Quantum Dynamical R-matrices and Quantum Integrable Systems

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MSc Mathematical Physics

Master Thesis

Quantum Dynamical R-matrices

and Quantum Integrable Systems

Author:

Supervisor:

Brinn Hekkelman

prof. dr. J. V. Stokman

Examination date:

August 29, 2016

Korteweg-de Vries Institute for Mathematics

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Abstract

Universal R-matrices and Drinfeld twisted universal R-matrices are studied as solutions to the quantum Yang-Baxter equation. Following a similar procedure, fusion opera-tors are used to define exchange operaopera-tors or quantum dynamical R-matrices that are solutions to the quantum dynamical Yang-Baxter equation. The quantum dynamical R-matrices are used to construct a set of transfer operators that describe a quantum in-tegrable system. An elaborate proof of the simultaneous diagonalizability of the transfer operators is provided. This work largely follows a structure outlined by Pavel Etingof.

Title: Quantum Dynamical R-matrices and Quantum Integrable Systems Author: Brinn Hekkelman, office@bhekkelman.com, 6345794

Supervisor: prof. dr. J. V. Stokman Second Examiner: dr. R. R. J. Bocklandt Examination date: August 29, 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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1 Introduction

The study of exactly solvable quantum systems, or quantum integrable systems, is one of the most influential and exciting fields in modern mathematical physics. A theory of exactly solvable quantum systems, referred to as the quantum inverse scattering method, was introduced by Faddeev in [Faddeev and Takhtadzhan, 1979]. A short introduction to the quantum inverse scattering method can be found in chapter 12 of [Essler et al., 2005]. The quantum Yang-Baxter equation lies at the basis of this theory of exactly solvable quantum systems.

The aim of this thesis is to define quantum integrable systems through the method of quantum inverse scattering, using solutions to the quantum dynamical Yang-Baxter equation referred to as quantum dynamical R-matrices. This process largely follows the structure outlined by Etingof in [Etingof and Latour, 2005].

The following three sections describe the contents of this thesis as presented in chapters 2, 3, and 4 respectively.

1.1 The Quantum Yang-Baxter Equation

As is well known, quantum groups Uq have a braided Hopf algebra structure (see

appendix sections A.3 and A.3.2). Universal R-matrices R ∈ Uq⊗ Uq satisfy the

quantum Yang-Baxter equation

(1.1) R12R13R23 = R23R13R12

on Uq⊗Uq⊗Uq. For U, V , W finite-dimensional representations of Uq, the operator

(1.2) RVW : V ⊗ W → V ⊗ W : (v ⊗ w ) 7→ R (v ⊗ w )

satisfies the quantum Yang-Baxter equation on U ⊗ V ⊗ W ; (1.3) R_UV12R_UW13 R_VW23 = R_VW23 R_UW13 R_UV12.

Further solutions to the quantum Yang-Baxter equation may be found by ‘twisting’ universal R-matrices. An invertible twist J ∈ Uq⊗ Uq satisfies the twist equation

(1.4) (∆ ⊗ id)(J) (J ⊗ 1) = (id ⊗∆)(J) (1 ⊗ J).

Any Hopf algebra can be twisted by J to yield a new Hopf algebra with a new universal R-matrix given by

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The twisted universal R-matrix RJ is again a solution to the quantum Yang-Baxter

equation. Note that if two twists J and J0 _{are gauge equivalent, i.e. there exists}

an invertible element x ∈ Uq such that

(1.6) J0 = ∆(x ) J (x-1⊗x-1_),

the twisted universal R-matrices RJ and RJ0 are equal up to a change of basis, and

are thus essentially the same solution to the quantum Yang-Baxter equation.

1.2 The Quantum Dynamical Yang-Baxter

Equation

Similar to finding solutions to the quantum Yang-Baxter equation on Uq⊗ Uq⊗ Uq

by twisting universal R-matrices, operators RVW may be twisted by a dynamical

twist called the fusion operator in an attempt to find solutions to the quantum dynamical Yang-Baxter equation on U ⊗ V ⊗ W .

A finite-dimensional representation V of Uq can be appended to an irreducible

Verma module Mλ by a unique intertwiner

(1.7) Φv_λ : Mλ → Mλ−wt v ⊗ V

defined to have the expectation value hΦv

λi = v ∈ V. These intertwiners can be

concatenated to ‘fuse’ two representations V and W of Uq together, resulting in

(1.8) Φv ,w_λ = (Φv_{λ−wt w} ⊗ id) Φw

λ : Mλ → Mλ−wt w −wt v ⊗ V ⊗ W

and the definition of an invertible dynamical twist called the fusion operator (1.9) JVW(λ) : V ⊗ W → V ⊗ W : v ⊗ w 7→ hΦv ,wλ i

that satisfies the dynamical twist equation (1.10) J_{U⊗V ,W}12,3 (λ) J12 UV(λ − h 3_{) = J}1,23 U,V ⊗W(λ) J 23 VW(λ).

The parameter λ is called the dynamical parameter.

Much like twisting a universal R-matrix R into a universal R-matrix RJ, an

operator RVW can be twisted into an operator

(1.11) RVW(λ) = JVW(λ)-1RWV21 J 21

WV(λ) : V ⊗ W → V ⊗ W

called the exchange operator. Since the exchange operator depends on the dy-namical parameter λ, it is a solution to the dydy-namical analog of the quantum

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1.3 Quantum Integrable Systems 3 Yang-Baxter equation on U ⊗ V ⊗ W ; (1.12) R_UV12(λ − h3) R_UW13 (λ) RVW23 (λ − h 1 ) = R_VW23 (λ) RUW13 (λ − h 2 ) R_UV12(λ), called the quantum dynamical Yang-Baxter equation. Because the exchange op-erator RVW(λ)is a solution to the dynamical analog of the quantum Yang-Baxter

equation, it is also referred to as a quantum dynamical R-matrix.

1.3 Quantum Integrable Systems

Quantum dynamical R-matrices may be used to construct a commuting set of operators that describe a quantum integrable system. One way to construct a suitable set of commuting operators is by considering operators DC associated to

central elements C ∈ Z(Uq). Taking V a finite-dimensional representation of Uq,

for every x ∈ Uq the unique operator Dx on functions f : h∗ → V [0] is defined by

(1.13) Dxtr |Mµ(Φ v µq 2λ_{) = tr |} Mµ(Φ v µx q 2λ_),

independent of the choice of Mµ, V and v ∈ V [0]. For central elements CW,

associated to finite-dimensional representations W of Uq, the set of operators DCW

commutes by definition. Moreover, the trace functions (1.14) Ψv(λ, µ) = tr |Mµ(Φ

v µq

2λ₎

satisfy the difference equations (1.15) DCWΨ

v_{(λ, µ) = χ}

W(q2(µ+ρ)) Ψv(λ, µ).

Using the quantum dynamical R-matrix RWV(λ), another set of operators DW on

functions f : h∗ _{→ V [0]}_{, called transfer operators, is defined as}

(1.16) DWf(λ) =

X

ν∈h∗

tr |W [ν]RWV(−λ − ρ)Tνf (λ).

Because of a remarkable relation between the transfer operators DW and the

op-erators DCW,

(1.17) DW =δq(λ) DCW δq(λ)

-1_,

where δq(λ)is a scaling factor, the transfer operators DW also commute. Moreover,

the trace function FV(λ, µ), which is a sum of trace functions Ψv(λ, µ) scaled by

the factor δq(λ), satisfies the difference equations

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In other words, the trace function FV(λ, µ)diagonalizes the set of commuting

trans-fer operators DW. This means that the set of algebraically independent transfer

operators {DΛ1, ... , DΛn−1}, with Λi the fundamental representations of Uq, define a

quantum integrable system.

From a physical perspective, the representation V is the quantum state space of the system and the operators DΛi are conserved quantities of the system.

Finally, two applications of the quantum integrable systems described are dis-cussed shortly. Firstly, quantum spin Calogero-Moser systems described by the Hamiltonian (1.19) H = 1 2∆h∗− X α∈Φ+ eαe−α e12α(λ)− e 1 2α(λ)2

emerge as the term of order ~2 (with q = e~

2) in the Taylor expansion of transfer

operators DW. Secondly, Macdonald operators are found to equal

(1.20) Mr =φk0(λ) -1_δ q(λ)-1DΛr δq(λ) φ k 0(λ), where φk

0(λ) is the vector-valued character, and thus allowing Macdonald

polyno-mials to be defined as a specific case of the trace functions Ψv_{(λ, µ), scaled by the}

vector-valued character φk 0(λ).

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5

2 Solutions to the QYBE

Quantum groups were introduced in the mid 1980’s by Drinfeld [Drinfeld, 1985, Drinfeld, 1986] and Jimbo [Jimbo, 1985, Jimbo, 1986] independently. Quantum groups appeared as the algebraic formulation of the work of physicists on the Yang-Baxter equation. The braided structure of quantum groups is closely related to solutions of the quantum Yang-Baxter equation, as is emphasised in this chapter.

2.1 Preliminaries

This section introduces the basic notation used throughout this thesis. Famil-iarity with semi-simple Lie algebras, Hopf algebras, and quantum groups will be assumed. A selection of known results that are used in this thesis is provided in the appendix for reference. A reader less familiar with the material may turn to [Humphreys, 1972] and [Kassel et al., 1997] for a detailed treatment of Lie algebras and quantum groups respectively.

Let g a semi-simple Lie algebra and h its Cartan subalgebra. The quantized universal enveloping algebra Uq(g) is the corresponding quantum group, and is

braided with a universal R-matrix R (see appendix remark A.35). Throughout this thesis, q ∈ C will be assumed not to be a root of unity. Verma modules of Uq(g) of highest weight λ ∈ h∗ will be denoted by Mλ, and are irreducible for

generic λ (see appendix definition A.27).

Of particular interest is g = sln, which will be the standard choice throughout

this thesis. This choice was made because quantum sln, or Uq(sln), is more practical

and insightful to work with. Moreover, the results for sln can still be generalized

to any semi-simple Lie algebra g without significant modification of the theory; see [Etingof and Latour, 2005].

Since many elements and operators will exist in tensor products of spaces and representations, a common notation is adopted to indicate on which components of tensor products these elements and operators act. An ordered series of super-script integers indicates in which component of the tensor product the respective components of the element or operator act. See appendix section B for further details and notation.

