Set theoretic Yang-Baxter&reflection equations and quantum group symmetries

Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for $R$-matrices being Baxterized solutions of the $A$-type Hecke algebra ${\cal H}_N(q=1)$. We show in the case of the reflection algebra that there exists a ``boundary'' finite sub-algebra for some special choice of ``boundary'' elements of the $B$-type Hecke algebra ${\cal B}_N(q=1, Q)$. We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the $B$-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the $B$-type Hecke algebra. These are universal statements that largely generalize previous relevant findings, and also allow the investigation of the symmetries of the double row transfer matrix.


Introduction
The Yang-Baxter equation and the R-matrix are central objects in the framework of quantum integrable systems. The Yang-Baxter equation was first introduced by Yang in [66] when investigating many particle systems with δ-type interactions and later in the celebrated work of Baxter, who solved the anisotropic Heisenberg magnet (XYZ model) [2]. The solution of the model by Baxter was achieved by implementing the so-called Q-operator method, a sophisticated approach leading to sets of functional relations known as T-Q relations, that provide information on the spectrum of the model. A different approach on the resolution of the spectrum of 1D statistical models is the Quantum Inverse Scattering (QISM) method, an elegant algebraic technique [43], that led directly to the invention of quasitriangular Hopf algebras known as quantum groups, which then formally developed by Jimbo and Drinfeld independently [24,38].
Drinfield [23] also suggested the idea of set-theoretic solutions to the Yang-Baxter equation, and since then a lot of research activity has been devoted to this issue (see for instance [34], [26]). Set-theoretical solutions and Yang-Baxter 1 maps have been investigated in the context of classical discrete integrable systems related also to the notion of Darboux-Bäcklund transformation [1,65,54]. Links between the set-theoretical Yang-Baxter equation and geometric crystals [27,3], or soliton cellular automatons [64,33] have been also revealed. Set-theoretical solutions of the Yang-Baxter equations have been investigated by employing the theory of braces and skew-braces. The theory of braces was established by W. Rump who developed a structure called a brace to describe all finite involutive set-theoretic solutions of the Yang-Baxter equation [56,57]. He showed that every brace provides a solution to the Yang-Baxter equation, and every non-degenerate, involutive set-theoretic solution of the Yang-Baxter equation can be obtained from a brace, a structure that generalizes nilpotent rings. Skew-braces were then developed in [32] to describe non-involutive solutions. Key links between set-theoretical solutions and quantum integrable systems and the associated quantum algebras were uncovered in [21].
Following the works of Cherednik [9] and Sklyanin [58], who introduced and studied the reflection equation, much attention has been focused on the issue of incorporating boundary conditions to integrable models. The boundary effects, controlled by the refection equation, shed new light on the bulk theories themselves, and also paved the way to new mathematical concepts and physical applications. The set-theoretical reflection equation together with the first examples of solutions first appeared in [5], while a more systematic study and a classification inspired by maps appearing in integrable discrete systems presented in [4]. Other solutions were also considered and used within the context of cellular automata [47]. In [62,41] methods coming from the theory of braces were used to produce families of new solutions to the reflection equation, and in [12] skew braces were used to produce reflections.
The outline of the paper. In this study we consider set-theoretic solutions of the Yang-Baxter and reflection equations coming from braces and we construct quantum spin chains with open boundary conditions through Sklyanin's double row transfer matrix [58]. We should mention that typical well studied solutions of the Yang-Baxter equation are the Yangians, expressed as R(λ) = P + λI, where P is the flip map: u ⊗ v → v ⊗ u. Here we consider more general classes of solutions of the Yang-Baxter equation that are expressed as R(λ) = P + λPř, whereř is a map that can be obtained for instance from a brace. Such solutions are of particular interest, given that in general they have no semi-classical analogue and as such they are distinctly different from the known quantum group solutions. Let us describe below in more detail what is achieved in each section: • In section 2 we present some basic background information. More precisely, in subsection 2.1 we review some background on R-matrices associated to non-degenerate, involutive, set-theoretic solutions of the Yang-Baxter equation as well as set-theoretic solutions of the reflection equation and some information on braces. Then in subsection 2.2 we provide a review on recent results on the connections of brace solutions of the Yang-Baxter equation and the corresponding quantum algebras and integrable quantum spin chains [21]. • In section 3 examples of set-theoretic R-matrices expressed as simple twists of known solutions via isomorhisms within the finite set {1, . . . , N } are presented. Based on these solutions we construct explicitly the associated "twisted" co-products by employing the finite set isomorphisms. We then move on to show that the generic brace solution of the Yang-Baxter equation can be obtained from the permutation operator via suitable Drinfeld twists [25]. Note that the properties of the brace structures are instrumental in deriving the form of the twist. Certain generalizations regarding the q-deformed case are also discussed. • In section 4 we focus on quadratic algebras, i.e. the reflection and twisted algebras [58,53].
(1) In subsections 4.1 and 4.2 we review some background information on reflection algebras and B-type Hecke algebras. More precisely, in subsection 4.1. we recall the links between the refection algebras and B-type Hecke algebras and the Baxterization process, whereas in subsection 4.2 we discuss set-theoretic representations of the Btype Hecke algebra by essentially reviewing some recent results on solutions of the set-theoretic reflection equation [62]. (2) In subsection 4.3 we derive the associated defining algebra relations for Baxterized solutions of the A-type Hecke algebra H N (q = 1), and we show in the case of the reflection algebra that there exist a finite sub-algebra for some special choice of "boundary" elements of the B-type Hecke algebra, which also turns out to be a symmetry of the double row transfer matrix for special boundary conditions as will be shown in subsection 5.2. • In section 5 we introduce open spin chains like systems and we focus on the study of the associated quantum group symmetries. We first review the construction of open quantum spin chains via the use of tensorial representations of the reflection algebras and the derivation of the double row transfer matrix. The findings of each subsection are described below.
(1) In subsection 5.1.we study the symmetries of the double row transfer matrix constructed from Baxterized solutions of the B-type Hecke algebra B N (q = 1, Q) . We first prove the key proposition of this study, i.e. we show that almost all the factors, but one, of the λ-series expansion of the open transfer matrix can be expressed in terms of the elements of the B-type Hecke algebra. Interestingly, when choosing special boundary conditions, the full open transfer matrix can be exclusively expressed in terms of elements of the A-type Hecke algebra.
