R-matrices for highest weight representations of sl_q(2,C) at roots of unity

9 Pages·0.12 MB·English

by  T. S. Hakobyan

R-matrices for highest weight representations
of sˆl (2,C) at roots of unity
q
3
9
9
1

n T.S. Hakobyan
a
Yerevan State University, Manukyan
J
st. 9, Yerevan, Armenia
8

2
and
v
5
2 A.G. Sedrakyan ∗
1 L
1 Laboratoire de Physique Th´eorique ENS APP †
1
Chemin de Bellevue BP 110, F - 74941 Annecy-le-Vieux Cedex, France
2
9
/
h
t
-
p
e Abstract
h
:
v
i We obtain the general formula for R-matrices of sˆlq(2,C) for the highest
X
weight representations both for general q and for q being a root of unity by gen-
r
a eralizing G.Gomez’s andSierra’s oneforsemiperiodic representations of sˆl (2,C)
q
at roots of unity. In a case of general q we obtain direct matrix form for the well
known Jimbo’s R-matrix. In a case of qN = 1 for semiperiodic and spin j < N−1
2
representations we obtain the new R-matrices which together with Jimbo’s one
obey the spectral parameter dependent Yang-Baxter equations.
L
ENS APP-A-406/92
October 1992
∗permanent address: YerevanPhysics Institute, Br. Alikhanian, st.2, Yerevan36,Armenia
†URA14-36duCNRS,associ´ee`al’E.N.S.deLyonetauL.A.P.P.(IN2P3-CNRS)d’Annecy-
le-Vieux
0
Quantum groups had been introduced by Drinfeld [1] and Jimbo [2]. Shortly
speaking, quantum group is some quasi-triangular Hopf algebra, parametrized by
some parameter q [1]. They play an important role in two-dimensional integrable
statistic models and conformal field theory.
For general q irreducible representations of quantum algebras are in one to
one correspondence with the representations of the classical algebras [2, 15]. For
q being a root of 1 this is not true. Irreducible representations in this case had
been considered in [7, 10, 11]. They are parametrized by some complex numbers.
Due to Hopf algebra structure one can consider an intertwiner on tensor prod-
ucts of two irreducible representations [1, 2]. In case of general q intertwiners
satisfy Yang-Baxter equations [1, 3], that is the integrability condition for the
two dimensional systems.
For q being a root of unity the standard expression of the universal R-matrix
[1] is ill defined because of singularities. In [9] Bazhanov and Stroganov showed
howthechiralPottsmodel[17,18]isrelatedtothetheoryofcyclicrepresentations
of the quantum algebra sˆl (2,C) at qN = 1. Intertwiners between semiperiodic
q
representations ofsl (2,C) hadbeen considered in[4]. In [12, 14] theintertwiners
q
between semiperiodic and spin j ≤ N−1 representations, where N is a minimal
2
integer, satisfying the condition qN = 1, had been considered. In [4] Gomez
and Sierra found an interesting recursive formula for intertwiners of semiperiodic
representations, having a reflection symmetry.
Inthis articlewegeneralize thisformula forthecase ofaffinesˆl (2,C)algebra.
q
It canbeused forthehighest weight representations of ordinarysl (2,C)algebra.
q
In a particular case of general q we obtain well known Jimbo’s R-matrix [2],
written in the matrix form. In case of q being an odd root of 1 by acting our
general formula to tensor product of semiperiodic representation and spin j ≤
N−1 representation, we obtain baxterization of intertwiners, considered in [12].
2
As a consequence we obtain new solutions of spectral parameter dependent Yang-
Baxter equations. We show also that the intertwiners, considered in [4, 5], are
particular cases of ones, obtained in [6] by a different approach.
Let us recall the definition of the affine quantum algebra sˆl (2,C). It is
q
generated by generators e ,f ,h , satisfying the following relations [2]:
i i i
[h ,e ] = a e [h ,f ] = −a f [h ,h ] = 0
i j ij j i j ij j i j
qhi −q−hi
[e ,f ] = δ [h ] := δ (1)
i j ij j q ij q −q−1
2 −2
(Here a = is Cartan matrix of sˆl (2,C), q is a parameter)
ij −2 2 q
!
