MSc Mathematics

Master Thesis

Heisenberg Spin Chains withBoundaries and Quantum Groups

Author: Supervisor:Tijmen Veltman dr. Jasper Stokman

Examination date:April 11, 2016

Korteweg-de Vries Institute forMathematics

Abstract

In this paper I will present a study of one-dimensional Heisenberg spin chains withboundaries. First I will explain how this system can be described using a representationof Uq(sl2). 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 theirproperties. Finally, we will study a subalgebra B ⊂ Uq(sl2) known as the q-Onsageralgebra. Using the properties of this subalgebra, we will be able to construct theseR-matrices and K-matrices explicitly for the lowest-dimensional case.

Title: Heisenberg Spin Chains with Boundaries and Quantum GroupsAuthor: Tijmen Veltman, tijmenveltman@student.uva.nl, 5928605Supervisor: dr. Jasper StokmanSecond Examiner: dr. Hessel PosthumaExamination date: April 11, 2016

Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamScience Park 105-107, 1098 XG Amsterdamhttp://kdvi.uva.nl

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Contents

1 Quantum Affine sl2 and its representations 61.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Quantum Affine sl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Finite Dimensional Representations . . . . . . . . . . . . . . . . . . . . . 81.4 Quantum Double Construction . . . . . . . . . . . . . . . . . . . . . . . 10

2 R-matrices 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 R-matrices: Existence And Construction . . . . . . . . . . . . . . . . . . 152.3 Example: R-Matrices For Two-Dimensional Representations . . . . . . . 202.4 Higher Dimensions And Fusion Of R-Matrices . . . . . . . . . . . . . . . 21

3 K-matrices 243.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 K-matrices: Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Higher Dimensions And Fusion Of K-Matrices . . . . . . . . . . . . . . . 26

4 The q-Onsager Algebra 274.1 The q-Onsager Algebra: Definitions . . . . . . . . . . . . . . . . . . . . . 274.2 Standard Finite-Dimensional Representations . . . . . . . . . . . . . . . 284.3 Tensor Product Representations . . . . . . . . . . . . . . . . . . . . . . . 314.4 Tensor Product Representations: continued . . . . . . . . . . . . . . . . 324.5 Fusion of K-matrices for the q-Onsager algebra . . . . . . . . . . . . . . 344.6 Example: reflection operators for dimension 2 . . . . . . . . . . . . . . . 34

Popular summary 35

Bibliography 38

3

Introduction

Our main object of study is the one-dimensional Heisenberg spin chain with a finitenumber 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 eitherinfinitely many particles or periodic boundary conditions. These systems have beenfound to be integrable by the quantum inverse scattering method. In our case, wetake a finite, open chain and introduce new boundary conditions that maintain theintegrability of the system. Here we observe how particles near the sides interact withthe boundaries and incorporate these interactions into our model.

We will study the state space of our system as a representation space of quantum affinesl2, i.e. a Uq(sl2)-module. Any interactions will be described by elements of quantumaffine sl2, acting as operators on (tensor products of) appropriate state spaces. Wewill 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 remainsintegrable.

The interactions will be governed by the intertwining operator R acting on the tensorproduct of two neighbouring particle spaces, and the reflection operator K whichdescribes the interaction of a boundary particle with the boundary of the system. Weare particularly interested in intertwining operators R and reflection operators K thatprovide 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, whereRij represents a crossing of strands i and j. We have the following equivalence for twobraids:

4

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 Vis 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 ourrepresentation we can find a solution R(z) that acts on V (z)⊗ V (z) as

a(z)

1 0 0 00 0 q−1 00 q−1 z−1(1− q−2) 00 0 0 1

and K(z) is represented as(

1 0

0 z2−λ21−z2λ2

): V (z)→ V (z−1)

where a(z) is a complex function and λ a nonzero complex parameter. Using themethod of fusion, these can be used to recursively construct R- and K-matrices forhigher dimensional representations. Finally, we will describe a method to find solutionsof the reflection equation using representations of the q-Onsager algebra, a subalgebraof Uq(sl2).

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1 Quantum Affine sl2 and itsrepresentations

1.1 Introduction

Our main object of study, the Heisenberg spin chain, describes the spin-spin interactionof particles in the shape of a one-dimensional lattice. Each particle has a copy of Cndescribing 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 bylinear operators on this Hilbert space.

As an example, for n = 2 the basic building blocks of these operators are the Paulimatrices:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)which form a basis of sl2(C) and act as linear operators on C2. More generally, thesethree generators and their relations define a Lie algebra sl2 (generally omitting thebase field C) which can act on Cn through a Lie algebra representation.

In this thesis we will enrich the structure of sl2 in several ways. Most notably, wewill introduce a quantum deformation, turning our Lie algebra into a quasitriangularHopf 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 sl2

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 insource [RSV16].

We take the affine algebra sl2 generated by elements hi, ei, fi, where i = 0, 1 and theirrelations:

6

[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 isthe 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 h := Ch0 ⊕ Ch1 is a Cartan subalgebra of sl2. We

define elements αi ∈(h)∗

(i = 0, 1) such that αi(hj) = aij .

Now to make a quantum deformation, take a number q = eη not equal to a rootof unity (that is, η is not of the form 2απi with α a rational number). We definethe quantum affine algebra Uq(sl2) as the unital associative algebra with generators

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

q0 = 1, qh+h′

= qhqh′

(h, h′ ∈ h)

qheiq−h = qαi(h)ei, qhfiq

−h = q−αi(h)fi ∀h ∈ h,

[ei, fj ] = δijqhi − q−hiq − q−1

,

3∑n=0

(−1)n

[n]q![3− n]q!eni eje

3−ni = 0 (i 6= j)

3∑n=0

(−1)n

[n]q![3− n]q!fni fjf

3−ni = 0 (i 6= j).

In this notation we use q-numbers, defined as

[n]q :=qn − q−n

q − q−1= qn−1 + qn−3 + . . .+ q1−n

and

[n]q! :=

n∏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 seth := h ⊕ Cd. Call the algebra generated by ei, fi, hi, d sl2. The extended quantumaffine algebra Uq(sl2) ⊃ Uq(sl2) is defined by the additional relations:

qh+h′

= qhqh′

(h, h′ ∈ h);

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qade0 = qae0qad, qadf0 = q−af0q

ad;

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

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

• A co-unit:ε(ei) = 0 = ε(fi), ε(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(sl2);hence we have a chain of Hopf subalgebras:

Uq(sl2) ⊂ Uq(sl2) ⊂ Uq(sl2).

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

ϕz(qλh0) = q−λh1 , ϕz(q

λh1) = 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 importantrole in constructing evaluation representations of Uq(sl2).