A result that will be used in many expressions is the identification of the Cartan subalgebra h with its dual h∗_{. The Killing form κ on g × g can be restricted to}

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h× h to yield a non-degenerate bilinear form

(2.1) κh = κ|h×h : h × h → C.

This map gives rise to an orthonormal basis {xi} of the Cartan subalgebra h, i.e.

κh(xi, xj) =δij, and the identifying isomorphism

(2.2) h → h∗ _{: x 7→}_κ h(x , ·).

2.1 Notation:

For each λ ∈ h∗_{, let} _{λ ∈ h denote the element for which}

(2.3) κh(λ, ·) = λ.

The element λ ∈ h may be conveniently written as

(2.4) λ =X i λ(xi) xi since (2.5) κh(λ, xj) = λ(xj) = X i λ(xi)κh(xi, xj) = κh X i λ(xi) xi, xj . Note that this means that µ(λ) = Piλ(xi)µ(xi) = λ(µ) for all λ, µ ∈ h

∗_.

2.2 The Quantum Yang-Baxter Equation

A universal R-matrix of Uq(sln)is a solution to the quantum Yang-Baxter equation.

2.2 Theorem (QYBE):

A universal R-matrix R of a Hopf algebra H satisfies the quantum Yang-Baxter equation on H ⊗ H ⊗ H:

(2.6) R12R13R23= R23R13R12.

Proof. Using relation 2 from the definition of the universal R-matrix, appendix definition A.3, then relation 1, and then relation 2 again provides

(2.7) R12_R13_R23_{= (R ⊗ 1) (∆ ⊗ id)(R)} = (∆op_{⊗ id)(R) (R ⊗ 1)} = (τ ⊗ id)(∆ ⊗ id)(R) R12 = (τ ⊗ id)(R13_R23_{) R}12 = R23_R13_R12_,

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2.3 Twisting 7 Now recall the definition of the tensor product of two representations (appendix definition A.13). For representations U, V , W of Uq(sln)write

(2.8) R_UV12 = RUV ⊗ id : U ⊗ V ⊗ W → U ⊗ V ⊗ W ,

with similar expressions for R23

VW and RUW13 .

2.3 Proposition:

Let U, V , W representations of Uq(sln), then

(2.9) R_UV12R_UW13 R_VW23 = R_VW23 R_UW13 R_UV12 on U ⊗ V ⊗ W .

Proof.

(2.10) R_UV12R_UW13 R_VW23 = (πU⊗ πV ⊗ πW)R12R13R23

since πU,πV,πW are linear. Now because R is a solution to the QYBE,

(2.11) (πU⊗ πV ⊗ πW)R12R13R23 = (πU⊗ πV ⊗ πW)R23R13R12

and the result follows. 2.4 Corollary:

The operator RVV is a solution to the QYBE on V ⊗ V ⊗ V .

The following section will take a closer look at finding solutions to the QYBE, based on the universal R-matrix R. Chapter 3 will instead consider the operator RWV, resulting in solutions to a dynamical analog of the QYBE.

2.3 Twisting

Theorem 2.2 shows that the universal R-matrix of Uq(sln) is a universal solution

to the QYBE. This implies that solutions to the QYBE can be found by finding universal R-matrices. One way to do this is through twisting.

Consider a Hopf algebra (H, µ, η, ∆, , S). 2.5 Definition:

An invertible element J ∈ H ⊗ H is called a twist if

(2.12) (∆ ⊗ id)(J) (J ⊗ 1) = (id ⊗∆)(J) (1 ⊗ J)

in H ⊗ H ⊗ H. This condition is also called the non-dynamical twist equation and will be abbreviated as

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2.6 Remark:

Note that applying the counit (id ⊗ ⊗ id) to equation 2.12 provides (2.14)

((id ⊗)∆ ⊗ id)(J) ((id ⊗ )J ⊗ 1) = (id ⊗( ⊗ id)∆)(J) (1 ⊗ ( ⊗ id)J) J ((id ⊗)J ⊗ 1) = J (1 ⊗ ( ⊗ id)J)

(id ⊗)J ⊗ 1 = 1 ⊗ ( ⊗ id)J

because of the counitality axiom 4 and the fact that J is invertible. Now applying another counit ( ⊗ id) shows

(2.15) ( ⊗ )(J) 1 = ( ⊗ id)(J)

where simply (_{⊗ )(J) ∈ C. Hence it may be assumed, without loss of generality,} that ( ⊗ )(J) = 1 and thus that ( ⊗ id)(J) = 1. By the same assumption, (id ⊗)(J) = 1. 2.7 Proposition: Writing J =P iai ⊗ bi, define ∆J : H → H ⊗ H and SJ : H → H as (2.16) ∆J(x ) = J-1∆(x ) J and SJ(x ) = p-1S (x ) p, with p = P

iS (ai) bi. Then (H,µ, η, ∆J,, SJ) is a Hopf algebra and is called the

twist of H by J, denoted as HJ.

Proof. To be a comultiplication, ∆J must satisfy the coassociativity axiom 3.

(2.17) (∆J ⊗ id)∆J(x ) = (∆J⊗ id)(J-1∆(x )J)

= (∆J⊗ id)(J-1)(∆J ⊗ id)(∆(x))(∆J ⊗ id)(J)

which becomes, using (∆J ⊗ id)(y ) = (J-1⊗1)(∆ ⊗ id)(y )(J ⊗ 1),

(2.18) (J-1⊗1)(∆ ⊗ id)(J-1_{)(∆ ⊗ id)(∆(x ))(∆ ⊗ id)(J)(J ⊗ 1).}

Now applying the condition for a twist, equation 2.12, and axiom 3, turns this into (2.19) (1 ⊗ J-1)(id ⊗∆)(J-1)(id ⊗∆)(∆(x ))(id ⊗∆)(J)(1 ⊗ J)

and then reduces to

(2.20) (id ⊗∆J)(J-1)(id ⊗∆J)(∆(x ))(id ⊗∆J)(J) = (id ⊗∆J)(J-1∆(x )J)

= (id ⊗∆J)∆J(x )

as required.

Considering remark 2.6, the counitality axiom 4 is trivially satisfied for since (2.21)

( ⊗ id)∆J(x ) = ( ⊗ id)(J-1∆(x )J)

= ( ⊗ id)(J-1_{)( ⊗ id)∆(x)( ⊗ id)(J)}

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2.3 Twisting 9 Finally, SJ must satisfy the convolution inverse axiom 5. However, before being

able to show this, it must be proved that the element p is indeed invertible. Writing J-1 ₌ P

ia 0 i ⊗ b

0

i and using the twist equation 2.12, remark 2.6, the convolution

inverse axiom 5 for S, and the Sweedler notation (see appendix section A.2.2), it is shown that (2.22) P ia 0 iS (b 0 i) p = P i ,ja 0 iS (b 0 i) S (aj) bj =P i ,j ,kak(bk) a 0 iS (ajbi0) bj =P i ,j ,kaka 0 iS (aj b0i)(bk) bj =P i ,j ,k,(bk)aka 0 iS (aj b0i) S (bk(1)) bk(2)bj =P i ,j ,k,(bk)aka 0 iS (bk(1)ajb0i) bk(2)bj

=µ(µ ⊗ id)(id ⊗S ⊗ id)(id ⊗∆)(J)(1 ⊗ J)(J-1⊗1) =µ(µ ⊗ id)(id ⊗S ⊗ id)(∆ ⊗ id)(J)(J ⊗ 1)(J-1_⊗1)

=µ(µ(id ⊗S)∆ ⊗ id)(J) =µ( ⊗ id)(J)

= 1

with a similar calculation showing that p Pia 0

iS (bi0) = 1, which means that the

element p is invertible with inverse

(2.23) p-1 = X

i

a_i0S (b0_i).

Now SJ may be shown to satisfy the convolution inverse axiom 5. Starting with

(2.24)

µ(SJ ⊗ id)∆J(x ) =µ(SJ⊗ id)(J-1∆(x )J)

=µ((p-1_{S (·) p ⊗ id)(J}-1_{∆(x )J)}

=µ((p-1_{⊗1)(S ⊗ id)(J}-1_{∆(x )J)(p ⊗ 1))}

= p-1_{µ((S ⊗ id)(J}-1_{∆(x )J)(p ⊗ 1))}

and writing out p, J, and ∆(x) explicitly yields

(2.25) (S ⊗ id)(J-1_{∆(x )J)(p ⊗ 1)} = (S ⊗ id) P j ,i ,(x ) (a0_j ⊗ b0 j)(x(1)⊗ x(2))(ai ⊗ bi) ! P k S (ak) bk ⊗ 1 = P j ,i ,(x ) S (a0_jx(1)ai) ⊗ bj0x(2)bi ! P k S (ak) bk ⊗ 1 = P k,j ,i ,(x ) S (ai) S (x(1)) S (a0j) S (ak) bk ⊗ bj0x(2)bi

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which means that (2.26) µ((S ⊗ id)(J-1_{∆(x )J)(p ⊗ 1)) =} P k,j ,i ,(x ) S (ai) S (x(1)) S (a0j) S (ak) bkbj0x(2)bi = P k,j ,i ,(x ) S (ai) S (x(1)) S (aka0j) bkbj0x(2)bi = P i ,(x ) S (ai) S (x(1))µ(S ⊗ id)(JJ-1) x(2)bi = P i ,(x ) S (ai) S (x(1)) x(2)bi =P i S (ai)µ(S ⊗ id)∆(x) bi =P i S (ai) bi(x) = p(x) and so finally (2.27) µ(SJ⊗ id)∆J(x ) = p-1p(x) = (x) as required. 2.8 Proposition:

Let H a braided Hopf algebra with universal R-matrix R and J a twist, then HJ

with universal R-matrix

(2.28) RJ =τ (J-1) R J

is again a braided Hopf algebra.

