Highest Weight $U_q[sl(n)]$ Modules and Invariant Integrable n-State Models with Periodic Boundary Conditions"

Abstract

The weights are computed for the Bethe vectors of an RSOS type model with periodic boundary conditions obeying $U_q[sl(n)]$ ($q=\exp(i\pi/r)$) invariance. They are shown to be highest weight vectors. The q-dimensions of the corresponding irreducible representations are obtained. Comment: 5 pages, LaTeX, SFB 288 preprint

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arXiv:hep-th/9406116v1 17 Jun 1994
Highest Weight Uq[sl(n)] Modules
and Invariant Integrable n-State Models
with Periodic Boundary Conditions
A. Zapletal∗†
M. Karowski
Institut f¨ ur Theoretische Physik
Freie Universit¨ at Berlin
Germany
Abstract
The weights are computed for the Bethe vectors of an RSOS type model with
periodic boundary conditions obeying Uq[sl(n)] (q = exp(iπ/r)) invariance. They
are shown to be highest weight vectors. The q-dimensions of the corresponding
irreducible representations are obtained.
In the last years considerable progress has been made on the ”quantum symmetry” of
integrable quantum chain models as the XXZ-Heisenberg model and its generalizations.
In [1] we constructed an slq(n) invariant RSOS type model with periodic boundary con-
ditions. In the present paper we prove for this model the highest weight property of
the Bethe states, calculate the weights and the q-dimensions of the representations and
classify the irreducible ones. For the case of open boundary conditions see e.g. [2], [3] and
[4].
The model of [1] is defined by the transfer matrix τ = τ(n)where
τ(k)(x,x(k)) = trq(T(k)(x,x(k)) =
?
α
qn+1−2α(T(k))α
α(x,x(k)),k = 1,...,n.(1)
The “doubled” monodromy matrix is given by
T(k)
0
(x,x(k)) =˜T(k)
0
·T(k)
0 (x,x(k)) = (R01...R0Nk)·(RNk0(xNk/x(k))...R10(x1/x(k))). (2)
∗Supported by DFG, Sonderforschungsbereich 288 ’Differentialgeometrie und Quantenphysik’
†e-mail: zapletal@omega.physik.fu-berlin.de
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For the purpose of the nested algebraic Bethe ansatz in addition to T (x) = T(n)(x) the
monodromy matrices for all k ≤ n are needed. The slq(k) R-matrix is given by
R(x) = xR−x−1PR−1P,R =
?
α?=β
Eαα⊗Eββ+q
?
α
Eαα⊗Eαα+(q−q−1)
?
α>β
Eαβ⊗Eβα,
(3)
The Yang-Baxter equation reads
R12(y/x)T1(x)R21T2(y) = T2(y)R12T1(x)R21(y/x).(4)
The model is quantum group invariant, i.e. the transfer matrix commutes with the gen-
erators of Uq[sl(n)]. These are obtained from the monodromy matrices T(x) in the limits
x to 0 or ∞ (up to normalizations)
T =