1.3 Finite Dimensional Representations

Our representation of Uq(sl2) is constructed based on the standard finite dimensionalrepresentation of Uq(sl2). For any n ∈ Z≥0, take an n+ 1-dimensional complex vectorspace V with basis {v0, v1, . . . , vn}. This becomes a left Uq(sl2)-module when we definethe following homomorphism πV : Uq(sl2)→ End(V ):

πV (qλh1)vi = qλ(n−2i)vi

πV (e1)vi = [i]q[n+ 1− i]qvi−1πV (f1)vi = vi+1

8

for i = 0, 1, . . . , n. Here we take v−1 = 0 = vn+1. We can now turn this into a repre-sentation of Uq(sl2) by composing with our algebra homomorphism ϕz. We define the

evaluation representation of Uq(sl2) as πVz := πV ◦ ϕz : Uq(sl2)→ End(V ).

Denote this evaluation representation as Vn(z), where z is the evaluation parameterand the representation homomorphism maps to an n + 1-dimensional complex vectorspace V . It is important to note that this representation is irreducible. Since e1 andf1 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 knowthat [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(sl2)-invariant subspaces. Therefore thisevaluation representation Vn(z) is irreducible.

Furthermore, we will frequently be studying tensor products of Uq(sl2)-modules. Theseare especially important when considering R-matrices (which act on these tensor prod-ucts) and fusion of R- and K-operators. The action of Uq(sl2) is defined as follows:

πV⊗Wz1,z2 : Uq(sl2)→ End(V ⊗W )

πV⊗Wz1,z2 (X) :=(πVz1 ⊗ π

Wz2

)◦∆(X).

We can extend this to multiple tensor products by setting

πV⊗W⊗Uz1,z2,z3 (X) :=(πVz1 ⊗ π

Wz2 ⊗ π

Uz3

)◦(

∆⊗ idUq(sl2)

)◦∆(X)

and so on. Note that (∆⊗ idUq(sl2)

)◦∆ =

(idUq(sl2)⊗∆

)◦∆

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 ∈ Cand n ∈ N. Two such q-strings S1 and S2 are said to be in general position if at leastone 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(sl2)-module if and only if the q-strings Sn1(a1), . . . , Snr(ar) are pairwisein general position.

For a full proof, we refer to the paper by Chari and Pressley ([CP91]).

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1.4 Quantum Double Construction

Following Drinfeld’s method as described in [Kas95], in this section we will exploitthe Hopf algebra structure of Uq(sl2) to realize it as a quantum double algebra. Thisstructure can then be used to construct a universal R-matix.

On the Cartan subalgebra h = Ch0 ⊕Ch1 ⊕Cd we define a nondegenerate symmetricbilinear pairing:

〈·, ·〉 : h× h→ C;

〈hi, hj〉 = aij ; 〈d, h0〉 = 1; 〈d, d〉 = 0 = 〈d, h1〉

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 ) := C〈e0, e1, qh|h ∈ h〉

Uq(sl−2 ) := C〈f0, f1, qh|h ∈ h〉.

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 bilinearfunction

ϕ : A×B → k

satisfying the following three compatibility conditions:

(i) (Co)units:

ϕ(a, 1) = ε(a);

ϕ(1, b) = ε(b).

(ii) (Co)multiplications:

ϕ(aa′, b) =∑(b)

ϕ(a, b(2))ϕ(a′, b(1));

ϕ(a, bb′) =∑(a)

ϕ(a(1), b)ϕ(a(2), b′).

(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) =∑(a)

a(1) ⊗ a(2)

and(∆⊗ id) ◦∆(a) =

∑(a)

a(1) ⊗ a(2) ⊗ a(3) = (id⊗∆) ◦∆(a).

The latter equality is valid thanks to the coassociative property of ∆. Note that, thanksto the universal property of the tensor product, any bilinear function ϕ : A × B → kfactors through A⊗B uniquely. Hence we can alternatively define a pairing as a linearfunction 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, qh

′) := q−〈h,h

′〉 (h, h′ ∈ h)

ϕ(ei, fj) := δij1

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 Uq(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) =∑(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

Dϕ(Uq(sl+2 ),Uq(sl

−2 ))

has the two-sided ideal I generated by

qh ⊗ 1− 1⊗ qh

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(where h ∈ h) as a Hopf ideal, and

Uq(sl2) ∼= Dϕ(Uq(sl+2 ),Uq(sl

−2 ))/I

as Hopf algebras, where the isomorphism is given by

ei 7→ ei ⊗ 1 + I,

fi 7→ 1⊗ fi + I,

qh 7→ qh ⊗ 1 + I.

We will use this quantum double structure in the next section to show the existenceof 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 introduceboundary properties to our system. Diagrammatically, this looks like the following:∣∣∣∣ V (z1)•

V (z2)• · · ·V (zN )•

∣∣∣∣Here V (zi) is the evaluation representation space describing the internal spin of par-ticle i for i = 1, . . . , N . To study interactions between particles i, j, we introduce alinear operator acting non-trivially on V (zi) and V (zj) only, and as the identity onall the other V -spaces. The interactions at the boundaries will be discussed later; fornow, 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 suchthat for all X ∈ H we have:

∆op(X)R = R∆(X)

where ∆ denotes the comultiplication and ∆op = PH,H ◦∆ its opposite (where PH,His the permutation operator acting on H ⊗H).

If we can find such a universal R-matrix for Uq(sl2), we can use a representation (likethe ones seen in the previous chapter) to let it act on a tensor space of the formV ⊗ V . This will serve as our interaction operator between two neighbouring spaces.Generally, we can write such an interaction operator as:

R =∑k

Ak ⊗Bk

where Ak, Bk ∈ End(V ). Indices will indicate on which tensor legs the operator isacting. Explicitly, for i < j we define:

Rij(zi, zj) :=∑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 onV (zi) and V (zj).

Analogously, we can define Rji(zj , zi) := PV (zj),V (zi) ◦ Rij(zi, zj) ◦ PV (zi),V (zj), wherePV,W is the permutation operator switching the spaces V and W inside the tensorproduct. We will later see that a suited choice of Rij(zi, zj) in fact only depends ofthe quotient of the two parameters zi/zj .

