Heisenberg Spin Chains with Boundaries and Quantum Groups

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MSc Mathematics

Master Thesis

Heisenberg Spin Chains with

Boundaries and Quantum Groups

Author: Supervisor:

Tijmen Veltman

dr. Jasper Stokman

Examination date:

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Abstract

In this paper I will present a study of one-dimensional Heisenberg spin chains with boundaries. First I will explain how this system can be described using a representation of Uq( ˆsl2). Then I will show how we can describe interactions in this system using

R-matrices. To study boundary effects, we will introduce K-matrices and describe their properties. Finally, we will study a subalgebra B ⊂ Uq( ˆsl2) known as the q-Onsager

algebra. Using the properties of this subalgebra, we will be able to construct these R-matrices and K-matrices explicitly for the lowest-dimensional case.

Title: Heisenberg Spin Chains with Boundaries and Quantum Groups Author: Tijmen Veltman, tijmenveltman@student.uva.nl, 5928605 Supervisor: dr. Jasper Stokman

Second Examiner: dr. Hessel Posthuma Examination date: April 11, 2016

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

Our main object of study is the one-dimensional Heisenberg spin chain with a finite number of particles and boundary conditions on each side. Here each pair of neigh-bouring particles has spin-spin interaction. This model is often studied with either infinitely many particles or periodic boundary conditions. These systems have been found to be integrable by the quantum inverse scattering method. In our case, we take a finite, open chain and introduce new boundary conditions that maintain the integrability of the system. Here we observe how particles near the sides interact with the boundaries and incorporate these interactions into our model.

We will study the state space of our system as a representation space of quantum affine sl₂, i.e. a Uq( ˆsl2)-module. Any interactions will be described by elements of quantum

affine sl2, acting as operators on (tensor products of) appropriate state spaces. We

will show that these operators can be chosen in such a way that they satisfy the so-called quantum Yang-Baxter equation, which in turn implies that the system remains integrable.

The interactions will be governed by the intertwining operator R acting on the tensor product of two neighbouring particle spaces, and the reflection operator K which describes the interaction of a boundary particle with the boundary of the system. We are particularly interested in intertwining operators R and reflection operators K that provide solutions to the quantum Yang-Baxter equation:

R12(z1/z2)R13(z1/z3)R23(z2/z3) = R23(z2/z3)R13(z1/z3)R12(z1/z2)

and the boundary Yang-Baxter equation, also known as the reflection equation: R12(z1/z2)K1(z1)R21(z2z1)K2(z2) = K2(z2)R12(z1z2)K1(z1)R21(z1/z2)

for complex parameters z1, z2, z3. These equations take their name from the

indepen-dent work of C. N. Yang from 1967 ([Yan67]), and R. J. Baxter from 1972 ([Bax72]). In one-dimensional systems like ours, the system is integrable if the Yang-Baxter equa-tions are satisfied. The equation also appears in the context of braids and knots, where Rij represents a crossing of strands i and j. We have the following equivalence for two

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This equivalence is also known as Reidemeister’s third move.

Similarly, the reflection equation expresses the following equivalence:

We will discuss these equivalences in more detail in later chapters.

These operators R and K act on (tensor products of) state spaces V (z), where V is a finite-dimensional complex vector space with representation parameter z ∈ C∗. For the most basic case where V is two-dimensional, we will show that under our representation we can find a solution R(z) that acts on V (z) ⊗ V (z) as

a(z)     1 0 0 0 0 0 q−1 0 0 q−1 z−1(1 − q−2) 0 0 0 0 1     and K(z) is represented as 1 0 0 _1−zz2−λ2_λ22 ! : V (z) → V (z−1)

where a(z) is a complex function and λ a nonzero complex parameter. Using the method of fusion, these can be used to recursively construct R- and K-matrices for higher dimensional representations. Finally, we will describe a method to find solutions of the reflection equation using representations of the q-Onsager algebra, a subalgebra of Uq( ˆsl2).

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1 Quantum Affine sl

₂

and its

representations

1.1 Introduction

Our main object of study, the Heisenberg spin chain, describes the spin-spin interaction of particles in the shape of a one-dimensional lattice. Each particle has a copy of Cn describing its spin state; the state of the full system lies in the Hilbert space (Cn)⊗N where N is the number of particles. Any observables of this system are represented by linear operators on this Hilbert space.

As an example, for n = 2 the basic building blocks of these operators are the Pauli matrices: σx = 0 1 1 0 , σy = 0 −i i 0 , σz= 1 0 0 −1

which form a basis of sl2(C) and act as linear operators on C2. More generally, these

three generators and their relations define a Lie algebra sl2 (generally omitting the

base field C) which can act on Cnthrough a Lie algebra representation.

In this thesis we will enrich the structure of sl2 in several ways. Most notably, we

will introduce a quantum deformation, turning our Lie algebra into a quasitriangular Hopf algebra. This will allow us to construct non-trivial intertwining operators R(z) for neighbouring particle spaces, serving as solutions to the Yang-Baxter equation ([Yan67], [Bax72]):

R12(z)R13(zw)R23(w) = R23(w)R13(zw)R12(z).

Here both sides are operators on a threefold tensor product space of the form A⊗B ⊗C, where the indices denote the two tensor spaces that the operator acts on. For instance, R12 acts nontrivally on A and B and acts as the identity on C.

1.2 Quantum Affine sl

2

There are several common notations and conventions when it comes to defining Quan-tum Affine sl2 and its representations. The following uses the conventions also used in

source [RSV16].

We take the affine algebra ˆsl2 generated by elements hi, ei, fi, where i = 0, 1 and their

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[hi, hj] = 0

[hi, ej] = aijej

[hi, fj] = −aijfj

[ei, fj] = δijhi

(ad(ei))3ej = 0 = (ad(fi))3fj.

Here ad denotes the adjoint representation, i.e. ad(x)y = [x, y]. Furthermore, δij is

the Kronecker delta (equal to 1 if i = j and to 0 otherwise) and the aij are entries

from the Cartan matrix 2 −2 −2 2

, functioning as ’weights’ on the space spanned by elements hi (i = 0, 1). This space ˆh:= Ch0⊕ Ch1 is a Cartan subalgebra of ˆsl2. We

define elements αi ∈ˆh ∗

(i = 0, 1) such that αi(hj) = aij.

Now to make a quantum deformation, take a number q = eη not equal to a root of unity (that is, η is not of the form 2απi with α a rational number). We define the quantum affine algebra Uq( ˆsl2) as the unital associative algebra with generators

qh = eηh (h ∈ ˆh), ei, fi (i = 0, 1) and defining relations:

q0= 1, qh+h0 = qhqh0 (h, h0 ∈ ˆh) qheiq−h = qαi(h)ei, qhfiq−h= q−αi(h)fi ∀h ∈ ˆh, [ei, fj] = δij qhi_{− q}−hi q − q−1 , 3 X n=0 (−1)n [n]q![3 − n]q! en_ieje3−ni = 0 (i 6= j) 3 X n=0 (−1)n [n]q![3 − n]q! f_infjfi3−n= 0 (i 6= j).

In this notation we use q-numbers, defined as [n]q:= qn− q−n q − q−1 = q n−1_{+ q}n−3_{+ . . . + q}1−n and [n]q! := n Y k=1 [k]q and [0]q! = 1.

We can further extend this algebra by adding a derivation element d to h, i.e. we set ˜

h _{:= h ⊕ Cd. Call the algebra generated by e}i, fi, hi, d ˜sl2. The extended quantum

affine algebra Uq( ˜sl2) ⊃ Uq( ˆsl2) is defined by the additional relations:

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qade0 = qae0qad, qadf0= q−af0qad;

qad commutes with e1, f1, qh (for h ∈ ˜h).

By the construction by Drinfeld and Jimbo (see [Kas95]), this quantum affine algebra is a Hopf algebra when we define:

• A co-unit: (ei) = 0 = (fi), (qh) = 1 • Comultiplication: ∆(ei) = ei⊗ 1 + q−hi ⊗ ei, ∆(fi) = fi⊗ qhi+ 1 ⊗ fi, ∆(qh) = qh⊗ qh • And an antipode: S(qh) = q−h, S(ei) = −q−hiei, S(fi) = −fiqhi for any h ∈ ˜h, i = 0, 1.

We can check that the algebra Uq(sl2), generated by elements qh1, e1, f1 is closed under

the Lie bracket and the Hopf algebra operations. The same goes for the algebra Uq( ˆsl2);

hence we have a chain of Hopf subalgebras:

Uq(sl2) ⊂ Uq( ˆsl2) ⊂ Uq( ˜sl2).

Moreover, we can construct a unique algebra homomorphism ϕz : Uq( ˆsl2) → Uq(sl2)

satisfying:

ϕz(qλh0) = q−λh1, ϕz(qλh1) = qλh1

ϕz(e0) = z−1f1, ϕz(e1) = z−1e1

ϕz(f0) = ze1, ϕz(f1) = zf1.

This homomorphism involves a parameter z ∈ C \ {0}, which will play an important role in constructing evaluation representations of Uq( ˆsl2).

1.3 Finite Dimensional Representations

Our representation of Uq( ˆsl2) is constructed based on the standard finite dimensional

representation of Uq(sl2). For any n ∈ Z≥0, take an n + 1-dimensional complex vector

space V with basis {v0, v1, . . . , vn}. This becomes a left Uq(sl2)-module when we define

the following homomorphism πV : Uq(sl2) → End(V ):

πV(qλh1_)v

i = qλ(n−2i)vi

πV(e1)vi = [i]q[n + 1 − i]qvi−1

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for i = 0, 1, . . . , n. Here we take v−1 = 0 = vn+1. We can now turn this into a

repre-sentation of Uq( ˆsl2) by composing with our algebra homomorphism ϕz. We define the

evaluation representation of Uq( ˆsl2) as πzV := πV ◦ ϕz : Uq( ˆsl2) → End(V ).

Denote this evaluation representation as Vn(z), where z is the evaluation parameter

and the representation homomorphism maps to an n + 1-dimensional complex vector space V . It is important to note that this representation is irreducible. Since e1 and

f1 cycle through all 1-dimensional subspaces of the form Cvi (and v−1 = 0 = vn+1),

for any v ∈ V we can find a k ∈ Z≥0 such that (e1)kv ∈ Cv0. After all, we know

that [i][n + 1 − i] 6= 0 as long as i = 1, 2, . . . , n. By then repeatedly applying f1,

we can reach any of the aforementioned one-dimensional subspaces, hence all of V . This means that V possesses no non-trivial Uq( ˆsl2)-invariant subspaces. Therefore this

evaluation representation Vn(z) is irreducible.