1−aij 1−a
(−1)ν ij e1−aij−νe eν = 0 (i 6= j)
ν i j i
" #
ν=0 q
X
1
1−1aj 1−a
(−1)ν ij f1−aij−νf fν = 0 (i 6= j) (2)
ν i j i
" #
ν=0 q
X
m [m]!
q
(Here := , [m] ! := [1] [2] ..[m] ).
" n #q [m−n]q![n]q! q q q q
Following [6] we take the extension of sˆl (2,C) by central elements z±1(i =
q i
0,1). There is the structure of Hopf algebra in this algebra:
∆(e ) = e ⊗k−1 +z ·k ⊗e ∆(k±1) = k±1 ⊗k±1
i i i i i i i i i (3)
∆(f ) = f ⊗k−1 +z−1 ·k ⊗f ∆(z±1) = z±1 ⊗z±1,
i i i i i i i i i
where ki := qh2i.
Note, that in [6] another comultiplication had been considered:
∆˜(e ) = e ⊗k−ǫi +z ·qǫi ⊗e ∆˜(qǫi) = qǫi ⊗qǫi
i i i i (4)
∆˜(f ) = f ⊗q−ǫi+1 +z−1 ·q−ǫi+1 ⊗f ∆˜(z±1) = z±1 ⊗z±1
i i i i i i i
where ǫ (i = 0,1) are the orthonormal basis in Cartan subalgebra of sˆl (2,C)
i q
ǫi−ǫi+1
and ki = q 2 ,ǫι+2 = ǫi.
The comultiplication (4) converts into (3) by changing: z ·e ⇒ e ,
i i
z−1 ·fi ⇒ fi, z2zi ⇒ zi, where z := qǫ0+2ǫ1 lies in the center of sˆlq(2,C).
From the representation of finite dimensional quantum algebra the parametri-
zed representation of its affine extension can be constructed. Let’s regognize this
method for sˆl (2,C). If e,f,k are the generators of sˆl (2,C), then there is the
q q
homomorphism ρ : sˆl (2,C) ⇒ sl (2,C) [2]:
x q q
ρ (e ) = x·f ρ (f ) = x−1 ·e ρ (k±1) = k∓1
x 0 x 0 x 0 (5)
ρ (e ) = e ρ (f ) = f ρ (k±1) = k±1
x 1 x 1 x 1
This homomorphism converts representations of sl (2,C) to parametrized repre-
q
sentation of the affine quantum algebra sˆl (2,C) with the central charge equal to
q
zero.
The Hopf algebra structure allows to consider the action of sˆl (2,C) on then-
q
sor products of representations. Let π and π be representations of sl (2,C) on
1 2 q
V andV respectively. Then, asit was mentioned above, π (x ) := π ◦ρ ,i = 1,2
1 2 i i i xi
are the representations of sˆl (2,C). The equivalence of tensor products π (x )⊗
q 1 1
π (x ) and π (x ) ⊗ π (x ) implies a relation between parameters of the repre-
2 2 2 2 1 1
sentations and means the existence of some intertwining operator Rˆ (x ,x )
π1π2 1 2
from V ⊗V into V ⊗V such that [1]):
1 2 2 1
Rˆ (x ,x )π (x )⊗π (x )(∆(g))
π1π2 1 2 1 1 2 2
= π (x )⊗π (x )(∆(g))Rˆ (x ,x ), (6)
2 2 1 1 π1π2 1 2
2
where g ∈ sˆl (2,C). We can take x = 1,x = x because Rˆ depends on x2 only.
q 2 1 x1
Then the equations (6) for g = f ,f can be represented in the following form:
0 1
Rˆ(x)·(f ⊗k−1 +z−1 ·k ⊗f) = (f ⊗k−1 +z−1 ·k ⊗f)·Rˆ(x)
1 1
Rˆ(x)·(xf ⊗k +z ·k−1 ⊗f) = (f ⊗k +xz ·k−1 ⊗f)·Rˆ(x) (7)
0 0
From (7) we obtain:
Rˆ(x)·(1⊗f) = [∆(f )·Rˆ(x)·x(1⊗k2)−∆(e )·Rˆ(x)]
1 0
×(xz−1 ·k ⊗k2 −z ·k−1 ⊗1)−1
1 0
Rˆ(x)·(f ⊗1) = [∆(f )·Rˆ(x)·(z z ·k−2 ⊗1)−∆(e )·Rˆ(x)]
1 0 1 0
×(z z ·k−2 ⊗k−1 −x(1⊗k))−1, (8)
0 1
where for simplicity we use the notations:
∆(f ) := π (1)⊗π (x)(∆(f )),∆(e ) := π (1)⊗π (x)(∆(e )).