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This system is integrable if these interaction operators satisfy the Yang Baxter Equa-tion (see also [CP94], section 7.5 for an in-depth discussion). This equation takes thefollowing 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 actsas 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 ofthe space V with respective spectral parameters z1, z2, z3. An interaction between twoparticles 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 particles 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 particles1 and 2 interacting, followed by 13, followed by 23. The right diagram has the sameinteractions in opposite order. Representing particle i by its evaluation representationspace V (zi), we precisely obtain the YBE. We can also eliminate one of the threeparameters 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, wecan define the monodromy matrix T (z0) ∈ End(W ⊗ V ⊗N ), where W is an auxiliaryspace isomorphic to V , as:

T (z0) := R01(z0/z1)R02(z0/z2) . . . R0N (z0/zN )

14

where (for i = 1, 2, . . . , N) R0i acts nontrivially on the 0th tensor component (being W ,our auxiliary space isomorphic to V ) and the ith (i.e. V (zi)). This operator representspairwise 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) T ′(w) ∈ End(W ⊗V ⊗n).As a consequence of the YBE for R-matrices, these monodromy matrices T, T ′ satisfy:

R(z/w)T (z)T ′(w) = T ′(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 spaceW :

τ(z) = TrW T (z) ∈ End(V ⊗N ).

Thanks to the so-called RTT -equation above, these transfer matrices commute fordifferent 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 theHamiltonian, which can then be diagonalised using the algebraic Bethe Ansatz. Forfurther details regarding this construction, see [Fad98].

To find solutions to the YBE, it is sufficient to find an operator R that satisfies thefollowing 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 referto 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 =∑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

15

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 ofa tensor product of representation spaces.

Explicitly, the formal sum takes the following form:

R = qc⊗d+d⊗c+∑r

i=1 xi⊗xi

(∑k

ak ⊗ ak)

where xi is an orthonormal basis of h, ak is a basis of Uq(n+) = C〈e0, e1〉 and ak is thecorresponding dual basis of Uq(n−) = C〈f0, f1〉. That is, ϕ(ak, a

m) = δkm.

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

The action of R under our evaluation representation cannot be defined, seeing asour representation homomorphism πVz is only defined on Uq(sl2) but not on Uq(sl2).However, we can define the truncated R-matrix:

R = q−c⊗d−d⊗cR = q∑r

i=1 xi⊗xi

(∑k

ak ⊗ ak)∈ Uq(sl

+

2 )⊗Uq(sl−2 ).

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

p(qλh1) = qλh1 ,

p(e1) = e1,

p(f1) = f1,

p(qλh0) = q−λh1 ,

p(e0) = f1,

p(f0) = e1.

Furthermore, define an inner automorphism Dz : Uq(sl2)→ Uq(sl2) as:

Dz(X) := qlog zlog q

dXq− 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 :Uq(sl2)→ Uq(sl2) from subsection 1.2. We now set:

R(z) := (Dz ⊗ id)R ∈ Uq(sl+

2 )⊗Uq(sl−2 ).

16

For this object, we have:

R(z) = (Dz ⊗ id)(R)

= (qlog zlog q

d ⊗ 1)R(q− log z

log qd ⊗ 1)

= (1⊗ q−log zlog q

d)∆op(q

log zlog q

d)Rqc⊗d+d⊗c∆(q

− log zlog q

d)(1⊗ q

log zlog q

d)

= (1⊗ q−log zlog q

d)R∆(q

log zlog q

d)qc⊗d+d⊗c∆(q

− log zlog q

d)(1⊗ q

log zlog q

d)

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

= (1⊗ q−log zlog q

d)Rqc⊗d+d⊗c(1⊗ q

log zlog q

d)

= (1⊗ q−log zlog q

d)R(1⊗ q

log zlog q

d)

= (id⊗Dz−1)(R).

This implies that we can write:

R(z1/z2) = (Dz1 ⊗Dz2)R.

Proposition 2.2.1. These R and R(z) have the following properties:

(i) R(z)q−c⊗d−d⊗c(Dz ⊗ id)(∆(X))qc⊗d+d⊗c = (Dz ⊗ id) (∆op(X)) R

(ii) (∆⊗ id)(R(z)) =(q−1⊗c⊗dR13(z)q

1⊗c⊗d)R23(z)

(iii) (id⊗∆)(R(z)) =(q−d⊗c⊗1R13(z)q

d⊗c⊗1)R12(z)

(iv) R12(z)q−1⊗c⊗dR13(zw)q1⊗c⊗dR23(w) = R23(w)q−d⊗c⊗1R13(zw)qd⊗c⊗1R12(z).

Proof. (i)

R(z)q−c⊗d−d⊗c(Dz ⊗ id)(∆(X))qc⊗d+d⊗c

=(Dz ⊗ id)(Rq−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)) R

Here we used the fact that R∆(X) = ∆op(X)R for all X ∈ Uq(sl2).

17

(ii)

(∆⊗ id)(R(z))

=(∆⊗ id)(id⊗Dz−1)(R)

=(id⊗ id⊗Dz−1)(∆⊗ id)(q−c⊗d−d⊗cR)

=(id⊗ id⊗Dz−1)(q−c⊗1⊗d−1⊗c⊗d−d⊗1⊗c−1⊗d⊗cR13R23

)(Here we used the identity (∆⊗ id)R = R13R23.

For more information, see [EFK98].)

=(id⊗ id⊗Dz−1)(q−1⊗c⊗dq−c⊗1⊗d−d⊗1⊗cR13q

1⊗c⊗dq−1⊗c⊗d−1⊗d⊗cR23

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

we have q−1⊗d⊗cR13 = R13q−1⊗d⊗c)

=(id⊗ id⊗Dz−1)(q−1⊗c⊗dR13q

1⊗c⊗dR23

)=q−1⊗c⊗dR13(z)q

1⊗c⊗dR23(z).

(iii) Somewhat analogously:

(id⊗∆)(R(z))

=(id⊗∆)(Dz ⊗ id)R

=(Dz ⊗ id⊗ id)(id⊗∆)q−c⊗d−d⊗cR

=(Dz ⊗ id⊗ id)(q−c⊗d⊗1−c⊗1⊗d−d⊗c⊗1−d⊗1⊗cR13R12

)=(Dz ⊗ id⊗ id)

(q−d⊗c⊗1q−c⊗1⊗d−d⊗1⊗cR13q

d⊗c⊗1q−d⊗c⊗1−c⊗d⊗1R12

)=(Dz ⊗ id⊗ id)

(q−d⊗c⊗1R13q

d⊗c⊗1R12

)=q−d⊗c⊗1R13(z)q

d⊗c⊗1R12(z).

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

R12R13R23 = R23R13R12

Writing out the definition of R:

qc⊗d⊗1+d⊗c⊗1R12qc⊗1⊗d+d⊗1⊗cR13q

1⊗c⊗d+1⊗d⊗cR23

=q1⊗c⊗d+1⊗d⊗cR23qc⊗1⊗d+d⊗1⊗cR13q

c⊗d⊗1+d⊗c⊗1R12

We will now apply the operator Dzw ⊗Dw ⊗ id left and right. When applied toR12, this operator is equal to:

(Dzw ⊗ id⊗ id)(id⊗Dw ⊗ id) = (Dzw ⊗ id⊗ id)(Dw−1 ⊗ id⊗ id) = Dz ⊗ id⊗ id .

