New Solutions to the Yang--Baxter Equation from Two--Dimensional Representations of $U_q(sl(2))$ at Roots of Unit

12 Pages·0.12 MB·English

by  M. ~Ruiz--Altaba

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0 New Solutions to the
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Yang–Baxter Equation from
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/ Two–Dimensional Representations
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of U (sl(2)) at Roots of Unit
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X M. Ruiz–Altaba
∗
r
a
D´ept. Physique Th´eorique, Universit´e de Gen`eve, CH–1211 Gen`eve 4
UGVA–DPT 1991/08–741
Abstract: We present particularly simple new solutions to the Yang–Baxter
equation arising from two–dimensional cyclic representations of quantum SU(2).
They are readily interpreted as scattering matrices of relativistic objects, and the
quantum group becomes a dynamical symmetry.
Supported in part by the Fonds National Suisse pour la Recherche Scientifique.
∗
1. Introduction: U (sℓ(2)) with ǫ4 = 1
ǫ
Quantum groups at roots of unit enjoy a beautiful representation theory [1,2,3,4,5,6]
whichhasbeenappliedsuccesfullytotheunderstandingofthechiralPottsmodel[7,8,2,9,10,11].
In this letter, we apply the general formalism of cyclic representations of U (sℓ(2)) to the
ǫ
somewhat degenerate case of ǫ4 = 1, i.e. q2 = 1. It complements the work for U (sℓ(2)) in
q
[3] of the p = 3 case and in [5] of the limit p , as well as the case p = 2 of U (sℓ(3)) in
q
→ ∞
[12].
When qp = 1, the center of U (sℓ(2)) contains not only the standard quadratic Casimir
ǫ
1
1 1 1
C = K FE + q K +K (1.1)
− − −
q 1
(cid:16) (cid:17)
−
butalsotheelementsFp,Ep andKp. Weworkinthecontourbasis[13]withco-multiplication
∆E = E 1+K E
⊗ ⊗
∆F = F 1+K F (1.2)
⊗ ⊗
∆K = K K
⊗
The generators E, F and K of U (sℓ(2)) satisfy the standard relations :
ǫ
2
EF qFE = 1 K
− −
EK qKE = 0 (1.3)
−
1
FK q KF = 0
−
−
Let us denote by ξ = (x,y,z) the eigenvalues of (Ep,Fp,Kp), and introduce the notation
x
µ =
1 z
−y (1.4)
ν =
1 z
−
Specializing to the case of interest, namely p = 2 (q = 1), and letting z = λ2, the
−
Casimir eigenvalue is then
2 √1 4µν
1
c = − − λ λ (1.5)
(cid:0) 2 (cid:1) − −
(cid:16) (cid:17)
The spectrum of U (sℓ(2)) with ǫ = eiπ/2 consists of a three–dimensional continuum of two–
ǫ
dimensional representations, labelled by ξ, with a singular (orbifold) point at z = 1, which
corresponds to the only regular representation in this theory, namely the identity.
The irreducible representations of the theory consist thus the identity operator and the
manifold of doublet cyclic representations. The latter constitute an intrinsically quantum
generalization of the customary and useful doublet irrep of SU(2), and the purpose of this
letter is to analyze some of their physical properties from the purely algebraic point of view.
Let er(ξ) (r = 0,1) be the basis for the cyclic representation πξ = π(x,y,z=λ2), defined
as follows (z = λ2 = 1):
6
1 √1 4µν
Ke0 = λe0 Fe0 = √ν(1 λ)e1 Ee0 = − − (1 λ)e1
− 2√ν −
(1.6)
1+√1 4µν
Ke1 = λe1 Fe1 = √ν(1+λ)e0 Ee1 = − (1+λ)e0
− 2√ν
In our choice of basis, we have implicitly assumed that ν = 0. The special class of represen-
6
tations with µ = 0 are called semi-cyclic: for them, e is a highest weight (Ee = 0) and yet
0 0
e is not a lowest weight (Fe = 0).
1 1
6
The seasoned reader is certainly struck by the appearence of anti-commutators in the
relations among the generators. Quite simply, when q2 = 1, q-commutators become anti-
commutators. For future reference, let us introduce a “fermionic” basis for U (sℓ(2)) when
ǫ
ǫ4 = 1:
1
b = E
1+K
(1.7)
1
b = F
†
1+K
The quantum algebra is then generated by K, b and b , with the following anticommutation
†
relations:
b,K = b ,K = 0
†
{ }
n o
b,b = 1
†
(1.8)
n o
b,b = 2µ
{ }
b ,b = 2ν
† †
n o
(µ and ν are c–numbers or, equivalently, operators proportional to the identity). We may
rescalethequadraticCasimirby(K2 1)togetthequantumrelativeofthenumber operator:
−
1
Q = K b b (1.9)
†
(cid:18) − 2(cid:19)
which commutes with K, b and b . A cyclic irreducible representation is characterized in
†
2
this language by e = and e = + with
0 1
|−i | i
K = λ
|±i ± |±i
1 √1 4µν
b = ∓ −
|±i 2√ν |∓i
b = √ν
†
|±i |∓i
Again, note that in the semi-cyclic case, µ = 0, we may think of + as akin to the ground
| i
state (it is annihilated by b).
