the dilute ghost lattice loop model, the BYBE for the top boundary is and the BYBE for the bottom boundary is (5.48) As in the fully-packed ghost model, we seek solutions to (5.47) for the top boundary face operator.By reflecting these solutions about a horizontal line and swapping parameters α i ↔ δ i , we can find the solutions to (5.48) for the bottom boundary face operator.The BYBEs are again homogeneous, so our solutions for the boundary face operator with parameter u will have overall scaling factors that are functions of u.
Similarly, the top boundary BYBE is also (5.47) in the one-boundary dilute ghost lattice model.The top boundary BYBE is actually contained in dGh 1  2 , so we will solve it within this algebra, using diagrams without a bottom boundary.Two solutions for the dilute bottom boundary BYBE without boundary connections are given in [18], so these can be used with the top boundary face operators found in Theorem 5.2 to construct commuting transfer tangles for the one-boundary dilute ghost lattice model, and thus this model is also Yang-Baxter integrable.The construction is as in Section 5.2.3, with the appropriate bottom boundary face operators. Let and, as we did for Gh 2 n , let (5.50) Theorem 5.2.Let the face operator for dGh 2 n be given by (5.37).Then the BYBE (5.47) admits five solutions for the boundary face operator (5.49).Solution I is  (5.71)  (5.86) Proof.Similar to the proof of Theorem 5.1, we consider the expression as a linear combination of the basis diagrams, and note that the BYBE (5.47) holds if and only if the coefficient of each basis diagram is zero.The exhaustive search for solutions to the BYBE is more complicated in the dilute ghost algebra than the ghost algebra, so we have separated it into cases.This is summarised in Figure 5. Expanding (5.87), there are 110 diagrams whose coefficients are not trivially zero.We begin by considering the diagrams with four strings connected to the boundary.These have coefficients of the form Figure 5: Flowchart showing the cases considered when solving the BYBE for the dilute ghost algebra.This is a binary tree, with cases numbered by the sequence of left (1) and right (2) steps to reach them from the top.For example, the case " Each vertex of the tree is also a hyperlink to the corresponding part of the proof.Note that P (σ, τ ; θ) = 2στ cos(θ)+ σ 2 − τ 2 sin(θ), and Q(σ, τ ) is a quartic polynomial in σ and τ , defined in (5.164).
we find respectively.For both of these to be zero, we need c 5 = 0. Hence c 2 = 0 implies c 5 = 0. Similarly, using the coefficients of (5.102) Both sides must equal some constant z ∈ C, so (5.103) Noting that c 5 , c 6 and z only ever appear as c 5 z and c 6 z, and that z is nonzero because g is nonzero, we may absorb z into c 5 and c 6 , equivalent to setting z = 1 above.Applying this, setting the coefficients of we find (5.111) for i = 5 and i = 6, respectively, and from the coefficients of we have for i = 5 and i = 6, respectively.We can rearrange each of these equations to find expressions for and then equate those expressions.Since c 5 and c 6 are not both zero, choose i ∈ {5, 6} such that c i is nonzero, and recall that b and η are nonzero.We can thus rearrange the resulting equation to find (5.115) The left and right sides are functions of u and v, respectively, so we may set both equal to some ρ ∈ C. Hence we have (5.116) Substituting this back into (5.113) and rearranging gives (5.117) We can then set both sides equal to some κ ∈ C, and find that Applying this, the coefficients of for i = 5 and i = 6, respectively.Since c 5 and c 6 are not both zero, setting this to zero gives ρ = 1 η .Applying this, the coefficients of all diagrams in the expansion of (5.87) become zero.Rescaling by a factor of η sin u − φ 2 , we observe that c 5 , c 6 and η only appear in this solution as c 5 η and c 6 η, so we will set ν = c 5 η and µ = c 6 η.This gives Solution I, for b nonzero, and µ and ν not both zero.to zero, we find (5.127)This is equivalent to with f 1 (x) = (a 1 (x) + a 2 (x)) sin(x) and f 2 (x) = (a 1 (x) − a 2 (x)) cos(x).It follows that f 1 and f 2 are both scalar multiples of some nonzero function h.Setting we find (5.132) Case 2.1: c 1 , c 2 , c 3 , c 4 not all zero ↑ Setting the coefficient of (5.133) equal to zero, we find (5.134) where we recall P (σ, τ ; θ) = 2στ cos(θ) + σ 2 − τ 2 sin(θ).If P (σ, τ ; −3φ) = 0, this can be rearranged to find h in terms of b.
Case 2.1.1:P (σ, τ ; −3φ) = 0 ↑ Rearranging (5.134) for h(u) yields Substituting this into c 1 c 3 = c 2 c 4 , we find If the coefficient of c 3 here is nonzero, this can be rearranged to find c 3 in terms of c 1 and c 2 .
Case 2.1.2: ( Since the left-and right-hand sides are functions of u and v, respectively, it follows that both sides are equal to some constant κ ∈ C. Thus (5.179) Substituting this into the coefficients of and setting them equal to zero yields (5.183) Hence or for some ρ ∈ C. Note that ρ = 0 because c 1 = 0.In the first case, we find (5.186) with c 1 c 3 = c 2 c 4 , and in the second, (5.187) For the cases c 2 = 0, c 3 = 0 and c 4 = 0, we instead use the coefficients of the three diagrams in (5.180), respectively, to find expressions similar to (5.179).Substituting these into the other three diagrams (including (5.177)), we similarly find that (5.184) or (5.185) must hold, and get the same expressions (5.186) and (5.187) for τ h(u), respectively.
If (5.185) holds, the coefficient of (5.188) becomes (5.189) Since ρ = 0, and cos v + 3φ become for i = 1, 2, 3, 4, respectively.Since c i is nonzero for some i ∈ {1, 2, 3, 4}, setting these coefficients to zero implies the last set of brackets is zero.Since the constant function 1 and tan u − 5φ 2 cot v + 3φ 2 + 1 are linearly independent, the coefficients of both of these functions must be zero.Hence we have  For this to be zero for all u, v, we must have P (σ, τ ; −3φ) = 0, and therefore σ = −τ tan 3φ 2 or σ = τ cot 3φ 2 .After some rescaling, these yield the solutions in Cases 2.2.1 and 2.2.2, respectively.These coincide with the dTL n BYBE solutions found by Batchelor and Yung in [18].Up to scaling, this is Solution II with µ = ν = 0. Note that allowing h = 0 would give the zero solution, and that Solution II becomes the zero solution when b = 0, so Solution II has now been fully accounted for.

