Twisted quantum affine algebras and solutions to the Yang-Baxter equation

Abstract

We construct spectral parameter dependent R-matrices for the quantized
enveloping algebras of twisted affine Lie algebras. These give new solutions to
the spectral parameter dependent quantum Yang-Baxter equation.

Full text

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arXiv:q-alg/9508012v2 7 Sep 1995
Twisted Quantum Affine Algebras
and Solutions to the Yang-Baxter Equation
Gustav W. Delius
Department of Mathematics, King’s College London
Strand, London WC2R 2LS, UK
e-mail: delius@mth.kcl.ac.uk
Mark D. Gould
Department of Mathematics, University of Queensland
Brisbane Qld 4072, Australia.
Yao-Zhong Zhang
Yukawa Institute for Theoretical Physics
Kyoto University, Kyoto 606, Japan
e-mail: yzzhang@yukawa.kyoto-u.ac.jp
February 9, 2008
Abstract
We construct spectral parameter dependent R-matrices for the quantized
enveloping algebras of twisted affine Lie algebras. These give new solutions
to the spectral parameter dependent quantum Yang-Baxter equation.
q-alg/9508012
KCL-TH-95-8
YITP/K-1119
0

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1Introduction
The solutions to the Yang-Baxter equation play a central role in the theory of
quantum integrable models [28, 23]. In statistical mechanics they are the Boltzman
weights of exactly solvable lattice models [3]. In quantum field theory they give the
exact factorizable scattering matrices [29]. For an introduction to the mathematical
aspects of the Yang-Baxter equation see e.g. [20].
The Yang-Baxter equation with spectral parameter has the form
R12(u)R13(uv)R23(v) = R23(v)R13(uv)R12(u).(1.1)
The Rab(u) are matrices which depend on a spectral parameter u and which act on
the tensor product of two vector spaces Vaand Vb
Rab(u) : Va⊗ Vb→ Va⊗ Vb.(1.2)
The products of R’s in (1.1) act on the space V1⊗ V2⊗ V3.
The mathematical framework for the construction of trigonometric solutions of
the quantum Yang-Baxter equation (1.1) is given by the quantum affine algebras
Uq(ˆL) introduced by Jimbo [18] and Drinfeld [11]. These are deformations of the
enveloping algebras U(ˆL) of affine Lie algebras [22]. Associated to any two finite-
dimensional irreducible Uq(ˆL)-modules V (λ) and V (µ) there exists a trigonometric
R-matrix Rλµ(u). Given three modules, the R-matrices for all pairs of these three
modules are a solution of (1.1). Many R-matrices of untwisted quantum affine alge-
bras have since been determined (see references in [7]), leading to a large number of
new quantum integrable models, quantum spin chains, exactly solvable lattice mod-
els and exact scattering matrices. The method has also been extended to quantum
affine superalgebras [8].
While it was clear from the beginning [11] that all (twisted and untwisted) affine
Lie algebras can be quantized and give rise to trigonometric R-matrices, the twisted
algebras have hardly been treated in the literature. The only R-matrices associ-
ated to twisted quantum affine algebras which we have found in the literature are
those associated to the vector representation of Uq(A(2)
Uq(A(2)
been found in [14] by “Baxterizing” the so-called dilute BWM algebra∗.) The
knowledge of these R-matrices has had many physical applications. They have for
example been used to obtain transfer matrices of solvable lattice models [25] or to
diagonalize quantum spin chain Hamiltonians on the periodic chain [27] and on the
open chain [1, 2]. To generalize these works to the models associated to higher di-
mensional representations it is necessary to know the corresponding R-matrices and
that is the topic of this paper.
l) and Uq(D(2)
l+1) [4, 19]. (The
2) R-matrix was found already in [17]; another R-matrix for Uq(D(2)
l+1) has
R-matrices are also needed to construct the S-matrices of quantum field theories
with quantum affine symmetries [10]. In particular the R-matrices associated to all
∗We thank Ole Warnaar for drawing our attention to the ref. [14]
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fundamental representations of A(2)
scattering matrices for the solitons in A(2)
2nwhich we construct in this paper will give the
2naffine Toda theory.