Another fundamental result is that that all elements of the of the B type Hecke algebra B N (1 = 1, Q) commute with a finite sub algebra of the reflection algebra. This then leads to another important proposition regarding the symmetry of the associate double row transfer matrix. These are universal results that largely extends earlier partial findings (see e.g. [55,19]), and are of particulate physical and mathematical significance. (2) In subsection 5.2 more symmetries of open transfer matrices associated to certain classes of set-theoretic solutions of the Yang-Baxter equation coming from braces are also discussed. The derivation of these symmetries is primarily based on the properties of the brace structures. Some of these symmetries generalize recent findings on periodic transfer matrices [21], while others are new. (3) In subsection 5.3 symmetries of the double row transfer matrix constructed from the special class of Lyubashenko's solutions are identified confirming also some of the findings of section 3.

Preliminaries
We present in this section some basic background information regarding settheoretic solutions of the Yang-Baxter and reflection equations and braces as well as a brief review on the recent findings of [21] on the links between set-theoretic solutions of the Yang-Baxter equation from braces and quantum algebras.
2.1. The set-theoretic Yang-Baxter equation. Let X = {x 1 , . . . , x N } be a set andř : X × X → X × X. Denotě We say that r is non-degenerate if σ x and τ y are bijective functions. Also, the solutions (X,ř) is involutive:ř(σ x (y), τ y (x)) = (x, y), (řř(x, y) = (x, y)). We focus on non-degenerate, involutive solutions of the set-theoretic braid equation: Let V be the space of dimension equal to the cardinality of X, and with a slight abuse of notation, letř also denote the R-matrix associated to the linearisation of r on V = CX (see [61] for more details), i.e.ř is the N 2 × N 2 matrix: where e x,y is the N ×N matrix: (e x,y ) z,w = δ x,z δ y,w . Then for theř-matrix related to (X,ř):ř(x, z|y, w) = δ z,σx(y) δ w,τy(x) . Notice that the matrixř : V ⊗ V → V ⊗ V satisfies the (constant) Braid equation: Notice also thatř 2 = I V ⊗V the identity matrix, becauseř is involutive.
For set-theoretical solutions it is thus convenient to use the matrix notation: Define also, r = Př, where P = x,y∈X e x,y ⊗ e y,x is the permutation operator, consequently r = x,y∈X e y,σx(y) ⊗ e x,τy(x) . The Yangian is a special case: r(x, z|y, w) = δ z,y δ w,x .
Let (X,ř) be a non-degenerate set-theoretic solution to the Yang-Baxter equation. A map k : X → X is a reflection of (X,ř) if it satisfieš We say that k is a set-theoretic solution to the reflection equation. We also say that k is involutive if k(k(x)) = x.
Using the matrix notation introduced above then the reflection matrix K is an N × N matrix represented as: and satisfies the constant reflection equation: Let us now recall the role of braces in the derivation of set-theoretic solutions of the Yang-Baxter equation. In [56,57] Rump showed that every solution (X,ř) can be in a good way embedded in a brace. Definition 2.1 (Proposition 4, [57]). A left brace is an abelian group (A; +) together with a multiplication · such that the circle operation a • b = a · b + a + b makes A into a group, and a · (b + c) = a · b + a · c.
In many papers, an equivalent definition is used [7] . The additive identity of a brace A will be denoted by 0 and the multiplicative identity by 1. In every brace 0 = 1. The same notation will be used for skew braces (in every skew brace 0 = 1).
Throughout this paper we will use the following result, which is implicit in [56,57] and explicit in Theorem 4.4 of [7]. [56,57,7]). It is known that for an involutive, non degenerate solution of the braid equation there is always an underlying brace (B, •, +), such that the maps σ x and τ y come from this brace, and X is a subset in this brace such thatř(X, where t is the inverse of σ x (y) in the circle group (B, •). Moreover, we can assume that every element from B belongs to the additive group (X, +) generated by elements of X. In addition every solution of this type is a non-degenerate, involutive set-theoretic solution of the braid equation.
We will call the brace B an underlying brace of the solution (X,ř), or a brace associated to the solution (X,ř). We will also say that the solution (X,ř) is associated to brace B. Notice that this is also related to the formula of settheoretic solutions associated to the braided group (see [26] and [30]).
The following remark was also discovered by Rump.
Remark 2.3. Let (N, +, ·) be an associative ring which is a nilpotent ring. For We focus here on brace solutions 1 of the YBE, given by (2.2) and the Baxterized solutions: (2.6)Ř(λ) = λř + I, where I = I X ⊗ I X and I X is the identity matrix of dimension equal to the cardinality of the set X. Let also R = PŘ, (recall the permutation operator P = x,y∈X e x,y ⊗ e y,x ), then the following basic properties for R matrices coming from braces were shown in [21]: Basic Properties. The brace R-matrix satisfies the following fundamental properties: where t1,2 denotes transposition on the fist, second space respectively, and recall N is the same as the cardinality of the set X. Let us also recall the connection of the brace representation with the A-type Hecke algebra.

Remark 2.5. The brace solutionř (2.2) is a representation of the A-type
Hecke algebra for q = 1. Indeed,ř satisfies the braid relation andř 2 = 1, which is shown by using the involution property. Also, the braid relation is satisfied by means of the brace properties (see also Theorem 2.2 and [21]).
The Quantum Algebra associated to braces. Given a solution of the Yang-Baxter equation, the quantum algebra is defined via the fundamental relation [28] (we have multiplied the familiar RTT relation with the permutation operator): where A 2 is the quantum algebra defined by (2.12). We shall focus henceforth on solutions associated to braces only given by (2.6), (2.2). The defining relations of the coresponding quantum algebra were derived in [21]: The quantum algebra associated to the brace R matrix (2.6), (2.2) is defined by generators L (m) z,w , z, w ∈ X, and defining relations The proof is based on the fundamental relation (2.12) and the form of the brace R-matrix (for the detailed proof see [21]). Recall also that in the index notation we defineŘ 12 =Ř ⊗ id A : I ⊗ e z,w ⊗ L z,w (λ). (2.14) The exchange relations among the various generators of the affine algebra are derived below via (2.12). Let us express L as a formal power series expansion L(λ) = ∞ n=0 L (n) λ n . Substituting expressions (2.6), and the λ −1 expansion in (2.12) we obtain the defining relations of the quantum algebra associated to a brace R-matrix (we focus on terms λ −n 1 λ −m 2 ): The latter relations immediately lead to the quantum algebra relations (2.13), after recalling: L x,y are the generators of the associated quantum algebra. The quantum algebra is also equipped with a co-product ∆ : A → A ⊗ A [28,24]. Indeed, we define (2.16) T 1;23 (λ) = L 13 (λ)L 12 (λ), 2 Notice that in L in addition to the indices 1 and 2 in (2.12) there is also an implicit "quantum index'n associated to A, which for now is omitted, i.e. one writes L 1n , L 2n .