18

Hence we obtain:

qc⊗d⊗1+d⊗c⊗1R12(z)qc⊗1⊗d+d⊗1⊗cR13(zw)q1⊗c⊗d+1⊗d⊗cR23(w)

=q1⊗c⊗d+1⊗d⊗cR23(w)qc⊗1⊗d+d⊗1⊗cR13(zw)qc⊗d⊗1+d⊗c⊗1R12(z)

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

qc⊗d⊗1+d⊗c⊗1+c⊗1⊗dR12(z)qd⊗1⊗c+1⊗d⊗cR13(zw)q1⊗c⊗dR23(w)

=qd⊗1⊗c+1⊗c⊗d+1⊗d⊗cR23(w)qc⊗1⊗d+c⊗d⊗1R13(zw)qd⊗c⊗1R12(z)

qc⊗d⊗1+d⊗c⊗1+c⊗1⊗dR12(z)(∆⊗ id)qd⊗cR13(zw)q1⊗c⊗dR23(w)

=qd⊗1⊗c+1⊗c⊗d+1⊗d⊗cR23(w)(id⊗∆)qc⊗dR13(zw)qd⊗c⊗1R12(z)

qc⊗d⊗1+d⊗c⊗1+c⊗1⊗d(∆op ⊗ id)qd⊗cR12(z)R13(zw)q1⊗c⊗dR23(w)

=qd⊗1⊗c+1⊗c⊗d+1⊗d⊗c(id⊗∆op)qc⊗dR23(w)R13(zw)qd⊗c⊗1R12(z)

qc⊗d⊗1+d⊗c⊗1+c⊗1⊗dqd⊗1⊗c+1⊗d⊗cR12(z)R13(zw)q1⊗c⊗dR23(w)

=qd⊗1⊗c+1⊗c⊗d+1⊗d⊗cqc⊗1⊗d+c⊗d⊗1R23(w)R13(zw)qd⊗c⊗1R12(z)

qd⊗c⊗1R12(z)R13(zw)q1⊗c⊗dR23(w) = q1⊗c⊗dR23(w)R13(zw)qd⊗c⊗1R12(z)

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

R12(z)q−1⊗c⊗dR13(zw)q1⊗c⊗dR23(w) = R23(w)q−d⊗c⊗1R13(zw)qd⊗c⊗1R12(z)

as desired.

Recall our definition πVz := πV ◦ ϕz. For two finite-dimensional complex vector spacesV and W we let

RVW (z/w) := (πVz ⊗ πWw )R : V ⊗W → V ⊗W.

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

RUV12 (x/z)RUW13 (x/w)RVW23 (z/w) = RVW23 (x/z)RUW13 (w/x)RUV12 (z/w)

19

where U is a finite-dimensional complex vector space, x is a complex parameter andboth sides of the equation are linear operators on U ⊗ V ⊗W . Here we use the factthat c acts as 0 in all evaluation representations (for further details, see [EFK98]).

Define RVW (z/w) := P VW RVW (z/w) : V ⊗W →W⊗V . Here P VW : V (z)⊗W (w)→W (w)⊗ V (z) denotes the permutation operator, i.e. P VW (v⊗ u) = (u⊗ v) for v ∈ V ,u ∈W . This tensor space is also a Uq(sl2)-module by defining πVWz,w := (πVz ⊗πWw )◦∆ :

Uq(sl2)→ End(V ⊗W ).

The R-matrix satisfies the intertwining property for Uq(sl2):

RVW (z/w)πVWz,w (X) = πWVw,z (X)RVW (z/w)

for all X ∈ Uq(sl2). If the tensor space V ⊗W is an irreducible Uq(sl2)-module, thenby Schur’s lemma we know that such an intertwiner is unique up to a scalar. Henceif z, w are such that the corresponding Uq(sl2)-module is irreducible, then the inter-twining property defines the R-matrix up to a scalar. Constructing a linear operatorV ⊗W → W ⊗ V that satisfies the intertwining property generally comes down tosolving a finite linear system, making finding an R-matrix a relatively easy task.

2.3 Example: R-Matrices For Two-DimensionalRepresentations

In the case where our representation space V is two- dimensional, being the simplestnon-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(sl2)-module V with evaluation representa-tion πVz . We have seen in the previous section that V (z1) ⊗ V (z2) is irreducible as aUq(sl2)-module if and only if z1z2 6= ±q. Hence for such a module, it suffices to construct

a Uq(sl2)-intertwiner. By Schur’s lemma, such an intertwiner will be unique up to ascalar and hence equal to a universal R-matrix acting on V (z1)⊗ V (z2).

Take W equal to V and let RVW (z1/z2) denote a Uq(sl2)-intertwiner in End(V ⊗W ).For RVW (z1/z2) to function as an intertwiner, we require:

RVW (z1/z2)πVWz1,z2(X) = πVWz2,z1(X)RVW (z1/z2)

for all X ∈ Uq(sl2). For qh1 in particular, we obtain:

πz1,z2(qh1)RVW (z1/z2)(v±⊗v±) = RVW (z1/z2)πz1,z2(qh1)(v±⊗v±) = q±2RVW (z1/z2)(v±⊗v±)

which means that RVW (z1/z2)v ∈ C(v+ ⊗ v+) if and only if v ∈ C(v+ ⊗ v+), andlikewise for C(v− ⊗ v−). Similarly, we have:

πz1,z2(qh1)RVW (z1/z2)(v±⊗v∓) = RVW (z1/z2)πz1,z2(qh1)(v±⊗v∓) = RVW (z1/z2)(v±⊗v∓)

which is only possible if RVW (z1/z2)(v± ⊗ v±) ∈ C(v+ ⊗ v−)⊕C(v− ⊗ v+). Thereforethe matrix takes the form:

RVW (z1/z2) =

a(z1/z2) 0 0 0

0 b(z1/z2) c(z1/z2) 00 d(z1/z2) e(z1/z2) 00 0 0 f(z1/z2)

20

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

Using the intertwining property

RVW (z1/z2)πVWz1,z2(X) = πVWz2,z1(X)RVW (z1/z2)

which holds for all X ∈ Uq(sl2), we obtain a set of linear relations for a, b, c, d, e, f withunique solution (up to a scalar function of z1/z2):

RVW (z1/z2) = a(z1/z2)

1 0 0 00 z1

z2(1− q−2) q−1( z1z2 − 1) 0

0 q−1( z1z2 − 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 otherintertwiner 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-operatorsusing 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 (recalltheorem 1.3.1), we can choose specific values where the product decomposes into atleast one subspace of higher dimension than the original representation spaces.