Proof. For RJ to be a universal R-matrix of HJ, it must satisfy the conditions from

definition A.3. The straightforward computation

(2.29) RJ∆J(x )(RJ)-1=τ (J-1) R J ∆J(x )J-1R-1τ (J) =τ (J-1_{) R ∆(x ) R}-1_{τ (J)} =τ (J-1_{) ∆}op_{(x )}_{τ (J)} =τ (J-1_{∆(x ) J)} = ∆Jop(x )

shows RJ to satisfy condition 1. For condition 2, another computation shows

(2.30)

(∆J ⊗ id)(RJ) = (∆J⊗ id)(τ (J-1) R J)

= (∆J⊗ id)(τ (J-1)) (∆J⊗ id)(R) (∆J ⊗ id)(J)

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2.3 Twisting 11 Rewriting

(2.31)

(J-1_{⊗1)(∆ ⊗ id)(τ (J}-1_{)) = (J}-1_{⊗1) (id ⊗τ )(τ ⊗ id)(id ⊗∆)(J}-1₎

= (id ⊗τ )(τ ⊗ id)(1 ⊗ J-1_{)(id ⊗∆)(J}-1₎

= (id ⊗τ )(τ ⊗ id)(J-1_{⊗1)(∆ ⊗ id)(J}-1₎

= (τ (J-1₎₎13_{(id ⊗τ )(τ ⊗ 1)(∆ ⊗ id)(J}-1₎

= (τ (J-1₎₎13_{(id ⊗τ )(∆}op_{⊗ id)(J}-1₎

so that

(2.32) (J-1⊗1) (∆ ⊗ id)(τ (J-1)) R13= (τ (J-1))13(id ⊗τ )(∆op⊗ id)(J-1) R13 = (τ (J-1₎₎13_R13_{(id ⊗τ )(∆ ⊗ id)(J}-1₎

and

(2.33) R23(∆ ⊗ id)(J)(J ⊗ 1) = R23(id ⊗∆)(J) (1 ⊗ J) = (id ⊗∆op_{)(J) R}23_J23_,

yield

(2.34) (∆J⊗ id)(RJ) = (τ (J-1))13R13(id ⊗τ )(∆ ⊗ id)(J-1) (id ⊗∆op)(J) R23J23

= (τ (J-1₎₎13_R13_{(id ⊗τ )(∆ ⊗ id)(J}-1_{)(id ⊗∆)(J) R}23_J23

where (2.35)

(∆ ⊗ id)(J-1)(id ⊗∆)(J) = (∆ ⊗ id)(J-1) (id ⊗∆)(J) (1 ⊗ J) (1 ⊗ J-1) = (∆ ⊗ id)(J-1) (∆ ⊗ id)(J) (J ⊗ 1) (1 ⊗ J-1) = (∆ ⊗ id)(1) (J ⊗ 1) (1 ⊗ J-1) = J12_(J-1₎23 and hence (2.36) (∆J⊗ id)(RJ) = (τ (J-1))13R13(id ⊗τ )J12(J-1)23 R23J23 = (τ (J-1₎₎13_R13_J13_(J-1₎32_R23_J23 = (τ (J-1₎₎13_R13_J13_{(τ (J}-1₎₎23_R23_J23 = (τ (J-1_{) R J)}13_{(τ (J}-1_{) R J)}23 = R_J13R_J23

as required. RJ is shown to satisfy condition 3 through a similar computation.

Proposition 2.8 shows that a twist J can be used to find a new universal R-matrix RJ, and thus a new solution to the QYBE, from a quantum group with

universal R-matrix R. 2.9 Corollary:

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twist of H by R-1 _{is H}op _{= (H,}_{µ, η, ∆}op_,_{, S}-1_{) with universal R-matrix R}21_.

Proof. By definition, R-1is invertible. Moreover, by definition A.3 and the QYBE,

equation 2.6, (2.37) (∆ ⊗ id)(R-1) (R-1⊗1) = (R13_R23₎-1_(R12₎-1 = (R12R13R23)-1 = (R23R13R12)-1 = (R13R12)-1(R23)-1 = (id ⊗∆)(R-1) (1 ⊗ R-1), which confirms that R-1 _{is a twist by definition 2.5.}

To find the twist of H by R-1_{, a calculation of the comultiplication and the}

antipode suffices. The comultiplication as defined in proposition 2.7 becomes (2.38) ∆R-1(x ) = R ∆(x ) R-1= ∆op(x ). Writing R-1₌P is 0 i ⊗ t 0

i, the element p becomes

(2.39) p =P iS (s 0 i) t 0 i =P iS (S-1(t 0 i) s 0 i) = S (u-1_),

where u is the Drinfeld element (appendix definition A.6). The antipode, then, is

(2.40) SR-1 = p-1S (x ) p = p-1S (x ) S (u-1) = p-1S (u-1x ) = p-1S (S−2(x ) u-1) = p-1S (u-1) S-1(x ) = S-1(x ).

Finally, proposition 2.8 states that

(2.41) RR-1 =τ (R) R R-1= R21

is the associated universal R-matrix. This completes the proof.

The following proposition shows how any twist J gives rise to other twists in a practical and straightforward way.

2.10 Proposition:

Given a twist J and an invertible element x ∈ H, (2.42) Jx = ∆(x ) J (x-1⊗x-1)

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2.3 Twisting 13 is also a twist.

Proof. Clearly, Jx is invertible with inverse Jx-1= (x ⊗ x )J-1∆(x-1). To verify the

non-dynamical twist equation, equation 2.12, consider

(2.43) (∆ ⊗ id)(Jx) (Jx ⊗ 1) = (∆ ⊗ id)(∆(x ) J (x-1_⊗x-1_{)) (∆(x ) J (x}-1_⊗x-1_{) ⊗ 1)} = (∆ ⊗ id)(∆(x ) J) (∆(x-1_{) ⊗ x}-1_{)(∆(x ) J (x}-1_⊗x-1_{) ⊗ 1)} = (∆ ⊗ id)(∆(x )) (∆ ⊗ id)(J) (J (x-1_⊗x-1_{) ⊗ x}-1₎ = (id ⊗∆)(∆(x )) (∆ ⊗ id)(J) (J ⊗ 1) (x-1_⊗x-1_⊗x-1₎ = (id ⊗∆)(∆(x )) (id ⊗∆)(J) (1 ⊗ J) (x-1_⊗x-1_⊗x-1₎ = (id ⊗∆)(∆(x ) J) (x-1_⊗∆(x-1_{x )) (1 ⊗ J (x}-1_⊗x-1₎₎ = (id ⊗∆)(∆(x ) J(x-1_⊗x-1_{) (1 ⊗ ∆(x ) J (x}-1_⊗x-1₎₎ = (id ⊗∆)(Jx) (1 ⊗ Jx),

which proves the proposition. 2.11 Corollary:

Setting J = Jtr _{= 1 ⊗ 1 shows that for any invertible x ∈ H,}

(2.44) J_xtr = ∆(x ) (x-1⊗x-1) is a twist.

Proposition 2.10 and corollary 2.11 povide a way to easily generate a plethora of twists. All that is required to find a new twist, and thus a new solution to the QYBE by proposition 2.8, is an invertible element of the Hopf algebra. How-ever, as the next proposition will demonstrate, the ‘new’ solution to the QYBE found in this way is isomorphic to the solution found through the original twist. Hence proposition 2.10 and corollary 2.11 merely generate trivial variations on the solutions of the QYBE.

2.12 Definition:

Two twists J and J0 are said to be gauge equivalent if there exists an invertible element x ∈ H such that

(2.45) J0 = ∆(x ) J (x-1⊗x-1_).

2.13 Proposition:

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Proof. Using the fact that J and J0 are gauge equivalent, definition 2.12, the rela-tion between HJ and HJ0 may be determined.

Recalling proposition 2.7, the comultiplications of HJ and HJ0 relate as

(2.46) ∆J0(y ) = (J0)-1∆(y ) J0 = (∆(x ) J (x-1⊗x-1₎₎-1_{∆(y ) ∆(x ) J (x}-1_⊗x-1₎ = (x ⊗ x ) J-1_{∆(x )}-1 _{∆(y ) ∆(x ) J (x}-1_⊗x-1₎ = (x ⊗ x ) J-1_∆(x-1_{y x ) J (x}-1_⊗x-1₎ = (x ⊗ x ) ∆J(x-1y x ) (x-1⊗x-1).

To compare the antipodes of HJ and HJ0, also defined in proposition 2.7, first

consider the relation between the elements p and p0_{. Writing J = P}

iai ⊗ bi and J0 =P iαi⊗ βi, the element p 0 _{may be written as} (2.47) p0 =P iS (αi)βi =P i ,(x )S (x(1)aix-1) x(2)bix-1 =P i ,(x )S (x -1_{) S (a} i) S (x(1)) x(2)bix-1 =P iS (x -1_{) S (a} i)(x) bix-1 =(x) S(x-1_{) p x}-1_.

Now the antipodes of HJ and HJ0 are seen to relate as

(2.48) SJ0(y ) = (p0)-1S (y ) p0 = ((x) S(x-1_{) p x}-1₎-1_{S (y )}_{(x) S(x}-1_{) p x}-1 = x p-1 _{S (x}-1₎-1_(x)-1_{S (y )}_{(x) S(x}-1_{) p x}-1 = x p-1 _{S (x ) S (y ) S (x}-1_{) p x}-1 = x p-1 S (x-1y x ) p x-1 = x SJ(x-1y x ) x-1.

Moreover, recalling proposition 2.8, the R-matrices of HJ and HJ0 relate as

(2.49) RJ0 =τ ((J0)-1) R J0 =τ ((∆(x) J (x-1_⊗x-1₎₎-1_{) R ∆(x ) J (x}-1_⊗x-1₎ = (x ⊗ x )τ (J-1_{) ∆}op_{(x )}-1 _∆op_{(x ) R J (x}-1_⊗x-1₎ = (x ⊗ x )τ (J-1_{) R J (x}-1_⊗x-1₎ = (x ⊗ x ) RJ(x-1⊗x-1).

The relations between the comultiplications, antipodes, and universal R-matrices of HJ and HJ0 show that there exists an isomorphism

(2.50) HJ → HJ0 : y 7→ x y x-1.

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2.3 Twisting 15 2.14 Corollary:

(2.51) HJtr ∼= H

Proof. Follows from proposition 2.13 and corollary 2.11. For J and J0 _{gauge equivalent the relation between R}

J and RJ0, as described in

the proof of proposition 2.13, shows that

(2.52) RJ120 R_J130 R_J230 = (x ⊗ x ⊗ x ) R_J12R_J13R_J23(x-1⊗x-1⊗x-1)

R_J230 R_J130 R_J120 = (x ⊗ x ⊗ x ) R_J23R_J13R_J12(x-1⊗x-1⊗x-1)

for some invertible element x. This confirms that these solutions to the QYBE are one and the same solution, up to a change of basis, as claimed before.