Furthermore, we will frequently be studying tensor products of Uq( ˆsl2)-modules. These

are especially important when considering R-matrices (which act on these tensor prod-ucts) and fusion of R- and K-operators. The action of Uq( ˆsl2) is defined as follows:

π_zV ⊗W₁_,z₂ : Uq( ˆsl2) → End(V ⊗ W )

π_zV ⊗W₁_,z₂ (X) := π_zV₁ ⊗ π_zW₂ ◦ ∆(X). We can extend this to multiple tensor products by setting

πV ⊗W ⊗U_z₁_,z₂_,z₃ (X) := π_zV₁ ⊗ π_zW₂ ⊗ πU_z₃ ◦∆ ⊗ id_U

q( ˆsl2)

◦ ∆(X) and so on. Note that

∆ ⊗ id_U q( ˆsl2) ◦ ∆ =id_U q( ˆsl2)⊗∆ ◦ ∆ thanks to the coassociativity property of the coproduct ∆.

The question rises whether or not these tensor representations of the form V (z1)⊗V (z2)

are irreducible. It turns out that this depends on the evaluation parameters z1 and z2.

We refer to a result by Vyjayanthi Chari and Andrew Pressley ([CP91].

First, define a q-string Sn(a) as the set {qn−1a, qn−3a, . . . , q1−na} ⊂ C where a ∈ C

and n ∈ N. Two such q-strings S1 and S2 are said to be in general position if at least

one of the following conditions holds:

(i) S1∪ S2 is not a q-string;

(ii) S1 ⊆ S2;

(iii) S2 ⊆ S1.

Theorem 1.3.1. (Chari, Pressley) A tensor product Vn1(a1) ⊗ · · · ⊗ Vnr(ar) is

irre-ducible as a Uq( ˆsl2)-module if and only if the q-strings Sn1(a1), . . . , Snr(ar) are pairwise

in general position.

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1.4 Quantum Double Construction

Following Drinfeld’s method as described in [Kas95], in this section we will exploit the Hopf algebra structure of Uq( ˜sl2) to realize it as a quantum double algebra. This

structure can then be used to construct a universal R-matix.

On the Cartan subalgebra ˜h_{= Ch}₀_{⊕ Ch}₁_{⊕ Cd we define a nondegenerate symmetric} bilinear pairing:

h·, ·i : ˜h× ˜h_{→ C;}

hh_i, hji = aij; hd, h0i = 1; hd, di = 0 = hd, h1i

where the aij are again the entries of the matrix

2 −2 −2 2

. We define two algebras (both of which are contained inside Uq( ˜sl2)) as follows:

Uq( ˜sl +

2) := Che0, e1, qh|h ∈ ˜hi

U_q( ˜sl−₂_{) := Chf}₀, f1, qh|h ∈ ˜hi.

To continue our construction we first define a Hopf pairing.

Definition 1.4.1. A Hopf pairing of two Hopf algebras A, B over a field k is a bilinear function

ϕ : A × B → k satisfying the following three compatibility conditions: (i) (Co)units: ϕ(a, 1) = (a); ϕ(1, b) = (b). (ii) (Co)multiplications: ϕ(aa0, b) =X (b) ϕ(a, b₍₂₎)ϕ(a0, b₍₁₎); ϕ(a, bb0) =X (a) ϕ(a(1), b)ϕ(a(2), b0). (iii) Antipodes: ϕ(S(a), b) = ϕ(a, S−1(b))

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Here we use the Sweedler notation, where we write ∆(a) =X (a) a(1)⊗ a(2) and (∆ ⊗ id) ◦ ∆(a) =X (a) a(1)⊗ a(2)⊗ a(3) = (id ⊗∆) ◦ ∆(a).

The latter equality is valid thanks to the coassociative property of ∆. Note that, thanks to the universal property of the tensor product, any bilinear function ϕ : A × B → k factors through A ⊗ B uniquely. Hence we can alternatively define a pairing as a linear function ϕ : A ⊗ B → k ([KRT97]).

We then have the following theorem by Drinfeld ([Kas95]):

• There exists a unique nondegenerate Hopf pairing ϕ : Uq( ˜sl + 2) ⊗ Uq( ˜sl − 2) → C such that: ϕ(qh, qh0) := q−hh,h0i (h, h0∈ ˜h) ϕ(ei, fj) := δij 1 q−1− q ϕ(ei, qh) = 0 = ϕ(qh, fi) (h ∈ ˜h, i = 0, 1).

With this pairing ϕ we can construct a so-called quantum double Dϕ(Uq( ˜sl + 2), Uq( ˜sl

− 2)).

This is a space isomorphic to Uq( ˜sl +

2) ⊗ Uq( ˜sl −

2), where the Hopf algebra structure is

determined by the following conditions:

• The canonical linear embeddings a 7→ a⊗1 and b 7→ 1⊗b of U_q( ˜sl+₂) resp. Uq( ˜sl − 2) into Uq( ˜sl + 2) ⊗ Uq( ˜sl −

2) are Hopf algebra morphisms.

• For a ∈ Uq( ˜sl +

2) and b ∈ Uq( ˜sl −

2), the multiplication is defined by:

(a ⊗ 1)(1 ⊗ b) = a ⊗ b, (1 ⊗ b)(1 ⊗ a) = X

(a),(b)

ϕ(S−1(a(1)), b(1))ϕ(a(3), b(3))a(2)⊗ b(2).

The next result is also due to Drinfeld (see also [KS97]):

• The corresponding quantum double D_ϕ(Uq( ˜sl

+ 2), Uq( ˜sl

− 2))

has the two-sided ideal I generated by

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(where h ∈ ˜h) as a Hopf ideal, and U_q( ˜sl₂) ∼= Dϕ(Uq( ˜sl + 2), Uq( ˜sl − 2))/I

as Hopf algebras, where the isomorphism is given by ei7→ ei⊗ 1 + I,

fi7→ 1 ⊗ fi+ I,

qh7→ qh⊗ 1 + I.

We will use this quantum double structure in the next section to show the existence of a universal R-matrix.

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2 R-matrices

2.1 Introduction

Consider a spin chain consisting of N particles. In later sections, we will also introduce boundary properties to our system. Diagrammatically, this looks like the following:

V (z1) • V (z•2) · · ·V (z•N)

Here V (zi) is the evaluation representation space describing the internal spin of

par-ticle i for i = 1, . . . , N . To study interactions between parpar-ticles i, j, we introduce a linear operator acting non-trivially on V (zi) and V (zj) only, and as the identity on

all the other V -spaces. The interactions at the boundaries will be discussed later; for now, we will focus on the ’bulk’ of the system where only pairwise interactions occur. For a Hopf algebra H, a universal R-matrix is an invertible element R ∈ H ⊗ H such that for all X ∈ H we have:

∆op(X)R = R∆(X)

where ∆ denotes the comultiplication and ∆op = PH,H◦ ∆ its opposite (where PH,H

is the permutation operator acting on H ⊗ H).

If we can find such a universal R-matrix for Uq( ˆsl2), we can use a representation (like

the ones seen in the previous chapter) to let it act on a tensor space of the form V ⊗ V . This will serve as our interaction operator between two neighbouring spaces. Generally, we can write such an interaction operator as:

R =X

k

Ak⊗ Bk

where Ak, Bk ∈ End(V ). Indices will indicate on which tensor legs the operator is

acting. Explicitly, for i < j we define:

Rij(zi, zj) :=

X

k

idV (z1)⊗ · · ·⊗idV (zi−1)⊗Ak⊗idV (zi+1)⊗ · · ·⊗idV (zj−1)⊗Bk⊗idV (zj+1)⊗ · · ·⊗idV (zN)

as an element in V (z1) ⊗ · · · ⊗ V (zN), where the operator acts non-trivially only on

V (zi) and V (zj).

Analogously, we can define Rji(zj, zi) := PV (zj),V (zi)◦ Rij(zi, zj) ◦ PV (zi),V (zj), where

PV,W is the permutation operator switching the spaces V and W inside the tensor

product. We will later see that a suited choice of Rij(zi, zj) in fact only depends of

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This system is integrable if these interaction operators satisfy the Yang Baxter Equa-tion (see also [CP94], secEqua-tion 7.5 for an in-depth discussion). This equaEqua-tion takes the following form:

R12(z1/z2)R13(z1/z3)R23(z2/z3) = R23(z2/z3)R13(z1/z3)R12(z1/z2).

Here both sides are linear operators acting on V (z1) ⊗ V (z2) ⊗ V (z3), where Rij acts

as R on the ith and jth components; on the third it acts merely as the identity. The physical interpretation consists of three particles, each represented by a copy of the space V with respective spectral parameters z1, z2, z3. An interaction between two

particles is given by the operator R acting on the tensor product of the two parti-cle spaces. In the situation of the YBE, we have three partiparti-cles interacting pairwise. The equation guarantees that we can, to an extent, interchange the order of interac-tion without altering the final result. Graphically, we have the following equivalence (reminiscent of Reidemeister’s third move):

Here we view time as the vertical axis, moving upward. On the left, we first see particles 1 and 2 interacting, followed by 13, followed by 23. The right diagram has the same interactions in opposite order. Representing particle i by its evaluation representation space V (zi), we precisely obtain the YBE. We can also eliminate one of the three

parameters and write:

R12(z)R13(zw)R23(w) = R23(w)R13(zw)R12(z).

Here we made the substitutions:

z = z1/z2

w = z2/z3

implying that zw = z1/z3.

Ultimately, our goal is to create an integrable system of operators τ (z) on V (z1) ⊗ · · · ⊗

V (zN) where these τ -operators commute for different spectral parameter values, i.e.

τ (z1)τ (z2) = τ (z2)τ (z1) for all z1, z2∈ C. To construct such a family of operators, we

can define the monodromy matrix T (z0) ∈ End(W ⊗ V⊗N), where W is an auxiliary

space isomorphic to V , as:

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where (for i = 1, 2, . . . , N ) R0iacts nontrivially on the 0th tensor component (being W ,

our auxiliary space isomorphic to V ) and the ith (i.e. V (zi)). This operator represents

pairwise interactions between the auxiliary space and each of the N particle spaces. We set T (z) ∈ End(W ⊗ V⊗n) and (defined in the same way) T0(w) ∈ End(W ⊗ V⊗n). As a consequence of the YBE for R-matrices, these monodromy matrices T, T0 satisfy:

R(z/w)T (z)T0(w) = T0(w)T (z)R(z/w)

where both sides are elements of End(W ⊗ W ⊗ V⊗n) and R acts on W ⊗ W .