10···
...
...
0
...
αE1
1
...
0
∗αEn−1 1







qW,T∞= q−W







1 −αF1
∗
01
...
... −αFn−1
0
...
01







, (5)
where α = q −q−1and the matrix W = diag{W1,...,Wn} contains the Uq[gl(n)] Cartan
elements. Analogously to eq. (5), we introduce as a limit of T(n)(x,x(n)) for x to 0
T =˜T · T,where
˜T = T−1
∞, (6)
here and in the following operators without argument denote these limits for x to 0.
We write the doubled monodromy matrices as k × k block-matrices of operators
T(k)(x) =
?
A(k)(x) B(k)(x)
C(k)(x)D(k)(x)
?
.(7)
We also introduce the reference states Φ(k)with C(k)(x)Φ(k)= 0. The eigenstates of
the transfer matrix τ(x) are the Bethe ansatz states Ψ = Ψ(n)obtained by the nested
procedure
Ψ(k)= B(k)
α1(x(k−1)
1
)...B(k)
αNk−1(x(k−1)
Nk−1)Φ(k)Ψ(k−1)
α
,(k = 2,...,n), Ψ(1)= 1.(8)
The sets of parameters x(k)
equations: for j = 1,...,Nkand k = 1,...,n − 1
j
= exp
?
iθ(k)
j
− (n − k)γ/2
?
(q = eiγ) fulfil the Bethe ansatz
q2+wn−k−wn−k+1
?
l=k±1
Nl
?
i=1
sinh1
2
?
?
θ(k)
j
θ(k)
j
− θ(l)
− θ(l)
i − iγ
?
?
sinh1
2
i + iγ
Nk
?
i=1
sinh1
2
?
?
θ(k)
j
θ(k)
j
− θ(k)
− θ(k)
i
+ 2iγ
?
sinh1
2
i
− 2iγ
? = −1(9)
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where, below, the wi = Nn−i+1− Nn−iwill turn out to be the weights of the state Ψ,
i.e. the eigenvalues of the Wi’s defined by eq. (5).
Theorem: The Bethe ansatz states are highest weight states, i.e.
Eiψ = 0(i = 1,...,n − 1).
The analogous statement for the case of open boundary conditions has been proved in [3].
Proof: First we prove Tα
(4) implies
βΨ = 0 and then Tα
βΨ = 0 for α > β. The Yang-Baxter relation
Tα
βBγ(x) = Bγ′(x)Tα′
β′ Rγ′α
γ′′α′Rβ′γ′′
βγ,(for α > 1, else see eq. (12)).(10)
We apply the technique of the nested algebraic Bethe ansatz and commute T through all
the B’s of Ψ in eq. (8)
Tα
βBα1(x1)...BαNn−1(xNn−1)Φ(k)Ψ(n−1)
α
= Bα1(x1)...BαNn−1(xNn−1)Φ(k)?
T(n−1)?α
βΨ(n−1)
α
.
(11)
Iterating this procedure β−1 times we arrive at C(k)
At this stage we use, as usual (see e.g. ref. [5]), the commutation rule
α′ Ψ(k)with k = n−β+1 and α′= α−β.
C(k)
α′ B(k)
γ(x) = q−1Rγ′α′
γα′′B(k)
γ′ (x)C(k)
α′′ + (1 − q−2)
?
(D(k))α′
γA(k)(x) − (D(k))α′
α′′(D(k))α′′
γ(x)
?
(12)
,
to prove that the Bethe ansatz equations (9) imply
C(k)
α′ Ψ(k)= 0(k = 2,...,n)and thereforeTα
βΨ = 0for α > β.(13)
Finally we show Tα
βΨ = 0 for α > β. We have with eqs. (5) and (13) for all β < n
Tn
βΨ =˜Tn
nTn
βΨ = 0(14)
and because˜Tn
previous row, where β < n − 1:
nis an invertible operator it follows that Tn
βΨ = 0. Now we consider the
Tn−1
β
Ψ =
?˜Tn−1
n−1Tn−1
β
+˜Tn−1
n
(T0)n
β
?
Ψ = 0,(15)
and therefrom along with the foregoing result we get (T0)n−1
Tα
weight property of the Bethe vectors.
Next we compute the weights of the Bethe ansatz states Ψ. We consider first
β
Ψ = 0. By iteration we find
βΨ = 0 for α > β and hence from eq. (5) EiΨ = 0 for all i, this proves the highest
Tα
αΨ =˜Tα
αTα
αΨ +
?
β>α
˜Tα
βTβ
αΨ. (16)
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Again shifting Tα
we arrive at A(k)Ψ(k)for k = n − α + 1. The Yang-Baxter relation (4) and eqs. (2) and
(3) imply
A(k)B(k)
αto the right as in eq. (11) we get the operator (T(n−1))α
α. By iteration
β(x) = q−2B(k)
β(x)A(k),A(k)Ψ(k)= q2NkΨ(k)
(17)
and therefore
A(k)Ψ(k)= q2(Nk−Nk−1)Ψ(k).(18)
From eqs. (5) and (6) we have