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3 Solutions to the QDYBE

Chapter 2 has shown that universal R-matrices of quantum groups are solutions to the QYBE. Moreover, further solutions to the QYBE were found by twisting the universal R-matrices. This chapter introduces a dynamical version of the twist, called the fusion operator, which depends on a dynamical parameter λ. Twisting a universal R-matrix using a fusion operator gives rise to an exchange operator or so-called quantum dynamical R-matrix that satisfies the dynamical analog of the QYBE; the quantum dynamical Yang-Baxter equation, or QDYBE for short.

3.1 Intertwining Operators

Corollary 2.4 shows that operators RVV on representations V of Uq(sln) are

solu-tions to the QYBE. Analogous to twisting universal R-matrices, operators RVW

may be twisted by a dynamical twist. A dynamical twist called the fusion operator is constructed from intertwining operators that ‘append’ representations of Uq(sln)

to Verma modules. 3.1 Definition:

Let V , W finite-dimensional representations of Uq(sln). An operator Φ : V → W

that commutes with the action of Uq(sln), i.e.

(3.1) Φ(xv ) = x Φ(v ) ∀ x ∈ Uq(sln), ∀ v ∈ V ,

is called an intertwining operator.

Now recall the definition of the Verma module (appendix definition A.26), and the definition of the tensor product of two representations (appendix definition A.8). For a finite-dimensional representation V of Uq(sln), consider the

intertwin-ing operator

(3.2) Φ : Mλ → Mµ⊗ V .

Intertwining operators of this form will be the main subject of interest here. 3.2 Definition:

Let V a finite-dimensional representation of Uq(sln) and λ, µ ∈ h∗ weights of V ,

then the map

(3.3) h·i : HomUq(sln)(Mλ, Mµ⊗ V ) → V : Φ 7→ (v

∗

µ⊗ id)(Φ vλ)

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3.1 Intertwining Operators 17 3.3 Proposition:

For an intertwiner Φ : Mλ → Mµ⊗ V ,

(3.4) hΦi ∈ V [λ − µ].

Proof. Using the fact that Φ is an intertwiner, definition 3.1, provides (3.5) KiΦ vλ = Φ Kivλ = Φ qλ(hi)_v λ = qλ(hi)_{Φ v} λ,

which implies that wt Φ vλ = λ. Now write Φ vλ = P_iai ⊗ bi with ai ∈ Mµ and

bi ∈ V, so that the expectation value of Φ becomes

(3.6) hΦi = (v∗ µ⊗ id)(Φ vλ) = (v_µ∗⊗ id)(P iai ⊗ bi) =P iv ∗ µ(ai) bi. Since v∗ µ ∈ M ∗ µ satisfies v ∗

µ(vµ) = 1 and vµ∗(w ) = 0 for wt w < µ, it follows that

(3.7) v_µ∗(ai) 6= 0 ⇔ wt ai =µ.

Because wt ai + wt bi = wt ai ⊗ bi = wt Φ vλ = λ, this means that vµ∗(ai) bi 6= 0 if

and only if wt bi =λ − µ, proving that Piv ∗

µ(ai) bi ∈ V [λ − µ].

3.4 Proposition:

If Mµ is irreducible, i.e. µ is generic, the expectation value map

(3.8) h·i : HomUq(sln)(Mλ, Mµ⊗ V ) → V [λ − µ]

is an isomorphism.

Proof. Recall the definition of the Verma module (appendix definition A.26), the one-dimensional representation λ of Uq(b+)(appendix proposition A.23), and the

restricted dual (appendix definition A.34). Applying appendix proposition A.25 and the fact that

(3.9) Hom(U, V ⊗ W ) ∼= Hom(V◦⊗ U, W ), yields (3.10) HomUq(sln)(Mλ, Mµ⊗ V ) ∼= HomUq(sln)(Uq(sln) ⊗Uq(b+)λ, Mµ⊗ V ) ∼ = HomUq(b+)(λ, Mµ⊗ V ) ∼ = HomUq(b+)(M ◦ µ⊗ λ, V ) ∼ = HomUq(b+)(M ◦ µ, V ⊗λ◦).

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Now consider the automorphism ω : Uq(sln) → Uq(sln) given by

(3.11) ω(Ei) = Fi ω(K±) = K∓ ω(Fi) = Ei

which, for a representation (W , πW) of Uq(sln), defines another representation

Wω _{= (W ,}_π

W ◦ ω) of Uq(sln). Applying the automorphism ω to the right hand

side of equation 3.10 results in (3.12) HomUq(b+)(M ◦ µ, V ⊗λ ◦ ) ∼= HomUq(b−)(M ◦ω µ , V ω_{⊗ λ),}

since λ◦ω _{= (−λ)}ω ₌ λ. Now consider the unique non-degenerated bilinear form,

called the Shapovalov form (see theorem 2.43 of [Etingof and Latour, 2005]) (3.13) Sµ : Mµ⊗ Mµ → C,

that satisfies Sµ(vµ, vµ) = 1 and Sµ(x v , w ) = Sµ(v , S (ω(x)) w ) for x ∈ Uq(sln),

v , w ∈ Mµ. Considering the definition of the dual representation (appendix

defini-tion A.12) on the restricted dual, the map φ : v 7→ Sµ(v , ·) on Mµ is seen to map

into M◦ω µ since

(3.14) φ(x v ) = Sµ(x v , ·) = Sµ(v , S (ω(x)) ·) = ω(x) φ(v ).

Because Mµis irreducible, this map φ is an injection. Moreover, because the weight

spaces of Mµ and Mµ◦ω have the same dimensions, φ is an isomorphism. Thus,

(3.15) HomUq(b−)(M ◦ω µ , Vω⊗ λ) ∼= HomUq(b−)(Mµ, V ω_{⊗ λ)} ∼ = HomUq(b−)(Uq(sln) ⊗Uq(b+)µ, V ω_{⊗ λ)} ∼ = HomUq(b−)(Uq(b−) ⊗Uq(h)µ, V ω_{⊗ λ)} ∼ = HomUq(h)(µ, V ω_{⊗ λ)} ∼ = HomUq(h)(µ ⊗ λ ◦ , Vω) ∼ = HomUq(h)(µ − λ, V ω₎ ∼ = HomUq(h)(λ − µ, V ) ∼ = V [λ − µ].

Now putting together equations 3.10, 3.11, and 3.15 yields (3.16) HomUq(sln)(Mλ, Mµ⊗ V ) ∼= V [λ − µ],

where the isomorphism is given by the map h·i, completing the proof. 3.5 Corollary:

If Mλ is irreducible, then for all v ∈ V , there exists a unique intertwining operator

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3.2 Fusion Operators 19

3.2 Fusion Operators

The intertwining operator Φv

λ introduced in corollary 3.5 provides a means to

‘append’ specific Verma modules to a representation of Uq(sln), yielding again a

representation of Uq(sln). Iterating this process allows multiple representations of

Uq(sln)to be ‘fused’ together.

3.6 Definition:

Let V , W finite-dimensional representations of Uq(sln) and v ∈ V , w ∈ W , then

(3.17) Φv ,w_λ = (Φv_{λ−wt w} ⊗ id) Φw

λ : Mλ → Mλ−wt w −wt v ⊗ V ⊗ W ,

for any genericλ, defines the composition of two intertwining operators. Note that the expectation value hΦv ,w

λ iof this composition is a bilinear function

in v and w. Therefore, there exists a linear operator V ⊗ W → V ⊗ W , of weight zero, that sends v ⊗ w to hΦv ,w

λ i.

3.7 Definition:

Let V , W finite-dimensional representations of Uq(sln), then the linear operator

(3.18) JVW(λ) : V ⊗ W → V ⊗ W : v ⊗ w 7→ hΦv ,w_λ i

is called the fusion operator.

Note that because JVW(λ)(v ⊗w ) = hΦv ,w_λ i, by corollary 3.5, Φv ,w_λ = Φ

JVW(λ)(v ⊗w )

λ .

3.8 Proposition:

The fusion operator JVW(λ) satisfies the following properties:

i. JVW(λ) has zero weight.

ii. JVW(λ) is lower triangular with respect to the weight decomposition, and has

ones on its diagonal. This means that JVW(λ)(v ⊗ w ) = v ⊗ w +P_ici ⊗ bi

where wt ci < wt v and wt bi > wt w for all v , w .

iii. JVW(λ) is invertible when defined.

Proof. The first property follows directly from proposition 3.4 and definition 3.6, which imply that for any weight µ, JVW(λ)maps (V ⊗ W )[µ] into itself.

For the second property, consider

(3.19) Φw_λ vλ = vλ−wt w ⊗ w +

X

i

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where necessarily wt ai < λ − wt w since vλ−wt w is a highest weight vector, and,

consequentially, wt bi > wt w since wt ai + wt bi =λ. Applying Φvλ−wt w ⊗ id yields

(3.20) (Φv_{λ−wt w} ⊗ id) Φw

λ vλ = vλ−wt w −wt v ⊗ v ⊗ w +

X

i

Φv_{λ−wt w} ai ⊗ bi + T

where the weight of the first component of T is lower than the highest weight λ − wt w − wt v. Now applying v∗

λ−wt w −wt v⊗ id ⊗ idto both sides provides, on the

left hand side,

(3.21) (v_{λ−wt w −wt v}∗ ⊗ id ⊗ id)((Φv

λ−wt w ⊗ id) Φ w

λ vλ) = hΦv ,wλ i

while on the right hand side this provides (3.22) vλ−wt w −wt v∗ (vλ−wt w −wt v)v ⊗ w + P i(v ∗ λ−wt w −wt v ⊗ id)(Φ v λ−wt wai) ⊗ bi = v ⊗ w +P ici ⊗ bi where wt ci = wt ai − (λ − wt w − wt v ) < λ − wt w − λ + wt w + wt v = wt v.

Equating the left and right hand sides results in (3.23) hΦv ,w

λ i = v ⊗ w +

X

i

ci ⊗ bi

where wt ci < wt v and wt bi > wt w, as required.

The third property follows from the second property.