The paper is organized as follows: In section 2 we review the necessary facts
about twisted Lie affine algebras [22]. In section 3 we discuss the quantized affine
algebras and give the equations which uniquely determine the R-matrices (Jimbo’s
equations). In section 4 we explain how to solve these equations. Our technique is
an extension of the tensor product graph method introduced in [30] and generalized
in [7]. We have obtained the R-matrices for the following twisted algebras and tensor
products:
Uq(A(2)
Uq(A(2)
Uq(D(2)
2l)
2l−1) l ≥ 3 V (kλ1) ⊗ V (rλ1)
l+1)l ≥ 2
l ≥ 2 V (λk) ⊗ V (λr)k + r ≤ l section 4.1
section 4.2
section 4.3V (kλl) ⊗ V (rλl)
(1.3)
Here λidenotes the i-th fundamental weight. We give some of the technical details in
appendices. In particular in appendix B we derive the tensor product decompositions
and branching rules which we need in the paper.
2Twisted affine Lie algebras
We recall the relevant information about twisted affine Lie algebras [22]. Let L be a
finite dimensional simple Lie algebra and σ a diagram automorphism of L of order k.
Associated to these one constructs the twisted affine Lie algebraˆL(k). In this paper
we will assume†k = 2. Let L0be the fixed point subalgebra under the diagram
automorphism σ. We recall that
L = L0⊕ L1,[Li,Lj] = L(i+j)mod2.(2.1)
L1gives rise to an irreducible L0-module under the adjoint action of L0. Let θ0be
its highest weight. In table 1 we list all the cases with k = 2. Below we restrict
ourselves to the three families and leave out the exceptional case for technical reasons
which will become apparent later.
We recall that L admits generators Ei,Fi,Hi, 0 ≤ i ≤ l, satisfying the defining
relations‡
[Hi,Ej] = (¯ αi, ¯ αj)Ej,
(adEi)1−aijEj= (adFi)1−aijFj= 0,
[Hi,Fj] = −(¯ αi, ¯ αj)Fj,[Ei,Fj] = δijHi,
i ?= j,(2.2)
where aij= 2(¯ αi, ¯ αj)/(¯ αi, ¯ αi) are the entries of the corresponding (twisted) Cartan
matrix ofˆL(2). Here the Ei,Fi,Hi, 1 ≤ i ≤ l, form the Chevalley generators for
†The only diagram automorphism which is not of order k = 2 is the triality of the Lie algebra
D4and we will not treat this case in this paper.
‡The rescaled generators E′
i=
?2/¯ α2
iEi, F′
i=
?2/¯ α2
iEi, H′
i= 2/¯ αiHisatisfy the more usual
commutation relations with the structure constants given by the Cartan matrix.
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LL0
θ0
A2l,
A2l−1, l ≥ 3 Cl
Dl+1,l ≥ 2 Bl
E6
l ≥ 1 Bl
2λ1= 2ǫ1
λ2= ǫ1+ ǫ2
λ1= ǫ1
λ4
F4
Table 1: Table of the finite dimensional simple Lie algebras L which posess a diagram
automorphism of order k = 2, their fixed point subalgebras L0and the highest weight
θ0of the adjoint L0-module L1. Here and in the rest of the paper we give weights
either as integer combinations of the fundamental weights λior alternatively we give
them in terms of the ǫiwhich form a basis of the root space of gl(n) into which we
imbed the other algebras, see Appendix A.
L0and the ¯ αi, (1 ≤ i ≤ l) are the simple roots of L0. E0∈ L1corresponds to the
minimal weight vector and thus has weight −θ0. It follows that ¯ α0= −θ0and that
H0= −?l
i=1aiHilies in the Cartan subalgebra H of L0. The integers aiare known
as the Kac labels ofˆL(2).
Throughout we let ( , ) be a fixed invariant bilinear form on L which induces a
corresponding invariant form ( , ) on H∗. A suitable choice for the invariant form
on L togeher with a realization of the simple generators is given in Appendix A for
completeness. With our choice we have
(Ei,Fj) = δij,(Hi,Hj) = (¯ αi, ¯ αj).(2.3)
We now introduce the corresponding twisted affine Lie algebraˆL(2)′which admits
the decomposition
ˆL(2)′=
?
m∈1
2Z
ˆLm⊕ Cc0,
ˆLm=
?
