algebraic bethe ansatz for deformed gaudin model
TRANSCRIPT
arX
iv:1
002.
4951
v1 [
nlin
.SI]
26
Feb
2010
Algebraic Bethe Ansatz for deformed
Gaudin model
N. Cirilo Antonio, ∗ N. Manojlovic † and A. Stolin ‡
∗Centro de Analise Funcional e Aplicacoes
Departamento de Matematica, Instituto Superior Tecnico
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
†Grupo de Fısica Matematica da Universidade de Lisboa
Av. Prof. Gama Pinto 2, PT-1649-003 Lisboa, Portugal
†Departamento de Matematica, F. C. T., Universidade do Algarve
Campus de Gambelas, PT-8005-139 Faro, Portugal
‡Departament of Mathematical Sciences
University of Goteborg, SE-412 96 Goteborg, Sweden
Abstract
The Gaudin model based on the sl2-invariant r-matrix with an extraJordanian term depending on the spectral parameters is considered. Theappropriate creation operators defining the Bethe states of the system areconstructed through a recurrence relation. The commutation relationsbetween the generating function t(λ) of the integrals of motion and thecreation operators are calculated and therefore the algebraic Bethe Ansatzis fully implemented. The energy spectrum as well as the correspondingBethe equations of the system coincide with the ones of the sl2-invariantGaudin model. As opposed to the sl2-invariant case, the operator t(λ) andthe Gaudin Hamiltonians are not hermitian. Finally, the inner productsand norms of the Bethe states are studied.
∗E-mail address: [email protected]†E-mail address: [email protected]‡E-mail address: [email protected]
1 INTRODUCTION
1 Introduction
A model of interacting spins in a chain was first considered by Gaudin [1, 2].In his approach these models were introduced as a quasi-classical limit of theintegrable quantum chains. Moreover, the Gaudin models were extended to anysimple Lie algebra, with arbitrary irreducible representation at each site of thechain [2].
In the framework of the Quantum Inverse Scattering Method (QISM) [3]–[8]the Gaudin models are based classical r-matrices, with loop algebras as under-lying dynamical symmetry algebras. Namely, Gaudin hamiltonians are relatedto classical r-matrices,
H(a) =N∑
b6=a
rab(za, zb). (1.1)
The condition of their commutativity [H(a), H(b)] = 0 is nothing else but theclassical Yang-Baxter equation
[rab(za, zb), rac(za, zc) + rbc(zb, zc)] + [rac(za, zc), rbc(zb, zc)] = 0 (1.2)
where r belongs to the tensor product g⊗g of a Lie algebra g, or its representa-tions and the indices fix the corresponding factors in the N -fold tensor product.Also, it is understood that the classical r-matrix has a unitarity property
r12(µ, ν) = −r21(ν, µ). (1.3)
Considerable attention has been devoted to Gaudin models based on theclassical r-matrices of simple Lie algebras [9]–[14] and Lie superalgebras [15, 16,17]. The spectrum and corresponding eigenfunctions were found using differentmethods such as coordinate and algebraic Bethe ansatz, separated variables, etc.Correlation functions for Gaudin models were explicitly calculated as combina-torial expressions obtained from the Bethe ansatz [2, 18]. In the case of the sl2Gaudin system, its relation to Knizhnik-Zamolodchikov equation of conformalfield theory [14, 19] or the method of Gauss factorization [11], provided alter-native approaches to computation of correlations. The non-unitary r-matricesand the corresponding Gaudin models have been studied recently, see [20] andthe references therein.
A classification of solutions to classical Yang-Baxter equation (1.2), withthe property (1.3), has been done in [21]. In particular, a class of rationalsolutions corresponding to the simple Lie algebras sln was discussed in [21] andthen further studied and generalised in [22, 23]. In the sl2 case, up to gaugeequivalence, only two such solutions exist: the sl2-invariant r-matrix with anextra Jordanian term (deformation)
rJξ (µ, ν) =c⊗2
µ− ν+ ξ(h⊗X+ −X+ ⊗ h), (1.4)
where c⊗2 = h ⊗ h + 2 (X+ ⊗X− +X− ⊗X+) denotes the quadratic tensorCasimir of sl2, and the sl2-invariant r-matrix with a deformation depending onthe spectral parameters
rξ(µ, ν) =c⊗2
µ− ν+ ξ
(
h⊗ (νX+)− (µX+)⊗ h)
. (1.5)
-2-
1 INTRODUCTION
The Gaudin model based on the r-matrix (1.4) was studied as the quasi-classical limit of the corresponding quantum spin chain [24, 25], which is adeformation of the XXX spin chain where the Jordanian twist is applied to theYang R-matrix [26, 27]. The algebraic Bethe Ansatz for this Gaudin model wasfully implemented in [28], following ideas in [15, 16, 29].
In the framework of the QISM, the classical Yang-Baxter equation guaran-tees for the Gaudin model the existence of integrals of motion in involution. Amore difficult problem is to determine the spectra and the corresponding eigen-vectors. The Algebraic Bethe Ansatz (ABA) is a powerful tool which yieldsthe spectrum and corresponding eigenstates in the applications of the QISMto the models for which highest weight type representations are relevant, likefor example Gaudin models, quantum spin systems, etc [4, 8, 10]. However theABA has to be applied on the case by case basis.
An initial step in the application of the Algebraic Bethe Ansatz to theGaudin model based on the r-matrix (1.5), is to define appropriate creationoperators which yield the Bethe states and consequently the spectrum of the cor-
responding Gaudin Hamiltonians. The operators B(k)M (µ1, . . . , µM ) used for this
model, introduced in [30], are defined through a recursive relation that general-izes the one used in [28] for the Gaudin model based on the r-matrix (1.4). The
commutation relations of the B(k)M (µ1, . . . , µM ) operators with the generators of
the loop algebra are straightforward to calculate, see lemma 2.2. The main step,lemma 2.3 below, in the implementation of the ABA is to calculate explicitlythe commutation relations between the creation operators BM (µ1, . . . , µM ) andthe transition matrix of this Gaudin model. The spectrum of the model is thena straightforward consequence of the previous result, theorem 3.1. Finally, itis important to stress that the dual generating function of integrals of motiont∗(λ) is not equal to t(λ) and hence these operators are not hermitian.
Besides determining the spectrum of the system, via ABA, another impor-tant problem is to calculate the inner product of the Bethe vectors. To thisend it is necessary to introduce the dual creation operators B∗
M (µ1, . . . , µM )obtained in the study of the dual Gaudin model based on the following classicalr-matrix
r∗(λ, µ) =1
λ− µ(h⊗h+2(X+⊗X−+X−⊗X+))+ξ(h⊗λX−−µX−⊗h). (1.6)
A method developed in [28] to determine the inner product of the Bethe vectorswill be used here. The main idea is to establish a relation between the creationoperators of the system and the ones in the sl2-invariant case, i.e. when ξ = 0.The advantage of this approach is that the cumbersome calculation involvingstrings of commutators between loop algebra generators becomes unnecessary.As opposed to the sl2-invariant case, the Bethe vectors ΨM (µ1, . . . , µM ) for thedeformed Gaudin model are not orthogonal for different M ’s.