˜Ti
i= Ti
i= qWi
andTi
i= q2Wi.(19)
and finally
q2WiΨ = q2wiΨwithwi= Nn−i+1− Nn−i.(20)
So any Bethe ansatz solution is characterized by a weight vector
w = (w1,...,wn) = (Nn− Nn−1,...,N2− N1,N1)(21)
with the usual highest weight condition
w1≥ ··· ≥ wn≥ 0.(22)
Here N = N(n)is the number of lattice sites and N(k)(k = n−1,...,1) is the number of
roots in the k-th Bethe ansatz level. The highest weight condition (22) may be shown as
usual. The result (21) is consistent with the “ice rule” fulfilled by the R-matrix (3). This
means that each operator B(k)
The q-dimension of a representation π with representation space V is obtained from
the “Markov trace” (see e.g. ref. [6])
α(x) reduces wkand lifts wαby one.
dimqπ = trV
?
q−?
i>j(Wj−Wi)?
.(23)
As is well known for the case of q being a root of unity the generators Eiand Fibecome
nilpotent
(Ei)r= (Fi)r= 0,q = exp(iπ/r),r = n + 2,n + 3,.... (24)
A highest weight module is equivalent to the corresponding one of sl(n), if this relation
does not concern it. These representations remain still irreducible and will be called good
ones. The other representations are called bad and up to special irreducible cases with
vanishing q-dimension they are reducible but not decomposable.
For the irreducible representations πwwith highest weight vector w eq. (23) gives
dimqπw=
?
α∈Φ+
[(w + g,α)]q
[(g,α)]q
=
?
i>j
[wj− wi+ i − j]q
[i − j]q
, (25)
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where
and g is the Weyl vector g =1
positive q-dimensions. From eq. (25) it follows that their weight patterns are restricted
in their length
w1− wn≤ r − n,
[x]q= (qx− q−x)/(q − q−1) is a q-number, Φ+denotes the set of positive roots
?
2
α∈Φ+α. The good representations are characterized by
(26)
The q-dimensions of bad representations vanish.
It is an interesting question, how these good representations are characterized in the
language of the Bethe ansatz. In ref. [4] it is shown for slq(2) that these are given by
all Bethe ansatz solutions with only positive parity strings (in the language of Takahashi
[7]). In a forthcoming paper we will show how this classification extends to q-symmetries
of higher rank.
References
[1] M. Karowski and A. Zapletal, Nucl. Phys. B (1994) in press.
[2] L. Mezincescu and R.I. Nepomechie, Mod. Phys. Lett. A6 (1991) 2497;
C. Destri and H.J. de Vega, Nucl. Phys. B361 (1992) 361;
A. F¨ orster and M. Karowski, Nucl. Phys. B408 (1993) 512;
H.J. de Vega and A. Gonz´ alez-Ruiz, Nucl. Phys. B417 (1994) 553.
[3] H.J. de Vega and A. Gonz´ alez-Ruiz, ‘Highest Weight Property for the SUq(n) Invari-
ant Spin Chains’, LPTHE-PAR 94/13 (1994).
[4] G. J¨ uttner and M. Karowski, ‘The “Good” Bethe Ansatz Solutions of Quantum
Group Invariant Heisenberg Models’, SFB 288 preprint (1994).
[5] H.J. de Vega, Int. J. Mod. Ph. A 4 (1989) 2063.
[6] N. Yu. Reshetikhin ‘Quantized Univ. Envel. Algebras, YBE and Invariants of Links
(Part I)’, LOMI E-4-87 (1987);
V. Pasquier and H. Saleur, Nucl. Phys. B330 (1990) 523;
N. Reshetikhin and V.G. Turaev, Invent. math. 103 (1991) 547;
J. Fuchs, ‘Affine Lie Algebras and Quantum Groups’, Cambridge Univ. Press (1992).
[7] M. Takahashi and M. Suzuki, Progr. Th. Ph., Vol. 48, No. 6b (1972) 2187.
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