Transfer tangle
As we did for the fully-packed model, we construct the transfer tangle

Discussion
In this paper, we constructed the ghost algebra Gh 2 n and the dilute ghost algebra dGh 2 n .These are two-boundary generalisations of the TL and dilute TL algebras, respectively, with three important properties.First, the algebras are associative.Second, their diagram bases include diagrams with odd numbers of strings connected to one or both boundaries.Third, associated with each boundary, they have a unital or non-unital subalgebra, respectively, that is isomorphic to the one-boundary TL algebra with two distinct boundary parameters.
In Section 1, we considered the algebra obtained by allowing diagrams with odd numbers of strings connected to each boundary, and applying the multiplication rules from TL 2 n .It was shown that such an algebra cannot have all three of our desired properties simultaneously.This was resolved by introducing bookkeeping devices called ghosts on the boundaries of our diagrams.The parity of the string endpoints on each boundary is now determined by counting ghosts as well as strings, and boundary arcs leave behind a ghost at each endpoint when removed.Associativity is then ensured by requiring the sum of the number of strings and ghosts on each boundary to be even.
We then introduced lattice loop models associated with the ghost algebra and the dilute ghost algebra.The bulk of the lattices is the same as the TL and dilute TL lattice models, respectively, so we used existing bulk face operators from these models that are known to satisfy the YBE, local inversion relations, and have crossing symmetry.The boundary face operators, however, needed to incorporate the new ghosts.Given the bulk face operators, we classified all boundary face operators satisfying the BYBEs for generic values of the algebra parameters.From these face operators, we built commuting families of transfer tangles, using the construction from [1].Some of our BYBE solutions for the ghost and dilute ghost algebras have multiple degrees of freedom, and potentially nonzero coefficients on all possible boundary triangles.It would thus be interesting to compute the Hamiltonians arising from these transfer tangles, and study their spectra, to learn whether these new boundary conditions lead to new physical behaviour.
The ghost algebra and the dilute ghost algebra are constructed to be cellular with respect to the anti-involution given by reflecting basis diagrams about a vertical line.Graham and Lehrer's paper [16] introduces the definition of cellular algebras, and also proves a number of results about their representation theory, provided they are finite-dimensional and defined over a field.In particular, the classification of finite-dimensional irreducible modules amounts to determining whether the discriminants of certain bilinear forms are zero.The bilinear forms in question are defined on cell modules; for Gh 2 n and dGh 2 n , these modules are spanned by the Gh 2 n -and dGh 2 n -half-diagrams, respectively, with a fixed number of defects, as defined in Appendix A. The action is defined similarly to the usual diagram multiplication.Computing these discriminants for all n and all defect numbers is highly non-trivial, given how fast the dimensions of these modules grow, but the author hopes to achieve it in future work.
Finally, in Appendix D, we considered generalisations of the ghost algebra and the dilute ghost algebra which are no longer cellular with respect to the reflection anti-involution.These are still associative and have parity-dependent boundary parameters, but can be further generalised.Indeed, the only purpose of the ghosts in our diagrams is to keep track of which strings should be numbered even or odd during multiplication.Thus, each string endpoint on each boundary could be labelled odd or even, and then we could remove the ghosts entirely.Similar algebras can then be constructed with any number of labels that may be applied to each boundary connection, and parameters assigned to boundary arcs with each possible pairing of labels.One could then find necessary relations on the parameters for these algebras to be cellular with respect to different anti-involutions, or determine whether their more general boundary conditions lead to even more interesting structure in the corresponding lattice models.
Proof.The argument is analogous to Proposition A.2, except we use Gh 1 n−v -half-diagrams with d defects, instead of Gh 2 n−v -half-diagrams with d defects.The number of such half-diagrams is the sum over j in the expression in Proposition A.3, but with n − v in place of n.