More concretely (here we use the reasoning and notation of [Jim89]), suppose we havea family of parametrised R-matrices:

RV V′(z) : V ⊗ V ′ → V ′ ⊗ V

satisfying the Yang-Baxter Equation in the following form:

(RV2V3(z)⊗I)(I⊗RV1V3(zw))(RV1V2(w)⊗I) = (I⊗RV1V2(w))(RV1V3(zw)⊗I)(I⊗RV2V3(z))(2.4.1)

as an equality in Hom(V1 ⊗ V2 ⊗ V3, V3 ⊗ V2 ⊗ V1). Note that, for z-independentsubspaces Wi ⊂ Vi such that:

RViVj (z)(Wi ⊗Wj) ⊂Wj ⊗Wi, (2.4.2)

equation 2.4.1 will still hold if we restrict each RViVj to Wi ⊗Wj .Furthermore, if we define:

RV⊗V′,V ′′(z) := (RV V

′(zz1)⊗I)(I⊗RV ′V ′′(zz2)) : V ⊗V ′⊗V ′′ → V ′′⊗V ⊗V ′ (2.4.3)

for fixed z1, z2, then equation 2.4.1 remains valid when we replace RV1Vi with RV1⊗V′1 ,Vi

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

RV′′,V⊗V ′(z) := (I ⊗ RV ′′V ′(zz1))(RV

′′V (zz2)⊗ I) (2.4.4)

21

such that the YBE still holds when we replace RViV3 with RVi,V3⊗V′3 for i = 1, 2.

In our case, we take the following subspace:

W = RV′V (z2/z1)(V

′ ⊗ V ) ⊂ V ⊗ V ′.

Equation 2.4.1 gives us:

RV⊗V′,V ′′(z)(W ⊗ V ′′)

= (by definition 2.4.3)

(RV V′′(zz1)⊗ I)(I ⊗ RV ′V ′′(zz2))(RV

′V (z2/z1)⊗ I)(V ′ ⊗ V ⊗ V ′′)= (by the YBE)

(I ⊗ RV ′V (z2/z1))(RV ′V ′′(zz2)⊗ I)(I ⊗ RV V ′′(zz1))(V ′ ⊗ V ⊗ V ′′)

⊂ V ′′ ⊗W.

This means that condition 2.4.2 is satisfied when we take:

W1 := RV′1V1(z2/z1)(V

′1 ⊗ V1) ⊂ V1 ⊗ V ′1 ;

W2 := V2;

W3 := V3.

Similarly, we can show that RV′′,V⊗V ′(z)(V ′′ ⊗W ) ⊂ W ⊗ V ′′. This space W is in

general not a proper subspace, but we will see that for specific choices of z2/z1, ourevaluation representation is reducible and W becomes a proper subspace of the tensorrepresentation space. In this case, we find non-trivial R-matrices:

RWV ′′(z) = RV⊗V′,V ′′(z)|W⊗V ′′ ;

RV′′W (z) = RV

′′,V⊗V ′(z)|V ′′⊗W .

Finally, we will perform this fusion process on our evaluation representations of Uq(sl2).Denote the representation space corresponding to spin k ∈ 1

2Z≥0 by V k, where V k =⊕2ki=0 v

ki . By abuse of notation, we also write Rk` instead of RV

kV `for the R-matrix

acting on V k ⊗ V `. Finally, let P k` : V k ⊗ V ` → V ` ⊗ V k denote the permutationoperator.

Proposition 2.4.1. We have the following splitting intertwiners:

(i) The linear map ιk : V k+ 12 ↪→ V

12 ⊗ V k defined by

ιk(vk+ 1

2n

)= q

12nv

120 ⊗ v

kn + q−

12(n−2k−1)[n]qv

121 ⊗ v

kn−1

defines a Uq(sl2)-intertwiner ιk : V k+ 12 (z) ↪→ V

12 (zq−k)⊗ V k(zq

12 ).

(ii) The linear map jk := P12kιk : V k+ 1

2 ↪→ V k ⊗ V12 defines a Uq(sl2)-intertwiner

jk : V k+ 12 (z) ↪→ V k(zq−

12 )⊗ V

12 (zqk).

22

Proof. The proof is a straightforward calculation that the intertwining relations holdfor X = qhi , ei, fi. We show the intertwining of ι and πz(e0) as an example:

ιkπk+ 1

2z (e0)v

k+ 12

n

=ιk(z−1f1 · v

k+ 12

n

)=z−1ιk

(vk+ 1

2n+1

)=z−1

(q

12(n+1)(v

120 ⊗ v

kn+1) + q−

12(n−2k)[n+ 1]q(v

121 ⊗ v

kn)

)=z−1

(q

12(n+1)(v

120 ⊗ v

kn+1) + qk−

12n q

n+1 − q−n−1

q − q−1(v

121 ⊗ v

kn)

)=z−1

(q

12(n+1)(v

120 ⊗ v

kn+1) + qk−

12n

(qn + q−1

qn − q−n

q − q−1

)(v

121 ⊗ v

kn)

)=z−1

(q

12(n+1)(v

120 ⊗ v

kn+1) +

(qk+

12n + qk−1−

12n q

n − q−n

q − q−1

)(v

121 ⊗ v

kn)

)=z−1

(qk+

12n(v

121 ⊗ v

kn) + q

12(n+1)(v

120 ⊗ v

kn+1) + q−

12 q−

12(n−2k−1)[n]qq

−1(v121 ⊗ v

kn)

)=(z−1qk(f1 ⊗ 1) + z−1q−

12 (qh1 ⊗ f1)

)·(q

12n(v

120 ⊗ v

kn) + q−

12(n−2k−1)[n]q(v

121 ⊗ v

kn−1)

)=

(π

12

zq−k ⊗ πkzq

12

)◦∆(e0)ι

kvk+ 1

2n .

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

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

(ιk ⊗ idV `)Rk+12,`(z) = R

12,`

13 (q−kz)Rk`23(q12 z)(ιk ⊗ idV `)

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

12 ⊗ V k ⊗ V `.

Proof. Due to the quasitriangular properties of R, we have:

((∆⊗ id)Rk+12,`(z1/z2)) = R

12,`

13 (q−kz1/z2)Rk`23(q

12 z1/z2).