The article is organized as follows. In section 2 we study the deformed
Gaudin model based on the r-matrix (1.5), emphasising the operators B(k)M and
their properties. Using some properties of the creation operators BM , namelythe commutation relations with the generating function of integrals of motiont(λ), in section 3 the spectrum and the Bethe vectors of the system are given.The dual creation operators are introduced in section 4 in order to obtain theexpressions for the inner products and the norms of Bethe states. The results of
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
the study are briefly summarised in the conclusions. The proofs of the lemmasare given in the appendix.
2 Algebraic Bethe Ansatz for deformed Gaudin
model
The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanianterm depending on the spectral parameters
rξ(λ, µ) =c⊗2
λ− µ+ ξp(λ, µ) =
1
λ− µ
(
h⊗ h+ 2(X+ ⊗X− +X− ⊗X+)
)
+ ξ
(
h⊗ (µX+)− (λX+)⊗ h
)
,
(2.1)
where (h,X±) are the standard sl2 generators
[h,X+] = 2X+, [X+, X−] = h, [h,X−] = −2X−, (2.2)
is considered. The matrix form of rξ(λ, µ) in the fundamental representation ofsl2 is given explicitly by [30]
rξ(λ, µ) =
1
λ− µµξ −λξ 0
0 −1
λ− µ
2
λ− µλξ
02
λ− µ−
1
λ− µ−µξ
0 0 01
λ− µ
, (2.3)
here λ, µ ∈ C are the so-called spectral parameters and ξ ∈ C is a deformationparameter. The next step in the Quantum Inverse Scattering Method [3]–[8] itis to introduce the L-operator. In the case of the Gaudin model the entries ofthe L-operator [9]
L(λ) =
(
h(λ) 2X−(λ)
2X+(λ) −h(λ)
)
(2.4)
are generators (h(λ), X±(λ)) of the corresponding loop algebra. The quasi-classical limit of the RTT-relations of the QISM is the matrix form of the loopalgebra relations, the so-called Sklyanin linear bracket [9]
[
L1(λ), L
2(µ)]
= −[
rξ(λ, µ), L1(λ) + L
2(µ)]
. (2.5)
Both sides of this relation have the usual commutators of the 4 × 4 matricesL1(λ) = L(λ) ⊗ 1, L
2(λ) = 1 ⊗ L(λ) and rξ(λ, µ), where 1 is the 2 × 2 identity
matrix.
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
The relation (2.5) is a compact matrix form of the corresponding commuta-tion relations between the loop algebra generators, explicitly given below
[h(λ), h(µ)] = 2ξ(
λX+(λ)− µX+(µ))
[
X−(λ), X−(µ)]
= −ξ(
µX−(λ) − λX−(µ))
,
[
X+(λ), X−(µ)]
= −h(λ)− h(µ)
λ− µ+ ξµX+(λ),
[
X+(λ), X+(µ)]
= 0, (2.6)
[
h(λ), X−(µ)]
= 2X−(λ) −X−(µ)
λ− µ+ ξµh(µ),
[
h(λ), X+(µ)]
= −2X+(λ)−X+(µ)
λ− µ.
Notice that by setting ξ = 0 the relations above define the loop algebra sl2[λ].In order to define a dynamical system besides the algebra of observables a
Hamiltonian should be specified. Due to the Sklyanin linear bracket (2.5) theelements
t(λ) =1
2TrL2(λ) = h2(λ) + 2(X+(λ)X−(λ) +X−(λ)X+(λ))
= h2(λ) − 2h′(λ) + 2(
2X−(λ) + ξλ)
X+(λ) (2.7)
generate an abelian subalgebra [9]
t(λ)t(µ) = t(µ)t(λ). (2.8)
One way to show (2.8) is to notice that from (2.5) and (2.7) follows that thecommutation relation between t(λ) and L(µ) can be written in the form [16, 28]
[t(λ), L(µ)] = [M(λ, µ), L(µ)] . (2.9)
Using (2.3-5) and (2.7) it is straightforward to calculate explicitly M(λ, µ)
M(λ, µ) = −Tr1
(
rξ(λ, µ)L1(λ))
−1
2Tr1
(
r2ξ (λ, µ))
=
−2h(λ)
λ− µ+ 2ξλX+(λ) −4
X−(λ)
λ− µ− 2ξµh(λ)
−4X+(λ)
λ− µ2h(λ)
λ− µ− 2ξλX+(λ)
−3
(λ − µ)21.
(2.10)
Thus the commutators between the generating function t(λ) and the generatorsof the loop algebra follow from (2.9-10)
[
t(λ), X+(µ)]
= 4X+(µ)h(λ)−X+(λ)h(µ)
λ− µ− 4 ξλX+(λ)X+(µ), (2.11)
[t(λ), h(µ)] = 8X−(µ)X+(λ)−X−(λ)X+(µ)
λ− µ− 4 ξ(1 + µh(λ))X+(µ), (2.12)
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
[
t(λ), X−(µ)]
= 4X−(λ)h(µ)−X−(µ)h(λ)
λ− µ+ 2ξ(1 + µh(λ))h(µ)
+ 4ξλX−(µ)X+(λ) + 2(ξµ)2X+(µ).
(2.13)
Preserving some generality, the representation space H of the loop algebra(2.5-6) can be a highest spin ρ(λ) representation with the highest spin vectorΩ+
X+(λ)Ω+ = 0, h(λ)Ω+ = ρ(λ)Ω+. (2.14)
The spectrum and the eigenstates of t(λ) can be studied in this general set-ting. In order to have a physical interpretation a local realization for which therepresentation space
H = V1 ⊗ · · · ⊗ VN (2.15)
is a tensor product of sl2-modules is used. Thus the L-operator is given by
L(λ) =
N∑
a=1
L0a(λ, za) =
N∑
a=1
(
1
λ− za
(
ha 2X−a
2X+a −ha
)
+ ξ
(
zaX+a −λha
0 −zaX+a
))
,
(2.16)where ha = πa(h) ∈ End(Va), X±
a = πa(X±) ∈ End(Va) and πa is an sl2-
representation whose representation space is Va, at a site a. For convenience,the generators Y (λ) = (h(λ), X±(λ)) of the loop algebra are given explicitly
h(λ) =N∑
a=1
(
ha
λ− za+ ξzaX
+a
)
,
X−(λ) =
N∑
a=1
(
X−a
λ− za−
ξ
2λha
)
, X+(λ) =
N∑
a=1
X+a
λ− za. (2.17)
In the case when ξ = 0 the system admits an sl2-symmetry algebra, theso-called global sl2 algebra whose generators are
Ygl =
N∑
a=1
Ya, (2.18)
with Y = (h,X±). Although the global sl2 algebra is not a symmetry algebraof the system when ξ 6= 0, its generators will be used below.