B Associativity
In this section, we prove that the dilute ghost ghost algebra dGh 2 n is associative.Since dGh 1 n , Gh 2 n and Gh 1 n are (unital, non-unital and non-unital) subalgebras of dGh 2 n , it follows that they are also associative.We start by introducing some terminology to describe diagram multiplication more formally, and then prove associativity in Theorem B.1 and Corollary B.2.
A dGh 2 n -pseudo-diagram consists of finitely many non-crossing strings drawn within the usual two-boundary rectangle of 2n nodes, and finitely many ghosts on each boundary, subject to the following requirements.Each node must have at most one string endpoint attached to it.Each string endpoint can be attached to a node or a boundary, or be left unattached.Strings may also form loops. Ghosts cannot be drawn on the ends of strings, and the number of ghosts plus the number of string endpoints attached to each boundary must be even.Two dGh 2 n -pseudo-diagrams are considered equal if their strings are equivalent under a continuous planar deformation that preserves the rectangle, up to sliding but not reordering ghosts and string endpoints, and their corresponding domains have the same number of ghosts, modulo 2.
For example, , , , are all dGh 2 6 -pseudo-diagrams.Note that the first two are not equal because planar deformations do not allow us to pull a loop over other strings.
Since the number of ghosts in each domain is only considered modulo 2, each dGh 2 n -pseudodiagram can be drawn with at most one ghost in each domain.Further, we can number the string endpoints and ghosts along each boundary, left-to-right, starting from 1.If two drawings of dGh 2 n -pseudo-diagrams are equal, then each string endpoint on the boundary might be assigned a different number in each drawing, but the number of string endpoints to its left is the same, and the number of ghosts in each domain to its left is the same modulo 2. Hence each string endpoint on the boundary has a well-defined parity.
Let • be the binary operation of concatenation on dGh 2 n -pseudo-diagrams, where the overlapping middle line from the two rectangles is removed.It is clear that this is associative.For example, if where #(ρ, x) is the number of strings in x that would be assigned the parameter ρ during multiplication in the dilute ghost algebra, as listed in Table 1.For example, with x and y as above, Proof.It suffices to show that multiplication of dGh 2 n -diagrams is associative.Let x, y and z be dGh 2 n diagrams.From (B.7), we have To show (B.12), recall that χ is determined by the number of loops and each kind of boundary arc in its dGh 2 n -pseudo-diagram argument.Since x, y and z are dGh 2 n -diagrams, not just pseudodiagrams, they have none of these features independently.There are three ways such objects can be formed from x, y and z in these products: boundary arcs we need to check that the parity of each endpoint is unchanged.Indeed, within a dGh 2 n -pseudo-diagram, the parity of each string endpoint on the boundary is well-defined.Since each dGh 2 n -pseudo-diagram has an even number of ghosts plus string endpoints on each boundary, concatenation changes the numbering by an even number, and so does not affect parity.Therefore χ( x • y • z) = χ(x • y • z).Similarly, each object of type 2 contributes a factor of its associated parameter to χ(y • z), while each object of type 1 or 3 contributes to χ(x • y • z ), so χ(x • y • z ) = χ(x • y • z).
The argument for (B.13) is analogous, since ζ is determined by the number of strings with at least one unattached endpoint.The only notable difference is that the three types of objects for ζ are formed from the strings or empty nodes of the relevant diagrams, instead of just the strings.