Remark 2.6. In the special caseř = P the Y(gl N ) algebra is recovered: The next natural step is the classification of solutions of the fundamental relation (2.12), for the brace quantum algebra. A first step towards this goal will be to examine the fundamental object L(λ) = L 0 + 1 λ L 1 , and search for finite and infinite representations of the respective elements. The fusion procedure [44] can be also exploited to yield higher dimensional representations of the associated quantum algebra. The classification of L-operators will allow the identification of new classes of quantum integrable systems, such as analogues of Toda chains or deformed boson models. A first obvious example to consider is associated to Lyubashenko's solutions, which are discussed in what follows. This is a significant direction to pursue and will be systematically addressed elsewhere.

Set-theoretic solutions as Drinfeld twists
In this section we first introduce some special cases of solutions of the braid equation that are immediately obtained from fundamental known solutions. We show in particular that a special class of solutions known as Lyubashenko's solutions [23] can be expressed as simple twists. Although the construction is simple it has significant implications on the associated symmetries of the braid solutions. We then move on to show that the generic brace solution of the Yang-Baxter equation (2.2) can be obtained from the permutation operator via a suitable Drinfeld twist [25], and we identify the specific form of the twist. Moreover, inspired by the isotropic case we provide a similar construction for the q-deformed analogue of Lyubashenko's solution.
Before we derive the Lyubashenko solution as a suitable twist we first introduce a useful Lemma.
Then any solution of the type (Lyubashenko's solution) can be obtained from the permutation operator P = x,y∈X e x,y ⊗ e y,x as Proof. The proof relies on the definitions of P, V, V −1 and the fundamental property e x,y e z,w = δ y,z e x,w : x,y∈X e σ(x),σ(y) ⊗ e y,x .
Note that r = Př, and consequently R = PŘ take a simple form for this class of solutions:
Before we present our findings on the symmetry of Lyubashenko'sř-matrix we first introduce a useful Lemma. Lemma 3.3. Let l x,y be the generators of the gl N algebra satisfying: The gl N algebra is equipped with a coproduct ∆ : gl N → gl N ⊗ gl N such that . By acting from the left with F (N ) and with (F N ) −1 from the right in the latter commutator we immediately obtain ∆ (N ) where we define the "twisted" co-products (i = 1, 2): Proof. This can be shown using the form of the special class of solutions (3.4). The permutation operator is gl N symmetric, i.e. where the co-products ∆(e x,y ) are defined in Lemma (3.3) (l x,y → e x,y ). Let V = x∈X e x,τ (x) , then (3.9) immediately follows from (3.11) and (3.5) after multiplying (3.11) from the left and right with V ⊗ I, and explicitly given by (3.10). Indeed, Ve x,y V −1 = e σ(x),σ(y) and V −1 e x,y V = e τ (x),τ (y) .
According to Lemma 3.3 ∆ i (e x,y ) also satisfy the gl N algebra relations, thusř (3.4) is gl N symmetric. In this particular case, as is clear from the computation above, two invertible linear maps are involved, F By iteration one derives the N co-products: ∆ The above expressions can be written in a compact form as: ∆ It was shown in [21] that the periodic Hamiltonian for systems built with Rmatrices associated to the Hecke algebra H N (q = 1) is expressed exclusively in terms of the A-type Hecke algebra elements. In the special case whereř = P, i.e. the Yangian the periodic transfer matrix is gl N symmetric. However, if we focus on the more general class of Lyubashenko's solutions of Proposition 3.2 and Corollary 3.4 we conclude that because of the existence of the termř N 1 (due to periodicity) [21], and also due to the form of the modified co-products (3.13), (3.14), the periodic Hamiltonian and in general the periodic transfer matrix is not gl N symmetric anymore. However, we shall be able to show in section 5 that for a special choice of boundary conditions not only the corresponding Hamiltonian is gl N symmetric, but also the double row transfer matrix. This means that the open spin chain enjoys more symmetry compared to the periodic one similarly to the q-deformed case [55,46,16,18,11]. It is therefore clear that from this point of view open spin chains are rather more natural objects to consider compared to the periodic ones. In [21] a systematic investigation of symmetries of the periodic transfer matrix for generic representations of the A-type Hecke algebra H N (q = 1) as well as for certain solutions of the Yang-Baxter equation coming from braces is presented.
With the following proposition we generalize the results on Lyubashenko's solutions. Specifically, we express the generic braceř-matrix (2.2) as a twist of the permutation operator. Drinfeld introduced [25] the"twisting" (or deformation) of a (quasi) triangular Hopf algebra that produces yet another (quasi) triangular (quasi) Hopf algebra (see also relevant [45,49]). Let us briefly recall the notion of a twist. LetŘ be the quantum group invariant matrix i.e. it commutes with the the respective quantum algebra [38,8]. We are focusing on the finite algebra g, specifically we are considering here the algebras gl N or U q (gl N ), although via the evaluation homomorphism one obtains the corresponding affine algebras, i.e. the Yangian Y(gl N ) or the affine U q ( gl N ) respectively [38,8]. Consider the fundamental representation π : g → End(C N ), the co-poducts ∆ : g → g ⊗ g and thě R-matrix satisfy linear intertwining relations: (π ⊗ π)∆(X)Ř =Ř (π ⊗ π)∆(X) for X ∈ g. Let also F ∈ End(C N ⊗ C N ), then theŘ matrix can be "twisted" as FŘF −1 , where F also satisfies a set of constraints dictated by the YBE. Given the linear intertwining relations and the twistedŘ-matrix, one derives the twisted co-products of the finite algebra as F (π ⊗ π)∆(X) F −1 (for a more detailed exposition on the notions of quasi-trinagular Hopf algebras and Drinfeld twists the interested reader is referred for instance to [8]).