L0(m), m ∈ Z
L1(m), m ∈ Z +1
2
(2.4)
with La(m) = {x(m)|x ∈ La}, a = 0,1 and c0a central charge. The Lie bracket is
given by
[x(m),y(n)] = [x,y](m + n) + m c0δm+n,0(x,y),[c0,x(m)] = 0.(2.5)
Here ( , ) is the fixed invariant bilinear form on L. Note thatˆL0= L0. A suitable
set of generators forˆL(2)′is given by
ei= Ei(0),
e0= E0(1/2),
hi= Hi(0),
h0= H0(0) + 1/2c0,
fi= Fi(0),1 ≤ i ≤ l,
f0= F0(−1/2).(2.6)
This algebra is extended toˆL(2)=ˆL(2)′⊕ Cd0 by the introduction of the level
operator d0satisfying
[d0,x(m)] = mx(m),[d0,c0] = 0.(2.7)
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As a Cartan subalgebra ofˆL(2)we take
ˆH = H(0) ⊕ Cc0⊕ Cd0. (2.8)
The weights forˆL(2)are of the form λ = (¯λ,cλ,dλ) where¯λ ∈ H∗and cλ,dλare the
eigenvalues of the central extension c0and the level operator d0respectively. The
simple roots corresponding to the set of simple generators in (2.6) are
αi= (¯ αi,0,0), 1 ≤ i ≤ l,α0= (−θ0,0,1
2).(2.9)
The invariant bilinear form onˆH∗is given by
(λ,µ) = (¯λ, ¯ µ) + cλdµ+ dλcµ. (2.10)
With this convention we have (αi,αj) = (¯ αi, ¯ αj), (0 ≤ i,j ≤ l) and our simple
generators satisfy the defining relations
[hi,ej] = (αi,αj)ej,
(adei)1−aijej= (adfi)1−aijfj= 0,
[hi,fj] = −(αi,αj)fj,[ei,fj] = δijhi,
i ?= j.(2.11)
Following Kac [22] it is useful to introduce the weights γ = (¯0,1,0) and δ = (¯0,0,1)
so that (γ,δ) = 1,(γ,γ) = (δ,δ) = 0. Our simple roots are then given by αi =
¯ αi, 1 ≤ i ≤ l, α0= −θ0+ 1/2δ.
We have an algebra homomorphism, called the evaluation map, evt: U(ˆL(2)) →
C[t,t−1] ⊗ U(L), with U(ˆL(2)),U(L) the enveloping algebras ofˆL(2),L respectively,
given by
evt(x(m)) = t2mx,evt(c0) = 0,evt(d0) =1
2td
dt,
(2.12)
and extended to all of U(ˆL(2)) in the natural way. Thus given a finite dimensional L-
module V carrying a representation π we have a correspondingˆL(2)module V (t) =
C[t,t−1] ⊗ V carrying the loop representation ˆ π given by
ˆ π = (1 ⊗ π)evt.
Below we consider the problem of quantizing such representations to give solutions
of the Yang-Baxter equation.
(2.13)
An important role will be played below by those irreducible L-modules which are
also irreducible under the L0subalgebra. We call these L0-irreducible modules. We
will see below that the loop representations built on L0-irreducible modules can all
be quantized. Such L0-irreducible modules appear to exist for the first three cases
in table 1 only and this is the reason why we are restricting to these cases. In table
2 we list for each of the three families the highest weights of all the L0-irreducible
irreps of L together with their highest weight with respect to L0.
In appendix B we show that the tensor product of any two such L0-irreducible
L-modules decomposes into a multiplicity free direct sum of irreducible L0-modules.
This is important because it implies that a solution to Jimbo’s equations will always
exist for such tensor products (see below).
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LL0
Bl
Cl
Bl
ΛΛ0
λk,
aλ1,
aλl,
A2l
A2l−1
Dl+1
λk,λ2l+1−k
aλ1
aλl,aλl+1
1 ≤ k ≤ l
a ∈ Z+
a ∈ Z+
Table 2: L0-irreducible irreps. Λ are the highest weights of the L0-irreducible irreps
of L and Λ0are the corresponding highest weights under L0.
3Twisted quantum affine algebras
Corresponding to the twisted affine algebraˆL(2)we have the twisted quantum affine
algebra Uq(ˆL(2)) with generators q±hi/2,ei,fi,d0, (0 ≤ i ≤ l) and defining relations
[hi,ej] = (αi,αj)ej,[hi,fj] = −(αi,αj)fj,
[d0,ei] =1
2δi,0ei,
[ei,fj] = δijqhi− q−hi
q − q−1,
1−aij
?