In section 2.3 a twist J, satisfying the non-dynamical twist equation 2.13, was used to find solutions to the QYBE. The fusion operator JVW(λ) satisfies a

dy-namical analog of equation 2.13. 3.9 Theorem:

Let U, V , W finite-dimensional representations of Uq(sln), then the fusion operator

satisfies the dynamical twist equation on U ⊗ V ⊗ W : (3.24) J_{U⊗V ,W}12,3 (λ) J12 UV(λ − h 3_{) = J}1,23 U,V ⊗W(λ) J 23 VW(λ),

using the notation J12

UV(λ − h3)(u ⊗ v ⊗ w ) = (JUV(λ − wt w )(u ⊗ v )) ⊗ w and

J_{U⊗V ,W}12,3 (λ) = J(U⊗V )W(λ).

Proof. Consider the iterated composition (3.25) (Φu_{λ−wt w −wt v} ⊗ id ⊗ id) (Φv λ−wt w ⊗ id) Φ w λ that maps (3.26) Mλ → Mλ−wt w −wt v −wt u⊗ U ⊗ V ⊗ W

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3.2 Fusion Operators 21 from two different perspectives. Firstly, write equation 3.25 as

(3.27) (Φu,v_{λ−wt w} ⊗ id) Φw λ = (Φ JUV(λ−wt w )(u⊗v ) λ−wt w ⊗ id) Φ w λ = ΦJUV(λ−wt w )(u⊗v ),w λ = ΦJU⊗V ,W(λ)(JUV(λ−wt w )(u⊗v )⊗w ) λ .

Secondly, write equation 3.25 as (3.28) (Φu λ−wt w −wt v⊗ id ⊗ id) Φ v ,w λ = (Φuλ−wt w −wt v ⊗ id ⊗ id) Φ JVW(λ)(v ⊗w ) λ = Φu,JVW(λ)(v ⊗w ) λ = ΦJU,V ⊗W(λ)(u⊗JVW(λ)(v ⊗w )) λ .

Taking the expectation value of equations 3.27 and 3.28, and equating them, yields (3.29)

hΦJU⊗V ,W(λ)(JUV(λ−wt w )(u⊗v )⊗w )

λ i = hΦ

JU,V ⊗W(λ)(u⊗JVW(λ)(v ⊗w ))

λ i

JU⊗V ,W(λ)(JUV(λ − wt w )(u ⊗ v ) ⊗ w ) = JU,V ⊗W(λ)(u ⊗ JVW(λ)(v ⊗ w ))

for all u ∈ U, v ∈ V and w ∈ W . Hence (3.30) J_{U⊗V ,W}12,3 (λ) J12 UV(λ − h 3_{) = J}1,23 U,V ⊗W(λ) J 23 VW(λ) on U ⊗ V ⊗ W as required.

3.2.1 The Universal Fusion Operator

The fusion operator JVW(λ)can be generalized to a universal fusion operator J(λ)

which specializes to the normal fusion operator JVW(λ) on every V ⊗ W . This

universal fusion operator J(λ) will be especially useful in chapter 4, when it is necessary to consider manipulations of operators without restricting to specific representations.

3.10 Theorem:

There is a unique J(λ) ∈ Uq(sln) ˆ⊗Uq(sln) (see appendix remark A.35) with the

properties from proposition 3.8 that satisfies the equation (3.31) J(λ)id ⊗qθ(λ)= R21q−Pixi⊗xi id ⊗qθ(λ)J(λ), where θ(λ) = 2(λ + ρ) −P ix 2

i. This unique solution J(λ) of zero weight is called

the universal fusion operator, which has the property that it specializes to the fusion operator JVW(λ) on V ⊗ W for V , W finite-dimensional representations of

Uq(sln).

The idea of the proof is to define an expression (3.32) F (λ) = (v_{λ−wt w −wt v}∗ ⊗ id ⊗ id)((Φv

λ−wt w ⊗ id) (u q

−2ρ_{⊗ id) Φ}w λ vλ),

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where u is the Drinfeld element (appendix definition A.6) and u q−2ρ _{is central in}

Uq(sln). F (λ) will then be computed in two different ways. First, it is shown that

u q−2ρ|Mλ = q

−λ(λ−2ρ)_id_{, i.e. it acts as straightforward multiplication, so that}

(3.33) F (λ) = q−(λ−wt w )(λ−wt w −2ρ)JVW(λ)(v ⊗ w ).

The second way to compute F (λ) is to pull the element u q−2ρ _{through the}

in-tertwiner Φv

λ−wt w and work out the expression from there. The two resulting

expressions for F (λ) are then equated to show that JVW(λ) is a solution to the

equation for every V , W . What is left is to show that there exists a solution J(λ) as in the theorem that specializes to JVW(λ) on every V , W .

A full proof of theorem 3.10 can be found in [Etingof and Schiffmann, 2002].

3.2.2 Example: U

q

(sl

2

)

The fusion operator JVV(λ)can be explicitly computed for finite-dimensional

repre-sentations V of Uq(sl2); for example the irreducible two-dimensional representation

V = V1. Appendix section A.3.3 states that this representation of highest weight

λ = 1equals V [+1] ⊕ V [−1] = C v+⊕ C v− under the actions

(3.34)

K±v+ = q±v+ K±v− = q∓v−

F v+ = v− F v− = 0

E v+ = 0 E v− = v+.

Now consider V ⊗ V with ordered basis consisting of: (3.35)

v+⊗ v+ of weight 2,

v+⊗ v− of weight 0,

v−⊗ v+ of weight 0,

v−⊗ v− of weight -2.

Proposition 3.8 states that the fusion operator is lower triangular with ones on the diagonal, and thus that JVV(λ) fixes each basis element except v+⊗ v−. This

leaves JVV(λ)(v+⊗ v−) = Φv+ λ+1⊗ idV Φ v− λ

to be determined. Corollary 3.5 and the implication of definition 3.2 determine

(3.36) Φv−

λ vλ = vλ+1⊗ v−+ψ(q, λ) F vλ+1⊗ v+

for Φv−

λ : Mλ+1+wt v− → Mλ+1⊗ V, where ψ is some unknown function of q and λ.

The next step then is to determine

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3.2 Fusion Operators 23 and (3.38) (v_λ∗⊗ idV)(Φv_λ+1+ F vλ+1) = (vλ∗⊗ idV)(∆(F ) Φv_λ+1+ vλ+1) = (v_λ∗⊗ idV)((F ⊗ 1 + K-1⊗F ) Φv_λ+1+ vλ+1) = (v_λ∗⊗ idV)((K-1⊗F ) Φv_λ+1+ vλ+1) = q−λF (v_λ∗⊗ idV)(Φvλ+1+ vλ+1) = q−λF v+ = q−λv−

These three results yield

(3.39) JVV(λ)(v+⊗ v−) = Φv+ λ+1⊗ idV Φ v− λ = (v_λ∗⊗ idV ⊗V) Φ v+ λ+1⊗ idV Φ v− λ vλ = (v_λ∗⊗ idV ⊗V) Φ v+ λ+1⊗ idV (vλ+1⊗ v−+ψ(q, λ) F vλ+1⊗ v+) = (v_λ∗⊗ idV ⊗V) Φ v+ λ+1vλ+1⊗ v−+ψ(q, λ) Φv_λ+1+ F vλ+1⊗ v+ = v+⊗ v−+ψ(q, λ) q−λv−⊗ v+

Finally, to determine ψ(q, λ) consider

(3.40) 0 = Φv− λ E vλ = ∆(E ) Φv− λ vλ = (E ⊗ K + 1 ⊗ E ) (vλ+1⊗ v−+ψ(q, λ) F vλ+1⊗ v+) = E vλ+1⊗ K v−+ vλ+1⊗ E v− +ψ(q, λ) (EF vλ+1⊗ K v++ F vλ+1⊗ E v+)

which, using relations specified in appendix equations A.43 and A.44 and equation 3.34, reduces to (3.41) 0 = 0 + vλ+1⊗ v++ψ(q, λ) K −K-1 q−q-1 + FE vλ+1⊗ qv++ 0 = vλ+1⊗ v++ψ(q, λ) qλ+1_−q−(λ+1) q−q-1 vλ+1⊗ q v+ =1 +ψ(q, λ)qλ+1_q−q−q−(λ+1)-1 q vλ+1⊗ v+ ⇒ 0 = 1 + ψ(q, λ)qλ+1_−q−(λ+1) q−q-1 q

and thus provides

(3.42) ψ(q, λ) = q − q

-1

q−(λ+1)_{− q}λ+1q -1_.

Substituting this result into the expression for JVV(λ)(v+⊗ v−)yields

(3.43) JVV(λ)(v+⊗ v−) = v+⊗ v−+

q − q-1

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The fusion operator JVV(λ) can now be written as the matrix (3.44) [JVV(λ)] =      1 0 0 0 0 1 0 0 0 _1−qq−q2(λ+1)-1 1 0 0 0 0 1     

with (v+⊗ v+, v+⊗ v−, v−⊗ v+, v−⊗ v−) the ordered basis of V ⊗ V .

3.3 Exchange Operators

Proposition 2.8 shows how a universal R-matrix, or solution to the QYBE, can be constructed by twisting another universal R-matrix. This suggests, in light of theorem 3.9, that twisting an operator RVW with the fusion operator, a dynamical

twist, might yield a solution to the quantum dynamical Yang-Baxter equation. 3.11 Definition:

Let V , W finite-dimensional representations of Uq(sln) and R its universal

R-matrix, then the operator

(3.45) RVW(λ) = JVW(λ)-1R21JWV21 (λ) : V ⊗ W → V ⊗ W

is called the exchange operator.

3.12 Theorem (QDYBE):

Let U, V , W finite-dimensional representations of Uq(sln), then the exchange

op-erator satisfies the quantum dynamical Yang-Baxter equation on U ⊗ V ⊗ W : (3.46) R_UV12 (λ − h3_{) R}13 UW(λ) R 23 VW(λ − h 1_{) = R}23 VW(λ) R 13 UW(λ − h 2_{) R}12 UV(λ)

Proof. Recall the dynamical twist equation, theorem 3.9, which states: (3.47) J_{U⊗V ,W}12,3 (λ) J12 UV(λ − h 3_{) = J}1,23 U,V ⊗W(λ) J 23 VW(λ),

and by permutation (see Notation B) implies (3.48) J_{U⊗W ,V}13,2 (λ) J13 UW(λ − h 2_{) = J}1,32 U,W ⊗V(λ) J 32 WV(λ), (3.49) J_{V ⊗W ,U}23,1 (λ) J23 VW(λ − h 1_{) = J}2,31 V ,W ⊗U(λ) J 31 WU(λ).