We now define the transfer matrix by taking the trace of T over the auxiliary space W :

τ (z) = TrWT (z) ∈ End(V⊗N).

Thanks to the so-called RT T -equation above, these transfer matrices commute for different spectral parameter values. Hence we can view the τ (z) as a generating func-tion for an infinite number of commuting charges. We can choose one of these as the Hamiltonian, which can then be diagonalised using the algebraic Bethe Ansatz. For further details regarding this construction, see [Fad98].

To find solutions to the YBE, it is sufficient to find an operator R that satisfies the following quasitriangularity properties:

(∆op(X))R = R∆(X) for any X ∈ Uq( ˜sl2); (2.1.1)

(∆ ⊗ id)R = R13R23; (2.1.2)

(id ⊗∆)R = R13R12; (2.1.3)

where ∆op is the opposite comultiplication, equal to P ◦ ∆ where P is the permu-tation operator. It follows that R then also satisfies the YBE. For a proof we refer to the course Quantum Groups and Knot Theory by Eric Opdam and Jasper Stokman.

2.2 R-matrices: Existence And Construction

Recall that we can view Uq( ˜sl2) as a quotient of a quantum double algebra, i.e.

Uq( ˜sl2) ∼= Dϕ(Uq( ˜sl + 2), Uq( ˜sl

− 2))/I

as Hopf algebras. The pairing ϕ on which this double is based is nondegenerate, hence for any basis {a1, a2, . . .} of Uq( ˜sl

+

2) we can find a dual basis {b1, b2, . . .} of Uq( ˜sl − 2), i.e.

ϕ(ai, bj) = δij. Consider the following formal sum:

˜ R =X

i

(ai⊗ 1) ⊗ (1 ⊗ bi)

where each summand is an element of Dϕ(Uq( ˜sl + 2), Uq( ˜sl

−

2))⊗2. This summation is

in fact independent of the chosen basis and cobasis, and it satisfies the three quasi-triangularity properties 2.1.1, 2.1.2 and 2.1.3. Hence this is a universal R-matrix for Dϕ(Uq( ˜sl

+ 2), Uq( ˜sl

−

2))⊗2. This means that its projection onto Uq( ˜sl2) ∼= Dϕ(Uq( ˜sl + 2), Uq( ˜sl

− 2))/I

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satisfies the properties that make it into a universal R-matrix for Uq( ˜sl2).

It should be noted that this expression of ˜R is a formal sum with infinitely many terms, hence it can only be seen as a formal R-matrix. However, using a suitable class of rep-resentations of Uq( ˜sl2) we are able to map this sum to a well-defined endomorphism of

a tensor product of representation spaces.

Explicitly, the formal sum takes the following form: ˜

R = qc⊗d+d⊗c+Pri=1xi⊗xi X

k

ak⊗ ak

!

where xi is an orthonormal basis of h, akis a basis of Uq(ˆn+) = Che0, e1i and ak is the

corresponding dual basis of Uq(ˆn−) = Chf0, f1i. That is, ϕ(ak, am) = δkm.

For more information, see Etingof et al. ([EFK98]), section 9.4.

The action of ˜R under our evaluation representation cannot be defined, seeing as our representation homomorphism πV_z is only defined on Uq( ˆsl2) but not on Uq( ˜sl2).

However, we can define the truncated R-matrix: ˆ R = q−c⊗d−d⊗cR = q˜ Pri=1xi⊗xi X k ak⊗ ak ! ∈ U_q( ˆsl+₂) ˆ⊗U_q( ˆsl−₂).

Define a homomorphism p : Uq( ˆsl2) → Uq(sl2) such that:

p(qλh1_{) = q}λh1_, p(e1) = e1, p(f1) = f1, p(qλh0_{) = q}−λh1_, p(e0) = f1, p(f0) = e1.

Furthermore, define an inner automorphism Dz: Uq( ˆsl2) → Uq( ˆsl2) as:

Dz(X) := q

log z log qd_Xq−

log z log qd_.

It is easy to check that this automorphism satisfies: Dz(e0) = z−1e0

Dz(f0) = zf0

Dz = id for all other generators.

Furthermore, we have ϕz = p ◦ Dz, where ϕz is our algebra homomorphism ϕz :

U_q( ˆsl₂) → Uq(sl2) from subsection 1.2. We now set:

ˆ R(z) := (Dz⊗ id) ˆR ∈ Uq( ˆsl + 2) ˆ⊗Uq( ˆsl − 2).

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For this object, we have: ˆ R(z) = (Dz⊗ id)( ˆR) = (qlog zlog qd⊗ 1) ˆ_R(q− log z log qd⊗ 1) = (1 ⊗ q−log zlog qd_)∆op_(q log z log qd_{) ˜}_Rqc⊗d+d⊗c_∆(q− log z log qd_{)(1 ⊗ q} log z log qd₎ = (1 ⊗ q−log zlog qd_{) ˜}_R∆(q log z log qd_)qc⊗d+d⊗c_∆(q− log z log qd_{)(1 ⊗ q} log z log qd₎

(seeing as ˜R∆(X) = ∆op(X) ˜R for all X ∈ Uq( ˜sl2))

= (1 ⊗ q− log z log qd_{) ˜}_Rqc⊗d+d⊗c_{(1 ⊗ q} log z log qd₎ = (1 ⊗ q− log z log qd_{) ˆ}_{R(1 ⊗ q} log z log qd₎ = (id ⊗D_z−1)( ˆR).

This implies that we can write: ˆ

R(z1/z2) = (Dz1 ⊗ Dz2) ˆR.

Proposition 2.2.1. These ˆR and ˜R(z) have the following properties: (i) ˆR(z)q−c⊗d−d⊗c(Dz⊗ id)(∆(X))qc⊗d+d⊗c= (Dz⊗ id) (∆op(X)) ˆR

(ii) (∆ ⊗ id)( ˆR(z)) =q−1⊗c⊗dRˆ13(z)q1⊗c⊗d ˆR23(z) (iii) (id ⊗∆)( ˆR(z)) = q−d⊗c⊗1Rˆ13(z)qd⊗c⊗1 ˆR12(z) (iv) ˆR12(z)q−1⊗c⊗dRˆ13(zw)q1⊗c⊗dRˆ23(w) = ˆR23(w)q−d⊗c⊗1Rˆ13(zw)qd⊗c⊗1Rˆ12(z). Proof. (i) ˆ R(z)q−c⊗d−d⊗c(Dz⊗ id)(∆(X))qc⊗d+d⊗c =(Dz⊗ id) ˆRq−c⊗d−d⊗c∆(X)qc⊗d+d⊗c =(Dz⊗ id) ˜R∆(X)qc⊗d+d⊗c =(Dz⊗ id) ∆op(X) ˜Rqc⊗d+d⊗c =(Dz⊗ id) (∆op(X)) ˆR

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(ii) (∆ ⊗ id)( ˆR(z)) =(∆ ⊗ id)(id ⊗D_z−1)( ˆR) =(id ⊗ id ⊗D_z−1)(∆ ⊗ id)(q−c⊗d−d⊗cR)˜ =(id ⊗ id ⊗D_z−1) q−c⊗1⊗d−1⊗c⊗d−d⊗1⊗c−1⊗d⊗cR˜13R˜23 (Here we used the identity (∆ ⊗ id) ˜R = ˜R13R˜23.

For more information, see [EFK98].) =(id ⊗ id ⊗Dz−1)

q−1⊗c⊗dq−c⊗1⊗d−d⊗1⊗cR˜13q1⊗c⊗dq−1⊗c⊗d−1⊗d⊗cR˜23

(since c is central and ˜R13 only acts on the first and third tensor leg,

we have q−1⊗d⊗cR˜13= ˜R13q−1⊗d⊗c) =(id ⊗ id ⊗D_z−1) q−1⊗c⊗dRˆ13q1⊗c⊗dRˆ23 =q−1⊗c⊗dRˆ13(z)q1⊗c⊗dRˆ23(z).

(iii) Somewhat analogously: (id ⊗∆)( ˆR(z)) =(id ⊗∆)(Dz⊗ id) ˆR =(Dz⊗ id ⊗ id)(id ⊗∆)q−c⊗d−d⊗cR˜ =(Dz⊗ id ⊗ id) q−c⊗d⊗1−c⊗1⊗d−d⊗c⊗1−d⊗1⊗cR˜13R˜12 =(Dz⊗ id ⊗ id) q−d⊗c⊗1q−c⊗1⊗d−d⊗1⊗cR˜13qd⊗c⊗1q−d⊗c⊗1−c⊗d⊗1R˜12 =(Dz⊗ id ⊗ id) q−d⊗c⊗1Rˆ13qd⊗c⊗1Rˆ12 =q−d⊗c⊗1Rˆ13(z)qd⊗c⊗1Rˆ12(z).