By representing these as linear operators on V12 (q−kz1)⊗ V k(q

12 z1)⊗ V `(z2) and pre-

composing with ιk ⊗ idV ` : V k+ 12 (z1)⊗ V `(z2)→ V

12 (q−kz1)⊗ V k(q

12 z1)⊗ V `(z2), we

obtain: ((∆⊗ id)Rk+

12,`)

(ιk ⊗ idV `) = R12,`

13 (q−kz1/z2)Rk`23(q

12 z1/z2)(ι

k ⊗ idV `)

or, after applying proposition 2.4.1 on the first two tensor legs:

(ιk ⊗ idV `)(Rk+

12,`(z1/z2)

)= R

12,`

13 (q−kz1/z2)Rk`23(q

12 z1/z2)(ι

k ⊗ idV `).

Setting z = z1/z2 gives the desired result.

23

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 ourspin chain. In some models, the chain is assumed to be circular, eliminating the needto specify these boundary conditions. In our case, we will set up reflectional walls oneach side, allowing the boundary particles to interact with themselves via these walls:

|V (z1)•

V (z2)• · · ·V (zN )• |

3.2 K-matrices: Properties

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

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

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 theinteraction of a particle with the boundary. For a particle with representation spaceV (z), this operator K is dependent on the parameter z. Moreover, since a reflectionchanges the direction of the particle, its parameter is flipped from z to z−1. Hence wehave reflection operators K(z) : V (z) → V (z−1). Analogously to the YBE, we wantto guarantee that two sets of reflections differing only chronologically yield the sameresult. Graphically, we require the following equivalence:

Source: [DM06].

24

Here one can see the vertical axis as the flow of time, moving upward. This translatesdirectly 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 (z−11 )⊗ V (z−12 )

). TheK(z) :

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

Like in the case of the general YBE, we can construct a family of operators whichcommute thanks to equality 3.2.1. This involves defining another reflection operatorK which satisfies a shifted version of equation 3.2.1:

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

2z1z2)K1(z1)R21(z2/z1).

Using these operators K and K and an auxiliary space W equal to V (similar to section2.1), 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] forthe details.

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

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

K(z)πVz (X) = πVz−1(X)K(z) (3.2.2)

for all X ∈ Uq(sl2). Indeed, this would imply:

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

=(πz−11⊗ πz−1

2)R(K(z1)⊗ id)P (πz−1

2⊗ πz1)RP (id⊗K(z2))

=(πz−11⊗ πz−1

2)R(K(z1)⊗ id)P (πz−1

2⊗ πz1)R(K(z2)⊗ id)P

=(πz−11⊗ πz−1

2)R(K(z1)⊗ id)P (K(z2)⊗ id)(πz2 ⊗ πz1)RP

=(πz−11⊗ πz−1

2)R(K(z1)⊗ id)(id⊗K(z2))P (πz2 ⊗ πz1)RP

=(πz−11⊗ πz−1

2)R(id⊗K(z2))(K(z1)⊗ id)P (πz2 ⊗ πz1)RP

=(id⊗K(z2))(πz−11⊗ πz2)R(K(z1)⊗ id)P (πz2 ⊗ πz1)RP

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

25

As usual, P stands for the permutation operator. We see that a sufficient property forK to satisfy the reflection equation would be the intertwining property above. In thiscase, we will construct an operator that satisfies equation 3.2.2 not for all X ∈ Uq(sl2),but for all X ∈ B ⊂ Uq(sl2), where B is the so-called q-Onsager algebra, a subalgebra

of Uq(sl2) that we will define later.

3.3 Higher Dimensions And Fusion Of K-Matrices

Recall our splitting intertwiners:

(i) The linear map ιk : V k+ 12 ↪→ V

12 ⊗ V k defined by

ιk(vk+ 1

2n

)= q

12nv

120 ⊗ v

kn + q−

12(n−2k−1)[n]qv

121 ⊗ v

kn−1

defines a Uq(sl2)-intertwiner ιk : V k+ 12 (z) ↪→ V

12 (zq−k)⊗ V k(zq

12 ).

(ii) The linear map jk := P12kιk : V k+ 1

2 ↪→ V k ⊗ V12 defines a Uq(sl2)-intertwiner

jk : V k+ 12 (z) ↪→ V k(zq−

12 )⊗ V

12 (zqk).

The following result is due to Reshetikhin, Stokman and Vlaar.[RSV16]

Suppose we have operators K12 (z) : V

12 (z) → V

12 (z−1) that satisfy the reflection

equation:

R12

12

21 (z1/z2)K121 (z1)R

12

12

12 (z2z1)K122 (z2) = K

122 (z2)R

12

12

21 (z1z2)K121 (z1)R

12

12

12 (z1/z2)

where the above is an identity in Hom(V

12 (z1)⊗ V

12 (z2), V

12 (z−11 )⊗ V

12 (z−12 )

). Then

for each k ∈ 12Z≥2, there exist unique linear operators Kk(z) : V k(z)→ V k(z−1) which

satisfy:

(i) the following recursive fusion formula:

jkKk+ 12 (z) = P

12kK

121 (zq−k)R

12k(z2q

12−k)Kk

2 (zq12 )ιk (3.3.1)

for all k ∈ 12Z≥1; and:

(ii) the reflection equation:

R`k21(z1/z2)Kk1 (z1)R

k`12(z2z1)K

`2(z2) = K`

2(z2)R`k21(z1z2)K

k1 (z1)R

k`12(z1/z2)

as an identity in Hom(V k(z1)⊗ V `(z2), V

k(z−11 )⊗ V `(z−12 )), for all k, ` ∈ 1

2Z≥1.

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

26

4 The q-Onsager Algebra

As seen in the last section, in order for our K(z)-operators to satisfy the boundaryYang-Baxter equation, a sufficient condition would be to satisfy equation 3.2.2. Notethat there are no nontrivial Uq(sl2)-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(sl2). For this

purpose, we introduce the q-Onsager Algebra: a coideal subalgebra of Uq(sl2) whoseproperties 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 propertythat V (z)|B and V (z−1)|B are isomorphic. It is a so-called tridiagonal algebra, theproperties of which have been studied extensively by Baseilhac, Terwilliger and Kolbamong others ([BB10], [Ter15], [IT09], [Kol14]).

4.1 The q-Onsager Algebra: Definitions

In its most general form, the q-Onsager Algebra B ⊂ Uq(sl2) 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(sl2).

Hence ∆(B) ⊂ B ⊗ Uq(sl2), meaning that B is a right coideal subalgebra of Uq(sl2).We can let B act on any finite-dimensional space V (z) through the usual evaluationrepresentation πVz . A straightforward but rather long and tedious calculation showsthat the generators satisfy:

B30B1 − [3]qB

20B1B0 + [3]qB0B1B

20 −B1B

30 = q(q + q−1)2c0[B1, B0]

B31B0 − [3]qB

21B0B1 + [3]qB1B0B

21 −B0B

31 = 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 theydetermine the structure of B uniquely. For a demonstration, we refer to [IT09].