C Cellularity
In this section, we show that Gh 1 n , Gh 2 n , dGh 1 n and dGh 2 n are cellular algebras, as defined by Graham and Lehrer in [16, §1].We reproduce the definition for convenience.We use the terms half-diagram, throughline and defect defined in Appendix A, as well as the injective map from pairs of half-diagrams to diagrams.We also use the notation χ, ζ, • and • from Appendix B.
Let A be an associative unital algebra over a commutative ring R with identity.Then A is a cellular algebra if it has a cell datum (A, M, C, * ), where 1. Λ is a partially ordered set, and for each λ ∈ Λ, M (λ) is a finite set such that C : λ∈Λ M (λ) × M (λ) → A is an injective map with image an R-basis of A. Proof.Let R = C and let A ∈ Gh 1 n , Gh 2 n , dGh 1 n , dGh 2 n .Let Λ = {0, 1, . . ., n} with the usual partial order ≤.For each λ ∈ Λ, let M (λ) be the set of all A-half-diagrams with λ defects.For λ ∈ Λ and S, T ∈ M (λ), let C λ S,T be the A-diagram with λ defects obtained by reflecting T about a vertical line and linking its defects to those of S.
Then from Appendix A, C : λ∈Λ M (λ) × M (λ) → A is injective, and its image is the set of all A-diagrams, which is indeed a C-basis for A.
Define * : A → A as the linear map that takes each A-diagram to its reflection about a vertical line.This is clearly bijective.To see that this is an anti-involution, consider the strings and ghosts on each boundary of each A-diagram.During multiplication these are numbered, and boundary arcs assigned parameters according to the parities of their endpoints.Recalling that the sum of the number of strings and ghosts on each boundary is even, the parity assigned to each string or ghost is swapped under the reflection * , but so is the left-to-right order.Hence, * maps each type of boundary arc to itself, except those that have both ends on the same boundary with the same parity, and top-to-bottom boundary arcs.These exceptional boundary arcs are assigned the same parameter as their images, and this ensures that * is an anti-involution.It follows from the definition of C that C λ S,T * = C λ T,S for all S, T ∈ M (λ) and λ ∈ Λ.
Finally, for part 3, consider the product of two A-diagrams C ν U,V and C λ S,T , where ν, λ ∈ Λ, U, V ∈ M (ν) and S, T ∈ M (λ).Note that C ν U,V C λ S,T is a scalar multiple of an A-diagram with at most min{λ, ν} throughlines.Thus if ν < λ, then C ν U,V C λ S,T ≡ 0 mod A(< λ), so we can take r C ν U,V (S, S ′ ) = 0 for all S ′ ∈ M (λ) in this case.If ν ≥ λ, it is possible that the A-diagram C ν U,V • C λ S,T , obtained by concatenating these diagrams and removing strings not connected to either side, has fewer than λ throughlines.Since C ν U,V C λ S,T is a scalar multiple of this diagram, it is congruent to 0 mod A(< λ), and thus we can take r C ν U,V (S, S ′ ) = 0 for all S ′ ∈ M (λ) in this case as well.If ν ≥ λ and C ν U,V • C λ S,T has λ throughlines, then since we can take does not depend on T because the only strings from T that may be connected to the left side of the resulting diagram are its throughlines, and these are cut to produce L C ν U,V C λ S,T .Hence in all cases, r C ν U,V is independent of T .For each a ∈ A, we can write for some f ν U,V ∈ C, and then r a (S ′ , S) = ν∈Λ U,V ∈M (ν) f ν U,V r C ν U,V (S ′ , S). (C.5) Since r C ν U,V does not depend on T , it follows that r a does not depend on T .Thus part 3 is satisfied, and A is a cellular algebra.
We note that the algebras Gh 1 n , Gh 2 n , dGh 1 n and dGh 2 n are finite-dimensional over the field C, so the representation theory results from [16, §3] apply to them.

Figure 1 :
Figure 1: Examples of basis diagrams from different diagram algebras.Figure 1d is included as an example of a two-boundary diagram that is not a basis diagram of TL 2 5 because it has an odd number of strings connected to each boundary.