Proposition 3.5. Letř = x,y∈X e x,σx(y) ⊗ e y,τy(x) be the brace solution of the Yang-Baxter equation (see also (2.2) and footnote 1 in page 6). Let also V k , k ∈ {1, . . . , N 2 } be the eigenvectors of the permutation operator P = x,y∈X e x,y ⊗ e y,x , andV k , k ∈ {1, . . . , N 2 } be the eigenvectors of the braceř matrix. Then thě r matrix can be expressed as a Drinfeld twist, such We divide our proof in three parts: (1) First we diagonalize the permutation operator. Letê j be the N dimensional column vectors with one at the j th position and zero elsewhere, then the (normalized) eigenvectors of the permutation operator are (x, y ∈ X): The first N 2 +N 2 eigenvectors have the same eigenvalue 1, while the rest eigenvectors have eigenvalue −1. Also it is easy to check that V k form an ortho-normal basis for the N 2 dimensional space. Indeed, (2) Second we diagonalize the braceř-matrix. First we observe thať r e x ⊗ e y = e σx(y) ⊗ e τy(x) ,ř e σx(y) ⊗ e τy(x) = e x ⊗ e y .
Then we find that the eigenvectors of theř matrix arê As in the case of the permutation operator theř matrix has the same eigenvalues 1 and −1 and the same multiplicities, N 2 +N 2 and N 2 −N 2 respectively. Hence, the two matrices are similar, i.e. there exists some F ∈ End(C N ⊗ C N ) (not uniquely defined) such thatř = F PF −1 .
(3) Our task now is to derive the explicit form of F . This is quite straightforward, indeed the eigenvalue problem for P (andř) reads as Note that ifř = F PF −1 (P the permutation operator) then r = Př = F (op) F −1 , where F (op) = PF P, and consequently the Baxterized solution (2.6) is given as Proof. The proof is straightforward as in Corollary 3.4 using the fact that the permutation operator is gl N symmetric.
Notice that here we identified the Drinfeld twist as a similarity transformation between the permutation operator and the brace solution. The twisted nco-product as well as the n form of F should be identified and the admissibility of the twist should be also examined. Also, issues on the co-associativity of the co-product need to be addressed. We already observe in the simple case of Lyubashenko's solutions that the co-associativity of the twisted co-products is not guaranteed. These are significant issues that are addressed in [22].
3.1. Parenthesis: the q-deformed case. We slightly deflect in this subsection from our main issue, which is the set-theoretic solutions of the Yang-Baxter equation, and briefly discuss the q-deformed case. Inspired by the special class of Lyubashenko's solutions, we generalize in what follows Proposition 3.2 and Corollary 3.4 in the case of the U q (gl N ) invariant representation of the A-type Hecke algebra [38]: x ⊗ e y,y + q.
Note that strictly speaking this solution is not a set-theoretic solution of the braid equation. Nevertheless, isomorphisms within the set of integers {1, . . . , N } can be still exploited to yield generalized solutions based on (3.15).
Proof. Let V = w e w,τ (w) , and V −1 = z e τ (z),z . We show by explicit computation that, . We then define, bearing in mind (3.17): which leads to (3.16). Also, g is a given representation of the A-type Hecke algebra, i.e.
By multiplying (3.19) with V ⊗ I ⊗ V −1 from the left and V −1 ⊗ I ⊗ V from the right, and also multiplying (3.20) with V ⊗ I from the left and V −1 ⊗ I from the right, and using the definition (3.18) we immediately conclude that G is also a representation of the A-type Hecke algebra (see also Lemma 3.1).
It will be useful for what follows to recall the basic definitions regarding the U q (gl N ) algebra [38]. Let be the Cartan matrix of the associated Lie algebra.
Definition 3.8. The quantum algebra U q (sl N ) has the Chevalley-Serre generators . . , N − 1} obeying the defining relations: and the q deformed Serre relations ǫi belongs to the center [38], and q ǫi , q ǫj = 0, q ǫi e j = q δi,j −δi,j+1 e j q ǫi , The N -fold co-product may be derived by using the recursion relations and as is customary, ∆ (2) = ∆ and ∆ (1) = id.
Let us now consider the fundamental representation of U q (gl N ) [38], π : and let us also introduce some useful notation: where we define the modified co-products (i = 1, 2): Proof. This can be shown in a straightforward manner from the properties of (3.16). Indeed, g (3.15) is U q (gl N ) invariant [38,39] (recall the fundamental representation (3.26)) wher Y ∈ e j,j+1 , e j+1,j , q ej,j and the co-products of the algebra elements are given in (3.24) (see also (3.26), (3.27)). We consider two invertible linear maps: F where the modified co-produts are defined as ∆ i (Y ) = F (2) i ∆(Y )(F (2) i ) −1 , and more specifically: , e j+1,j (3.32) and explicitly given by (3.29).
The coproducts ∆(Y ) satisfy the U q (gl N ) relations, then via Explicit expressions for the modified N co-products are then given as: where ξ F n ∈ e F n (j),F n (j+1) , e F n (j+1),F n (j) . The above expressions can be written in a compact form as: ∆  [22]).
Some general comments are in order here. We should note that set-theoretic solutions from braces have no semi-classical analogue [21], thus they are fundamentally different from the known Yangian solutions or the q-deformed solutions of the YBE associated to gl N or U q (gl N ) [38,8,52]. This is evident even in the simple case of Lyubashenko's solution (please see Proposition 3.2 and simple examples 1 and 2 in page 9), recall r = V −1 ⊗ V ⇒ R(λ) = λV −1 ⊗ V + P, where V = x,∈X e x,τ (x) (more generally due to Proposition 3.5, R(λ) = λF (op) F −1 + P and F (op) = PF P). Such R-matrices can not be expressed as 1 + r (1) + ... (up to an overall multiplicative function f (λ)), given the form of V (or F explicitly given in [22]), a fact that makes our construction distinct compared to the known examples of quantum algebras (quasi triangular Hopf algebras) as described for instance by Drinfeld in [24] (a detailed analysis on these issues is presented in [22]). In this spirit it would be also very interesting to consider general twists, in analogy to Proposition 3.5, for the q-deformed case as well as the corresponding quantum groups and make possible connections with the theory of braces.