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3.3 Exchange Operators 25 equation 3.46 becomes (3.50) R12 UV(λ − h3) RUW13 (λ) RVW23 (λ − h1) = J12 UV(λ − h3)-1R21JVU21(λ − h3) JUW13 (λ)-1R31JWU31 (λ) J23 VW(λ − h1)-1R32JWV32 (λ − h1).

Rewriting equations 3.47 and 3.49 as

(3.51) J_UV12 (λ − h3)-1= J_VW23 (λ)-1J_{U,V ⊗W}1,23 (λ)-1J_{U⊗V ,W}12,3 (λ), (3.52) J_VU21(λ − h3_{) = J}21,3 V ⊗U,W(λ) -1_J2,13 V ,U⊗W(λ) J 13 UW(λ), (3.53) J_VW23 (λ − h1₎-1_{= J}31 WU(λ)-1J 2,31 V ,W ⊗U(λ) -1_J23,1 V ⊗W ,U(λ), (3.54) J_WV32 (λ − h1_{) = J}32,1 W ⊗V ,U(λ) -1_J3,21 W ,V ⊗U(λ) J 21 VU(λ)

and substituting those into equation 3.50 yields

(3.55) R_UV12(λ − h3_{) R}13 UW(λ) RVW23 (λ − h1) = J23 VW(λ)-1J 1,23 U,V ⊗W(λ) -1_J12,3 U⊗V ,W(λ) R 21_J21,3 V ⊗U,W(λ) -1_J2,13 V ,U⊗W(λ) × J13 UW(λ) JUW13 (λ)-1R31JWU31 (λ) JWU31 (λ)-1 × J2,31 V ,W ⊗U(λ)-1J 23,1 V ⊗W ,U(λ) R32J 32,1 W ⊗V ,U(λ)-1J 3,21 W ,V ⊗U(λ) JVU21(λ) = J23 VW(λ)-1J 1,23 U,V ⊗W(λ)-1J 12,3 U⊗V ,W(λ) J 12,3 U⊗V ,W(λ)-1R21 × J2,13 V ,U⊗W(λ) R31J 2,31 V ,W ⊗U(λ)-1 × R32_J32,1 W ⊗V ,U(λ) J 32,1 W ⊗V ,U(λ)-1J 3,21 W ,V ⊗U(λ) JVU21(λ) = J23 VW(λ)-1J 1,23 U,V ⊗W(λ)-1R21R31R32J 3,21 W ,V ⊗U(λ) JVU21(λ).

Here part 1 of definition A.3 was used to find R12_J12,3

U⊗V ,W(λ) = J 21,3

V ⊗U,W(λ) R12.

Analogously, the right hand side of equation 3.46 becomes (3.56) R23 VW(λ) RUW13 (λ − h2) RUV12(λ) = J23 VW(λ)-1R32JWV32 (λ) JUW13 (λ − h2)-1R31JWU31 (λ − h2) J12 UV(λ)-1R21JVU21(λ). Rewriting equation 3.48 as (3.57) J_UW13 (λ − h2₎-1_{= J}32 WV(λ) -1_J1,32 U,W ⊗V(λ) -1_J13,2 U⊗W ,V(λ), (3.58) J_WU31 (λ − h2_{) = J}31,2 W ⊗U,V(λ) -1_J3,12 W ,U⊗V(λ) J 12 WV(λ)

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and substituting those into equation 3.56 yields (3.59) R23 VW(λ) RUW13 (λ − h2) RUV12(λ) = J23 VW(λ)-1R32JWV32 (λ) JWV32 (λ)-1J 1,32 U,W ⊗V(λ)-1 × J_{U⊗W ,V}13,2 (λ) R31_J31,2 W ⊗U,V(λ)-1 × J_{W ,U⊗V}3,12 (λ) J12 WV(λ) J 12 UV(λ) -1_R21_J21 VU(λ) = J_VW23 (λ)-1_J1,23 U,V ⊗W(λ) -1 _R32 × R31_J31,2 W ⊗U,V(λ) J 31,2 W ⊗U,V(λ) -1 × R21_J3,21 W ,U⊗V(λ) J 21 VU(λ) = J23 VW(λ)-1J 1,23 U,V ⊗W(λ) -1_R32_R31_R21_J3,21 U,W ⊗V(λ) J 21 VU(λ)

Now because of the QYBE, equation 2.6, the expressions in 3.55 and 3.59 are equal, leading to:

(3.60) R_UV12 (λ − h3) R_UW13 (λ) RVW23 (λ − h 1 ) = R_VW23 (λ) RUW13 (λ − h 2 ) R_UV12(λ) as required. 3.13 Corollary:

The exchange operator RVV(λ) is a solution to the QDYBE on V ⊗ V ⊗ V , and is

called a quantum dynamical R-matrix.

Note that the approach taken to find solutions to the QDYBE in this chapter is slightly different from the approach taken in chapter 2. In chapter 2, a twist is defined as an invertible element satisfying the twist equation. These twists are then used to twist universal R-matrices and find new solutions to the QYBE. In this chapter, however, one specific dynamical twist called the fusion operator is shown to satisfy the dynamical twist equation. Twisting a universal R-matrix by this dynamical twist results in a solution to the QDYBE. Alternatively, mimicking chapter 2, a general dynamical twist could be defined as satisfying the dynamical twist equation and then used to find general solutions to the QDYBE.

3.3.1 Example: U

q

(sl

2

)

Like the fusion operator, the exchange operator RVV(λ)can be explicitly computed

for finite-dimensional representations of Uq(sl2). Consider again the irreducible

two-dimensional representation V = V1 of Uq(sl2), described in appendix section

A.3.3 and section 3.2.2.

The universal R-matrix R|V ⊗V can be computed using the general expression for

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3.3 Exchange Operators 27 represention V the expression reduces to:

(3.61) R|V ⊗V = q 1 2(h⊗h) 1 X n=0 (q − q-1)n(En⊗ Fn) . Simple calculations yield

(3.62) R|V ⊗V(v+⊗ v+) = q 1 2v₊⊗ v₊ R|V ⊗V(v+⊗ v−) = q− 1 2v₊⊗ v₋ R|V ⊗V(v−⊗ v+) = q− 1 2v₋⊗ v₊+ q− 1 2 (q − q-1) v₊⊗ v₋ R|V ⊗V(v−⊗ v−) = q 1 2v₋⊗ v₋

so that R|V ⊗V can be expressed in matrix form as

(3.63) [R|V ⊗V] =      q12 0 0 0 0 q−12 q− 1 2(q − q-1) 0 0 0 q−12 0 0 0 0 q12     

with (v+⊗ v+, v+⊗ v−, v−⊗ v+, v−⊗ v−) the ordered basis of V ⊗ V .

Since JVV(λ)has already been determined, in equation 3.44, RVV(λ)can now be

determined by straightforward matrix multiplication:

(3.64) [RVV(λ)] = [JVV(λ)]-1R21|V ⊗V JVV21(λ) to find (3.65) [RVV(λ)] =        1 0 0 0 0 1 _1−qq−q2(λ+1)-1 0 0 _1−qq−q−2(λ+1)-1 (q2(λ+1)_−q2₎₍_q2(λ+1)_−q−2₎ (1−q2(λ+1)₎2 0 0 0 0 1       

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4 Quantum Integrable Systems

In this chapter, the exchange operators or quantum dynamical R-matrices intro-duced in section 3.3 are used to construct a set of commuting operators that describes a quantum integrable system. This quantum integrable system will give rise to the quantum spin Calogero-Moser system, treated in section 4.4, and Mac-donald operators, treated in section 4.5.

A quantum integrable system is a quantum system whose time-evolution can be exactly solved. Since time-evolution is governed by the Schrödinger equation, one way of describing a quantum integrable system involves diagonalizing the energy function or Hamiltonian of the system. This requires a maximal set of commuting linear operators that is simultaneously diagonalized. See [Caux and Mossel, 2011] for a discussion on the definition of quantum integrable systems.

4.1 Transfer Matrix Construction

This section presents a variation on the construction of a quantum integrable system as outlined in chapter 6 of [Stokman, 2016].

Recall, from proposition 2.3, that the operator RWV : W ⊗ V → W ⊗ V for

W , V representations of Uq(sln)is a solution to the QYBE, and consider the tensor

representation V⊗n−1 _{= V}

(1)⊗ · · · ⊗ V(n−1) of Uq(sln).

4.1 Definition:

For W a representation of Uq(sln), the operator

(4.1) TW = RWV12 (1)· · · R

1n

WV(n−1) : W ⊗ V

⊗n−1_{→ W ⊗ V}⊗n−1

is called the monodromy matrix associated to RWV.

4.2 Definition: The operator (4.2) TW = tr |WTW = tr |W R_WV12 (1)· · · R 1n WV(n−1) : V⊗n−1 → V⊗n−1 is called a transfer matrix of the system.

The transfer matrices TW can be thought of as acting on the quantum state

space of the system. 4.3 Theorem:

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4.2 Quantum Inverse Scattering 29 and satisfy

(4.3) TU⊗W = TUTW.

From a physical perspective, the transfer matrices TW represent conserved

quan-tities on the quantum state space V⊗n−1_{of the system. Taking Λ}

i the fundamental

representations of Uq(sln) (see appendix definition A.32), the set of transfer

ma-trices {TΛ1, ... , TΛn−1} describes a quantum integrable system

4.4 Remark:

In [Stokman, 2016], the operators RWV depend on a so-called spectral parameter.

Instead of choosing different representations W for the transfer matrices to find a set of commuting operators, the spectral parameter may be varied to find a com-muting set of transfer matrices with fixed W . This leads to a quantum integrable system known as the Heisenberg spin chain of n − 1 lattice sites. The spin at the i -th site is then described by the representation V(i ).