α 3 )
ψ add top boundary a d d b o t t o m b o u n d a r y ad d gh os ts ad d em pt y no de s (a) Commutative diagram relating notable unital and non-unital subalgebras of Gh 2 n and dGh 2 n .

a d d b o t t o m b o u n d a r y ad d gh os ts ad d em
pt y no de s (b) Commutative diagram relating notable unital subalgebras of Gh 2 n and dGh 2 n .

Figure 2 :
Figure 2: Commutative diagrams summarising the maps from Section 4, with blue notes indicating their 3D layout.Solid and dashed arrows are unital and non-unital homomorphisms, respectively.Hooked arrows are injections.The map ψ is defined in Section 4, paragraph three.
.22) Since one side depends only on u and the other only on v, both sides are independent of u and v.So each b j is a scalar multiple of b i .Hence there exists a nonzero function b such that, for each k ∈ {1, 2, 3, 4}, we can write b k (u) = c k b(u) (5.23)for some c k ∈ C.This also holds if b i = 0 for all i ∈ {1, 2, 3, 4}, in which case c k = 0 for all k ∈ {1, 2, 3, 4}.
88) for i, j ∈ {1, 2, 3, 4}.Analogously to the Gh 2 n case, this implies b k (u) = c k b(u) for some c k ∈ C and nonzero b : C → C, for each k ∈ {1, 2, 3, 4}.Applying this to all coefficients, the next four simplest coefficients are for the diagrams , , , ,

) so c 2 c 5 =
c 4 c 6 = 0, or the expression in the larger brackets is zero.Now, suppose c 2 = 0. Then the coefficients of ,

For c 1 ,
c 2 , c 3 or c 4 nonzero, setting the coefficient of each of these diagrams to zero implies 0

3 and c 4
are not all zero, consider the case where c 1 = 0. Then we can set the coefficient of the diagram (5.177) equal to zero, and rearrange to find

2 and sin v + 3φ 2 are
linearly independent, this expression is nonzero.Hence we cannot have a solution with (5.185), and so we must have c 1 c 3 = c 2 c 4 .This means we can writec 1 = m 1 m 2 , c 2 = m 1 m 3 , c 3 = m 3 m 4 , c 4 = m 2 m 4 ,(5.190) for some m 1 , m 2 , m 3 , m 4 ∈ C. Since c 1 , c 2 , c 3 and c 4 are not all zero, m 2 and m 3 are not both zero, and m 1 and m 4 are not both zero.The coefficients of the diagrams , , ,

5 )
Define a function ζ on dGh 2n -pseudo-diagrams by ζ(x) = 1 if each string endpoint in x is attached to a node or boundary, and ζ(x) = 0 otherwise.Note that loops do not have endpoints, so ζ(x) = 1 does not imply that x has no loops.Indeed, with x and y as above,ζ(x) = 1, ζ(y) = 0 and ζ(x • y) = 0.For each dGh 2 n -pseudo-diagram x, let x be the dGh 2 n -diagram obtained by removing all strings from x except those that are attached to a node at one end, and a node or boundary at the other.A ghost must be left at the position of each removed string endpoint on the boundary, though we may choose to draw the resulting diagram with at most one ghost in each domain.With x and y as above,2  n -diagrams x and y, we havexy = χ(x • y) ζ(x • y) x • y .(B.7)Theorem B.1.The dilute ghost algebra dGh 2 n is associative.
)where L(x) is the left A-half-diagram obtained by cutting the A-diagram x in half along a vertical line.Observe that the boundary arcs and loops formed in the concatenation C ν U,V • C λ S,T can only use strings from the A-half-diagrams V and S, soχ C ν U,V • C λ S,Tis a function of V and S only.A string meets an empty node in this concatenation if and only if a string from V meets an empty node of S or vice versa, so ζ C ν U,V • C λ S,T is also a function of V and S only.Note also that L C ν U,V C λ S,T

Table 1 :
Table of boundary arcs and their associated parameter values in the ghost algebra.

Table 2 :
Table of dimensions of Gh 1 n , Gh 2 n , dGh 1 n and dGh 2 n for small n.
B.2.The ghost algebra Gh 2 n , the one-boundary ghost algebra Gh 1 n , and the oneboundary dilute ghost algebra dGh 1 n are associative.Proof.Since Gh 2 n , Gh 1 n and dGh 1 n are non-unital, non-unital and unital subalgebras of dGh 2 n , respectively, they inherit associativity from dGh 2 n .