A representation of the model with the above Gaudin realization is obtained
by considering at each site a an sl2 irreducible representation π(ℓa)a , with rep-
resentation space V(ℓa)a , with highest weight ℓa corresponding to the highest
weight vector ωa ∈ V(ℓa)a
X+a ωa = 0 and haωa = ℓa ωa.
Thus the representation space H (2.15) of the loop algebra (2.5-6) is
H = V(ℓ1)1 ⊗ · · · ⊗ V
(ℓN )N , (2.19)
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
with the highest spin vector (2.14)
Ω+ = ω1 ⊗ · · · ⊗ ωN (2.20)
and the corresponding highest spin
ρ(λ) =
N∑
a=1
ℓa
λ− za. (2.21)
For the r-matrix (2.1) the Gaudin Hamiltonians are
H(a) =N∑
b6=a
rab(za, zb) =N∑
b6=a
(
c⊗2 (a, b)
za − zb+ ξ
(
zbhaX+b − zaX
+a hb
)
)
, (2.22)
where c⊗2 (a, b) = ha⊗hb+2(X+
a ⊗X−b +X−
a ⊗X+b ), and they can be obtained as
the residues of the generating function t(λ) at the points λ = za, a = 1, . . . , Nusing the expansion
t(λ) =1
2TrL2(λ) =
N∑
a=1
(
c2(a)
(λ− za)2+
2H(a)
λ− za
)
+2ξ(1−hgl)X+gl+ξ2
N∑
a,b=1
zazbX+a X+
b ,
(2.23)here c2(a) = h2
a +2ha +4X−a X+
a is the sl2 Casimir at site a. As it follows from(1.6), both the Gaudin Hamiltonians H(a) (2.22) and the generating functiont(λ) (2.23) are non-hermitian operators.
The first step in the Algebraic Bethe Ansatz is to define appropriate cre-ation operators that yield the Bethe states and consequently the spectrum ofthe generating function t(λ). The creation operators used in the sl2-invariantGaudin model coincide with one of the L-matrix entry [2, 9]. However, in thepresent case these operators are non-homogeneous polynomials of the generatorX−(λ). It is convenient to define a more general set of operators in order tosimplify the presentation.
Definition 2.1. Given integers M and k ≥ 0, let µ = µ1, . . . , µM be a set ofcomplex scalars. Define the operators
B(k)M (µ) =
(
X−(µ1) + kξµ1
) (
X−(µ2) + (k + 1)ξµ2
)
· · ·(
X−(µM ) + (M + k − 1)ξµM
)
=
M+k−1∏
n=k→
(
X−(µn−k+1) + n ξµn−k+1
)
, (2.24)
with B(k)0 = 1 and B
(k)M = 0 for M < 0.
The following lemma describes some properties of the B(k)M (µ) operators
which will be used later on.
Lemma 2.1. Some useful properties of B(k)M (µ1, . . . , µM ) operators are:
i) B(k)M (µ1, . . . , µM ) = B(k)
m (µ1, . . . , µm)B(k+m)M−m (µm+1, . . . , µM ), m = 1, . . . ,M
= B(k)1 (µ1)B
(k+1)M−1 (µ2, . . . , µM )
= B(k)M−1(µ1, . . . , µM−1)B
(M+k−1)1 (µM );
(2.25)
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
ii) The operators B(k)M (µ1, . . . , µM ) are symmetric functions of their argu-
ments;
iii) B(k)M (µ) = B
(k−1)M (µ) + ξ
∑Mi=1 µiB
(k)M−1(µ
(i)).
The notation used above is the following. Let µ = µ1, . . . , µM be a set ofcomplex scalars, then
µ(i1,...,ik) = µ \ µi1 , . . . , µik
for any distinct i1, . . . , ik ∈ 1, . . . ,M.The Algebraic Bethe Ansatz requires the commutation relations between
the generators of the loop algebra and the B(k)M (µ) operators. To this end the
notation for the Bethe operators are introduced
βM (λ;µ) = h(λ) +∑
µ∈µ
2
µ− λ, (2.26)
where µ = µ1, . . . , µM−1 and λ ∈ C \ µ. In the particular case M = 1,
β1(λ; ∅) = h(λ) and will be denoted by β1(λ). The required commutators aregiven in the following lemma.
Lemma 2.2. The commutation relations between the generators h(λ), X±(λ)
and the B(k)M (µ1, . . . , µM ) operators are given by
h(λ)B(k)M (µ) = B
(k)M (µ)h(λ) + 2
M∑
i=1
B(k)M (λ ∪ µ(i))−B
(k)M (µ)
λ− µi
+ ξ
M∑
i=1
B(k+1)M−1 (µ(i))
(
µiβM (µi;µ(i))− 2k
)
; (2.27)
X+(λ)B(k)M (µ) = B
(k)M (µ)X+(λ)− 2
M∑
i,j=1i<j
B(k+1)M−1 (λ ∪ µ(i,j))
(λ− µi)(λ− µj)
−M∑
i=1
B(k+1)M−1 (µ(i))
(
βM (λ;µ(i))− βM (µi;µ(i))
λ− µi
− ξµiX+(λ)
)
;
(2.28)
X−(λ)B(k)M (µ) = B
(k)M (µ)
(
X−(λ) +Mξλ)
− ξ
M∑
i=1
µiB(k)M (λ ∪ µ(i))
= B(k)M+1(λ ∪ µ)− ξ
M∑
i=1
µiB(k)M (λ ∪ µ(i)). (2.29)
It is important to notice that the creation operators that yield the Bethe
states of the system are the operators B(k)M (µ), defined in (2.24), in the case
when k = 0. The creation operators will be denoted by BM (µ) = B(0)M (µ).
These so-called B-operators are essentially based on those proposed by Kulishin [29] and then used in the implementation of the ABA for the Gaudin model
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
based on the r-matrix (1.4) [28]. A recursive relation defining the creationoperators follows from i) of the lemma 2.1
BM (µ) = BM−1(µ(M))
(
X−(µM ) + (M − 1)ξµM
)
. (2.30)
It may be useful to write down explicitly first few B-operators
B0 = 1, B1(µ) = X−(µ), B2(µ1, µ2) = X−(µ1)X−(µ2) + ξµ2X
−(µ1)
B3(µ1, µ2, µ3) = X−(µ1)X−(µ2)X
−(µ3) + 2ξµ3X−(µ1)X
−(µ2)
+ ξµ2X−(µ1)X
−(µ3) + 2ξ2µ2µ3X−(µ1).