(iv) We start with the Yang-Baxter Identity for ˜R: ˜

R12R˜13R˜23= ˜R23R˜13R˜12

Writing out the definition of ˜R:

qc⊗d⊗1+d⊗c⊗1Rˆ12qc⊗1⊗d+d⊗1⊗cRˆ13q1⊗c⊗d+1⊗d⊗cRˆ23

=q1⊗c⊗d+1⊗d⊗cRˆ23qc⊗1⊗d+d⊗1⊗cRˆ13qc⊗d⊗1+d⊗c⊗1Rˆ12

We will now apply the operator Dzw⊗ Dw⊗ id left and right. When applied to

ˆ

R12, this operator is equal to:

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Hence we obtain:

qc⊗d⊗1+d⊗c⊗1Rˆ12(z)qc⊗1⊗d+d⊗1⊗cRˆ13(zw)q1⊗c⊗d+1⊗d⊗cRˆ23(w)

=q1⊗c⊗d+1⊗d⊗cRˆ23(w)qc⊗1⊗d+d⊗1⊗cRˆ13(zw)qc⊗d⊗1+d⊗c⊗1Rˆ12(z)

Since c is central, we can again switch certain factors to obtain:

qc⊗d⊗1+d⊗c⊗1+c⊗1⊗dRˆ12(z)qd⊗1⊗c+1⊗d⊗cRˆ13(zw)q1⊗c⊗dRˆ23(w) =qd⊗1⊗c+1⊗c⊗d+1⊗d⊗cRˆ23(w)qc⊗1⊗d+c⊗d⊗1Rˆ13(zw)qd⊗c⊗1Rˆ12(z) qc⊗d⊗1+d⊗c⊗1+c⊗1⊗dRˆ12(z)(∆ ⊗ id)qd⊗cRˆ13(zw)q1⊗c⊗dRˆ23(w) =qd⊗1⊗c+1⊗c⊗d+1⊗d⊗cRˆ23(w)(id ⊗∆)qc⊗dRˆ13(zw)qd⊗c⊗1Rˆ12(z) qc⊗d⊗1+d⊗c⊗1+c⊗1⊗d(∆op⊗ id)qd⊗cRˆ12(z) ˆR13(zw)q1⊗c⊗dRˆ23(w) =qd⊗1⊗c+1⊗c⊗d+1⊗d⊗c(id ⊗∆op)qc⊗dRˆ23(w) ˆR13(zw)qd⊗c⊗1Rˆ12(z) qc⊗d⊗1+d⊗c⊗1+c⊗1⊗dqd⊗1⊗c+1⊗d⊗cRˆ12(z) ˆR13(zw)q1⊗c⊗dRˆ23(w) =qd⊗1⊗c+1⊗c⊗d+1⊗d⊗cqc⊗1⊗d+c⊗d⊗1Rˆ23(w) ˆR13(zw)qd⊗c⊗1Rˆ12(z) qd⊗c⊗1Rˆ12(z) ˆR13(zw)q1⊗c⊗dRˆ23(w) = q1⊗c⊗dRˆ23(w) ˆR13(zw)qd⊗c⊗1Rˆ12(z)

Multiplying on the left by q−1⊗c⊗d−d⊗c⊗1 and using again that c is central:

ˆ

R12(z)q−1⊗c⊗dRˆ13(zw)q1⊗c⊗dRˆ23(w) = ˆR23(w)q−d⊗c⊗1Rˆ13(zw)qd⊗c⊗1Rˆ12(z)

as desired.

Recall our definition π_zV := πV ◦ ϕ_z. For two finite-dimensional complex vector spaces V and W we let

ˆ

RV W(z/w) := (π_zV ⊗ πW_w ) ˆR : V ⊗ W → V ⊗ W.

This is a well-defined invertible operator, which indeed only depends on the quotient z/w as we saw previously. This turns identity (iv) of theorem 2.2.1 into the following form of the quantum Yang-Baxter Equation:

ˆ

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where U is a finite-dimensional complex vector space, x is a complex parameter and both sides of the equation are linear operators on U ⊗ V ⊗ W . Here we use the fact that c acts as 0 in all evaluation representations (for further details, see [EFK98]). Define ˇRV W(z/w) := PV WRˆV W(z/w) : V ⊗W → W ⊗V . Here PV W : V (z)⊗W (w) → W (w) ⊗ V (z) denotes the permutation operator, i.e. PV W(v ⊗ u) = (u ⊗ v) for v ∈ V , u ∈ W . This tensor space is also a Uq( ˆsl2)-module by defining πz,wV W := (πzV ⊗ πwW) ◦ ∆ :

Uq( ˆsl2) → End(V ⊗ W ).

The R-matrix satisfies the intertwining property for Uq( ˆsl2):

ˇ

RV W(z/w)πV W_z,w (X) = πW V_w,z (X) ˇRV W(z/w)

for all X ∈ Uq( ˆsl2). If the tensor space V ⊗ W is an irreducible Uq( ˆsl2)-module, then

by Schur’s lemma we know that such an intertwiner is unique up to a scalar. Hence if z, w are such that the corresponding Uq( ˆsl2)-module is irreducible, then the

inter-twining property defines the R-matrix up to a scalar. Constructing a linear operator V ⊗ W → W ⊗ V that satisfies the intertwining property generally comes down to solving a finite linear system, making finding an R-matrix a relatively easy task.

2.3 Example: R-Matrices For Two-Dimensional

Representations

In the case where our representation space V is two- dimensional, being the simplest non-trivial case, we can make this R-matrix explicit as an element in End(V (z1) ⊗

V (z2)). Here we denote by V (z) the Uq( ˆsl2)-module V with evaluation

representa-tion πV_z. We have seen in the previous section that V (z1) ⊗ V (z2) is irreducible as a

Uq( ˆsl2)-module if and only if z_z1₂ 6= ±q. Hence for such a module, it suffices to construct

a Uq( ˆsl2)-intertwiner. By Schur’s lemma, such an intertwiner will be unique up to a

scalar and hence equal to a universal R-matrix acting on V (z1) ⊗ V (z2).

Take W equal to V and let ˇRV W(z1/z2) denote a Uq( ˆsl2)-intertwiner in End(V ⊗ W ).

For ˇRV W(z1/z2) to function as an intertwiner, we require:

ˇ RV W(z1/z2)πzV W1,z2(X) = π V W z2,z1(X) ˇR V W_(z 1/z2)

for all X ∈ Uq( ˆsl2). For qh1 in particular, we obtain:

πz1,z2(q

h1_{) ˇ}_RV W_(z

1/z2)(v±⊗v±) = ˇRV W(z1/z2)πz1,z2(q

h1_)(v

±⊗v±) = q±2RˇV W(z1/z2)(v±⊗v±)

which means that ˇRV W(z1/z2)v ∈ C(v+⊗ v+) if and only if v ∈ C(v+ ⊗ v+), and

likewise for C(v−⊗ v−). Similarly, we have:

πz1,z2(q

h1_{) ˇ}_RV W_(z

1/z2)(v±⊗v∓) = ˇRV W(z1/z2)πz1,z2(q

h1_)(v

±⊗v∓) = ˇRV W(z1/z2)(v±⊗v∓)

which is only possible if ˇRV W(z1/z2)(v±⊗ v±) ∈ C(v+⊗ v−) ⊕ C(v−⊗ v+). Therefore

the matrix takes the form:

ˇ RV W(z1/z2) =     a(z1/z2) 0 0 0 0 b(z1/z2) c(z1/z2) 0 0 d(z1/z2) e(z1/z2) 0 0 0 0 f (z1/z2)    

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where both a(z1/z2) and f (z1/z2) are nonzero.

Using the intertwining property ˇ RV W(z1/z2)πzV W1,z2(X) = π V W z2,z1(X) ˇR V W_(z 1/z2)

which holds for all X ∈ Uq( ˆsl2), we obtain a set of linear relations for a, b, c, d, e, f with

unique solution (up to a scalar function of z1/z2):

ˇ RV W(z1/z2) = a(z1/z2)     1 0 0 0 0 z1 z2(1 − q −2₎ _q−1₍z1 z2 − 1) 0 0 q−1(z1 z2 − 1) 1 − q −2 ₀ 0 0 0 1     .

For each value of z1/z2 and each complex function a(z1/z2) this yields an intertwiner.

Since V (z1) ⊗ V (z2) is irreducible, by Schur’s lemma we can conclude that no other

intertwiner exists. Hence we have found the explicit expression of the universal R-matrix acting on V (z1) ⊗ V (z2).

2.4 Higher Dimensions And Fusion Of R-Matrices

We would now like to extend our intertwining operators to higher dimensional modules. The fusion method allows us to recursively construct higher dimensional R-operators using reducibility of certain evaluation representations. The basic principle is to con-struct a tensor product of fundamental representations and decompose this product. Even though such a tensor product is irreducible for generic parameter values (recall theorem 1.3.1), we can choose specific values where the product decomposes into at least one subspace of higher dimension than the original representation spaces. More concretely (here we use the reasoning and notation of [Jim89]), suppose we have a family of parametrised R-matrices:

ˇ

RV V0(z) : V ⊗ V0 → V0⊗ V satisfying the Yang-Baxter Equation in the following form:

( ˇRV2V3_{(z)⊗I)(I⊗ ˇ}_RV1V3_{(zw))( ˇ}_RV1V2_{(w)⊗I) = (I⊗ ˇ}_RV1V2_{(w))( ˇ}_RV1V3_{(zw)⊗I)(I⊗ ˇ}_RV2V3_(z))

(2.4.1) as an equality in Hom(V1 ⊗ V2 ⊗ V3, V3 ⊗ V2 ⊗ V1). Note that, for z-independent

subspaces Wi⊂ Vi such that:

ˇ

RViVj_(z)(W

i⊗ Wj) ⊂ Wj⊗ Wi, (2.4.2)

equation 2.4.1 will still hold if we restrict each ˇRViVj _{to W}

i⊗ Wj. Furthermore, if we define: ˇ RV ⊗V0,V00(z) := ( ˇRV V0(zz1) ⊗ I)(I ⊗ ˇRV 0_V00 (zz2)) : V ⊗ V0⊗V00→ V00⊗V ⊗V0 (2.4.3)

for fixed z1, z2, then equation 2.4.1 remains valid when we replace ˇRV1Vi with ˇRV1⊗V

0 1,Vi

for i = 2, 3. Similarly, we can define: ˇ

RV00,V ⊗V0(z) := (I ⊗ ˇRV00V0(zz1))( ˇRV

00_V

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such that the YBE still holds when we replace ˇRViV3 _{with ˇ}_RVi,V3⊗V30 for i = 1, 2.

In our case, we take the following subspace:

W = ˇRV0V(z2/z1)(V0⊗ V ) ⊂ V ⊗ V0.

Equation 2.4.1 gives us: ˇ RV ⊗V0,V00(z)(W ⊗ V00) = (by definition 2.4.3) ( ˇRV V00(zz1) ⊗ I)(I ⊗ ˇRV 0_V00 (zz2))( ˇRV 0_V (z2/z1) ⊗ I)(V0⊗ V ⊗ V00)

= (by the YBE) (I ⊗ ˇRV0V(z2/z1))( ˇRV 0_V00 (zz2) ⊗ I)(I ⊗ ˇRV V 00 (zz1))(V0⊗ V ⊗ V00) ⊂ V00⊗ W.

This means that condition 2.4.2 is satisfied when we take: W1 := ˇRV 0 1V1(z₂/z₁)(V0 1⊗ V1) ⊂ V1⊗ V10; W2 := V2; W3 := V3.