4.2 Quantum Inverse Scattering

This section introduces a set of algebraically independent commuting operators DΛi with common eigenvectors given by the trace function FV(λ, µ), that describes

a quantum integrable system on a quantum state space V . The first part of this section demonstrates a method for finding a set of operators DC based on central

elements C ∈ Z(Uq(sln)). It is easily shown that these operators DC form a set

of commuting difference operators. In the second part of this section, operators DW associated to finite-dimensional representations W of Uq(sln), called transfer

operators, are constructed from the exchange operators or quantum dynamical R-matrices RWV(λ). The operators DC and DW are found to relate to each other,

implying that the operators DW also form a set of commuting difference operators.

The subset of operators DΛi is then found to describe a quantum integrable system.

Section 4.3 provides an elaborate proof of how the operators DC and DW relate

to each other, proving the main theorem for the operators DW.

4.2.1 Commuting Operators and Central Elements

A quantum integrable system may be described by a maximal set of commuting operators that share common eigenfunctions given by a trace function. One way to construct such a set of commuting operators, using central elements of Uq(sln),

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4.5 Definition:

Define the ring of difference operators (4.4) D = nD = X

ν∈h∗

A(ν)Tν

almost all A(ν) = 0 o

acting on functions f on h∗, where (Tνf )(λ) = f (λ + ν) and the A(ν) are

mero-morphic functions on h∗, with multiplication given by

(4.5) (A(ν)Tν) (B(µ)Tµ) = A(ν) B(µ + ν)Tµ+ν.

Now fix a representation V of Uq(sln)with V [0] 6= 0.

4.6 Theorem:

Let v ∈ V [0], and µ generic. Then for every x ∈ Uq(sln) there exists a unique

operator Dx ∈ D ⊗ Uq(sln) such that

(4.6) Dxtr |Mµ(Φ v µq 2λ ) = tr |Mµ(Φ v µx q 2λ ),

independent of the choice of representations Mµ, V and v ∈ V [0].

Proof. See chapter 6 of [Etingof and Kirillov, 1994].

Considering the operators defined in theorem 4.6 for central elements of Uq(sln)

leads to an important corollary. 4.7 Corollary:

If C ∈ Z(Uq(sln)), then the trace function Ψv(λ, µ) = tr |Mµ(Φ

v

µq2λ), v ∈ V [0],

satisfies the difference equation

(4.7) DCΨv(λ, µ) = CµΨv(λ, µ),

where Cµ∈ C(q) such that C |Mµ = Cµ· id.

Proof. The corollary follows easily from theorem 4.6;

(4.8) DC Ψv(λ, µ) = DCtr |Mµ(Φ v µq2λ) = tr |Mµ(Φ v µC q2λ) = tr |Mµ(Φ v µCµq2λ) = Cµ tr |Mµ(Φ v µq2λ) = CµΨv(λ, µ).

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4.2 Quantum Inverse Scattering 31 For any finite-dimensional representation W of Uq(sln), a central element CW can

be defined. The associated operators DCW will play an important role in describing

the quantum integrable system. 4.8 Theorem:

For U, W a finite-dimensional representations of Uq(sln), the element

(4.9) CW = (id ⊗ tr |W)(R21R(1 ⊗ q2ρ))

is central in Uq(sln) and satisfies

(4.10) CU⊗W = CUCW

and

(4.11) CW|Mµ =χW(q

2(µ+ρ)_{) id .}

Proof. See section 4.3.1. 4.9 Corollary:

The operator DCW satisfies

(4.12) DCU⊗W = DCUDCW = DCW DCU,

and the trace functions Ψv_{(λ, µ), v ∈ V [0], satisfy the difference equation}

(4.13) DCWΨ

v

(λ, µ) = χW(q2(µ+ρ)) Ψv(λ, µ).

Proof. Follows from corollary 4.7 and theorem 4.8.

4.2.2 Transfer Operators and the Trace Function

A transfer operator is constructed from an exchange operator or quantum dynam-ical R-matrix RWV(λ), similar to the construction of a transfer matrix from an

operator RWV as shown in definition 4.2.

Again, fix a finite-dimensional representation V of Uq(sln) with V [0] 6= 0. From

a physical perspective, this representation V is the quantum state space of the system, resembling the quantum integrable system described in section 4.1. 4.10 Definition:

Let W a finite-dimensional representation of Uq(sln), then the transfer operator

DW ∈ D ⊗ Uq(sln) is defined on functions f : h∗ → V [0] as

(4.14) DWf(λ) =

X

ν∈h∗

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The transfer operators DW have a remarkable relation to the operators DCW that

were constructed from central elements CW in section 4.2.1.

4.11 Proposition: (4.15) DW =δq(λ) DCW δq(λ) -1_, where (4.16) δq(λ) = Y α∈Φ+ (qα(λ) − q−α(λ)) = q2ρ(λ) Y α∈Φ+ (1 − q−2α(λ)).

Proof. See section 4.3 and section 4.3.6 in particular.

Proposition 4.11 is an amazing result and implies that the transfer operators DW satisfy the same properties, stated in corollary 4.9, as the operators DCW. In

particular, the set of transfer operators DW has common eigenfunctions given by

a normalized trace function. This trace function is constructed as a sum of the trace functions Ψv_{(λ, µ)}_{defined in corollary 4.7, defining}

(4.17) ΨV(λ, µ) = X i Ψvi _{⊗ v}∗ i ∈ V [0] ⊗ V ∗ [0],

where the vi form a basis of V [0] and the vi∗ are dual to the vi. The summed trace

function ΨV(λ, µ) will play an important role in the proof of the main theorem.

However, the trace function that provides common eigenfunctions of the transfer operators DW requires scaling by the factor δq(λ).

4.12 Definition:

Forλ, µ ∈ h∗_, _{µ generic, the trace function F}

V(λ, µ) is defined as

(4.18) FV(λ, µ) = ΨV(λ, −µ − ρ) δq(λ).

4.13 Theorem (Main Theorem):

The trace function FV(λ, µ) satisfies the difference equations

(4.19) DW FV(λ, µ) = χW(q−2¯µ) FV(λ, µ),

where the transfer operators DW act in the FV(·,µ) : h∗ → V [0] component.

Proof. Follows from proposition 4.11; see section 4.3.6.

Theorem 4.13 shows that the transfer operators DW form a maximal set of

commuting operators with common eigenfunctions given by the trace function FV(λ, µ).

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4.3 Proof of the Main Theorem 33 Now consider the transfer operators DΛ1, ... , DΛn−1, where the Λi are the

fun-damental representations of Uq(sln). By proposition 4.11 and corollary 4.9, the

transfer operators DΛ1, ... , DΛn−1 form a set of algebraically independent

commut-ing operators. Moreover, because of corollary 4.9 and appendix proposition A.33, every transfer operator DW may be written as an algebraic combination of transfer

operators DΛi. From a physical perspective, the transfer operators DΛ1, ... , DΛn−1

represent conserved quantities of the system. This means that the set of transfer operators {DΛ1, ... , DΛn−1}describes a quantum integrable system on the quantum

state space V .

4.3 Proof of the Main Theorem

This section largely follows the structure of sections 7.5, 7.6, 7.7, and 7.8 of [Etingof and Latour, 2005] and sections 1 and 2 of [Etingof and Varchenko, 2000]. The proof of theorem 4.13 (and proposition 4.11) follows the procedure outlined in section 4.2.1. It is extensive, but the computations are worked out in detail and organized into lemmas so that it should be relatively easy to follow.

4.3.1 Central Elements

The goal of this section is to determine the central elements CW ∈ Z(Uq(sln))

related to finite-dimensional representations W of Uq(sln), which were introduced

in [Drinfeld, 1990]. Interestingly, these central elements are constructed using universal R-matrices.

4.14 Proposition: Let x ∈ H, then

i. P

(a)a(1)x S (a(2)) = (a)x for all a ∈ H,

ii. x ∈ Z(H), are equivalent.

Proof. Suppose ii., then (4.20) X (a) a(1)x S (a(2)) = x X (a) a(1)S (a(2)) = x(a)1 = (a)x.

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Now suppose i. and let a ∈ H, then (4.21) xa = x ( ⊗ id)∆(a) = xP (a)(a(1))a(2) =P (a)(a(1))x a(2) =P (a)a(1)x S (a(2)) a(3) =P (a)a(1)(a(2))x = (id ⊗)∆(a)x = ax 4.15 Definition:

An H-bimodule of an algebra H is a vector space V with left and right actions H ⊗ V → V : h ⊗ v 7→ hv and V ⊗ H → V : v ⊗ h 7→ vh that are compatible, i.e. (hv )h0 = h(vh0).

Note that an H-bimodule is the same as an H ⊗ Hop_-module.

4.16 Proposition:

Let V an H-bimodule and v ∈ V , then i. P

(a)a(1)v S (a(2)) =(a)v for all a ∈ H,

ii. va = av for all a ∈ H, are equivalent.

Proof. Use the proof of proposition 4.14 and replace x by v. 4.17 Definition:

A linear functional_{θ : H → C such that θ(xy) = θ(yS}2_{(x )) for all x , y ∈ H is called}

a quantum trace. 4.18 Proposition:

Let _{θ : H → C a quantum trace and z ∈ H ⊗ H such that ∆(a)z = z∆(a) for all} a ∈ H, then C = (id ⊗θ)z ∈ Z(H).

Proof. Consider the H-bimodule V = H ⊗ H with H ⊗ V → V : a ⊗ v 7→ ∆(a)v and V ⊗ H → V : v ⊗ a 7→ v∆(a). Then for all a ∈ H,

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4.3 Proof of the Main Theorem 35 This means proposition 4.16 can be applied to find

(4.23)

(a)C = (id ⊗θ) (a)z = (id ⊗θ)P (a)a(1)z S (a(2)) = (id ⊗θ)P (a)∆(a(1)) z ∆(S (a(2))) = (id ⊗θ)P (a)(a(1)⊗ a(2)) z (S (a(4)) ⊗ S (a(3))) = (id ⊗θ)P (a)(a(1)⊗ 1) z (S(a(4)) ⊗ S (a(3))S2(a(2))) = (id ⊗θ)P (a)(a(1)⊗ 1) z (S(a(4)) ⊗ S (S (a(2))a(3))) = (id ⊗θ)P

(a)(a(1)⊗ 1) z (S(a(3)) ⊗ S ((a(2))1))

= (id ⊗θ)P

(a)(a(1)⊗ 1) z (S(a(2)) ⊗ 1)

=P

(a)a(1)(id ⊗θ)(z) S(a(2))

=P

(a)a(1)C S (a(2))

which, by proposition 4.14, implies that C ∈ Z(H) as required. 4.19 Corollary:

Let W a finite-dimensional representation of Uq(sln), then

(4.24) CW = (id ⊗ tr |W)(R21R(1 ⊗ q2ρ))

is a central element of Uq(sln).