i 2 1 i i 2 1 i
If π and π are the highest weight representations with highest vectors v(1)
1 2 c
andv(2) respectively andV ⊗V decomposesintodirect sumofpairwisenonequiv-
c 1 2
alent irreducible representations, then Rˆ(x)(v(1) ⊗v(2)) = α·(v(2) ⊗v(1)), where
c c c c
α ∈ C can be equated to 1. In this case we can use (8) to obtain the recursive
formula for Rˆ(x)(fr1v(1) ⊗fr2v(2)). In fact, using the equalities:
c c
1⊗k2 fr1v(1) ⊗fr2v(2) = λ q−2r2 fr1v(1) ⊗fr2v(2) ,
c c (2) c c
k(cid:16)−2 ⊗1(cid:17)(cid:16)fr1v(1) ⊗fr2v(2)(cid:17) = λ−1 q2r2 f(cid:16)r1v(1) ⊗fr2v(2) (cid:17),
c c (1) c c
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)
we obtain from (8):
Rˆ(x)(fr1 ⊗fr2)v(1) ⊗v(2) = xλ2 q−2(r2−1)∆(f )−∆(e ) Rˆ(x)
c c (2) 1 0
×(fr1 ⊗fr2−1)(q−2r2−r1−2xz−h1k ⊗k2 −qr1z k−1 ⊗1)−1v(1i) ⊗v(2)
1 0 c c
(9)
Rˆ(x)(fr1 ⊗fr2)v(1) ⊗v(2) = λ−2 q2(r1−1)(z z ⊗1)∆(f )−∆(e )
c c (1) 0 1 1 0
×Rˆ(x)(fr1−1 ⊗fr2)(q2r1+r2−2hz z k−2 ⊗k−1 −q−r2x·1⊗k)−1v(1)i⊗v(2)
0 1 c c
Here λ (i = 1,2) are values of k on highest vectors v(i). Using this and denoting
(i) c
by z(i) the values of central elements z (j = 0,1) on V , we obtain by induction
j j i
from (10):
(z(1))−r1
R(x)(fr1 ⊗ fr2)v(1) ⊗v(2) = 1
c c r1+r2−1 qι(λ λ )−1z(1) −q−ι(λ λ )xz(1)−1
ι=0 1 2 0 1 2 c
r1−1 Q (cid:16) (cid:17)
× λ−1qι1z(2)z(2)∆(f )−λ q−ι1∆(e )
(1) c 1 1 (1) 0
ιY1=0h i
r2−1
× λ−1qι2∆(e )−xλ q−ι2∆(f ) (v(2) ⊗v(1)) (10)
(2) 0 (2) 1 c c
ιY2=0h i
3
It can be proved that this R-matrix commutes also with ∆(f ) and ∆(e )
0 1
(As in example 3 the condition on parameters of representations follows from the
equality of central elements in tensor products).
The formula (10) generalizes the results of [4] in case of spectral parameter
depending R-matrix and any highest weight representation of quantum algebra
sl (2,C). In the particular case of x = 1,z = 1, the formula (10) determines the
q i
reflection symmetric intertwiner for sl (2,C), constructed in [4]. Note, however,
q
that we used generators, slightly differing by ones of sl (2,C), so there are some
q
differences in formulas. We kept track by the method of [4] in our derivation.
Consider now some examples.
Example 1. First we consider the case of general q. We put z = 1. The
i
irreducible representations of quantum algebra for general q are the deformations
of representations of corresponding classic algebra and characterized by half in-
teger highest weight. As it was proved by in ref. [3], Rˆ(x) exists and is unique.