1−aij
?
where
[hi,hj] = 0,
[d0,fi] = −1
2δi,0fi,[d0,hi] = 0,
k=0
(−1)ke(1−aij−k)
i
eje(k)
i
= 0, i ?= j,
k=0
(−1)kf(1−aij−k)
i
fjf(k)
i
= 0, i ?= j,(3.1)
e(k)
i
=
ek
i
[k]qi!, f(k)
i
=
fk
i
[k]qi!,
[k]q=qk− q−k
q − q−1,[k]q! =
k?
n=1
[n]q,qi= q
1
2(αi,αi).(3.2)
Uq(ˆL(2)) is a quasi-triangular Hopf algebra with coproduct ∆ and antipode S given
by
∆(ei) = q−hi/2⊗ ei+ ei⊗ qhi/2,
∆(fi) = q−hi/2⊗ fi+ fi⊗ qhi/2, S(fi) = −q−1
∆(q±hi/2) = q±hi/2⊗ q±hi/2, S(q±hi/2) = q∓hi/2,
∆(d0) = 1 ⊗ d0+ d0⊗ 1, S(d0) = −d0.
S(ei) = −qiei,
i fi,
(3.3)
Throughout¯R denotes the universal R-matrix of Uq(ˆL(2)) which by definition
satisfies
¯R∆(a) = ∆T(a)¯R, ∀a ∈ Uq(ˆL(2)),
(1 ⊗ ∆)¯R =¯R13¯R12,(∆ ⊗ 1)¯R =¯R13¯R23
(3.4)
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where ∆T(a) is the opposite coproduct. A direct consequence of the above relations
is that¯R satisfies the quantum Yang-Baxter equation
¯R12¯R13¯R23=¯R23¯R13¯R12
(3.5)
Note that the generators q±hi/2,ei,fi, (1 ≤ i ≤ l) generate the quantum algebra
Uq(L0) which is a quasi-triangular Hopf subalgebra of Uq(ˆL(2)). We denote its uni-
versal R-matrix by R.
We shall see below that any minimal irrep V0(λ) of Uq(L0) can be affinized to
give rise to an irrep of Uq(ˆL(2)). To perform such an affinization it is necassary
and sufficient to find operators πλ(e0) and πλ(f0) acting on V0(λ) which satisfy the
required defining relations (3.1) of Uq(ˆL(2)).
We define an automorphism Dtof Uq(ˆL(2)) by
Dt(ei) = tδi0ei, Dt(fi) = t−δi0fi, Dt(hi) = hi.(3.6)
Given any two minimal irreps πλand πµof Uq(L0) and their affinizations to irreps
of Uq(ˆL(2)), we obtain a one-parameter family of representations ∆u
V0(λ) ⊗ V0(µ) defined by
∆u
λµof Uq(ˆL(2)) on
λµ(a) = πλ⊗ πµ((Du⊗ 1)∆(a)),∀a ∈ Uq(ˆL(2)),(3.7)
where u is the spectral parameter. We define the spectral parameter dependent
R-matrix
Rλµ(u) = (πλ⊗ πµ)
?
(Du⊗ 1)¯R
?
.(3.8)
It follows from (3.5) that this R-matrix gives a solution to the spectral parameter
dependent Yang-Baxter equation (1.1). From the defining property (3.4) of the
universal R-matrix one derives the equations
Rλµ(u)∆u
λµ(a) = (∆T)u
λµ(a)Rλµ(u)(3.9)
which, because the representations ∆u
mine Rλµ(u) up to a scalar function of u. These are the Jimbo equations for twisted
affine algebras.
λµare irreducible for generic u, uniquely deter-
As in [7] we normalize Rλµ(u) such that
ˇRλµ(u)ˇRµλ(u−1) = I and R(0) = πλ⊗ πµ(R),
where R is the R-matrix of Uq(L0) andˇRλµ(u) = P Rλµ(u) with P : V0(λ)⊗V0(µ) →
V0(µ) ⊗ V0(λ) the usual permutation operator.