Similarly, we can show that ˇRV00,V ⊗V0(z)(V00⊗ W ) ⊂ W ⊗ V00_{. This space W is in}

general not a proper subspace, but we will see that for specific choices of z2/z1, our

evaluation representation is reducible and W becomes a proper subspace of the tensor representation space. In this case, we find non-trivial R-matrices:

ˇ

RW V00(z) = ˇRV ⊗V0,V00(z)|W ⊗V00;

ˇ

RV00W(z) = ˇRV00,V ⊗V0(z)|V00_⊗W.

Finally, we will perform this fusion process on our evaluation representations of Uq( ˆsl2).

Denote the representation space corresponding to spin k ∈ 1₂Z≥0 by Vk, where Vk =

L2k

i=0vki. By abuse of notation, we also write Rk` instead of RV

k_V`

for the R-matrix acting on Vk⊗ V`_{. Finally, let P}k` _{: V}k_{⊗ V}` _{→ V}`_{⊗ V}k _{denote the permutation}

operator.

Proposition 2.4.1. We have the following splitting intertwiners: (i) The linear map ιk: Vk+12 ,→ V

1 2 ⊗ Vk defined by ιk vk+ 1 2 n = q12nv 1 2 0 ⊗ vkn+ q −1 2(n−2k−1)[n]_qv 1 2 1 ⊗ vkn−1 defines a Uq( ˆsl2)-intertwiner ιk: Vk+ 1 2(z) ,→ V 1 2(zq−k) ⊗ Vk(zq 1 2).

(ii) The linear map jk := P12kιk : Vk+ 1 2 ,→ Vk⊗ V 1 2 defines a U_q( ˆsl₂)-intertwiner jk: Vk+12(z) ,→ Vk(zq− 1 2) ⊗ V 1 2(zqk).

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Proof. The proof is a straightforward calculation that the intertwining relations hold for X = qhi_{, e}

i, fi. We show the intertwining of ι and πz(e0) as an example:

ιkπk+ 1 2 z (e0)v k+1₂ n =ιk z−1f1· v k+1₂ n =z−1ιk vk+ 1 2 n+1 =z−1 q12(n+1)(v 1 2 0 ⊗ v k n+1) + q −1 2(n−2k)[n + 1]_q(v 1 2 1 ⊗ v k n) =z−1 q12(n+1)(v 1 2 0 ⊗ v k n+1) + qk− 1 2nq n+1_{− q}−n−1 q − q−1 (v 1 2 1 ⊗ v k n) =z−1 q12(n+1)(v 1 2 0 ⊗ v k n+1) + qk− 1 2n qn+ q−1q n_{− q}−n q − q−1 (v 1 2 1 ⊗ v k n) =z−1 q12(n+1)(v 1 2 0 ⊗ vn+1k ) + qk+12n+ qk−1− 1 2nq n_{− q}−n q − q−1 (v 1 2 1 ⊗ vkn) =z−1 qk+12n(v 1 2 1 ⊗ v k n) + q 1 2(n+1)(v 1 2 0 ⊗ v k n+1) + q −1 2q− 1 2(n−2k−1)[n]_qq−1(v 1 2 1 ⊗ v k n) =z−1qk(f1⊗ 1) + z−1q− 1 2(qh1 ⊗ f₁) · q12n(v 1 2 0 ⊗ v k n) + q− 1 2(n−2k−1)[n]_q(v 1 2 1 ⊗ v k n−1) = π 1 2 zq−k⊗ πk zq12 ◦ ∆(e₀)ιkvk+ 1 2 n .

Using the ι-intertwiner in particular, we find the following relation between R-matrices of different dimensions.

Proposition 2.4.2. For k, ` ∈ 1₂Z≥0 we have the fusion formula

(ιk⊗ id_V`)Rk+ 1 2,`(z) = R 1 2,` 13 (q −k_z)Rk` 23(q 1 2z)(ιk⊗ id V`)

where the equality holds as an identity of linear maps Vk+12 ⊗ V`→ V 1

2 ⊗ Vk⊗ V`.

Proof. Due to the quasitriangular properties of R, we have: ((∆ ⊗ id)Rk+12,`(z₁/z₂)) = R 1 2,` 13(q −k_z 1/z2)R23k`(q 1 2z₁/z₂).

By representing these as linear operators on V 12(q−kz₁) ⊗ Vk(q 1 2z₁) ⊗ V`(z₂) and pre-composing with ιk⊗ id_V` : Vk+ 1 2(z₁) ⊗ V`(z₂) → V 1 2(q−kz₁) ⊗ Vk(q 1 2z₁) ⊗ V`(z₂), we obtain: (∆ ⊗ id)Rk+12,` (ιk⊗ id_V`) = R 1 2,` 13 (q −k_z 1/z2)Rk`23(q 1 2z₁/z₂)(ιk⊗ id_V`)

or, after applying proposition 2.4.1 on the first two tensor legs: (ιk⊗ id_V`) Rk+12,`(z₁/z₂) = R 1 2,` 13 (q −k z1/z2)Rk`23(q 1 2z₁/z₂)(ιk⊗ id V`).

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3 K-matrices

3.1 Introduction

Using R-matrices, we can now very well describe spin interactions between neighbour-ing particles. However, this fails to describe what happens at the two boundaries of our spin chain. In some models, the chain is assumed to be circular, eliminating the need to specify these boundary conditions. In our case, we will set up reflectional walls on each side, allowing the boundary particles to interact with themselves via these walls:

|V (z•1)V (z•2)· · ·V (z•N)|

3.2 K-matrices: Properties

We have already seen that, in order to have an integrable system, our R-matrices need to satisfy the quantum Yang-Baxter equation:

Physically, this equation corresponds to the equivalence of two different orders of inter-action for three particles. This suffices for the bulk of our system; at the boundaries, however, the situation is different. Here we introduce an operator K to account for the interaction of a particle with the boundary. For a particle with representation space V (z), this operator K is dependent on the parameter z. Moreover, since a reflection changes the direction of the particle, its parameter is flipped from z to z−1. Hence we have reflection operators K(z) : V (z) → V (z−1). Analogously to the YBE, we want to guarantee that two sets of reflections differing only chronologically yield the same result. Graphically, we require the following equivalence:

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Here one can see the vertical axis as the flow of time, moving upward. This translates directly to the reflection equation (also known as the boundary Yang-Baxter equation):

R12(z2/z1)K1(z1)R21(1/z1z2)K2(z2) = K2(z2)R12(1/z1z2)K1(z1)R21(z1/z2) (3.2.1)

where both sides are elements in Hom V (z1) ⊗ V (z2) → V (z1−1) ⊗ V (z −1

2 ). The K(z) :

V (z) → V (z−1) are the reflection matrices; their index refers to the tensor leg on which they are acting. For more information, see also [DN02].

Like in the case of the general YBE, we can construct a family of operators which commute thanks to equality 3.2.1. This involves defining another reflection operator

˜

K which satisfies a shifted version of equation 3.2.1:

R12(z2/z1) ˜K1(z1)R21(1/q2z1z2) ˜K2(z2) = ˜K2(z2)R12(q2z1z2) ˜K1(z1)R21(z2/z1).

Using these operators K and ˜K and an auxiliary space W equal to V (similar to section 2.1), we define operators t(z) as

t(z) = tr0

K(z)R01(z1/z0) . . . R0N(zN/z0) ˜K(z)R0N(zN/z0) . . . R01(z1/z0)

where we take the trace over the auxiliary space W . These operators form a commu-tative family; that is:

[t(z1), t(z2)] = 0

for all z1, z2. The commutativity is a direct result of the reflection equations above.

For the sake of brevity we will not discuss this proof here; see for example [Skl87] for the details.

Hence if we can find matrices R, K that satisfy the YBE as well as the reflection equation, we can construct an integrable system of operators for the entirety of the Heisenberg spin chain with boundaries.

The reflection equation is complicated to solve in the general case. It is certainly satisfied if we can find K(z) ∈ Hom(V (z), V (z−1)) such that:

K(z)π_zV(X) = π_zV−1(X)K(z) (3.2.2)

for all X ∈ Uq( ˆsl2). Indeed, this would imply:

ˆ R12(z2/z1)K1(z1) ˆR21(1/z1z2)K2(z2) =(π_z−1 1 ⊗ π_z−1 2 ) ˆR(K(z1) ⊗ id)P (πz −1 2 ⊗ π_z₁) ˆRP (id ⊗K(z2)) =(π_z−1 1 ⊗ πz −1 2 ) ˆR(K(z1) ⊗ id)P (πz −1 2 ⊗ πz1) ˆR(K(z2) ⊗ id)P =(π_z−1 1 ⊗ πz −1 2 ) ˆR(K(z1) ⊗ id)P (K(z2) ⊗ id)(πz2⊗ πz1) ˆRP =(π_z−1 1 ⊗ πz −1 2 ) ˆR(K(z1) ⊗ id)(id ⊗K(z2))P (πz2 ⊗ πz1) ˆRP =(π_z−1 1 ⊗ π_z−1 2 )R(id ⊗K(z2))(K(z1) ⊗ id)P (πz2 ⊗ π_z₁) ˆRP =(id ⊗K(z2))(π_z−1 1 ⊗ π_z₂) ˆR(K(z1) ⊗ id)P (πz2 ⊗ πz1) ˆRP =K2(z2) ˆR12(1/z1z2)K1(z1) ˆR21(z1/z2)

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As usual, P stands for the permutation operator. We see that a sufficient property for K to satisfy the reflection equation would be the intertwining property above. In this case, we will construct an operator that satisfies equation 3.2.2 not for all X ∈ Uq( ˆsl2),

but for all X ∈ B ⊂ Uq( ˆsl2), where B is the so-called q-Onsager algebra, a subalgebra

of Uq( ˆsl2) that we will define later.

3.3 Higher Dimensions And Fusion Of K-Matrices

Recall our splitting intertwiners:

(i) The linear map ιk: Vk+12 ,→ V 1 2 ⊗ Vk defined by ιk vk+ 1 2 n = q12nv 1 2 0 ⊗ v k n+ q −1 2(n−2k−1)[n]_qv 1 2 1 ⊗ v k n−1 defines a Uq( ˆsl2)-intertwiner ιk : Vk+ 1 2(z) ,→ V 1 2(zq−k) ⊗ Vk(zq 1 2).