27

For any Uq(sl2)-module M , we can construct a B-module (denoted by M |B) by restrict-

ing the action of Uq(sl2) to elements in the q-Onsager algebra B ⊂ Uq(sl2). Supposethat, like in the construction in the previous section, we have B-intertwiners

Kk(z) : V k(z)|B → V k(z−1)|B;

K`(z) : V `(z)|B → V `(z−1)|B

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

for Kk to be a B-intertwiner we require

Kk(z)πVz (Q) = πVz−1(Q)Kk(z) ∈ Hom(V k(z), V k(z−1)) (4.1.1)

for all Q ∈ B (and similarly for K l).

Seeing as B is a right coideal subalgebra, i.e. ∆(B) ⊂ B ⊗ Uq(sl2), this implies thatboth

R`k21(z2/z1)Kk2 (z2)R

k`12(z

−11 z−12 )K`

1(z1)

andK`

1(z1)R`k21(z

−11 z−12 )Kk

2 (z2)Rk`12(z1/z2)

(both sides of the reflection equation) are B-intertwiners for the modules

V k(z1)⊗ V `(z2)|B → V k(z−11 )⊗ V `(z−12 )|B.

In order to use Schur’s lemma for V k(z1) ⊗ V `(z2)|B later on, we would like that,for fixed ci, si (which determine B), V k(z1)⊗ V `(z2) is irreducible as a B-module forgeneric values of z1, z2. This will automatically allow us to conclude that the reflectionequation 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 requiredfor these modules to be irreducible.

4.2 Standard Finite-Dimensional Representations

Consider the evaluation representations πVz : Uq(sl2)→ End(V (z)) as seen in previous

sections. We can use this same map πVz to let B ∈ U(sl2) act on an evaluation repre-sentation space V (z). In order to prove uniqueness (up to a scalar) of our intertwiningand reflection operators, we would like to again use Schur’s lemma. For this we requireV (z) to be irreducible as a B-module, at least for generic values of z1, z2.

Letting B act on V =⊕n

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)

]2q

−[

1

2(n+ 1− 2k)

]2q

.

28

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 s1q

n−4

. . .... s1q

2−n −c1z−1q−n[n]q[1]q0 . . . 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 s0q

4−n

. . .... s0q

n−2 z[n]q[1]q0 . . . −c0z−1qn−2 s0q

n

We can find eigenvalues of πz(B1) following [Koe96], section 5. Take the algebraUq(su(2)) as generated by elements A,B,C,D with the following relations:

AD = 1 = DA, AB = qBA, AC = q−1CA, BC − CB =A2 −D2

q − q−1.

This algebra acts on W = ⊕nk=0Cek through representation tn : Uq(su(2))→ End(W )as:

tn(A)ek = q12n−kek; tn(D)ek = qk−

12nek

tn(B)ek =√

[n− k + 1]q[k]qek−1

tn(C)ek =√

[n− k]q[k + 1]qek+1

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

Xσ = iq12B − iq−

12C − [σ]q(A−D) ∈ Uq(su(2))

Proposition 5.2 of [Koe96] allows us to find explicit eigenvalues and eigenvectors of

tn(XσA) = iq12 tn(BA)− iq−

12 tn(CA)− [σ]q(t

n(A)2 − idW )

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

µj(σ) = [−2j − σ]q.

29

We now want to choose nonzero coefficients αk ∈ C such that ek = αkvk and tn(YσA)ek =πz(B1)ek for all k = 0, 1, . . . , n. This gives us:(

iq12 tn(BA)− iq−

12 tn(CA)− [σ]qt

n(A)2)ek = αkπz(B1)vk

which breaks down to:

iq12 q

12n−k√

[n− k + 1]q[k]qαk−1 = −αkc1z−1qn−2k[k]q[n− k + 1]q

[σ]qqn−2kαk = αks1q

n−2k

−kq−12 q

12n−k√

[n− k]q[k + 1]qαk+1 = αkz

which, in turn, forces:

αk−1αk

= ic1z−1q−

12+ 1

2n−k√

[k]q[n− k + 1]q

[σ]q = s1αkαk+1

= −ic1z−1q−12+ 1

2n−k√

[k + 1]q[n− k]q.

Combining the first and last equation gives:

ic1z−1q−

32+ 1

2n−k√

[k + 1]q[n− k]q =αkαk+1

= −ic1z−1q−12+ 1

2n−k√

[k + 1]q[n− k]q

resulting in c1 = −q.

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

k=0 vn,jk (σ)ek of the oper-

ator tn(YσA), corresponding to eigenvalues

µj(σ) = [−2j − σ]q.

The coefficients vn,jk are known explicitly:

vn,jk = Cn,j(σ)ik−nqσ(n−k)q12n−kn−k−1

((q2n; q−2)n−k(q2; q2)n−k

) 12

×∞∑m=0

(q2k−2n; q2)m(q2j−n; q2)m(−q−2j−n−2σ; q2)m(q−2n; q2)m(q2; q2)m

q2m

where the constant is given by

Cn,j(σ) = q12n+j

((qn; q−1) 1

2n−j

(q; q) 12n−j

) 12 (

1 + q−4j−2σ

1 + q−2σ

) 12 (

(−q2−2σ; q2) 12n−j(−q

2+2σ; q2) 12n+j

)− 12.

Here we use the shifted factorial notation:

(a; q)k :=k−1∏i=0

(1− aqi).

With our explicit form of πz(B0) it is possible to show (through straightforward buttedious 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 isirreducible as a B-module.

We now turn to tensor product representations of the form (V (z1)⊗ V (z2)) |B.

30

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 ◦ ϕz1 ⊗ πV ◦ ϕz2) ◦∆. We usethe ordered basis given by:

v+ ⊗ v+ =

1000

; v+ ⊗ v− =

0100

; v− ⊗ v+ =

0010

; v− ⊗ v− =

0001

.

This gives us the following linear operators:

π(B1) =

q2s1 −c−10 qz−12 −c−10 q2z−11 0

z2 s1 0 −c−10 z−11

qz1 0 s1 −c−10 qz−12

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 −c0z−11 −c0q−1z−12 q2s0

.

Both of these operators give a similar eigenspace decomposition; in both cases, thevector 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 qz−12 0 r + 2q3 − s1(q2 − 1)

√c0r r + 2q3 + s1(q

2 − 1)√c0r

s1(q2 − 1) qz2 z2

(c0s1(q

2 − 1)−√c0r)

z2(c0s1(q

2 − 1) +√c0r)

0 −z1 qz1(q2−1)√c0

(c0s1(q

2 − 1)−√c0r) qz1

(q2−1)√c0

(c0s1(q

2 − 1) +√c0r)

qz1 0 2c0qz1z2 2c0qz1z2

where r = c0s

21(q

2−1)2−4q3. Note that the third and fourth column differ only by thesign of

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

eigenvalues 12

(s1(q

2 + q−2)± (1 + q−2)√c−10 r

).