2. Intertwiners and Yang–Baxter equation: general solution
The intertwiner R(ξ ,ξ ) between two cyclic representations effects a braiding and can
1 2
be thought of as a 2 2 scattering matrix:
→
r′r′
er1(ξ1)⊗er2(ξ2) = Rr11r22(ξ1,ξ2)er1′(ξ2)⊗er2′(ξ1) (2.1)
Quasi–triangularity requires [R,∆] = 0, i.e.
r′′r′′ r′r′ r′′r′′ r′r′
R 1 2(ξ ,ξ )∆ (g) 1 2 = ∆ (g) 1 2R 1 2(ξ ,ξ ) (2.2)
r′r′ 1 2 ξ1,ξ2 r1r2 ξ2,ξ1 r′r′ r1r2 1 2
1 2 1 2
for any g U (sℓ(2)). Specializing to g = E2 and g = F2, we see that if the intertwiner
q
∈
R(ξ ,ξ ) is to exist, then ξ and ξ are constrained to lie on the same spectral variety [2]:
1 2 1 2
x x
1 2
= = µ
1 λ2 1 λ2
− 1 − 2 (2.3)
y y
1 2
= = ν
1 λ2 1 λ2
− 1 − 2
with arbitrary µ,ν C and λ2 = 1.
∈ i 6
The main result we discuss in this letter is that, forξ , ξ , ξ on thesame spectral variety
1 2 3
there exists an R–matrix R(ξ ,ξ ) satisfying (2.2) and the Yang–Baxter equation
i j
Rr1′r2′(ξ ,ξ )Rs3r3′(ξ ,ξ )Rs1s2(ξ ,ξ ) = Rr2′r3′(ξ ,ξ )Rr1′s1(ξ ,ξ )Rs2s3(ξ ,ξ )
s1s3 2 3 s2r3 1 3 r1r2 1 2 s1s3 1 2 r1s2 1 3 r2r3 2 3
s1X,s2,s3 s1X,s2,s3
(2.4)
The explicit form of the R–matrix intertwiner satisfying the Yang–Baxter equation is
3
the following (we show only the non–zero entries):
00
R (ξ ,ξ ) = 1
00 1 2
(1 λ )(1+λ )
01 1 2
R (ξ ,ξ ) = Ω −
01 1 2 1
Ω λ λ Ω
2 1 2 1
−
λ Ω λ Ω
10 1 2 2 1
R (ξ ,ξ ) = −
01 1 2
Ω λ λ Ω
2 1 2 1
−
λ Ω λ Ω (2.5)
01 2 2 1 1
R (ξ ,ξ ) = −
10 1 2
Ω λ λ Ω
2 1 2 1
−
(1+λ )(1 λ )
10 1 2
R (ξ ,ξ ) = Ω −
10 1 2 2
Ω λ λ Ω
2 1 2 1
−
Ω λ λ Ω
11 1 1 2 2
R (ξ ,ξ ) = −
11 1 2
Ω λ λ Ω
2 1 2 1
−
Here, Ω = Ω(ξ ) are the values of an arbitrary function of the labels of the cyclic represen-
i i
tation. Note that what really appears is only the ratio Ω /Ω = Ω(λ )/Ω(λ ).
1 2 1 2
The R–matrix (2.5) with arbitary Ω(ξ) enjoys two remarkable properties. First, it is
normalized:
r′r′ r′ r′
R 1 2(ξ,ξ) = δ 1δ 2 (2.6)
r1r2 r1 r2
Second, it is unitary:
′ ′ ′′ ′′ ′′ ′′
r r r r r r
R 1 2(ξ ,ξ )R 1 2(ξ ,ξ ) = δ 1δ 2 (2.7)
r1r2 1 2 r′r′ 2 1 r1 r2
1 2
rX′r′
1 2
Let us stress some important differences between the above solution for q2 = 1 and the
generic (qp = 1, p 3) semi-cyclic (y = 0) situation [3,5]. Firstly, this is the only case in
≥
which an arbitrary function Ω appears, i.e. there is a whole family of spectral–dependent
R–matrices. Secondly, the R–matrix (2.5) does not involve the parameters µ = x /(1 z )
i i
−
nor ν = y /(1 z ) explicitly, although of course ξ and ξ must lie on the same spectral
i i 1 2
−
variety, i.e. share common values for µ and ν, for R(ξ ,ξ ) to exist at all. (A dependence
1 2
of R on µ and ν could be introduced through Ω.) Thirdly, the R–matrix (2.5) conserves the
“quantum isospin” exactly
r′,r′
R 1 2(ξ ,ξ ) = 0 = r +r = r +r (2.8)
r1,r2 1 2 6 ⇒ 1 2 1′ 2′
instead of just modulo 2. In particular, R00 and R11 are always zero, so if we interpret
11 00
the R–matrix entries as Boltzmann weights, we are dealing with a six–vertex model. These
three observations are not unrelated, as will become clearer below.