As a particular case of lemma 2.2 the commutation relations between loop al-gebra generators h(λ), X±(λ) and the B-operators can be obtained by settingk = 0 in (2.31)-(2.33)
h(λ)BM (µ) = BM (µ)h(λ)
+M∑
i=1
(
2BM (λ ∪ µ(i))−BM (µ)
λ− µi
+ ξµiB(1)M−1(µ
(i))βM (µi;µ(i))
)
,
(2.31)
X+(λ)BM (µ) = BM (µ)X+(λ) − 2
M∑
i,j=1i<j
B(1)M−1(λ ∪ µ(i,j))
(λ− µi)(λ − µj)
−M∑
i=1
B(1)M−1(µ
(i))
(
βM (λ;µ(i))− βM (µi;µ(i))
λ− µi
− ξµiX+(λ)
)
,
(2.32)
X−(λ)BM (µ) = BM (µ)(
X−(λ) +Mξλ)
− ξ
M∑
i=1
µiBM (λ ∪ µ(i)). (2.33)
The main step in the implementation of the ABA is to calculate the com-mutation relations between the creation operators, in this case the B-operators,and the generating function of the integrals of motion t(λ). The complete ex-pression of the required commutator is given in the lemma below and is basedon the results established previously in this section. This will be crucial fordetermining the spectrum of the system and thus constitutes one of the mainresults of the paper.
Lemma 2.3. The commutation relations between the generating function of theintegrals of motion t(λ) and the B-operators are given by
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
t(λ)BM (µ) = BM (µ)
t(λ) −M∑
i=1
4h(λ)
λ− µi
+
M∑
i<j
8
(λ− µi) (λ− µj)+ 4MξλX+(λ)
+ 4
M∑
i=1
BM (λ ∪ µ(i))
λ− µi
βM (µi;µ(i))
+ 2ξ
M∑
i=1
B(1)M−1(µ
(i))(µih(λ) + 1)βM (µi;µ(i))
+ 4ξ
M∑
i,j=1i6=j
µi
B(1)M−1(λ ∪ µ(i,j))−B
(1)M−1(µ
(i))
λ− µj
βM (µi;µ(i))
+ ξ2M∑
i,j=1i6=j
µiB(2)M−2(µ
(i,j))(
µj βM−1(µj ;µ(i,j))− 2
)
βM (µi;µ(i))
+ 2ξ2M∑
i=1
µ2iB
(1)M−1(µ
(i))X+(µi).
(2.34)
Proof. In the case when M = 1 the commutator between the operators t(λ)and B1(µ) = X−(µ) has been calculated already, see (2.13). In this calculationthe matrix M(λ, µ) (2.10) has been used. It is straightforward to check thatsubstituting M = 1 into the equation (2.34) yields (2.13).
The next step is to outline the proof in the case when M > 1. The startingpoint in the calculation of the commutator [t(λ), BM (µ)] has to be the expression(2.7), so that
[t(λ), BM (µ)] = [h(λ), [h(λ), BM (µ)]] + 2 [h(λ), BM (µ)]h(λ)− 2d
dλ[h(λ), BM (µ)]
+ 2(
2X−(λ) + ξλ) [
X+(λ), BM (µ)]
+ 4[
X−(λ), BM (µ)]
X+(λ).
(2.35)
Notice that the terms on the right hand side of the equation above involveonly the commutators between the generators of the loop algebra and the B-operators. Each of them will be calculated separately, but previously an aux-iliary result is established. Using the commutator (2.31) the following auxiliarexpression
[
h(λ), BM (λ ∪ µ(i))]
= 2d
dλBM (λ ∪ µ(i)) + ξλB
(1)M−1(µ
(i))βM (λ;µ(i)) (2.36)
+
M∑
i6=j
(
2X−(λ)B
(1)M−1(λ ∪ µ(i,j))−BM (λ ∪ µ(i))
λ− µj
+ ξµjB(1)M−1(λ ∪ µ(i,j))βM (µj ;λ ∪ µ(i,j))
)
,
is obtained. Then the first term on the right hand side of the equation (2.35)can be calculated using the expression (2.31) for the commutator [h(λ), BM (µ)],
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2 ALGEBRAIC BETHE ANSATZ FOR DEFORMED GAUDIN MODEL
the result above (2.36) and the relations (2.27)
[h(λ), [h(λ), BM (µ)]] = BM (µ)M∑
i<j
8
(λ− µi)(λ − µj)
− 4
M∑
i=1
(
BM (λ ∪ µ(i))−BM (µ)
(λ− µi)2−
ddλ
BM (λ ∪ µ(i))
λ− µi
)
+ 8M∑
i<j
X−(λ)B(1)M−1(λ ∪ µ(i,j))
(λ− µi)(λ − µj)− 4
M∑
i=1
BM (λ ∪ µ(i))
λ− µi
M∑
j 6=i
2
λ− µj
+ 4ξλ
M∑
i<j
B(1)M−1(λ ∪ µ(i,j))
(λ− µi)(λ− µj)+ 2ξ
M∑
i=1
B(1)M−1(µ
(i))λβM (λ;µ(i))− µiβM (µiµ
(i))
λ− µi
+ξ
M∑
i6=j
µi
(
4B
(1)M−1(λ ∪ µ(i,j))−B
(1)M−1(µ
(i))
λ− µj
+ ξB(2)M−2(µ
(i,j))(
µj βM−1(µj ;µ(i,j))− 2
)
)
βM (µi;µ(i))
+ 2ξ2M∑
i=1
µiB(1)M−1(µ
(i))(λX+(λ)− µiX+(µi)). (2.37)
The commutator [h(λ), BM (µ)], the equation (2.31), is also used to determinethe next two terms on the right hand side of the equation (2.35)
2 [h(λ), BM (µ)]h(λ) = BM (µ)
M∑
i=1
−4 h(λ)
λ− µi
+ 4
M∑
i=1
BM (λ ∪ µ(i))
λ− µi
h(λ)
+ 2ξ
M∑
i=1
µiB(1)M−1(µ
(i))h(λ)βM (µi;µ(i))− 4ξ2
M∑
i=1
µiB(1)M−1(µ
(i))(λX+(λ) − µiX+(µi)),
(2.38)
and after diferentiating (2.31)
− 2d
dλ[h(λ), BM (µ)] = 4
M∑
i=1
BM (λ ∪ µ(i))−BM (µ)
(λ− µi)2−
ddλ
BM (λ ∪ µ(i))
λ− µi
.
(2.39)The forth term on the right hand side of the equation (2.35) can be obtaineddirectly using the expression for the commutator [X+(λ), BM (µ)] given in (2.32)
2(
2X−(λ) + ξλ) [
X+(λ), BM (µ)]
= −4M∑
i=1
BM (λ ∪ µ(i))βM (λ;µ(i))− βM (µi;µ
(i))
λ− µi
− 8
M∑
i<j
X−(λ)B(1)M−1(λ ∪ µ(i,j))
(λ− µi)(λ− µj)− 2ξλ
M∑
i=1
B(1)M−1(µ
(i))βM (λ;µ(i))− βM (µi;µ
(i))
λ− µi
− 4ξλ
M∑
i<j
B(1)M−1(λ ∪ µ(i,j))
(λ− µi)(λ− µj)+ 4ξ
M∑
i=1
µiBM (λ ∪ µ(i))X+(λ)
+ 2ξ2λ
M∑
i=1
µiB(1)M−1(µ
(i))X+(λ). (2.40)
-11-
3 SPECTRUM AND BETHE VECTORS OF THE MODEL
Finally, the last term on the right hand side of the equation (2.35) is determinedusing the equation (2.33)
4[
X−(λ), BM (µ)]
X+(λ) = 4M ξλBM (µ)X+(λ)−4 ξ
M∑
i=1
µiBM (λ∪µ(i))X+(λ).