In order for the equation (3.9) to hold for all a ∈ Uq(ˆL(2)) it is sufficient that it
holds for all a ∈ Uq(L0) and in addition for the extra generator e0. The relation for
e0reads explicitly
(3.10)
Rλµ(u)
?
uπλ(e0) ⊗ πµ(qh0/2) + πλ(q−h0/2) ⊗ πµ(e0)
uπλ(e0) ⊗ πµ(qh0/2) + πλ(q−h0/2) ⊗ πµ(e0)
?
=
??
Rλµ(u),(3.11)
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or equivalently
ˇRλµ(u)
?
uπλ(e0) ⊗ πµ(qh0/2) + πλ(q−h0/2) ⊗ πµ(e0)
πµ(e0) ⊗ πλ(qh0/2) + uπµ(q−h0/2) ⊗ πλ(e0)
?
=
??ˇRλµ(u).(3.12)
4Solutions to Jimbo’s equations
With V0(λ) and V0(µ) denoting two minimal irreps of Uq(ˆL(2)) we write the tensor
product decomposition into irreducible Uq(L0)-modules as
V0(λ) ⊗ V0(µ) =
?
ν
V0(ν)(4.1)
and note that there are no multiplicities in this decomposition for the cases which
we are considering (c.f. Appendix B). We let Pλµ
V0(λ) ⊗ V0(µ) onto V0(ν) and set
ˇPλµ
ν
ν
be the projection operator of
ν =ˇRλµ(1)Pλµ
= Pµλ
ν
ˇRλµ(1).(4.2)
We may thus write
ˇRλµ(u) =
?
ν
ρν(u)ˇPλµ
ν,ρν(1) = 1.(4.3)
Following our previous approach [7], the coefficients ρν(u) may be determined ac-
cording to the recursion relation
ρν(u) =qC(ν)/2+ ǫνǫν′uqC(ν′)/2
uqC(ν)/2+ ǫνǫν′ qC(ν′)/2ρν′(u),(4.4)
which holds for any ν ?= ν′for which
Pλµ
ν
?
πλ(e0) ⊗ πµ(qh0/2)
?
Pλµ
ν′ ?= 0.(4.5)
Here C(ν) is the eigenvalue of the universal Casimir element of L0on V0(ν) and ǫν
denotes the parity of V0(ν) ⊆ V0(λ) ⊗ V0(µ), (c.f. Appendix C).
To graphically encode the recursive relations between the different ρνwe intro-
duce the Twisted Tensor Product Graph˜Gλµassociated to the tensor product
module V0(λ) ⊗ V0(µ). The nodes of this graph are given by the highest weights
ν of the Uq(L0)-modules occuring in the decomposition (4.1) of the tensor product
module. There is an edge between two nodes ν ?= ν′iff (4.5) holds.
Given a tensor product module and its decomposition, it is not in general an easy
task to determine the twisted tensor product graph because in order to determine
between which nodes of the graph relation (4.5) holds requires detailed calculations.
We therefore introduce the Extended Twisted Tensor Product Graph˜Γλµ
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which has the same set of nodes as the twisted tensor product graph (TPG) but has
an edge between two vertices ν ?= ν′whenever
V0(ν′) ⊆ V0(θ0) ⊗ V0(ν)
and
(4.6)
ǫνǫν′ =
?
+1
−1
if V0(ν) and V0(ν′) are in the same irrep of L
if V0(ν) and V0(ν′) are in different irreps of L.
(4.7)
The conditions (4.6) and (4.7) are necessary conditions for (4.5) to hold and therefore
the twisted TPG is contained in the extended twisted TPG. To see why (4.6) is a
necessary condition for (4.5) one must realize that e0⊗qh0/2is the lowest component
of a tensor operator corresponding to V0(θ0), see [30] for details. The necessity of
(4.7) follows from the following fact derived in Appendix C: Two vertices ν ?= ν′
connected by an edge in the twisted TPG (i.e., for which (4.5) is satisfied) must have
the same parity if V0(ν) and V0(ν′) belong to the same irreducible L-module while
they must have opposite parities if they belong to different irreducible L-modules.
While the extended twisted TPG will always include the twisted TPG, it will
in general have more edges. Only if the extended twisted TPG is a tree are we
guaranteed that it coincides with the twisted TPG.
Note: Unlike the untwisted case [7], we may now get an edge between ν and ν′of
the same parity. This gives rise to a twisted TPG which may be topologically quite
different to the untwisted TPG.