(ii) The linear map jk := P12kιk : Vk+ 1 2 ,→ Vk⊗ V 1 2 defines a U_q( ˆsl₂)-intertwiner jk: Vk+12(z) ,→ Vk(zq− 1 2) ⊗ V 1 2(zqk).

The following result is due to Reshetikhin, Stokman and Vlaar.[RSV16] Suppose we have operators K12(z) : V

1

2(z) → V 1

2(z−1) that satisfy the reflection

equation: R 1 2 1 2 21 (z1/z2)K 1 2 1(z1)R 1 2 1 2 12 (z2z1)K 1 2 2(z2) = K 1 2 2(z2)R 1 2 1 2 21 (z1z2)K 1 2 1(z1)R 1 2 1 2 12 (z1/z2)

where the above is an identity in Hom V12(z₁) ⊗ V 1 2(z₂), V 1 2(z−1 1 ) ⊗ V 1 2(z−1 2 ) . Then for each k ∈ 1₂Z≥2, there exist unique linear operators Kk(z) : Vk(z) → Vk(z−1) which

satisfy:

(i) the following recursive fusion formula: jkKk+12(z) = P 1 2kK 1 2 1(zq −k_)R1 2k(z2q 1 2−k)Kk 2(zq 1 2)ιk (3.3.1)

for all k ∈ 1₂Z≥1; and:

(ii) the reflection equation:

R`k₂₁(z1/z2)K1k(z1)Rk`12(z2z1)K2`(z2) = K2`(z2)R`k21(z1z2)K1k(z1)Rk`12(z1/z2)

as an identity in Hom Vk(z1) ⊗ V`(z2), Vk(z1−1) ⊗ V`(z −1

2 ), for all k, ` ∈ 12Z≥1.

This construction is similar to the fusion procedure used to construct higher-dimensional R-matrices. For a full proof of the result above we refer to [RSV16].

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4 The q-Onsager Algebra

As seen in the last section, in order for our K(z)-operators to satisfy the boundary Yang-Baxter equation, a sufficient condition would be to satisfy equation 3.2.2. Note that there are no nontrivial Uq( ˆsl2)-intertwiners V (z)| > V (z−1) since for generic z,

V (z) and V (z−1) are irreducible and inequivalent. Therefore one needs to look at in-tertwiners V (z)|B → V (z−1)|B for a smaller coideal subalgebra B of Uq( ˆsl2). For this

purpose, we introduce the q-Onsager Algebra: a coideal subalgebra of Uq( ˆsl2) whose

properties make it a useful substitute for the complete algebra in this context. Specifi-cally, it is a coideal subalgebra that is as large as possible, with the additional property that V (z)|B and V (z−1)|B are isomorphic. It is a so-called tridiagonal algebra, the

properties of which have been studied extensively by Baseilhac, Terwilliger and Kolb among others ([BB10], [Ter15], [IT09], [Kol14]).

4.1 The q-Onsager Algebra: Definitions

In its most general form, the q-Onsager Algebra B ⊂ Uq( ˆsl2) is the subalgebra

gener-ated by the elements

Bi = fi− cieiqhi+ siqhi

for fixed parameters ci, si∈ C, where i = 0, 1. Note that we have:

∆(Bi) = ∆(fi− cieiqhi+ siqhi)

= fi⊗ qhi+ 1 ⊗ fi+ siqhi⊗ qhi − cieiqhi⊗ qhi− ci1 ⊗ eiqhi

= Bi⊗ qhi + 1 ⊗ Bi− si1 ⊗ qhi ∈ B ⊗ Uq( ˆsl2).

Hence ∆(B) ⊂ B ⊗ Uq( ˆsl2), meaning that B is a right coideal subalgebra of Uq( ˆsl2).

We can let B act on any finite-dimensional space V (z) through the usual evaluation representation πV_z. A straightforward but rather long and tedious calculation shows that the generators satisfy:

B₀3B1− [3]qB02B1B0+ [3]qB0B1B02− B1B30 = q(q + q−1)2c0[B1, B0]

B₁3B0− [3]qB12B0B1+ [3]qB1B0B12− B0B31 = q(q + q −1

)2c1[B0, B1].

Here we again use the q-number notation:

[n]q:=

qn_{− q}−n

q − q−1 .

One can alternatively define the q-Onsager algebra by only these relations, as they determine the structure of B uniquely. For a demonstration, we refer to [IT09].

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For any Uq( ˆsl2)-module M , we can construct a B-module (denoted by M |B) by

restrict-ing the action of Uq( ˆsl2) to elements in the q-Onsager algebra B ⊂ Uq( ˆsl2). Suppose

that, like in the construction in the previous section, we have B-intertwiners Kk(z) : Vk(z)|B→ Vk(z−1)|B;

K`(z) : V`(z)|B→ V`(z−1)|B

for all k ∈ 1₂Z≥1. This intertwining property is regarded on Vk(z) as a B-module, i.e.

for Kk to be a B-intertwiner we require

Kk(z)πV_z(Q) = π_zV−1(Q)Kk(z) ∈ Hom(Vk(z), Vk(z−1)) (4.1.1)

for all Q ∈ B (and similarly for Kl_).

Seeing as B is a right coideal subalgebra, i.e. ∆(B) ⊂ B ⊗ Uq( ˆsl2), this implies that

both R`k₂₁(z2/z1)K2k(z2)Rk`12(z −1 1 z −1 2 )K1`(z1) and K₁`(z1)R`k21(z−11 z −1 2 )K k 2(z2)Rk`12(z1/z2)

(both sides of the reflection equation) are B-intertwiners for the modules Vk(z1) ⊗ V`(z2)|B→ Vk(z1−1) ⊗ V`(z

−1 2 )|B.

In order to use Schur’s lemma for Vk_(z

1) ⊗ V`(z2)|B later on, we would like that,

for fixed ci, si (which determine B), Vk(z1) ⊗ V`(z2) is irreducible as a B-module for

generic values of z1, z2. This will automatically allow us to conclude that the reflection

equation holds up to a multiplicative constant, thanks to Schur’s lemma. In the fol-lowing subsections, we will investigate several B-modules and the conditions required for these modules to be irreducible.

4.2 Standard Finite-Dimensional Representations

Consider the evaluation representations πV

z : Uq( ˆsl2) → End(V (z)) as seen in previous

sections. We can use this same map π_zV to let B ∈ U ( ˆsl₂) act on an evaluation repre-sentation space V (z). In order to prove uniqueness (up to a scalar) of our intertwining and reflection operators, we would like to again use Schur’s lemma. For this we require V (z) to be irreducible as a B-module, at least for generic values of z1, z2.

Letting B act on V =Ln

k=0Cvk, we get:

πz(B1)vk= zvk+1− c1qn−2k[k]q[n + 1 − k]qvk−1+ s1qn−2kvk

πz(B0)vk= [k]q[n + 1 − k]qvk−1− c0q2k−nz−1vk+1+ s0q2k−n

where k = 0, 1, . . . , n and we take v−1 = 0 = vn+1. Notice that we can write:

[k]q[n+1−k]q= (qn+1− 2 + q−n−1) − (qn+1−2k− 2 + q2k−n−1₎ (q − q−1₎2 = 1 2(n + 1) 2 q − 1 2(n + 1 − 2k) 2 q .

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Hence in matrix form these operators are given by: πz(B1) =            s1qn −c1z−1qn−2[1]q[n]q 0 . . . 0 z s1qn−2 −c1z−1qn−4[2]q[n − 1]q ... 0 z s1qn−4 . .. .. . s1q2−n −c1z−1q−n[n]q[1]q 0 . . . z s1q−n            and πz(B0) =            s0q−n z[1]q[n]q 0 . . . 0 −c₀z−1q−n s0q2−n z[2]q[n − 1]q ... 0 −c0z−1q2−n s0q4−n . .. .. . s0qn−2 z[n]q[1]q 0 . . . −c₀z−1qn−2 _s 0qn           

We can find eigenvalues of πz(B1) following [Koe96], section 5. Take the algebra

U_q(su(2)) as generated by elements A, B, C, D with the following relations:

AD = 1 = DA, AB = qBA, AC = q−1CA, BC − CB = A

2_{− D}2

q − q−1 . This algebra acts on W = ⊕n_k=0Cek through representation tn : Uq(su(2)) → End(W )

as: tn(A)ek= q 1 2n−ke_k; tn(D)e_k= qk− 1 2ne_k tn(B)ek= q [n − k + 1]q[k]qek−1 tn(C)ek= q [n − k]q[k + 1]qek+1

where en+1 = 0 = e−1. Furthermore, for σ ∈ R we define:

Xσ = iq

1

2B − iq− 1

2C − [σ]_q(A − D) ∈ U_q(su(2))

Proposition 5.2 of [Koe96] allows us to find explicit eigenvalues and eigenvectors of tn(XσA) = iq

1

2tn(BA) − iq− 1

2tn(CA) − [σ]_q(tn(A)2− id_W)

where the eigenvalues have the form

λj(σ) = [−2j − σ]q+ [σ]q

for j = −1₂n, −1₂n + 1, . . . ,1₂n. Equivalently, we can define Yσ := Xσ− [σ]qD to see

that tn(YσA) has eigenvalues

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We now want to choose nonzero coefficients αk∈ C such that ek= αkvkand tn(YσA)ek =

πz(B1)ek for all k = 0, 1, . . . , n. This gives us:

iq12tn(BA) − iq− 1 2tn(CA) − [σ]_qtn(A)2 ek = αkπz(B1)vk

which breaks down to: iq12q 1 2n−k q [n − k + 1]q[k]qαk−1= −αkc1z−1qn−2k[k]q[n − k + 1]q [σ]qqn−2kαk= αks1qn−2k −kq−12q 1 2n−k q [n − k]q[k + 1]qαk+1= αkz

which, in turn, forces: αk−1 αk = ic1z−1q− 1 2+ 1 2n−k q [k]q[n − k + 1]q [σ]q= s1 αk αk+1 = −ic1z−1q− 1 2+ 1 2n−k q [k + 1]q[n − k]q.

Combining the first and last equation gives: ic1z−1q− 3 2+ 1 2n−k q [k + 1]q[n − k]q = αk αk+1 = −ic1z−1q− 1 2+ 1 2n−k q [k + 1]q[n − k]q resulting in c1 = −q.