(2.41)The last step in to proof of the equation (2.34) is a straightforward addition
of all the necessary terms (2.37)-(2.41) calculated above.
Computationally the equation (2.34) is the crucial step of the ABA. In thenext section this result will be used to determine the spectrum of the system.
At the very end of this section a property of the B-operators relevant forthe calculation of the inner products and the norms of the Bethe states is es-tablished. Namely, using the realization (2.17), the relation between the B-operators, given by the recurrent relation (2.30), and the corresponding onesfor the sl2-invariant Gaudin model, the case when ξ = 0, is obtained
BM (µ) =
M−1∑
k=0
ξkM∑
j1<···<jM−k
BM−k(µj1 , . . . , µjM−k)0 p
(M−k)k
(
µ(j1,...,jM−k))
+ξM pM (µ)
(2.42)where
BM−k(µj1 , . . . , µjM−k)0 = BM−k(µj1 , . . . , µjM−k
)|ξ=0
and the operators p(j)i (µ1, . . . , µi) are given by
p(j)i (µ1, . . . , µi) = µ1(−
hgl
2+ j) · · ·µi(−
hgl
2+ i + j − 1), (2.43)
with pi(µ1, . . . , µi) = p(0)i (µ1, . . . , µi) and p
(i)0 = 1.
In the next section it will be shown how the full implementation of theAlgebraic Bethe Ansatz is based on the properties of the creation operatorsBM (µ) obtained here.
3 Spectrum and Bethe vectors of the model
In this section the spectrum and the Bethe vectors of the Gaudin model based onthe r-matrix (1.5) will be determined by the ABA. The first step of the ABA isthe observation that the the highest spin vector Ω+ (2.20) is an eigenvector of thegenerating function of integrals of motion t(λ), due to (2.7), (2.14) and (2.21).Then Bethe vectors ΨM (µ) are given by the action of the B-operators on thehighest spin vector ΨM (µ) = BM (µ)Ω+. Finally the spectrum of the systemis obtained as a consequence of the commutation relations between t(λ) andBM (µ) lemma 2.3, (2.34). The unwanted terms coming from the commutatorare annihilated by the Bethe equations on the parameters µ = µ1, . . . , µMas well as the condition X+(λ)Ω+ = 0. Thus the implementation of AlgebraicBethe Ansatz in this case can be resumed in the following theorem.
Theorem 3.1. The highest spin vector Ω+ (2.20) is an eigenvector of the op-erator t(λ)
t(λ)Ω+ = Λ0(λ)Ω+ (3.1)
-12-
3 SPECTRUM AND BETHE VECTORS OF THE MODEL
with the corresponding eigenvalue
Λ0(λ) = ρ2(λ)− 2ρ′(λ) =
N∑
a=1
2
λ− za
N∑
b6=a
ℓaℓb
za − zb
+
N∑
a=1
ℓa(ℓa + 2)
(λ− za)2. (3.2)
Furthermore, the action of the B-operators on the highest spin vector Ω+ yieldsthe Bethe vectors
ΨM (µ) = BM (µ)Ω+, (3.3)
so thatt(λ)ΨM (µ) = ΛM (λ;µ)ΨM (µ), (3.4)
with the eigenvalues
ΛM (λ;µ) = ρ2M (λ;µ)− 2∂ρM
∂λ(λ;µ) and ρM (λ;µ) = ρ(λ) −
M∑
i=1
2
λ− µi
,
(3.5)provided that the Bethe equations are imposed on the parameters µ = µ1, . . . , µM
ρM (µi;µ(i)) =
N∑
a=1
ℓa
µi − za−
M∑
j 6=i
2
µi − µj
= 0, i = 1, . . . ,M. (3.6)
Proof. The first step in the proof of the Theorem 3.1 is to show the highestspin vector Ω+ (2.20) is an eigenvector of the generating function of integrals ofmotion t(λ). Using (2.7), (2.14) and (2.7) one gets
t(λ)Ω+ =(
h2(λ)− 2h′(λ) + 2(
2X−(λ) + ξλ)
X+(λ))
Ω+
= (h2(λ)− 2h′(λ))Ω+ = Λ0(λ)Ω+
with the eigenvalue Λ0(λ) = (ρ2(λ) − 2ρ′(λ)) and the function ρ(λ) is given inthe equation (2.21). The next step is an important observation that the actionof the operator t(λ) on the Bethe vectors ΨM (µ) (3.3) is given by
t(λ)ΨM (µ) = t(λ)BM (µ)Ω+ = Λ0(λ)ΨM (µ) + [t(λ), BM (µ)] Ω+.
From the equation above it is evident that the main step in the calculation ofthe action of the generating function of the integrals of motion on the Bethevectors is the commutator between the operator t(λ) and the creation operatorsBM (µ). From the Lemma 2.3 follows
t(λ)ΨM (µ)=
Λ0(λ) −M∑
i=1
4ρ(λ)
λ− µi
+
M∑
i<j
8
(λ − µi)(λ − µj)
ΨM (µ)
+ 4
M∑
i=1
ΨM (λ ∪ µ(i))
λ− µi
ρM (µi;µ(i)) + 2ξ
M∑
i=1
ρM (µi;µ(i))×
×
(1 + µiρ(λ))Ψ(1)M−1(µ
(i)) + 2µi
M∑
i6=j
Ψ(1)M−1(λ ∪ µ(i,j))−Ψ
(1)M−1(µ
(i))
λ− µj
+ ξ2M∑
i=1
µi
M∑
j 6=i
(
µjρM−1(µj ;µ(i,j))− 2
)
Ψ(2)M−2(µ
(i,j))
ρM (µi;µ(i)).
(3.7)
-13-
3 SPECTRUM AND BETHE VECTORS OF THE MODEL
The unwanted terms in (3.7) are all proportional to the function ρM (µi;µ(i))
and thus vanish when the Bethe equations ρM (µi;µ(i)) = 0, (3.6) are imposed
on the parameters µ = µ1, . . . , µM. Then the equation (3.7) has the requiredform (3.4) with the eigenvalues
ΛM (λ;µ) = Λ0(λ)−M∑
i=1
4ρ(λ)
λ− µi
+
M∑
i<j
8
(λ− µi)(λ− µj)= ρ2M (λ;µ)−2
∂ρM
∂λ(λ;µ).
(3.8)
It is of interest to notice that the spectrum of the system (3.5) and the Betheequations (3.6) are the same as in the sl2-invariant case.
Remark 3.1. The fact that the spectrum of the system (3.5) is the same as thespectrum of the sl2-invariant Gaudin model follows from the relation betweenthe generating functions of the integrals of motion of the two systems
t(λ) = t(λ)0 + 2ξ(
h(λ)0X+gl +X+
gl − λhglX+(λ)
)
+ ξ2(X+gl)
2, (3.9)
where X+gl =
∑Na=1 zaX
+a , h(λ)0 = h(λ)|ξ=0 and t(λ)0 = t(λ)|ξ=0 is the gener-
ating function of the integrals of motion in the sl2-invariant case.