Co-ideals: reflection & twisted algebras
We introduce two, in principle distinct, quadratic algebras associated to the classification of boundary conditions in quantum integrable models. To define these quadratic algebras in addition to the R-matrix we also need to introduce the K-matrix, which physically describes the interaction of particle-like excitations displayed by the quantum integrable system, with the boundary of the system. The K-matrix satisfies [9,58,53]: where we define in general A 21 = P 12 A 12 P 12 . We make two distinct choices forR, which lead to the two district quadratic algebras: notice N 2 is the Coxeter number for gl N . In the self-conjugate cases e.g. in the case of e.g. sl 2 , U q (sl 2 ) or so n , sp n Rmatrices R(λ) ∼ C 1 R t1 12 (−λ − c)C 1 , for some matrix C : C 2 = I , i.e. the R-matrix is crossing symmetric, and the two algebras, twisted and refection, coincide. The constant c is associated to the Coxeter number of the corresponding algebra. It is worth noting that these algebras are linked to two distinct types of integrable boundary conditions, extensively studied in the context of A (1) N −1 affine Toda field theories [10,13,20], and quantum spin chains [58] associated to gl N , U q (gl N ), and gl(N |M) algebras [14], [50], [16]- [19].
we consider the boundary Yang-Baxter or reflection equation [9,58], expressed in the braid form (4.4) As in the case of the Yang-Baxter equation, where representations of the A-type Hecke algebra are associated to solutions of the Yang-Baxter equation [38], via the Baxterization process, representations of the B-type Hecke algebra provide solutions of the reflection equation [48,15].
We focus here on the case where q = 1 and Q arbitrary, and consider the brace solutions (2.2) as representation of the Hecke elements g l . We can solve the quadratic relation (4.5) together with (4.7) to provide representation of the G 0 element. Then via Baxterization we are able to identify suitable solutions of the reflection equation. It is obvious that the identity is a solution of the relations (4.5), (4.7), and hence of the reflection equation.

Remark 4.2. Let b =
x,z∈X b z,w e z,w be a representation of the G 0 element of the B-type Hecke algebra andř is the set-theoretic solution given in (2.2). Representations of G 0 can be identified.
Indeed, let us solve the quadratic relation (4.5) The LHS of the latter equation leads to subject to:ŷ = τ y (x), whereas the RHS gives: Study of the fundamental relations above for any brace solution will lead to admissible representations for G 0 .
Note that in the special case that b z,w = δ w,k(z) , where k : X → X satisfies k(k(x)) = x (Q = 1), and some extra conditions that are discussed in the subsequent subsection, one recovers set-theoretic reflections (see also next subsection and [62] for a more detailed discussion). In general, the full classification of representations of the B-type Hecke algebra using the braceř-matrix (2.2) is an important problem itself, which however will be left for future investigations.
We also define (4.14) whereĉ is an arbitrary constant, κ = Q − Q −1 and I the N × N identity matrix.

4.2.
Set-theoretic representations of B-type Hecke algebras. In this section we further investigate connections between the B-type Hecke algebra and the set-theoretic reflection equation, and give some specific examples of representations of Hecke algebras that correspond to set-theoretic reflections. (1) k : X → X is a solution to the set-theoretic reflection equation for the solution (X,ř):ř where K [1] (x, y) = (k(x), y). (2) k : X → X is a solution to the following version of the reflection equation considered in [62] for the solution (X,ř ′ ): Proof. Observe thatř is non-degenerate, hence maps σ x , τ y are bijections. Consequently,ř is non-degenerate. Let P : X × X → X × X be defined as usually as P (x, y) = (y, x) for x, y ∈ X. Observe thatř ′ = PřP , indeed PřP (x, y) = Př(y, x) = P (σ y (x), τ x (y)) =ř ′ (x, y). Notice thatř ′ is involutive: which immediately leads toř It remains to check thatř ′ is also a solution to the braid equation. For this purpose let us introduce, in the index notation, P 13 : P 13 (x, y, z) = (z, y, x), it then follows that P 13 (ř × id X )P 13 = id X ×ř ′ and P 13 (id X ×ř)P 13 =ř ′ × id X . This is easily shown, indeed P 13 (ř × id X )P 13 (x, y, z) = P 13 (ř × id X )(z, y, x) = P 13 (σ z (y), τ y (z), x) = (x, τ y (z), σ z (y)) = (id X ×ř ′ )(x, y, z). Similarly, we show that P 13 (id X ×ř)P 13 =ř ′ × id X . By acting on the braid equation forř with P 13 from the left and right it then immediately follows thatř ′ also satisfies the braid relation.
Examples of functions k satisfying the reflection equation related to braces can be found in [62,41,12]. Recall that this set-theoretical version of the reflection equation together with the first examples of solutions first appeared in the work of Caudrelier and Zhang [5] Notice that the element of the Hecke algebra can be used to construct c-number K-matrices satisfying equation (4.4), provided that Q = 1. Hence, by Lemma 4.5, constant K-matrices can be obtained from involutive set-theoretic solutions to the reflection equation. In particular, involutive τ -equivariant functions give c-number solutions of the parameter dependent equation (4.4), and every linear combination over C of such K-matrices is also a constant K-matrix, and hence gives a solution to equation (4.4) (by Theorem 5.6 [62] applied with interchanging σ and τ ).
As an application of Lemma 4.5 we obtain: Proposition 4.6. Let (X,ř) be an involutive, non-degenerate solution of the braid equation. Letř = x,y∈X e x,σx(y) ⊗ e y,τx(y) , and let g n = I ⊗n−1 ⊗ř ⊗ I ⊗N −n−1 . Let b = x∈X e x,k(x) for some function k : X → X such that k(k(x)) = x for all x ∈ X. Then b ⊗ I is a representation of the G 0 element of the B-type Hecke algebra (together withř used for representation of elements g n ) if and only if τ τy(x) (k(σ x (y))) = τ τy(k(x)) (k(σ k(x) (y))).