Proof. Recall that S2_{(x ) = q}2ρ_{x q}−2ρ _{(appendix proposition A.16). The function}

(4.25) θ(x) = tr |W(x q2ρ)

is a quantum trace by definition 4.17 since (4.26)

θ(xy ) = tr |W(xy q2ρ)

= tr |W(y q2ρx )

= tr |W(yS2(x ) q2ρ)

=θ(yS2_{(x )).}

Because R21_{R∆(x ) = R}21_∆op_{(x )R = ∆(x )R}21_R_{, proposition 4.18 can be applied}

to provide that

(4.27) (id ⊗θ)R21_{R ∈ Z(U} q(sln))

and hence that

(4.28) (id ⊗ tr |W)(R21R(1 ⊗ q2ρ)) ∈ Z(Uq(sln))

as required.

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For all representations V , W of Uq(sln),

(4.29) CV ⊗W = CV CW.

Proof. Writing out the definition of CV ⊗W as given in corollary 4.19 becomes

(4.30)

CV ⊗W = (id ⊗ tr |V ⊗ tr |W)((id ⊗∆)(R21)(id ⊗∆)(R)(1 ⊗ q2ρ⊗ q2ρ))

= (id ⊗ tr |V ⊗ tr |W)(R21R31R13R12(1 ⊗ q2ρ⊗ q2ρ))

= (id ⊗ tr |V)(R21(CW ⊗ 1)R12(1 ⊗ q2ρ)).

Now, since CW is central, this further reduces to

(4.31) CV ⊗W = CW(id ⊗ tr |V)(R21R12(1 ⊗ q2ρ))

= CWCV = CV CW

as required.

4.21 Theorem:

On the Verma module Mµ,

(4.32) CW|Mµ =χW(q

2(µ+ρ)

) id .

Proof. First note that central elements C ∈ Z(Uq(sln)) always act as a constant

on Verma modules as these are cyclic, i.e. they are generated by a highest weight vector vµ.

Now consider the central element CW. Looking at the explicit expression for a

universal R-matrix as given in section 8.3 of [Klimyk and Schmüdgen, 1997], an R-matrix R of Uq(sln) can be written as

(4.33) R = qPixi⊗xi ₊X

j

aj ⊗ bj

where wt aj > 0 and wt bj < 0. This means that on a highest weight vector vµ,

(4.34) CWvµ= (id ⊗ tr |W)(R21R(1 ⊗ q2ρ)) vµ = (id ⊗ tr |W)(q P ixi⊗xi _qPixi⊗xi_{(1 ⊗ q}2ρ_{)) v} µ = tr |W(q2 P iµ(xi)xi_q2ρ_{) v} µ = tr |W(q2(µ+ρ)) vµ =χW(q2(µ+ρ)) vµ.

Again, because CW is central and Mµ is cyclic, it follows that

(4.35) CW x =χW(q2(µ+ρ)) x

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4.3 Proof of the Main Theorem 37

4.3.2 Difference Equation

Now that a suitable central element of Uq(sln)has been found in CW, the next step

is to apply the corresponding operator DCW to the trace function ΨV(λ, µ). On

the one hand, by corollary 4.7, this will equal C|MµΨV(λ, µ). On the other hand,

using theorem 4.6 will lead to an explicit expression for the operator DCW.

Translating to the transfer operator DW and the trace function FV(λ, µ)

satisfy-ing the difference equations as they appear in theorem 4.13 will be a trivial effort later on in section 4.3.6.

4.22 Notation:

The expressions found in this section live in either Mµ ⊗ V ⊗ V∗, Mµ ⊗ V ⊗

V∗ ⊗ Uq(sln), or Mµ ⊗ V ⊗ V∗ ⊗ Uq(sln) ⊗ Uq(sln). Adopting the notation used

in [Etingof and Latour, 2005], the components of such expressions will be labelled with sub- or superscript 0, 1, 1∗, 2, and 3 respectively.

Applying DCW to the trace function ΨV(λ, µ) and writing out its definition,

definition 4.17, and the action of DCW from theorem 4.6 yields

(4.36) DCW ΨV(λ, µ) = DCW Ψ 11∗ V (λ, µ) = DCW P iΨ vi,1_{(λ, µ) ⊗ v}∗ i = DCW P itr |0(Φ vi,01 µ q02λ) ⊗ vi∗ =P itr |0(Φ vi,01 µ CW q02λ) ⊗ vi∗. Setting (4.37) ΦV ,011_µ ∗ = X i (Φvi,01 µ ⊗ v ∗ i ) : Mµ → Mµ⊗ V ⊗ V∗,

using the fact that CW is central, and writing out its definition from corollary 4.19,

The proof of theorem 4.13 proceeds by further rewriting this expression for the action of DCW on the trace function ΨV(λ, µ). This is done in three steps, each

formulated as a proposition and treated in a seperate section to maintain overview. The proof of each of these three propositions depends on a series of lemmas, and is given at the end of each respective section.

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4.3.3 First Proposition

Part of the expression in equation 4.38 still contains R-matrix coefficients that will need to be reworked. The lemmas in this section represent steps taken to arrive at an expression that is based only on the trace function ΨV and a modified version

of the universal fusion operator. 4.23 Definition:

The operator

(4.39) J (λ) = J(−λ − ρ + 1 2(h

1_{+ h}2₎₎

in Uq(sln) ⊗ Uq(sln) is called the modified fusion operator.

The final result of this section is formulated in the following proposition. 4.24 Proposition: (4.40) DCW ΨV(λ, µ) = tr |W2 (µ23◦ S3) q₂−2λJ3,12_{(λ) J}12_{(λ −} 1 2h 3_{) ×} Ψ11∗ V (λ + 1 2h 2₋ 1 2h 3_,_{µ) J}32_(λ)-1 _q2λ 2 q2ρ

To get to the proof of proposition 4.24, start by considering the expression on the right hand side of equation 4.38;

(4.41) DCW ΨV(λ, µ) = tr |W2(tr |0(R 20_R02_ΦV ,011∗ µ q 2λ 0 ) q 2ρ 2 ).

Comparing to the right hand side of proposition 4.24, it is clear that an equality may be found by considering only the factor

(4.42) tr |0(R20R02ΦV ,011

∗

µ q 2λ 0 ).

After a small reformulation, using appendix proposition A.4, this becomes (4.43) (µ23◦ S3) tr |0(R20(R03)-1ΦV ,011 ∗ µ q 2λ 0 ) . Now since (4.44) tr |0 R20(R03)-1ΦV ,011 ∗ µ q2λ0 = tr |0 (R03)-1ΦV ,011 ∗ µ q02λR20 = tr |0 (R03)-1ΦV ,011 ∗ µ q −2λ 2 R20q02λq22λ = q₂−2λtr |0 (R03)-1ΦV ,011 ∗ µ R20q02λq22λ = q₂−2λtr |0 ΦV ,011 ∗ µ R20q02λ(R03)-1q22λ

it suffices to show that (4.45) tr |0 ΦV ,011 ∗ µ R20q2λ0 (R03)-1 = J3,12(λ) J12_{(λ −} 1 2h 3_{) Ψ}11∗ V (λ +12h 2₋ 1 2h 3_,_{µ) J}32_(λ)-1_.

(45)

4.3 Proof of the Main Theorem 39 Defining XV(λ, µ) = tr |0(ΦV ,011

∗

µ R20q02λ(R03)-1), this requirement becomes

(4.46) XV(λ, µ) = J3,12(λ) J12(λ − 1₂h3) Ψ11 ∗ V (λ + 12h 2₋1 2h 3 ,µ) J32(λ)-1. Moving the two outer modified fusion terms from the right hand side of equation 4.46 to the left, and defining YV(λ, µ) = J3,12(λ)-1 XV(λ, µ) J32(λ), rewrites the

requirement as (4.47) YV(λ, µ) = J12(λ − 1₂h3) Ψ11 ∗ V (λ + 1 2h 2₋ 1 2h 3_,_µ).

This equation can now be shown to hold by studying the functions XV and YV.

4.25 Lemma:

(4.48) XV(λ, µ) = R12,3q32λXV(λ, µ) q3−2λ(R 23₎-1

Proof. The lemma follows from a straightforward calculation using the definition of ΦV ,011∗

µ , the cyclic property of the trace, and properties of the universal R-matrix.

(4.49) XV(λ, µ) = tr |0(ΦV ,011 ∗ µ R20q02λ(R03)-1) = tr |0((R03)-1 ΦV ,011 ∗ µ R20q02λ) = tr |0(R13(R01,3)-1 ΦV ,011 ∗ µ R20q02λ) = R13 tr |0(ΦV ,011 ∗ µ (R03)-1 R20q2λ0 ) = R13 tr |0(ΦV ,011 ∗ µ R23R20(R03)-1(R23)-1 q02λ) = R13R23 tr |0(ΦV ,011 ∗ µ R20(R03)-1 q02λ) (R23)-1 = R12,3 _{tr |} 0(ΦV ,011 ∗ µ R20q02λq32λ(R03)-1 q −2λ 3 ) (R23)-1 = R12,3_q2λ 3 tr |0(ΦV ,011 ∗ µ R20q02λ(R03)-1) q −2λ 3 (R23)-1 = R12,3_q2λ 3 XV(λ, µ) q3−2λ(R23)-1.

This calculation makes use of (4.50) (R03)-1 = R13(R13)-1(R03)-1 = R13(R03R13)-1 = R13_{(((∆ ⊗ id)(R))}013₎-1 = R13_(R01,3₎-1_, (4.51) (R03₎-1 _R20 _{= R}23_(R23₎-1_(R03₎-1 _R20 = R23_(R03_R23₎-1 _R20 = R23(((∆ ⊗ id)(R))023)-1 R20 = R23R20(((∆op⊗ id)(R))203₎-1 = R23R20(R23R03)-1 = R23R20(R03)-1(R23)-1,

Quantum Dynamical R-matrices and Quantum Integrable Systems

MSc Mathematical Physics

Master Thesis