To obtain the spectrum of the Gaudin Hamiltonians H(a) (2.22) one uses thepole expansion (2.23) of the generating function t(λ) and the previous theorem.
Corollary 3.1. The Bethe vectors ΨM (µ) (3.3) are eigenvectors of the GaudinHamiltonians (2.22)
H(a)ΨM (µ) = E(a)M ΨM (µ), (3.10)
with the corresponding eigenvalues
E(a)M =
N∑
b6=a
ℓaℓb
za − zb−
M∑
i=1
2ℓaza − µi
. (3.11)
Although from the corollary above it follows that, under some conditions,the spectra of the Gaudin Hamiltonians H(a) are real, these operators are nothermitian. It is straightforward to check that, once all the parameters of thesystem are real, the corresponding dual operators are
(H(a))∗ =
N∑
b6=a
(
c⊗2 (a, b)
za − zb+ ξ(zbhaX
−b − zahbX
−a )
)
. (3.12)
Analogously, the generating functions of the integrals of motion t(λ) is also nothermitian
(t(λ))∗ = t(λ)0 + 2ξ
(
X−glh(λ)0 +X−
gl − λX−(λ)hgl −ξλ
2
2h2gl
)
+ ξ2(X−gl)
2,
(3.13)
where λ is the complex conjugate of λ and X−gl =
∑N
a=1 zaX−a . The dual creation
operators will be discussed in the next section where the inner product of theBethe vectors will be studied.
-14-
4 INNER PRODUCT AND NORM OF THE BETHE VECTORS
4 Inner product and norm of the Bethe vectors
Correlation functions of the XXX and XXZ spin chains have been extensivelystudied [18, 6]. Using the Gauss factorisation Sklyanin calculated explicitlya large class of correlators, including the inner product and the norm of theBethe vectors, of the sl2-invariant Gaudin model [11]. In [28] the inner productof the Bethe vectors of the Gaudin model based on the r-matrix (1.4) have beencalculated through a relation with the corresponding ones in the sl2-invariantcase. The main advantage of this approach is that the cumbersome calcula-tion involving strings of commutators between loop algebra generators becomesunnecessary. For the system under study a similar approach is developed below.
Throughout this section it is assumed that the local parameters za are realnumbers, i.e. za = za for all a = 1, . . . , N . The Hilbert structure in therepresentation space H (2.19) is chosen so that the following duality relationsfor the local operators (X+
a )∗ = X−a , (X−
a )∗ = X+a and h∗
a = ha are valid.Therefore the dual operators of the loop algebra generators are given by
(h(λ))∗ =
N∑
a=1
(
ha
λ− za+ ξzaX
−a
)
, (4.1)
(X+(λ))∗ =
N∑
a=1
X−a
λ− za, (4.2)
(X−(λ))∗ =
N∑
a=1
(
X+a
λ− za−
ξλ
2ha
)
. (4.3)
It is straightforward to see that the dual creation operators are defined asthe following ordered products
B∗M (λ1, . . . , λM ) =
(
(X−)∗(λM )+(M−1)ξλM
)
· · ·
(
(X−)∗(λ2)+ξλ2
)
(X−)∗(λ1),
(4.4)where
(X−)∗(λ) =(
X−(λ))∗
. (4.5)
For example
B∗1(λ) = X+(λ)−
ξλ
2hgl, (4.6)
and similarly
B∗2 (λ1, λ2) = X+(λ1)X
+(λ2) + ξλ2
(
−hgl
2+ 1
)
X+(λ1)
+ ξλ1
(
−hgl
2+ 1
)
X+(λ2) + ξ 2λ1λ2
(
−hgl
2
)(
−hgl
2+ 1
)
.
(4.7)
In other words the creation operators defined above in the equation (4.4) cor-respond to the Gaudin model based on the dual r-matrix (1.6).
In order to establish a relation between the inner products of the Bethevectors (3.3) and the corresponding ones in the sl2-invariant case it is of interest
-15-
4 INNER PRODUCT AND NORM OF THE BETHE VECTORS
to consider the property (2.42) of the B-operators, established at the end of thesection 2, in its dual form
B∗M (λ) =
M−1∑
k=0
ξ k
M∑
i1<···<iM−k
p(M−k)k
(
λ(i1,...,iM−k))
B∗M−k(λi1 , . . . , λiM−k
)0+ξM pM (λ)
(4.8)where
B∗N (λ1, . . . , λN )0 = B∗
N (λ1, . . . , λN )|ξ=0 = X+(λ1)X+(λ2) · · ·X
+(λN ). (4.9)
Notice that the dual creation operators (4.4) do not annihilate the highestspin vector Ω+ (2.20)
B∗M (λ1, . . . , λM )Ω+ = ξM pM (λ)Ω+ = ξMpM (λ1, . . . , λM )Ω+, (4.10)
here pM (λ1, . . . , λM ) = λ1 · · ·λM pM with
pM =(
−ρgl
2
)(
−ρgl
2+ 1)
· · ·(
−ρgl
2+M − 1
)
and ρgl =
N∑
a=1
ℓa. (4.11)
The following lemma yields an explicit relation between the inner productsof the Bethe vectors of the deformed Gaudin model and the corresponding onesin the sl2-invariant case. Notice that the property of self-conjugacy of solutionsto Bethe equations [31] is used below.
Lemma 4.1. Let M1,M2 be two natural numbers and λ = λ1, . . . , λM1, µ =µ1, . . . , µM2 be sets of complex scalars, closed under conjugation, i.e. λ = λ
and µ = µ. Then the inner product between the Bethe vectors (3.3) is given by
〈ΨM1(λ)|ΨM2(µ)〉 = ξM1ξM2 pM1(λ)pM2(µ)+
M1−1∑
k1=0
M2−1∑
k2=0
ξ k1ξ k2⟨
φM1, k1(λ)|φM2, k2(µ)⟩
,
(4.12)where
φM,k(λ) =M∑
16i1<···<iM−k
p(M−k)k
(
λ(i1,...,iM−k))
ΨM−k
(
λi1 , . . . , λiM−k
)
0,
(4.13)with ΨM (·)0 = ΨM (·)|ξ=0 and
p(j)i (λ1, . . . , λi)Ω+ = p
(j)i (λ1, . . . , λi)Ω+
=(
λ1(−ρgl
2+ j) · · ·λi(−
ρgl
2+ i+ j − 1)
)
Ω+.(4.14)
Proof. Let ΨM1(λ) and ΨM2(µ) be two Bethe vectors. Then,
〈ΨM1(λ)|ΨM2(µ)〉 = 〈BM1(λ)Ω+ |BM2(µ)Ω+〉 = 〈Ω+ |B∗M1
(λ)BM2(µ)Ω+〉.(4.15)
-16-
4 INNER PRODUCT AND NORM OF THE BETHE VECTORS
From the eqns. (2.42) and (4.8) follows
B∗M1
(λ)BM2(µ) = ξM1 pM1(λ)BM2 (µ) + ξM2 B∗M1
(λ)pM2(µ)− ξM1ξM2 pM1(λ)pM2(µ)
+
M1−1∑
k1=0
M2−1∑
k2=0
ξ k1ξ k2
M1∑
i1<···<iM1−k1
M2∑
j1<···<jM2−k2
p(M1−k1)k1
(
λ(i1,...,iM1−k1)))
×
×B∗M1−k1
(λi1 , . . . , λiM1−k1)0 BM2−k2(µj1 , . . . , µjM2−k2
)0 p(M2−k2)k2
(
µ(j1,...,jM2−k2))
.