Proof. This follows immediately from Lemma 4.5 and Theorem 1.8 from [62], when we interchange σ with τ .
Let (X,ř) be an involutive, non-degenerate solution of the braid equation where we denoteř(x, y) = (σ x (y), τ y (x)), and let k : X → X be a function. We say that k is τ -equivariant if for every x, y ∈ X we have τ x (k(y)) = k(τ x (y)).
It was shown in [62] that every function k : X → X satisfying k(σ x (y)) = σ x (k(y)) satisfies the set-theoretic reflection equation. By interchanging σ with τ and applying Lemma 4.5 we get: x∈X e x,k(x) for some τ -equivariant function k : X → X such that k(k(x)) = x for all x ∈ X. Then b ⊗ I is a representation of the G 0 element of the B-type Hecke algebra (together withř used for representation of elements g n in this Hecke algebra).
Examples of τ -equivariant functions can be defined by fixing x, y ∈ X and defining for k(r) = τ z (y) for r = τ z (x) (provided that τ v (x) = x implies τ v (y) = y for every v ∈ X). In [41] Kyriakos Katsamaktsis used central elements to construct G(X, r) equivariant functions, his ideas also allow to define τ -equivariant functions in an analogous way-as k(x) = τ c (x), where c is central.

4.3.
Reflection & twisted algebras. We shall discuss in more detail now the two distinct algebras associated to the quadratic equation (4.1). A solution of the quadratic equation (4.1) is of the form [58,53] (4.17) where L(λ) ∈ End(C N )⊗A satisfies the RTT relation (2.12) and K(λ) ∈ End(C N ) is a c-number solution of the quadratic equation (4.1) (for some R(λ) ∈ End(C N ⊗ C N ), solution of the Yang-Baxter equation). We also define (in the index notation (see also Footnote 2, page 7)) Twisted algebra. (4.18) The quadratic algebra B defined by (4.1) is a left co-ideal of the quantum algebra A for a given R-matrix (see also e.g. [58,13,18]), i.e. the algebra is endowed with a co-product ∆ : B → B ⊗ A [58]. Indeed, we define (in the index notation) where K(λ|θ 1 ) is given in (4.17) and in the index notation θ 2 )), then via expression (4.19): where the elements K k,l (λ|θ 1 ) can be also re-expressed in terms of the elements of the c-number matrix K and L when considering the realization (4.17). In our analysis in the subsequent section, we shall be primarily focusing on tensor representations of K and on the special case: L(λ) → R(λ),L(λ) →R(λ) and for the rest of the present subsection and subsections 5.1-5.3 we shall be considering R(λ) = λPř + P, whereř provides a representation of the A-type Hecke algebra H N (q = 1) and P is the permutation operator.
z,w are the generators of the quadratic algebra defined by (4.1). The exchange relations among the quadratic algebra generators are encoded in: whereř * andP are defined in (4.21), (4.22).
Notice that due to (4.17) in the case of the reflection algebra K (0) ∝ I when the c-number matrix K ∝ I. For m = 1 equation (4.28) provides the defining relations of a finite sub-algebra of the reflection algebra generated by K (1) x,y . x,y is the gl N algebra. Moreover, traces of K (m) commute with the elements K (1) x,y , (4.29) K (1) x,y , tr 1 (K (m) Proof. For the special class of solutionsř 12 which reads for the matrix elements as: K (m) x,y =K (m) τ (x),τ (y) . The latter commutator implies that K (0) is a c-number matrix (i.e. the entries of K (0) are c-numbers). Also, (4.28) becomes Given that K (0) is a c-number matrix we conclude that expression (4.30) for m = 1 provides a closed algebra formed by the elements of K (1) . For m = 1 and for K (0) ∝ I (4.30) gives the gl N exchange relations (up to an overall multiplicative factor, which can be absorbed by rescalling the generators). See also relevant results on tensor realizations of the sub-algebra in Corollary 5.17.

Open quantum spin chains & associated symmetries
We consider in what follows spin-chain like representations, i.e. we are focusing on tensor representations of the quadratic algebra (4.1) (see also (4.17)): We introduce the open monodromy matrix T 0;12...N (λ|{θ i }) ∈ End((C N ) ⊗(N +1) ) [58], which provides a tensor representation of (4.1): where {θ i } := {θ 1 , . . . , θ N } and the monodromy matrix T 0;12...N (λ|{θ i }) ∈ End((C N ) ⊗(N +1) ) is given by and satisfies (2.12). Also,T 0;12...N (λ|{θ i }) = T −1 0;12...N (−λ|{θ i }) in the case of the reflection algebra andT 0;12...N (λ|{θ i }) = T t0 0;12...N (−λ − N 2 |{θ i }) in the case of twisted algebra. We shall consider henceforth in expression (5.2) θ i = 0, i ∈ {1, . . . , N }. Such a choice is justified by the fact that we wish to construct local Hamiltonians, based on the fact that R(0) ∝ P (P the permutation operator), as will be transparent in the next subsection. The fact that the monodromy matrix T satisfies the RTT relation and K is a c-number solution of the refection equation guarantee that the modified monodromy T also satisfies the reflection equation, The elements of the modified monodromy matrix are T x,y (λ) = ∆ (N ) (K x,y (λ)) (see also discussion in the first paragraph of subsection 4.3). We also define the open or double row transfer matrix [58] as whereK is a solution of a dual quadratic equation 3 (4.1). Note that for historical reasons the space indexed by 0 is usually called the auxiliary space, whereas the spaces indexed by 1, 2, . . . , N are called quantum spaces. Notice also that the quantum indices are suppressed in the definitions of T,T and T for brevity.