It also satisfies Yang-Baxter equations on π (xy)⊗π (x)⊗π (1) [1, 2]:
1 2 3
(Rˆ(x)⊗id)(id⊗Rˆ(xy)(Rˆ(y)⊗id) = (id⊗Rˆ(y))(Rˆ(xy)⊗id)(id⊗Rˆ(x)) (11)
From the uniqueness of R-matrix it follows that constructed above operator
(10) coincides with Jimbo’s R-matrix in [2]. The later, which had been repre-
sented by means of projecting operators, can be also represented in the explicit
form:
Rˆ(x) v(j1) ⊗v(j2) = r1+r2−1 qι−j1−j2 −xq−(ι−j1−j2) −1
r1 r2
ιY=0 (cid:16) (cid:17)
r1−1
× qι1−j1∆(f )−q−(ι1−j1)∆(e )
1 0
ιY1=0h i
r2−1
× qι2−j2∆(e )−xq−(ι2−j2)∆(f ) v(j2) ⊗v(j1) , (12)
0 1 c c
ιY2=0h i(cid:16) (cid:17)
where v(j) := fkv(j).
k c
Example 2. Let q be a root of 1 and N be the minimal integer, such that
qN = 1. For simplicity we consider the case of odd N only. Then eN,fN,kN lie in
the center of sl (2,C) [10, 11]. It follows from this that in this case appear new
q
irreducible representations that are called periodic or cyclic [10, 11]. They are N-
dimensional and parametrized by 3 complex numbers. The factorized S-matrix
of sl (N,C)-generalized chiral Potts model has been found first by Bazhanov et
q
all in ref.[16]. In the article [6] Date, Jimbo, Miki and Miwa by means of [16]
constructed intertwining operators (6) for the minimal cyclic representation of
the gˆl (N,C) affine algebra with central extension. They also proved the Yang-
q
Baxter equations between them for parameters of corresponding representations
lying on some algebraic curve.
4
Recall that the minimal cyclic representation of sl (2,C) has the following
q
form:
π (e)w = yqµ1+µ0[2µ −m] w π (k)w = qµ1−µ0−m−1w
ξ m 1 q m−1 ξ m m
(13)
π (f)w = y−1q−µ1−µ0[2µ +m+2] w π (z ) = ci (i = 0,1),
ξ m 0 q m+1 ξ i ci+1
wherec := c ,w := w ,m = 0,1...N−1,andξ := qµ0,qµ1,y, c1 isaparameter
2 0 N 0 c2
of the representation. (cid:16) (cid:17)
After affinization of(13) andimposing comultiplication (3)we obtainminimal
representation of sˆl (2,C), considered in [6]. (Here we use slightly different basis
q
and comultiplication). Recall that inertwining operator: V ⊗V′ ⇒ V′⊗V exist
ξ ξ ξ ξ
if ξ and ξ′ lie on algebraic curve, which is parametrized by 2 complex varieties s
and s′:
uN = s−λ vN = s−µ
i i i i
(14)
u′N = s′ −λ v′N = s′ −µ (i = 0,1),
i i i i i
where:
u q2µi v′ 1 u′
i = , i−1 = q2µic , x = ωN i,
u′ c v i i i v′
i i i−1 i
(Here x := y,x := x). The moduli parameters γ := (ω ,λ ,µ ) are fixed for
1 0 y i i i
different representations (µ is not the same variable as in (13)).
i
Ifµ = 0thenπ (e)w = 0,andperiodicrepresentationconvertsintosemiperi-
1 ξ 0
odic one. It follows from (14) that µ = λ in γ in this case. So, the R-matrix,
0 1
which intertwines semiperiodic representations, is a particular case of R-matrix,
intertwining periodic ones. 1 (For µ = λ , as it can be verified, there are no
0 1
singularities in R-matrix.) Thus the formula (10) for R matrix can be used for
semiperiodic representations. In the case of x = c = z = 1 we obtain R-matrix,
i i
considered in [4].
Example 3. Consider now the tensor product V ⊗V , where by V we denote
ξ j ξ
the semiperiodic representation with parameter ξ = (qµ,λ) and by V the spin j
j
one, 2j +1 < N. We take the following basis for V [12]:
ξ
π (e)w(ξ) = [2µ−m+1] [m] w π (k)w(ξ) = qµ−mw
ξ m q q m−1 ξ m m
(15)
π (f)w(ξ) = w ,m = 0,..N −2 π (f)w(ξ) = λw
ξ m m+1 ξ N−1 0
We shall show that Rˆ(x), obtained using (10) for all z(i) = 1, is an intertwiner
j
from V ⊗V into V ⊗V of sˆl (2,C) (see (6)).