(4.16)
Replacing (4.16) into (4.15), and taking into account the action of B∗M1
(λ) andBM2(µ) on the highest spin vector Ω+
〈ΨM1(λ)|ΨM2(µ)〉 = ξM1ξM2 pM1(λ)pM2(µ) +
M1−1∑
k1=0
M2−1∑
k2=0
ξ k1ξ k2 ×
×M1∑
i1<···<iM1−k1
M2∑
j1<···<jM2−k2
p(M1−k1)k1
(
λ(i1,...,iM1−k1)))
p(M2−k2)k2
(
µ(j1,...,jM2−k2))
×
×
⟨
ΨM1−k1(λi1 , . . . , λiM1−k1)0
∣
∣
∣
∣
ΨM2−k2(µj1 , . . . , µjM2−k2)0
⟩
.
(4.17)
Given that the inner product 〈·|·〉 is a bilinear and Hermitian form and also thatthe set λ is closed to conjugation, the right hand side of (4.17) can be expressedas in (4.12).
As an illustration of the formula (4.12) of the previous lemma two particularcases are given below. In the simplest cases M1 = M2 = 1 the equation (4.12)reads
〈Ψ1(λ)|Ψ1(µ)〉 = 〈Ψ1(λ)0|Ψ1(µ)0〉+ |ξ|2 (p1)2λµ. (4.18)
Also, in the case when M1 = M2 = 2 the expression (4.12) gives
〈Ψ2(λ1, λ2)|Ψ2(µ1, µ2)〉 = 〈Ψ2(λ1, λ2)0|Ψ2(µ1, µ2)0〉
+ |ξ|2(p(1)1 )2 〈λ2Ψ1(λ1)0 + λ1Ψ1(λ2)0|µ2Ψ1(µ1)0 + µ1Ψ1(µ2)0〉
+ |ξ|4 (p2)2λ1λ2µ1µ2.
(4.19)
The main result of this section is the equation (4.12) of the lemma 4.1 wherethe inner products of the Bethe vectors of the deformed Gaudin model are ex-pressed in terms of the sl2-invariant inner products calculated by Sklyanin [11].A substantial advantage of this approach is that the result was obtained withoutdoing the cumbersome calculation involving the commutators (2.6) between loopalgebra generators. As it can be seen in the examples given above (4.18-19), theright hand sides involve only the sl2-invariant inner products 〈Ψ1(λ)0|Ψ1(µ)0〉and 〈Ψ2(λ1, λ2)0|Ψ2(µ1, µ2)0〉, which were given explicitly in [11].
-17-
6 ACKNOWLEDGMENTS
The lemma 4.1 yields an expression of the norm of the Bethe vectors of themodel under study.
Corollary 4.1. The norms of Bethe vectors ΨM (µ) are given by
‖ΨM (µ)‖2 = |ξ|2M |µ|2(pM )2 +
M−1∑
k=0
|ξ|2k‖φM,k(µ)‖2
(4.20)
where |µ|2= |µ1|2· · · |µM |2 and φM,k(µ) are defined in (4.13).
The corollary above reveals the full strength of our approach in the factthat the expression of the norm of the Bethe vectors again involves only thesl2-invariant inner products given in [11].
5 Conclusions
The deformed Gaudin model based on the sl2-invariant r-matrix with a defor-mation depending on the spectral parameters (1.5) is studied. The usual Gaudin
realisation of the model is introduced and the B-operators B(k)M (µ1, . . . , µM ) are
defined as non-homogeneous polynomials of the generator X−(λ) of the corre-sponding loop algebra. These operators are symmetric functions of their argu-ments and they satisfy certain recursive relations with explicit dependency onthe quasi-momenta µ1, . . . , µM . The creation operators BM (µ1, . . . , µM ) whichyield the Bethe vectors of the system form their proper subset. The commuta-tion relations between the B-operators and the generators of the loop algebra aregiven. Moreover, the commutator of the creation operators with the transitionmatrix of the Gaudin model under study is calculated explicitly. Based on theprevious result the spectrum of the system is determined. It turns out that thespectrum of the system and the corresponding Bethe equations coincide withthe ones of the sl2-invariant model. However, contrary to the sl2-invariant case,the generating function of integrals of motion and the corresponding GaudinHamiltonians are not hermitian.
In order to obtain expressions for the inner product and the norm of theBethe vectors the dual creation operators B∗
M (µ1, . . . , µM ) are introduced. Themethod used in the study of the inner product of the Bethe vectors is based onthe relation between the creation operators of the system and the ones in the sl2-invariant case. The advantage of this approach is that the cumbersome calcula-tion involving strings of commutators between loop algebra generators becomesunnecessary. The final expression of the inner products and norms of the Bethestates is given in terms of the corresponding ones of the sl2-invariant Gaudinmodel. As opposed to the sl2-invariant case, the Bethe vectors ΨM (µ1, . . . , µM )for the deformed Gaudin model are not orthogonal for different M ’s.
6 Acknowledgments
We acknowledge stimulating and illuminating discussions with Professor P. P.Kulish. This work was supported by the FCT projects PTDC/MAT/69635/2006and PTDC/MAT/99880/2008.
-18-
A PROOFS OF LEMMAS
Appendix
A Proofs of lemmas
The proofs of the lemmas, not included in the main text, are detailed in thisappendix.
Proof of the lemma 2.1
i) The recurrence relations (2.25) are evident from the expression (2.24) in thedefinition 2.1.ii) Given the natural number k, and according to the definition (2.24) withM = 2
B(k)2 (µ1, µ2) = X−(µ1)X
−(µ2)+(k+1) ξµ2X−(µ1)+k ξµ1X
−(µ2)+k(k+1)ξ2µ1µ2.
(A.1)
Using (2.6), it is straightforward to verify that B(k)2 (µ1, µ2) = B
(k)2 (µ2, µ1).
Moreover, it is clear from (2.25) and (A.1) that
B(k)M (µ1, . . . , µi, µi+1, . . . , µM ) = B
(k)i−1(µ1, . . . , µi−1)B
(k+i−1)2 (µi, µi+1)B
(k+i+1)M−i−1 (µi+2, . . . , µM )
= B(k)M (µ1, . . . , µi+1, µi, . . . , µM ),
therefore B(k)M (µ1, . . . , µM ) is a symmetric function.
iii) Given the natural number k, a direct consequence of the expression (A.1) isthat
B(k)2 (µ1, µ2) = B
(k−1)2 (µ1, µ2) + ξ(µ2B
(k)1 (µ1) + µ1B
(k)1 (µ2)).