We will impose a relation (4.4) for every edge in the extended twisted TPG.
Because the extended TPG will in general have more edges than the unextended
twisted TPG, we will be imposing too many relations. These relations may be
inconsistent and we are therefore not guaranteed a solution. If however a solution
exists, then it must be the unique correct solution to Jimbo’s equations.
As seen below, for the minimal cases we are considering, the extended twisted
TPG is always consistent and thus will always give rise to a solution of the quantum
Yang-Baxter equation.
Throughout we adopt the convenient notation
?a?±=1 ± xqa
x ± qa,(4.8)
so that the relation (4.4) may be expressed as
ρν(u) =
?C(ν′) − C(ν)
2
?
ǫνǫν′
ρν′(u).(4.9)
We will now determine the R-matrices for any tensor product of any two L0-
irreducible representations for all the three families of twisted quantum affine alge-
bras Uq(A(2)
2l),Uq(A(2)
2l−1) and Uq(D(2)
l+1).
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4.1R-matrices for Uq(A(2)
2l)
This is the case of the first line in table 1, i.e. L = A2l= sl(2l + 1), L0= Bl=
so(2l + 1) and θ0= 2ǫ1= 2λ1.
The defining (vector) irrep V (λ1) of Uq(L) is undeformed. By this we mean that
the representation matrices for the fundamental generators are the same as those
in the classical case, i.e. they are independent of q. It is also a minimal irrep, i.e.
V (λ1) = V0(λ1) is also irreducible as a module of Uq(L0). Furthermore it is affiniz-
able, i.e. it carries a representation of Uq(A(2)
undeformed, i.e. π(e0) and π(f0) are given by the classical expressions.
2l). Also this affinized representation is
We have the corresponding twisted TPG for V (λ1) ⊗ V (λ1)
++
~
0
~
2λ1
~
λ2
−
(4.10)
where ± indicate the parities. This is quite different to the untwisted TPG
+−
~
0
~
λ2
~
2λ1
+
(4.11)
Since λ2is an extremal node on the twisted TPG it follows that V0(λ2) is affiniz-
able, i.e. it too carries an irrep of Uq(A(2)
TPGs and finite dimensional irreps of quantum affine algebras see [9]). More gen-
erally we have the following twisted TPG for for V (λ1) ⊗ V (λk), k < l
++
2l) (for a discussion of the relation between
~~~
λk−1
λ1+ λk
λk+1
−
(4.12)
so that λk+1is an extremal node and hence, by recursion, each of the fundamental
irreps V0(λk), 1 ≤ k ≤ l, is affinizable. Again the above twisted TPG (4.12) is
different to the untwisted one which is
+−
~~~
λk−1
λk+1
λ1+ λk
+
(4.13)
Note: In the untwisted case V0(λk), k > 1 does not occur on extremal nodes of
any TPG and can therefore not be shown to be affinizable to a representation of
Uq(ˆL(1)
seen above, it is nevertheless affinizable to a representation of the twisted algebra
Uq(ˆL(2)).
0). In fact it is generally not affinizable [9, 7] in the untwisted sense. But, as
Now for 1 ≤ k ≤ r ≤ l we have the tensor product decomposition
V0(λk) ⊗ V0(λr) =
k
?
a=0
a
?
c=0
V0(λc+ λd)(4.14)
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where
d =
?
k + r − 2a + c
2l + 1 − (k + r − 2a + c) for 2a − c < r + k − l
for 2a − c ≥ r + k − l
(4.15)
which is multiplicity free (c.f. Appendix B). The corresponding extended twisted
tensor product graph is consistent and quite different in topology to the the extended
untwisted TPG (which is inconsistent). We illustrate this below with the case r+k ≤
l, k ≤ r. In this case d = k + r − 2a + c and we have the extended twisted TPG
depicted in figure 1.
~~
~
@
~
@
~
+
~
−
~
@
+
~
~
@
~
@
+
~
@
−
~
@
+
~
@@
+
~
@@
−
?
?
?
?
?
?
??
?
?
?
??
@
@
@
@
@
@
?
?
?
?
?
@
@
?
@
@
?
?
@
@
?
?
?
?
@
@
@
@
?
?
?@
@
??
λr
λr−1 λr−2
···λr−k+1λr−k
λr+1
λr+kλr+k−1
···
λk
λk−1
λk−2
...