To prove integrability of the open spin chain, constructed from the brace Rmatrix and the corresponding K-matrices we make use of the two important properties for the R-matrix, i.e. the unitarity and crossing-unitarity (2.7) and (2.8) respectively. Indeed, using the fact that T andK satisfy the quadratic and dual equations (4.1), and also R satisfies the fundamental properties (2.8), (2.9) it can be shown that (see [58,17] for detailed proofs on the commutativity of the open transfer matrices associated to both reflection and twisted algebras): We focus henceforth on the reflection algebra only, and we investigate the symmetries associated to the open transfer matrix for generic boundary conditions. The main goal in the context of quantum integrable systems is the derivation of the eigenvalues and eigenstates of the transfer matrix. This is in general an intricate task and the typical methodology used is the Bethe ansatz formulation, or suitable generalizations [43,29]. In the algebraic Bethe anastz scheme the symmetries of the transfer matrices and the existence of a reference state are essential components. When an obvious reference state is not available, which is the typical scenario when considering set-theoretic solutions, certain Bethe ansatz generalizations can be used. Specifically, the methodology implemented by Faddeev and Takhtajan in [29] to solve the XYZ model, based on the application of local gauge (Darboux) transformations at each site of the spin chain can be used. The Separation of Variables technique, introduced by Sklyanin [59], and recently further developed for open quantum spin chains [42], can also be employed, in particular when addressing the issue of Bethe ansatz completeness, but also as a further consistency check. Moreover, we plan to generalize the findings of [49] on the role of Drinfeld twists in the algebraic Bethe ansatz, for set-theoretic solutions. This will lead to new significant connections, for instance with generalized Gaudin-type models.

5.1.
Symmetries of the open transfer matrix. We shall prove in what follows some fundamental Propositions that will provide significant information on the symmetries of the double row transfer matrix (5.3). Note that henceforth we considerK ∝ I in (5.3).
Let us first prove a useful lemma for the braceř matrix. Proof. Let (X,ř) be our underlying set-theoretic solution. Recall thať r = x,y∈X e x,σx(y) ⊗ e y,τy(x) . Observe that where (x, y) ∈ W if and only if y = τ y (x). Notice that if (x, y) ∈ W then x = τ −1 y (y). Observe that τ −1 y (y) is always in the set X (because our sets are finite so the inverse of map τ is some power of map τ ), so for each y there exist x such that (x, y) ∈ W . This implies that that for each y in X there is exactly one x in X such that (x, y) is in W , we will denote this x as x [y] . This implies that tr 0 (ř n0 ) = y∈X e x [y] ,σx [y] (y) . We notice that σ x [y] (y) = x [y] , it follows from the fact that (x [y] , y) is in W . Consequently, tr 0 (ř n0 ) = y∈X e x [y] ,x [y] . We notice further that if (x, y) in W and (x, z) in W then y = z, so for each x there is exactly one y such that (x, y) is in W . Therefore, (where z equals elements x [y] for different y). Hence, that tr 0 (ř n0 ) = I where recall I is the identity matrix of dimension equal to the cardinality of X.
In the proof of Proposition 4.1 in [21] all the members of the expansion of the monodromy T (k) , were computed using the notation introduced above and the definition of the monodromy, and were expressed as: T (N −k) 0 = T (N −k−1) + r N 0 T (N −k) P 01 Π, and similarly:T Let us also express the c-number K-matrix (4.14) (derived up to an overall constant) as: K(λ) = λb + I, whereb = c b − κ 2 I (see also (4.14)), and recall hereK = I. Also, in accordance to the expansion of the monodromy matrix in the previous section we express the modified monodromy as a formal series expansion: λ k , then each term of the expansion is expressed as: After taking the trace and using the fact the tr 0 (ř N 0 ) = I we conclude for the first term of the expression (5.5) above: Analogous expression is derived for the second term in (5.5), given thatb → I and k + l = n in the expression above. The first three terms of (5.6) are clearly expressed only in terms of the elements of the B-type Hecke algebrař nn+1 , b 1 (recallb = c b − κ 2 I ). Let us focus on the last term: and [n k−1 , n j ) : 1 ≤ n k−1 , < ... < n j < N − 1, and [n k−1 , n j ] : 1 ≤ n k−1 , < ... < n j = N −1. The last three terms above (B, C, D) lead to the following expressions, after using the braid relation, involution and the fact that tr 0 (ř N 0 ) = I: The terms above clearly they depend only onř nn+1 , b 1 . Let us now focus on the more complicated first term of (5.7), and consider: .
It is thus clear that the factor tr 0 ř N 0 Ař N 0 is also expressed in terms of the elementsř nn+1 and b 1 . Indeed, then all the factors t (k) , k ∈ {1, . . . , 2N + 1} are expressed in terms ofř nn+1 , b 1 . However, the term 1NřN 0 can not be expressed in the general case in terms ofř nn+1 , b 1 . Notice that in the special case where b = I we obtain t (0) ∝ I ⊗N The local Hamiltonian of the system for instance is given by the following explicit expression We prove below a useful Lemma: , introduced on Proposition 5.2, satisfy the following relations with the A-type Hecke algebra H N (q = 1) elementsř nn+1 : Proof. The proof is straightforward for T (0) ,T (0) due to the form of T (0) ,T (0) and the use of the braid relation. For T (1) ,T (1) the proof is a bit more involved. Let us focus on T (1) acting oň r nn+1 , which can be explicitly expressed as Using the braid relations and the fact thatř 2 = I ⊗2 , we show that: Ař nn+1 = r n−1n A, Bř nn+1 =ř n−1n D, Cř nn+1 =ř n−1n C and Dř nn+1 =ř n−1n B, which immediately lead to T (1)ř nn+1 =ř n−1n T (1) , n ∈ {2, . . . N − 1}. The proof forT (1) is in exact analogy, so we omit the details here for brevity.
For the rest of the section we focus on representations of the the B-type Hecke algebra B N (q = 1, Q = 1).
Proposition 5.4. Let R(λ) = λPř + P, and K(λ) = λcb + I (c is an arbitrary constant), whereř and b provide a representation of the B-type Hecke algebra B N (q = 1, Q = 1), and P is the permutation operator. The elements of T (i) , i ∈ {0, 1}, introduced on Proposition 5.2, commute with the B-type Hecke algebra B N (q = 1, Q = 1) generators: Proof. We first write down explicitly the elements T (0) and T (1) . Recall that and from the proof of Proposition 5.2: (0) . Using Lemma 5.3 and the expressions just above we conclude: T (i) ,ř nn+1 = 0, n ∈ {1, . . . N − 2}, i ∈ {0, 1}. Moreover, using the quadratic relation of the B-type algebrař 12 b 1ř12 b 1 = b 1ř12 b 1ř12 and the form of T (0) we show that T (0) , b 1 = 0, while use of the braid relation and the form of T (0) lead to It now remains to show that T (1) ,ř N −1N = T (1) , b 1 = 0, the proof of the latter is more involved. Indeed, let us first focus on T (1) , b 1 , it is convenient in this case to express the first two terms of T (1) as a = a 1 + a 2 and b 1ř23 ...ř N −1NřN 0 . Using the quadratic relationř 12 b 1ř12b1 = b 1ř12 b 1ř12 , and the fact that b 2 = I we show that: We lastly focus on T (1) ,ř N −1N , it is convenient in this case as well to express the first two terms of T (1) as a =â 1 +â 2 and b =b 1 +b 2 , where we definê 1ř12 ...ř N −2N −1řN 0 . Using the braid relation and the fact thatř 2 = I ⊗2 we show that: And this concludes our proof.