If the expression iii) of the lemma 2.1 is assumed to be true for some M ≥ 2,we have, using (2.25),
B(k)M+1(µ) =
(
B(k−1)M (µ(M+1)) + ξ
M∑
i=1
µiB(k)M−1(µ
(i,M+1))
)
(
B(M+k−1)1 (µM+1) + ξµM+1
)
= B(k−1)M+1 (µ) + ξµM+1 B
(k−1)M (µ(M+1)) + ξ
M∑
i=1
µi
(
B(k)M (µ(i)) + ξµM+1 B
(k)M−1(µ
(i,M+1)))
= B(k−1)M+1 (µ) + ξ
M∑
i=1
µiB(k)M (µ(i)) + ξµM+1B
(k)M (µ(M+1)) = B
(k−1)M+1 (µ) + ξ
M+1∑
i=1
µiB(k)M (µ(i)).
Proof of the formula (2.27) of the lemma 2.2.Let k be a natural number. From the definition of the loop algebra (2.6) andthe definition 2.1 it is immediately clear that
[
h(λ), B(k)1 (µ)
]
=[
h(λ), X−(µ)]
= 2B
(k)1 (λ) −B
(k)1 (µ)
λ− µ+ ξ
(
β1(µ)− 2k)
,
(A.2)Assume that the expression (2.27) of the lemma 2.2 is true for some M ≥ 1
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A PROOFS OF LEMMAS
then[
h(λ), B(k)M+1(µ)
]
=[
h(λ), B(k)M (µ(M+1))
]
B(M+k)1 (µM+1) +B
(k)M (µ(M+1))[h(λ), B
(M+k)1 (µM+1)]
=
M∑
i=1
(
2B
(k)M+1(λ ∪ µ(i))−B
(k)M+1(µ)
λ− µi
+ ξB(k+1)M (µ(i))
(
µiβM (µi;µ(i,M+1))− 2k
)
)
+ ξ
M∑
i=1
µiB(k+1)M−1 (µ(i,M+1))[h(µi), B
(M+k)1 (µM+1)] +B
(k)M (µ(M+1))[h(λ), B
(M+k)1 (µM+1)]
= 2
M+1∑
i=1
B(k)M+1(λ ∪ µ(i))−B
(k)M+1(µ)
λ− µi
+ ξ
M∑
i=1
B(k+1)M (µ(i))
(
µiβM (µi;µ(i,M+1)) +
2µi
µM+1 − µi
− 2k
)
+ ξB(k+1)M (µ(M+1))
(
−2k +
M∑
i=1
2µM+1
µi − µM+1
)
+ ξ
(
B(k)M (µ(M+1)) + ξ
M∑
i=1
µiB(k+1)M−1 (µ(i,M+1))
)
h(µM+1)
=
M+1∑
i=1
(
2B
(k)M+1(λ ∪ µ(i))−B
(k)M+1(µ)
λ− µi
+ ξB(k+1)M (µ(i))
(
µiβM+1(µi;µ(i))− 2k
)
)
,
where the equation (A.2) and the relation (2.25) of the lemma 2.1, are usedwhen appropriate.Proof of the formula (2.28) of the lemma 2.2.Let k be a natural number. From the definition 2.1 and the commutators (2.6)it is clear that[
X+(λ), B(k)1 (µ)
]
=[
X+(λ), X−(µ)]
= −β1(λ) − β1(µ)
λ− µ+ ξµX+(λ), (A.3)
Assume that the expression (2.28) of the lemma 2.2 is true for some M ≥ 1then[
X+(λ), B(k)M+1(µ)
]
=[
X+(λ), B(k)M (µ(M+1))
]
B(M+k)1 (µM+1) +B
(k)M (µ(M+1))
×[
X+(λ), B(M+k)1 (µM+1)
]
= −M∑
i=1
B(k+1)M (µ(i))
βM (λ;µ(i,M+1))− βM (µi;µ(i,M+1))
λ− µi
− 2
M∑
i<j
B(k+1)M (λ ∪ µ(i,j))
(λ− µi)(λ− µj)+ ξ
M∑
i=1
µiB(k+1)M (µ(i))X+(λ)
+
(
B(k)M (µ(M+1)) + ξ
M∑
i=1
µiB(k+1)M−1 (µ(i,M+1))
)
[
X+(λ), B(M+k)1 (µM+1)
]
−M∑
i=1
B(k+1)M−1 (µ(i,M+1))
[
h(λ), B(M+k)1 (µM+1)
]
−[
h(µi), B(M+k)1 (µM+1)
]
λ− µi
= −M∑
i=1
B(k+1)M (µ(i))
βM+1(λ;µ(i))− βM+1(µi;µ
(i))
λ− µi
− 2
M∑
i<j
B(k+1)M (λ ∪ µ(i,j))
(λ− µi)(λ− µj)
−B(k+1)M (µ(M+1))
βM+1(λ;µ(M+1))− βM+1(µM+1;µ
(M+1))
λ− µM+1
+ ξ
M+1∑
i=1
µiB(k+1)M (µ(i))X+(λ)− 2
M∑
i=1
B(k+1)M (λ ∪ µ(i,M+1))
(λ− µi)(λ − µM+1)
-20-
A PROOFS OF LEMMAS
= −M+1∑
i=1
B(k+1)M (µ(i))
(
βM+1(λ;µ(i))− βM+1(µi;µ
(i))
λ− µi
− ξµiX+(λ)
)
− 2M+1∑
i<j
B(k+1)M (λ ∪ µ(i,j))
(λ− µi)(λ− µj)
where the equations (A.2), (A.3) and the relation (2.25) of the lemma 2.1, areused when appropriate.
Proof of the formula (2.29) in the lemma 2.2.Let k be a natural number. From the definition 2.1 and the commutators (2.6)it is clear that
[
X−(λ), B(k)1 (µ)
]
=[
X−(λ), X−(µ)]
= ξ(
λB(k)1 (µ) − µB
(k)1 (λ)
)
. (A.4)
Assum that the expression (2.29) of the lemma 2.2 is true for some M ≥ 1 then
[
X−(λ), B(k)M+1(µ)
]
=[
X−(λ), B(k)M (µ(M+1))
]
B(M+k)1 (µM+1) +B
(k)M (µ(M+1))
×[
X−(λ), B(M+k)1 (µM+1)
]
=
(
MξλB(k)M (µ(M+1))− ξ
M∑
i=1
µiB(k)M (λ ∪ µ(i,M+1))
)
×B(M+k)1 (µM+1) + ξB
(k)M (µ(M+1))(λB
(M+k)1 (µM+1)− µM+1B
(M+k)1 (λ))
= (M + 1)ξλB(k)M+1(µ)− ξ
M+1∑
i=1
µiB(k)M+1(λ ∪ µ(i)),
where the equation (A.4) and the relation (2.25) of the lemma 2.1, are usedwhen appropriate.
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