Using this eigenbasis, one can calculate

E−11 π(B0)E1

to see how π(B0) acts on the eigenvectors of π(B1). While the resulting matrix istoo 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. Thisimplies 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) acton π(B0)vi, we can obtain different ratios of v3 and v4 (seeing as they have differenteigenvalues). Hence, through linear combinations of actions of π(B0) and π(B1) on anyvi, we can obtain non-zero elements in at least subspaces Cv4, Cv3 and Cv1 ⊕ Cv2. A

31

B-stable subspace of V can then only exist in the form of C(av1 + bv2)⊕Cv3⊕Cv4 forsome constants a and b. Again, using the expressions above it is possible to explicitlycalculate constraints on a and b for this to be the case, leading to specific constraintson the parameters z1, z2. In the generic case however, we can conclude that there existsno 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 andTerwilliger ([Ter15] and [IT09]). Let A denote the associative C-algebra with 1 definedby 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 theq-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(sl2)

defined by

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

ϕs,t(z∗) = sf0q

h0 + s−1e1 + st−1qh0 + s−1tq−h0 .

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

Bδ−1

∼= Aϕs,t−→ Uq(sl2)

ϕVz−→ End(V (z))

allowing us to view the action of B on any representation space V (z) of Uq(sl2) throughthis morphism chain. Recall our notation Vn(z) for the evaluation representation wherez is the evaluation parameter and Vn is the irreducible Uq(sl2)-module of dimensionn+1. One can check that, for appropriately chosen parameters s, t, the representationof B by the composition ϕVz ◦ ϕs,t ◦ δ−1 coincides with the standard representationgiven by ϕVz (B). The advantage of viewing this representation through A is that thisallows 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(sl2)-module if and onlyif the q-strings Sn1(a1), . . . , Snr(ar) are pairwise in general position.

32

Here we call two such q-strings S1 and S2 in general position if at least one of thefollowing conditions holds:

(i) S1 ∪ S2 is not a q-string;

(ii) S1 ⊆ S2;(iii) S2 ⊆ S1.

Furthermore, we call two multi-sets {Sni(ai)}mi=1, {Sn′i(a′i)}m

′i=1 equivalent if m = m′

and there exists a permutation σ of {1, 2, . . . ,m} and εi ∈ {±1} such that ni = n′σ(i)and aεii = a′σ(i) for i = 1, 2, . . . , n. This means that the multi-sets {Sni(a

εii )}mi=1 and

{Sn′i(a′i)}m

′i=1 are equal. A multi-set Σ := {Sni(ai)}mi=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 willdenote as d (for more information on this definition, see [ITT01]). We then have thefollowing theorem from Ito and Terwilliger ([IT10], Theorem 1.17):

Theorem 4.4.1. For a Uq(sl2)-module V = Vn1(z1)⊗· · ·⊗Vnr(zr) and nonzero s, t ∈ C,the representation

r⊗i=1

πVni ◦ ϕzi

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, z−1i ) 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(V k(z1)⊗ V `(z2)

)|B →

(V k(z−11 )⊗ V `(z−12 )

)|B:

R`k21(z2/z1)Kk2 (z2)R

k`12(z

−11 z−12 )K`

1(z1)

andK`

1(z1)R`k21(z

−11 z−12 )Kk

2 (z2)Rk`12(z1/z2),

both sides of the reflection equation. We know(V k(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 toa constant. Hence we can solve a linear intertwining relation (see equation 3.2.2) forB to find solutions to the reflection equation (see also [DM06]).

33

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 fusionmethod guarantees that our recursively constructed higher-dimensional K-matricesretain the intertwining property. Suppose we have B-intertwiners Kk(z) : V k(z) →V k(z−1) for a certain k ∈ 1

2Z≥2, as well as for k = 12 .

Analogously to subsection 3.3, we can obtain the following recursive fusion equality:

ιkKk+ 12 (z) = P k

12Kk

2 (zq−k)Rk12 (z2qk−

12 )Kk

1 (zq12 )jk.

Writing R := PR we can write the right-hand side of this equation as

K122 (zq−k)Rk

12 (z2qk−

12 )Kk

1 (zq12 )jk

which we can see to be a B-intertwiner

V k+ 12 (z)|B →

(V

12 (z−1q−k)⊗ V k(z−1q

12 ))|B.

Hence the left-hand side of the fusion formula tells us that the fused K-operator isalso a B-intertwiner:

Kk+ 12 (z) : V k+ 1

2 (z)|B → V k+ 12 (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 dimV = 2 is of special importance as it plays an important role in theprocess 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 + s0zz−2−z2q−q−1

q(z−2−z2)q−q−1 qs1z + s0z

−1

)where f(z) is a complex function.

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Popular summary

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

| 1• 2• . . .N• |

Each of these particles has a spin state, describing its spin (a type of intrinsic angularmomentum) 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 spacedenoted by V . The state of the entire system can be seen as an element in the tensorproduct of these separate spaces:

V ⊗N = V ⊗ V ⊗ · · · ⊗ V

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

πz : Uq(sl2)→ End(V )

Here, πz is a homomorphism (depending on a complex parameter z) that maps elementsin an algebra Uq(sl2) to endomorphisms of V . This effectively turns the state space into

a so-called Uq(sl2)-module, denoted as V (z) (where z is the same complex parameteras featured in the homomorphism πz). We can define similar representations for tensorproducts 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 = −JN∑j=1

σjσj+1 − hN∑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 theirstate 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 notchange the result inasmuch as:

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Here we view time as the vertical axis, moving upward. On the left, we first seeparticles 1 and 2 interacting, followed by 13, followed by 23. The right diagram hasthe 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:

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

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 oursystem, we require the following equivalence:

Here one can see the vertical axis as the flow of time, moving upward. On the left, wehave interactions 2-1, 1-boundary, 1-2, 2-boundary; on the right we have the same inthe opposite order. This translates to the following reflection equation (also known asthe 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 interactionwith the boundary. Their index refers to the tensor leg on which they are acting.

We use several properties of the algebra Uq(sl2) to explicitly construct R- and K-operators that satisfy these requirements. Specifically, we use irreducibility of cer-tain Uq(sl2)-modules and B-modules (where B is a subalgebra of Uq(sl2) called the

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q-Onsager algebra). This means that a space W has no subspaces that are left intactunder the action of Uq(sl2) (apart from the trivial subspaces {0} and W itself). Thisallows us to reduce the Yang-Baxter equations to a simpler equation which we cansolve explicitly.

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