ξ j j ξ q
1We use the unicity arguments of R-matrix which follows from the irreducibility of tensor
product in sˆl (2,C)
q
5
First we note that Rˆ(x) commute with ∆(f ) and ∆(e ). To see this it’s
1 0
enough to check:
N−1 qι−µ∆(f )−q−(ι−µ∆(e ) (v(j) ⊗v(µ))
ι=0 1 0 c c = λ(v(j) ⊗v(µ)) (16)
Q h N−1(qι−µ−j −xq−(ι−µi−j)) c c
ι=0
Q
2j qι−j∆(e )−xq−(ι−j)∆(f ) (v(j) ⊗v(µ))
ι=0 0 1 c c
= 0 (17)
Q h 2j (qι−µ−j −xq−(ι−µ−ij))
ι=0
(16) can be proven usingQcommutativity of ∆(f ) and ∆(e ), Gauss binomial
1 0
formula:
N−1 N N
(1−q2ιa) = (−1)ι qι(ι−1)aι (= 1+aNfor our q),
1
" #
ι=0 ι=0 q
Y X
the equations ∆(f )N = fN ⊗k−N +kN ⊗FN,∆(e )N = fN ⊗kN +xNk−N ⊗fN,
1 0
the fact that fN| ≡ 0, k2N| ≡ id and (14). To prove (17) we consider it for
Vj Vj
general q. Because of existence and uniqueness of an R-matrix for half integer
values of µ the equation (17) is valid for such µ. It is evident, that the numerator
in (17) can be represented as a linear combination of v(j) ⊗v(µ),p+m = 2j +1,
p m
with the coefficients, which are µ-independent polynomials on qµ. Because of
vanishing such polynomials on infinite numbers of points qhalfinteger they are
vanished trivially. So, (17) is valid.
Now recall the fact that V ⊗V as a sl (2,C) module is a fully reducible. It
ξ j q
decomposes into 2j + 1 semiperiodic representations V (i),i = 1..2j + 1 ([12]).
ξ
From this it follows that ∆(e ) and ∆(f ) also commute with Rˆ(x). Indeed, if
1 0
vµi) is highest weight of V (i) then considering the half integer values of µ as
c ξ
above it can be proven that ∆(f )Rˆ(x)v(µi) = ∆(e )Rˆ(x)v(µi) = 0. Then:
0 c 1 c
∆(e )Rˆ(x)∆(fk)v(µi) = ∆(e )∆(fk)Rˆ(x)v(µi) = Rˆ(x)∆(e )∆(f )v(µi)
1 1 c 1 1 c 1 1 c
∆(f )Rˆ(x)∆(ek)v(µi) = ∆(f )∆(ek)Rˆ(x)v(µi) = Rˆ(x)∆(f )∆(e )v(µi)
0 0 c 0 0 c 0 0 c
So, we proved the Rˆ(x) is an intertwiner from V ⊗V to V ⊗V of affine sˆl (2,C).
ξ j j ξ q
The Yang-Baxter equations on V ⊗V ⊗V can be proven using the irreducibility
ξ j j
of tensor product as in [6].
At the end we write the components of Rˆ(x) : V ⊗ V ⇒ V ⊗ V for N =
ξ j j ξ
3,j = 1.
2
[2µ] (−µ+ 1)
Rˆ00(x) = 1 Rˆ10(x) = − Rˆ01(x) = 2 x
00 10 (−µ− 1) 10 (−µ− 1)
2 x 2 x
(µ− 1) x
Rˆ10(x) = 2 x Rˆ01(x) = Rˆ02(x) = −
01 (−µ− 1) 01 11 (−µ− 1)
2 x 2 x
6
(µ− 3) x(1)
Rˆ11(x) = 2 x Rˆ02(x) = x
11 (−µ− 1) 20 (−µ− 1) (−µ+ 3)
2 x 2 x 2 x
x[µ− 3]+(µ− 1) (1−x) αx
Rˆ11(x) = 2 2 x Rˆ00(x) =
20 (−µ− 1) (−µ+ 1) 21 (−µ− 1)
2 x 2 x 2 x
x[2µ−3]+(µ− 3) (−µ+ 1)
Rˆ12(x) = 2 x 2 x
21 (−µ− 1)(−µ+ 3)
2 2 x
Here we omitted the index q in [n] and used the notation:
q
qn −xq−n
(n) :=
x
q −q−1
We would like to thank V.Bazhanov for a comments.
Note added. When this work was finished and submited for a publication
B.M.McCoy kindly informed us that similar results was obtained by I.T.Ivanov
and D.B.Uglov in the article Phys.Lett. A 167 (1992) 459, for which we express
our gratitudes to him.
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