4
The arbitrary function Ω is quite akin to an affinization parameter. This interpretation
is reinforced by the fact that only the ratio Ω /Ω appears in the R–matrix, which can be
1 2
decomposed as follows
1/2 1/2
λ λ Ω1 − R(λ ,λ ;0) λ λ Ω1 R(λ ,λ ; )
1 2Ω2 1 2 − 1 2Ω2 1 2 ∞
R(λ ,λ ;Ω /Ω ) = (cid:16) (cid:17) (cid:16) (cid:17) (2.9)
1 2 1 2
1/2 1/2
λ λ Ω1 − λ λ Ω1
1 2Ω2 − 1 2Ω2
(cid:16) (cid:17) (cid:16) (cid:17)
or, less symmetrically but more succinctly, as
1 Ω
1
R(λ ,λ ;Ω /Ω ) = R(λ ,λ ;0) λ λ R(λ ,λ ; ) (2.10)
1 2 1 2 1 2 1 2 1 2
1−λ1λ2ΩΩ21 (cid:20) − Ω2 ∞ (cid:21)
We may thus view R(λ ,λ ;0) as the basic R–matrix from which the above family is built,
1 2
because R(λ ,λ ; ) = R(λ ,λ ;0) 1. Remarkably, the R–matrix R(λ ,λ ;0) can be ob-
1 2 1 2 − 1 2
∞
tained with the help of contour techniques in the semi-cyclic case (µ = 0) supplemented
with the rule that F2 = 0. It is thus apparent that R(λ ,λ ;0) is a slight generalization
1 2
11
of the usual R22, the R–matrix for the regular doublet representation of quantum SU(2)
with qp = 1 (p 3). In fact, R(λ ,λ ;0) has been considered in [4]. Let us conclude this
1 2
≥
detour by noting that we may not give a contour representation of the general R(λ ,λ ;Ω):
1 2
since it is a linear combination of R(λ ,λ ;0) and R(λ ,λ ;0) 1, it consists of a piece due
1 2 1 2 −
to braiding by π and another by π.
−
3. Clebsch–Gordan coefficients and crossing symmetry
We may consider the tensor product of two cyclic representations π , π on the same
ξ1 ξ2
spectral variety, parametrized by (µ,ν). The result is a direct sum of two cyclic representa-
tions π again on the same variety, with
ξ±
λ = λ λ
1 2
± ±
and
2 2 2
x = µ(1 λ ) = x +x λ = x +x λ
1 2 1 2 1 2
± − ±
and similarly for y. It is easy to check that
e (λ λ ) = e (λ ) e (λ )
0 1 2 0 1 0 2
⊗
λ (1+λ ) 1+λ
1 2 1
e (λ λ ) = − e (λ ) e (λ )+ e (λ ) e (λ )
1 1 2 1 1 0 2 0 1 1 2
1 λ λ ⊗ 1 λ λ ⊗
1 2 1 2
− − (3.1)
1 λ λ (1 λ )
1 1 2
e ( λ λ ) = − e (λ ) e (λ )+ − e (λ ) e (λ )
0 1 2 1 1 0 2 0 1 1 2
− 1 λ λ ⊗ 1 λ λ ⊗
1 2 1 2
− −
e ( λ λ ) = e (λ ) e (λ )
1 1 2 1 1 1 2
− ⊗
5
Thus, the non–zero quantum Clebsch–Gordan coefficients are
K0 0 λ1λ2 = 1
λ1 λ2 0
1+λ
K0 1 λ1λ2 = 1
λ1 λ2 1 1 λ λ
1 2
−
λ (1+λ )
K1 0 λ1λ2 = − 1 2
λ1 λ2 1 1 λ λ
1 2
− (3.2)
λ (1 λ )
K0 1 −λ1λ2 = 1 − 2
λ1 λ2 0 1 λ λ
1 2
−
1 λ
K1 0−λ1λ2 = − 1
λ1 λ2 0 1 λ λ
1 2
−
K1 1−λ1λ2 = 1
λ1 λ2 1
For completeness, we note also the non–zero inverse quantum Clebsch–Gordan coefficients:
K˜λ1 λ2 0 = 1
0 0 λ1λ2
1 λ
K˜λ1 λ2 1 = − 1
0 1 λ1λ2 1+λ λ
1 2
λ (1 λ )
K˜λ1 λ2 1 = − 1 − 2
1 0 λ1λ2 1+λ λ
1 2
(3.3)
λ (1+λ )
K˜λ1 λ2 0 = 1 2
0 1 λ1λ2 1+λ λ
− 1 2
1+λ
K˜λ1 λ2 0 = 1
1 0 λ1λ2 1+λ λ
− 1 2
K˜λ1λ2 1 = 1
1 1 λ1λ2
−
Imaginenowdecomposing ei(±λ1λ2)withKλj11 jλ22±iλ1λ2,andthenbraidingtheresultwith
R(λ ,λ ;Ω /Ω ). We may compare the result of these two actions, RK12, with the single
1 2 1 2
immediate decomposition into ej1(λ2) ⊗ ej2(λ1) with the help of Kλj12 jλ21±iλ1λ2, i.e. K21.