0
λ1
···
+
Figure 1: The extended twisted TPG for Uq(A(2)
(k ≤ r, r+k ≤ l) . The nodes correspond to representations whose highest weight is
given by the sum of the weight labeling the column and the weight labeling the row.
The ± indicate the parity. The parities are equal along the northwest-southeast
diagonals and they alternate along the northeast-southwest diagonals.
2l) for the product V0(λk) ⊗ V0(λr)
To see that the extended twisted TPG in figure 1 is consistent, consider a typical
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closed loop:
~
~~
@
~
@
@
@
@
@@
?
?
?
?
?
?@
@
@
@
@
?
?
−
?
?
??+
+
−
λc+ λd
λc−1+ λd−1
λc−1+ λd+1
λc−2+ λd
(4.16)
where we have indicated the relative parities of the vertices. Using the fact that on
V0(λc+ λd) the universal Casimir element of L0takes the eigenvalue
Cc,d= (c + d)(2l + 2 − c) − (d + 1)(d − c),
it is easily seen that
(4.17)
Cc,d− Cc−1,d−1= Cc−1,d+1− Cc−2,d= 2(2l + 3 − c − d),
Cc,d− Cc−1,d+1= Cc−1,d−1− Cc−2,d= 2(d − c + 2).
This implies that the extended twisted TPG is consistent, i.e. that the recursion
relations (4.4) give the same result independent of the path along which one recurses.
(4.18)
We are now in a position to write down our solution to Jimbo’s equation and
thus to the quantum Yang-Baxter equation arising from the above extended twisted
TPG:
ˇRλk,λr(u) =
k
?
a=0
a
?
c=0
k−1
?
i=a
?k + r − 2i?−
a−c
?
j=1
?n − r − k + 2j?+ˇPλk,λr
λc+λk+r−2a+c
(4.19)
4.2R-matrices for Uq(A(2)
2l−1)
This is the case of the second line in table 1, i.e. L = A2l−1= sl(2l), L0= Cl= sp(2l)
and θ0= ǫ1+ ǫ2= λ2.
Starting with the vector irrep V0(λ1) of Uq(L0) (and also of Uq(L)) we have the
following twisted TPG for V (λ1) ⊗ V (λ1)
−
~
0
~
λ2
~
2λ1
−+
(4.20)
which has quite a different topology to the untwisted TPG
−
~
0
~~
λ2
2λ1
+−
(4.21)
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Because V0(2λ1) appears as an extremal node on the twisted TPG (4.20), it is
affinizable. Continuing in this way it is easily seen that V0(aλ1) is affinizable for
any positive integer a. We have the following Uq(L0)-module decomposition of the
tensor product of any two such representations
V0(kλ1) ⊗ V0(rλ1) =
k
?
a=0
a
?
b=0
V0((k + r − 2a)λ1+ bλ2),k ≤ r.(4.22)
@
+
@
@
@
@
−
@
@
@
@
+
@
@
@
+
@
@
@
@
−
@@
+
@
@
@
@@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@@
~
~~
~~~
~~~
~~~~~
~
···
···
(r + k)λ1
(r + k − 2)λ1
(r + k − 4)λ1
(r − k + 2)λ1
(r − k)λ1
λ2
2λ2
(k − 1)λ2
kλ2
...
...
0
Figure 2: The extended twisted TPG for Uq(A(2)
V (rλ1). The nodes correspond to modules whose highest weight is the sum of the
weight labeling the column and the weight labeling the row. Some of the parities
are indicated below the nodes. The parities are the same along the horizontal and
they alternate along the vertical.
2l−1) for the tensor product V (kλ1)⊗
The corresponding extended twisted TPG is shown in figure 2. To see that this
graph is consistent we have to consider the closed loops of the form
~
@
+
~
@
+
~
−
~
−
@
@
@
@ @
@
@
@
@
@
cλ1+ (d − 1)λ2
cλ1+ dλ2
(c − 2)λ1+ (d + 1)λ2
(c − 2)λ1+ dλ2
(4.23)
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where we have indicated the relative parities of the vertices. Using the fact that on
V0(cλ1+ dλ2) the quadratic Casimir element of L0takes the eigenvalue
Cc,d= (c + d)(n + c + d) + (n − 2 + d)d,
it is easily seen that the above loop is consistent:
(4.24)
Cc,d− Cc,d−1= Cc−2,d+1− Cc−2,d= 2(n − 2 + c + 2d),
Cc,d− Cc−2,d+1= Cc,d−1− Cc−2,d= 2c. (4.25)
We can now read off the R-matrix from the extended twisted TPG
ˇRkλ1,rλ1(u) =
k
?
a=0
a
?