We obtain an orthonormal basis of eigenvectors vn,j(σ) =Pn

k=0v n,j

k (σ)ek of the

oper-ator tn(YσA), corresponding to eigenvalues

µj(σ) = [−2j − σ]q.

The coefficients vn,j_k are known explicitly:

vn,j_k = Cn,j(σ)ik−nqσ(n−k)q12n−kn−k−1 (q 2n_{; q}−2₎ n−k (q2_{; q}2₎ n−k 1₂ × ∞ X m=0 (q2k−2n; q2)m(q2j−n; q2)m(−q−2j−n−2σ; q2)m (q−2n_{; q}2₎ m(q2; q2)m q2m where the constant is given by

Cn,j(σ) = q12n+j (qn_{; q}−1₎ 1 2n−j (q; q)1 2n−j !1₂ 1 + q−4j−2σ 1 + q−2σ 1₂ (−q2−2σ; q2)1 2n−j(−q 2+2σ_{; q}2₎ 1 2n+j −1₂ . Here we use the shifted factorial notation:

(a; q)k:= k−1

Y

i=0

(1 − aqi).

With our explicit form of πz(B0) it is possible to show (through straightforward but

tedious calculations) that, for generic c0, these vn,j(σ) are not eigenvectors of πz(B0)

and that in fact, only for specific values of c0 do the operators πz(B1) and πz(B0)

share a non-trivial invariant subspace of V . Hence for generic parameter values, V is irreducible as a B-module.

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4.3 Tensor Product Representations

Another representation of interest is the tensor space V (z1) ⊗ V (z2), where V is

two-dimensional. We again use the representation π := (πV ◦ ϕ_z₁⊗ πV _{◦ ϕ}

z2) ◦ ∆. We use

the ordered basis given by:

v+⊗ v+ =     1 0 0 0     ; v+⊗ v− =     0 1 0 0     ; v−⊗ v+ =     0 0 1 0     ; v−⊗ v−=     0 0 0 1     .

This gives us the following linear operators:

π(B1) =     q2s1 −c−1₀ qz₂−1 −c−1₀ q2z−1₁ 0 z2 s1 0 −c−10 z −1 1 qz1 0 s1 −c−10 qz −1 2 0 q−1z1 z2 q−2s1     and π(B0) =     q−2s0 z2 q−1z1 0 −c0q−1z−1₂ s0 0 qz1 −c0q−2z−11 0 s0 z2 0 −c₀z₁−1 −c₀q−1z₂−1 q2s0     .

Both of these operators give a similar eigenspace decomposition; in both cases, the vector space C2⊗ C2 _{decomposes into one two-dimensional eigenspace and two}

one-dimensional eigenspaces. For π(B1), a basis of eigenvectors is given by the matrix:

E1 =     c−1₀ qz₂−1 0 r + 2q3_{− s} 1(q2− 1) √ c0r r + 2q3+ s1(q2− 1) √ c0r s1(q2− 1) qz2 z2 c0s1(q2− 1) − √ c0r z2 c0s1(q2− 1) + √ c0r 0 −z₁ qz1 (q2₋₁₎√_c 0 c0s1(q 2_{− 1) −}√_c 0r _qz₁ (q2₋₁₎√_c 0 c0s1(q 2_{− 1) +}√_c 0r qz1 0 2c0qz1z2 2c0qz1z2    

where r = c0s21(q2− 1)2− 4q3. Note that the third and fourth column differ only by the

sign of √r. The first two columns both have eigenvalue s1; the others have respective

eigenvalues 1₂ s1(q2+ q−2) ± (1 + q−2) q c−1₀ r . Using this eigenbasis, one can calculate

E₁−1π(B0)E1

to see how π(B0) acts on the eigenvectors of π(B1). While the resulting matrix is

too big and too complicated to fit on this page, Wolfram Mathematica shows that, for generic values of c0, s1, z1, z2, this matrix has rank 4 and contains no zeroes. This

implies that for any eigenvector vi (where i = 1, 2, 3, 4) of π(B1), we have π(B0)vi =

a1v1 + a2v2+ a3v3 + a4v4 for generally non-zero a1, a2, a3, a4. By letting π(B1) act

on π(B0)vi, we can obtain different ratios of v3 and v4 (seeing as they have different

eigenvalues). Hence, through linear combinations of actions of π(B0) and π(B1) on any

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B-stable subspace of V can then only exist in the form of C(av1+ bv2) ⊕ Cv3⊕ Cv4 for

some constants a and b. Again, using the expressions above it is possible to explicitly calculate constraints on a and b for this to be the case, leading to specific constraints on the parameters z1, z2. In the generic case however, we can conclude that there exists

no proper subspace of V that is invariant under the action of both π(B0) and π(B1).

Hence V (z1) ⊗ V (z2) is irreducible as a representation of the q-Onsager Algebra B.

4.4 Tensor Product Representations: continued

For higher-dimensional tensor product representations, we can use results by Ito and Terwilliger ([Ter15] and [IT09]). Let A denote the associative C-algebra with 1 defined by the generators z and z∗, subject to the relations

z3z∗− [3]_qz2z∗z + [3]qzz∗z2− z∗z3 = (q2− q−2)2[z∗, z],

z∗3z − [3]qz∗2zz∗+ [3]qz∗zz∗2− zz∗3 = (q2− q−2)2[z, z∗].

Terwilliger proves in [Ter15] (Proposition 12.12) that there exists an injective homo-morphism δ : A → B such that δ(A) = B0 and δ(z∗) = B1, where we take

Bi = fi− q−1(q − q−1)2eiqhi+ siqhi.

for i = 0, 1. In other words, c0 = c1 = q−1(q − q−1)2. Since B0, B1 generate the

q-Onsager algebra, this homomorphism must also be surjective. It follows that A ∼= B. Following [IT09], we construct the following algebra homomorphism:

ϕs,t: A → Uq( ˆsl2)

defined by

ϕs,t(z) = −q−1(q − q−1)2(se0+ s−1f1qh1) + stqh0 + s−1t−1q−h0

ϕs,t(z∗) = sf0qh0+ s−1e1+ st−1qh0 + s−1tq−h0.

Terwilliger shows (Proposition 1) that this algebra homomorphism exists for all nonzero s, t ∈ C and is furthermore injective. Hence we have the following morphisms:

B δ −1 ∼ = A−→ Uϕs,t _q( ˆsl₂) ϕ V z −→ End(V (z))

allowing us to view the action of B on any representation space V (z) of Uq( ˆsl2) through

this morphism chain. Recall our notation Vn(z) for the evaluation representation where

z is the evaluation parameter and Vn is the irreducible Uq( ˆsl2)-module of dimension

n + 1. One can check that, for appropriately chosen parameters s, t, the representation of B by the composition ϕV_z ◦ ϕs,t◦ δ−1 coincides with the standard representation

given by ϕV_z(B). The advantage of viewing this representation through A is that this allows to see under what conditions this representation is irreducible.

We recall theorem 1.3.1:

A tensor product Vn1(a1) ⊗ · · · ⊗ Vnr(ar) is irreducible as a Uq( ˆsl2)-module if and only

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Here we call two such q-strings S1 and S2 in general position if at least one of the

following conditions holds:

(i) S1∪ S2 is not a q-string;

(ii) S1 ⊆ S2;

(iii) S2 ⊆ S1.

Furthermore, we call two multi-sets {Sni(ai)}

m

i=1, {Sn0_i(a0i)}m

0

i=1 equivalent if m = m0

and there exists a permutation σ of {1, 2, . . . , m} and i ∈ {±1} such that ni = n0_σ(i)

and ai

i = a0σ(i) for i = 1, 2, . . . , n. This means that the multi-sets {Sni(a

i i )}mi=1 and {S_n0 i(a 0 i)}m 0

i=1 are equal. A multi-set Σ := {Sni(ai)}

m

i=1 is said to be strongly in general

position if any multi-set of q-strings equivalent to Σ is also in general position. The elements B0 and B1 form a so-called TD pair with a common dimension we will

denote as d (for more information on this definition, see [ITT01]). We then have the following theorem from Ito and Terwilliger ([IT10], Theorem 1.17):

Theorem 4.4.1. For a Uq( ˆsl2)-module V = Vn1(z1)⊗· · ·⊗Vnr(zr) and nonzero s, t ∈ C,

the representation r O i=1 πVni _{◦ ϕ}_z i

of A is irreducible if and only if:

1. the multi-set {S(ni, zi)}ri=1 of q-strings is strongly in general position,

2. none of −s2, −t2 belong to S(ni, zi) ∪ S(ni, zi−1) for any i (1 ≤ i ≤ r),

3. none of the four scalars ±st, ±st−1 equals qi _{for any i ∈ Z with |i| ≤ d − 1.}

Through the isomorphism δ : A → B, this in turn gives us an irreducible representa-tion V of B.

Recalling subsection 4.1, we have two intertwiners for Vk(z1) ⊗ V`(z2) |B→ Vk(z1−1) ⊗ V`(z −1 2 ) |B: R`k₂₁(z2/z1)K2k(z2)Rk`12(z −1 1 z −1 2 )K ` 1(z1) and K₁`(z1)R`k21(z −1 1 z −1 2 )K k 2(z2)R12k`(z1/z2),

both sides of the reflection equation. We know Vk(z1) ⊗ V`(z2) |B to be irreducible

as a B-module by theorem 4.4.1. It follows from Schur’s lemma that the two inter-twiners above must be equal up to a scalar. This proves the reflection equation up to a constant. Hence we can solve a linear intertwining relation (see equation 3.2.2) for B to find solutions to the reflection equation (see also [DM06]).

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4.5 Fusion of K-matrices for the q-Onsager algebra

Recall the fusion procedure described in subsection 3.3. We can see that this fusion method guarantees that our recursively constructed higher-dimensional K-matrices retain the intertwining property. Suppose we have B-intertwiners Kk(z) : Vk(z) → Vk(z−1) for a certain k ∈ 1₂Z≥2, as well as for k = 1₂.

Analogously to subsection 3.3, we can obtain the following recursive fusion equality: ιkKk+12(z) = Pk 1 2Kk 2(zq −k )Rk12(z2qk− 1 2)Kk 1(zq 1 2)jk.

Writing ˇR := P R we can write the right-hand side of this equation as K 1 2 2(zq −k ) ˇRk12(z2qk− 1 2)Kk 1(zq 1 2)jk

which we can see to be a B-intertwiner Vk+12(z)|_B → V12(z−1q−k) ⊗ Vk(z−1q 1 2) |_B.