Remark 5.7. The twisted co-products for the finite algebra generated by the element of T (1) , in the special case b = I can be expressed as follows, after recalling the notation introduced in the proof of Proposition 5.2: where r = Př,r =řP and P the permutation operator. After using expression (5.13), the brace relation and recalling that r = Př,r =řP, we have (5.14) Note that explicit expressions of the above co-products for (5.14) can be computed for the brace solution. We shall derive in the next subsection the co-products associated to Lyubashenko's solutions recovering the twisted co-products of Corollary 3.4.
An interesting direction to pursue is the derivation of analogous results in the case of the twisted algebras extending the findings of [11] on the duality between twisted Yangians and Brauer algebras [52] to include set-theoretic solutions. We aim at examining whether the corresponding transfer matrix can be expressed in terms of the elements of the Brauer algebra, and also check if the elements of the Brauer algebra commute with a finite sub-algebra of the twisted algebra. These findings will have significant implications on the symmetries of open transfer matrices providing valuable information on their spectrum.

5.2.
More examples of symmetries. In this subsection we present examples of symmetries of the double row transfer matrix partly inspired by the symmetries in [21], but also some new ones. Let (X,ř) be a set-theoretic solution, as usually we denoteř(x, y) = (σ x (y), τ y (x)). In all the examples below we assume that the solution (X,ř) is involutive, non-degenerate and finite. Also, we always assume thatK = I in (5.3).
The following Lemma is similar to Proposition 4.11 in [21], but here for the double row transfer matrix, we obtain a stronger result: Proof. Similarly as in the proof of Proposition 4.11 from [21] it can be shown that e ⊗N xi,xj commutes withř nn+1 , ∀n ∈ {1, . . . , N − 1}. The result now immediately follows from Proposition 5.2 and from the fact that for b = I, we have T (0) = I ⊗(N +1) and t (0) = I ⊗N .
We also present the following new examples of symmetries, different to the ones derived in [21]. Let us first introduce some invariant subsets of a set-theoretic solution. Let (X,ř) be an involutive, non-degenerate set-theoretic solution.
Definition 5.11. Let (X,ř) be a finite set-theoretic solution of the braid equation and let Y ⊆ X. Denoteř(x, y) = (σ x (y), τ y (x)). We say that Y is a σ-equivariant set if whenever x, y ∈ Y then σ x (y) and τ y (x) ∈ Y . To show thatř is a bijective function on Y × Y observe thatř has the zero kernel on X ⊗ X, so is injective on Y ⊗ Y . Notice thatř(Y ⊗ Y ) ⊆ Y ⊗ Y since Y is σ-equivariant set. Becauseř : Y ⊗ Y → Y ⊗ Y is injective thenř(Y ⊗ Y ) has the same cardinality as Y ⊗ Y , henceř : Y ⊗ Y → Y ⊗ Y is surjective and hence bijective.
Remark 5.13. We we choose σ-equivariant subsets of X which have pairwise empty intersections we get similar algebra of symmetries as in the previous Lemma.
Definition 5.14. Let z ∈ X. By the orbit of z we will mean the smallest set Y ⊆ X such that z ∈ Y and σ x (y) ∈ Y and τ x (y) ∈ Y , for all y ∈ Y, x ∈ X.
We have also the following symmetries: Lemma 5.15. Let (X,ř) be an involutive, non degenerate solution of the braid equation and let Q 1 , . . . , Q t be orbits of X. Define W p1,...,pt,q1,...,qt = {i 1 , i 2 , . . . , i n , j 1 , j 2 , . . . , j n : exactly p i elements among i 1 , i 2 , . . . , i n belong to the orbit Q i and exactly q i elements among j 1 , j 2 , . . . , j n belong to the orbit Q i for every i ≤ t}.
Proof. It follows from the fact that if (X,ř) is an involutive, non-degenerate settheoretic solution of the Braid equation then r(Q i , Q j ) ⊆ (Q i , Q j ) and it is a bijective map, for every i, j ≤ t.  x,y = 0, x, y ∈ X.
Proof. Recall from the notation introduced in the proof of Proposition 5.2 that T (0) = r 0N ...r 01 cb 0r01 ...r 0N ,r = PrP. Then using the fact that in the special case of Lyubashenko's solutions, r 0n = V −1 0 V n , we can explicitly write T (0) = V −N 0 cb 0 V N 0 , which is a c-number matrix and t (0) = tr 0 (cb 0 ), which immediately leads to t (0) , T (1) x,y = 0, and via Propositions 5.2 and 5.4 we arrive at (5.15). x,y of (5.13) are twisted co-products of gl N , and hence the corresponding double row transfer matrix t(λ) is gl N symmetric.
Proof. Recall that in the special case where b = I the quantity T (1) is given in (5.13). In the case of the special solutions of Proposition 3.2 recall thatř n0 = V −1 0 P 0n V 0 , then expression (5.13) simplifies to Recall also from Proposition 3.2 that V = x∈X e σ(x),x and P = x,y∈X e x,y ⊗ e y,x , then T (1) can be explicitly expressed as (we set for simplicity 2c = 1) The latter expression immediately provides the elements T (1) x,y = ∆ (N ) 1 (e σ(y),σ(x) ), where the twisted N co-product ∆ (N ) 1 of gl N is defined in Corollary 3.4, expression (3.13). Then due to Corollary 5.6 we deduce that the corresponding double row transfer matrix t(λ) is gl N symmetric And with this we conclude our proof (compare also with the results in Corollary 4.10 for K (0) = I).