סPkλ1,rλ1
b=0
(a−b)
?
(k+r−2a)λ1+bλ2.
i=1
?n + k + r − 2i?+
a
?
j=1
?k + r + 2 − 2j?−
(4.26)
Again it can be seen that the above eigenvalues ofˇR(u) are quite different to those
arising from the untwisted case and thus give rise to new solutions to the quantum
Yang-Baxter equation.
4.3 R-matrices for Uq(D(2)
l+1)
This is the case of L = Dl+1= so(2l + 2), L0= Bl= so(2l + 1) and θ0= ǫ1= λ1.
We set n = 2l + 1.
The fundamental spinor irrep V0(λl), λl=1
and affinizable and also carries the spinor irreps V (λl) and V (λl+1) of Uq(L). We
have the tensor product decomposition, for a ∈ Z+,
l−1
?
with the corresponding twisted TPG
2(ǫ1+···+ǫl) of Uq(L0) is undeformed
V0(λl) ⊗ V0(aλl) =
k=0
V0(λk+ (a − 1)λl) ⊕ V0((a + 1)λl)(4.27)
zz
µ1
zzzz
µ2
···
(a + 1)λl
···
(a − 1)λl
+−−++
(4.28)
where we have indicated the parity of the vertices and introduced µi= (a−1)λl+λl−i.
Note that his graph has a very different topology from the corresponding untwisted
TPG given in Fig. 1 of [7]. Because (a+1)λlis on an extremal node of this twisted
TPG it then follows by induction that V0(aλl) is affinizable for any positive integer
a. The eigenvalue of the universal Casimir element of L0on V0(λk+ (a − 1)λl) is
given by
C (λk+ (a − 1)λl) = k(n + a − k − 1) +1
C ((a + 1)λl) = l(n + a − l − 1) +1
4l(a − 1)(n + a − 2), 1 ≤ k ≤ l − 1,
4l(a − 1)(n + a − 2)(4.29)
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Using this we can read off the R-matrix from the twisted TPG (4.28):
ˇRλl,aλl(u) =
l−1
?
k=0
ρk(u)ˇPλl,aλl
λk+(a−1)λl+ˇPλl,aλl
(a+1)λl
(4.30)
with the eigenvalues ρk(u) given by
ρk(u) =
l−k
?
i=1
?1
2(a − 1) + i
?
(−1)i.(4.31)
We now proceed to the general case V0(aλl) ⊗ V0(bλl), 0 ≤ a ≤ b ∈ Z. In view
of appendix B we now have the multiplicity-free tensor product decomposition
V0(aλl) ⊗ V0(bλl) =
?
Λ
V0(Λ + (b − a)λl),(4.32)
where the sum is over all dominant weights Λ = (Λ1,Λ2,···,Λl) satisfying
a ≥ Λ1≥ Λ2≥ ··· ≥ Λl≥ 0, Λi∈ Z,
each such weight appearing exactly once.
(4.33)
The extended twisted TPG will typically be l-dimensional and we can therefore
not draw a diagramm for it. However to determine wether it is consistent, it is
sufficient to look at closed loops of the form (labeling the vertices by the weight Λ
since a and b are here fixed and thus redundant labels):
~
~~
@
~
@
@
@
@
@@
?
?
?
?
?
?@
@
@
@
@
?
?
?
?
??
Λ
Λ − ǫi
Λ − ǫj
Λ − ǫi− ǫj
(4.34)
To show that all the edges in this loop really exist, i.e. that (4.5) is satisfied, one
uses the following theorem proven in [12].
Theorem 1 Suppose λ,µ are dominant weights and ν is a weight Weyl group con-
jugate to λ. If µ+ν is dominant then V (µ+ν) occurs exactly once in V (λ)⊗V (µ).
To obtain the relative (to the top vertex) parities of the vertices of the closed
loops it is necessary to determine which irreps of so(2l+1) belong to the same irrep
of so(2l+2) (which will all have the same parity). As seen in appendix B, two such
14