Hence the left-hand side of the fusion formula tells us that the fused K-operator is also a B-intertwiner: Kk+12(z) : Vk+ 1 2(z)|_B → Vk+ 1 2(z−1)|_B.

This allows us to construct reflection operators for representation spaces of any di-mension based on the case V 12. We will discuss this specific case in the following

subsection.

4.6 Example: reflection operators for dimension 2

As we mentioned before, in order for our reflection operator K(z) to satisfy the re-flection equation, we would like to find a K(z) ∈ Hom(V (z), V (z−1)) such that K(z) satisfies the intertwining relation given by equation 4.1.1. Like with our R-matrices, the case where dim V = 2 is of special importance as it plays an important role in the process of fusion. Therefore we will explicitly calculate K(z) in this case.

We write V = Cv+⊕ Cv−. Let K(z) =

a(z) b(z) c(z) d(z)

, where the entries are complex functions. Checking the commutation relation for X = B1 acting on v+left and right,

we find that

c(z) = qb(z) and s1(q − q−1)c(z) = z−1a(z) − zd(z).

Doing the same for B0, we obtain

c(z) = −c0q−1b(z) and s0(q − q−1)c(z) = c0q−1(za(z) − z−1d(z)).

We see that qb(z) = −c0q−1b(z), hence c0 = −q2 (assuming c(z) is not identically zero,

a necessity to find non-trivial K(z)). Solving the remaining linear system gives us:

K(z) = f (z) · qs1z −1_{+ s} 0z z −2_−z2 q−q−1 q(z−2−z2₎ q−q−1 qs1z + s0z−1 !

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Popular summary

The main object of study in this thesis is the one-dimensional Heisenberg spin chain with boundaries. This can be seen as a finite set of magnetic particles in a row, with a wall on each side:

|•1 2• . . . N• |

Each of these particles has a spin state, describing its spin (a type of intrinsic angular momentum) on a quantum level. This state is an element of a state space, which (for each particle separately) can be given by an n-dimensional complex vector space denoted by V . The state of the entire system can be seen as an element in the tensor product of these separate spaces:

V⊗N = V ⊗ V ⊗ · · · ⊗ V

When investigating the properties of this system, we need to define operators acting on the space V⊗N. For this, we define a representation πz with representation space

V :

πz : Uq( ˆsl2) → End(V )

Here, πzis a homomorphism (depending on a complex parameter z) that maps elements

in an algebra Uq( ˆsl2) to endomorphisms of V . This effectively turns the state space into

a so-called Uq( ˆsl2)-module, denoted as V (z) (where z is the same complex parameter

as featured in the homomorphism πz). We can define similar representations for tensor

products of vector spaces, resulting in a tensor product module of the form: V (z1) ⊗ V (z2) ⊗ · · · ⊗ V (zN).

The Hamiltonian for this system can be given by:

H = −J N X j=1 σjσj+1− h N X j=1 σj

where each σ is a spin matrix for a certain particle; J and h are constants.

An interaction between two neighbouring particles can be described by acting on their state spaces with an operator R(z1/z2) acting as a linear operator on V (z1) ⊗ V (z2).

Here we make the requirement that the order in which particles interact does not change the result inasmuch as:

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Here we view time as the vertical axis, moving upward. On the left, we first see particles 1 and 2 interacting, followed by 13, followed by 23. The right diagram has the same interactions in opposite order, which we require to yield the same result. Representing an interaction between particle i and j by Rij(zi/zj), this implies that:

This is the so-called quantum Yang-Baxter equation (see [Yan67] and [Bax72]). Similarly, for two particles interacting with each other and with a boundary of our system, we require the following equivalence:

Here one can see the vertical axis as the flow of time, moving upward. On the left, we have interactions 2-1, 1-boundary, 1-2, 2-boundary; on the right we have the same in the opposite order. This translates to the following reflection equation (also known as the boundary Yang-Baxter equation):

R12(z2/z1)K1(z1)R21(1/z1z2)K2(z2) = K2(z2)R12(1/z1z2)K1(z1)R21(z1/z2)

where the K(z) : V (z) → V (z−1) are linear operators corresponding to interaction with the boundary. Their index refers to the tensor leg on which they are acting. We use several properties of the algebra Uq( ˆsl2) to explicitly construct R- and

K-operators that satisfy these requirements. Specifically, we use irreducibility of cer-tain Uq( ˆsl2)-modules and B-modules (where B is a subalgebra of Uq( ˆsl2) called the

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q-Onsager algebra). This means that a space W has no subspaces that are left intact under the action of Uq( ˆsl2) (apart from the trivial subspaces {0} and W itself). This

allows us to reduce the Yang-Baxter equations to a simpler equation which we can solve explicitly.

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Bibliography

[Yan67] C.N. Yang. “Some exact results for the many-body problem in one dimen-sion with delta-function interaction”. In: Phys. Rev. Lett. (1967), pp. 1312– 1314.

[Bax72] R.J. Baxter. “Partition function of the eight vertex lattice model”. In: Ann. Pys. (1972), pp. 193–228.

[Skl87] E. K. Sklyanin. “Boundary conditions for integrable systems”. In: VIIIth international congress on mathematical physics (Marseille, 1986). World Sci. Publishing, Singapore, 1987, pp. 402–408.

[Jim89] Michio Jimbo. “Introduction to the Yang-Baxter equation”. In: Internat. J. Modern Phys. A 4.15 (1989), pp. 3759–3777. issn: 0217-751X. doi: 10.

1142/S0217751X89001503. url: http://dx.doi.org/10.1142/S0217751X89001503. [CP91] Vyjayanthi Chari and Andrew Pressley. “Quantum affine algebras”. In:

Comm. Math. Phys. 142.2 (1991), pp. 261–283. issn: 0010-3616. url: http: //projecteuclid.org/euclid.cmp/1104248585.

[CP94] Vyjayanthi Chari and Andrew Pressley. A guide to quantum groups. Cam-bridge University Press, CamCam-bridge, 1994, pp. xvi+651. isbn: 0-521-43305-3. [Kas95] Christian Kassel. Quantum groups. Vol. 155. Graduate Texts in Mathemat-ics. Springer-Verlag, New York, 1995, pp. xii+531. isbn: 0-387-94370-6. doi: 10.1007/978-1-4612-0783-2. url: http://dx.doi.org/10.1007/978-1-4612-0783-2.

[Koe96] H. T. Koelink. “Askey-Wilson polynomials and the quantum SU(2) group: survey and applications”. In: Acta Appl. Math. 44.3 (1996), pp. 295–352. issn: 0167-8019.

[KRT97] Christian Kassel, Marc Rosso, and Vladimir Turaev. Quantum groups and knot invariants. Vol. 5. Panoramas et Synthèses [Panoramas and Syntheses]. Société Mathématique de France, Paris, 1997, pp. vi+115. isbn: 2-85629-055-8.

[KS97] Anatoli Klimyk and Konrad Schm¨udgen. Quantum groups and their repre-sentations. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997, pp. xx+552. isbn: 3-540-63452-5. doi: 10.1007/978-3-642-60896-4. url: http://dx.doi.org/10.1007/978-3-642-60896-4.

[EFK98] Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov Jr. Lectures on representation theory and Knizhnik-Zamolodchikov equations. Vol. 58. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998, pp. xiv+198. isbn: 0-8218-0496-0. doi: 10 . 1090 / surv/058. url: http://dx.doi.org/10.1090/surv/058.

[Fad98] L. D. Faddeev. “How the algebraic Bethe ansatz works for integrable mod-els”. In: Sym´etries quantiques (Les Houches, 1995). North-Holland, Ams-terdam, 1998, pp. 149–219.

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[ITT01] Tatsuro Ito, Kenichiro Tanabe, and Paul Terwilliger. “Some algebra related to P - and Q-polynomial association schemes”. In: Codes and association schemes (Piscataway, NJ, 1999). Vol. 56. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc., Providence, RI, 2001, pp. 167– 192.

[DN02] G. W. Delius and Rafael I. Nepomechie. “Solutions of the boundary Yang-Baxter equation for arbitrary spin”. In: J. Phys. A 35.24 (2002), pp. L341– L348. issn: 0305-4470. doi: 10.1088/0305-4470/35/24/102. url: http: //dx.doi.org/10.1088/0305-4470/35/24/102.

[DM06] G.W. Delius and N.J. MacKay. “Affine quantum groups”. In: Elsevier (2006). url: https://arxiv.org/abs/math/0607228v1.

[IT09] Tatsuro Ito and Paul Terwilliger. “The q-Onsager algebra”. In: (2009). url: https://arxiv.org/abs/0904.2895v1.

[BB10] Pascal Baseilhac and Samuel Belliard. “Generalized q-Onsager algebras and boundary affine Toda field theories”. In: Lett. Math. Phys. 93.3 (2010), pp. 213–228. issn: 0377-9017. doi: 10.1007/s11005- 010- 0412- 6. url: http://dx.doi.org/10.1007/s11005-010-0412-6.

[IT10] Tatsuro Ito and Paul Terwilliger. “The augmented tridiagonal algebra”. In: Kyushu J. Math. 64.1 (2010), pp. 81–144. issn: 1340-6116. doi: 10.2206/ kyushujm.64.81. url: http://dx.doi.org/10.2206/kyushujm.64.81. [Kol14] Stefan Kolb. “Quantum symmetric Kac-Moody pairs”. In: Adv. Math. 267

(2014), pp. 395–469. issn: 0001-8708. doi: 10.1016/j.aim.2014.08.010. url: http://dx.doi.org/10.1016/j.aim.2014.08.010.

[Ter15] Paul Terwilliger. “The q-Onsager algebra and the positive part of Uq( ˆsl2)”.

In: (2015). url: https://arxiv.org/abs/1506.08666.

[RSV16] Nicolai Reshetikhin, Jasper Stokman, and Bart Vlaar. “Boundary quantum Knizhnik-Zamolodchikov equations and fusion”. In: Ann. Henri Poincar´e 17.1 (2016), pp. 137–177. issn: 14240637. doi: 10 . 1007 / s00023 014 -0395-4. url: http://dx.doi.org/10.1007/s00023-014--0395-4.

Heisenberg Spin Chains with Boundaries and Quantum Groups

MSc Mathematics

Master Thesis