The two operations are related by a factor, depending only on the label of the composed
representation, λ λ , but not on any of the quantum group indices i,j ZZ :
1 2 2
± ∈
R(λ1,λ2;Ω1/Ω2)ij11 ij22Kλi11iλ22±kλ1λ2 = Φ±(λ1,λ2)Kλj12 jλ21±kλ1λ2 (3.4)
and
11
Φ (λ ,λ ) = 1 , Φ (λ ,λ ) = R (λ ,λ ;Ω /Ω ) (3.5)
+ 1 2 1 2 11 1 2 1 2
−
It is interesting to note that a single unique particular choice of the arbitrary function
Ω allows us to set Φ = 1, namely
−
Ω /Ω = 1 (3.6)
1 2
6
Then, the R–matrix R(λ ,λ ;1) is given merely by the mismatch in the decompositions of
1 2
e(λ ) e(λ ) and e(λ ) e(λ ) into cyclic irreps, without any extra phase factors:
1 2 2 1
⊗ ⊗
Rr1′r2′(λ ,λ ;1) = K˜λ1λ2r3Kr1′r2′λ3 (3.7)
r1r2 1 2 r1r2λ3 λ2λ1r3
X
λ3,r3
Equation (3.7) embodies the fulfillment of the bootstrap program. This R–matrix is the
only one, among the one–parameter family of semi-cyclic intertwiners (2.5) satisfying the
Yang–Baxter equation, which does enjoy the “crossing”–symmetry
r′r′ r′r′
R 1 2(ξ ,ξ ;1) = R 2 1(ξ ,ξ ;1) (3.8)
r1r2 1 2 r2r1 2 1
This noteworthy property is crucial for the interpretation of R as a scattering matrix.
Compare, again, with the case of qp = 1 (p 3): there, the unique intertwiner satisfying
≥
Yang–Baxter always has crossing symmetry. The crossing–summetric R–matrix (3.7) with
the particular choice Ω(ξ) = 1 is thus the natural extension to p = 2 of the general semi-
cyclic solutions. The family of R–matrices affinized by Ω(ξ) is peculiar to q = 1, but we
−
shall now study in detail the particular crossing–symmetric R–matrix with Ω = 1.
4. The particular solution: soliton interpretation
The non–zero entries of the crossing–symmetric R(ξ ,ξ ;1) = R(ξ ,ξ ) are, explicitly,
1 2 1 2
00
R (ξ ,ξ ) = 1
00 1 2
(1 λ )(1+λ )
01 1 2
R (ξ ,ξ ) = −
01 1 2
1 λ λ
1 2
−
λ λ
10 1 2
R (ξ ,ξ ) = −
01 1 2
1 λ λ
1 2
− (4.1)
λ λ
01 2 1
R (ξ ,ξ ) = −
10 1 2
1 λ λ
1 2
−
(1+λ )(1 λ )
10 1 2
R (ξ ,ξ ) = −
10 1 2
1 λ λ
1 2
−
11
R (ξ ,ξ ) = 1
11 1 2
Due to the crossing symmetry (3.8), in addition to the already noted unitarity and
normalization properties (2.6,2.7), the “semi-cyclic” R–matrix admits a clear interpretation
as a solitonic S–matrix: we may picture the two states e (λ) and e (λ) as localized around
0 1
each one of the two potential minima, with λ a label very much like relativistic velocity.
7
Introduce the “relative velocity”
λ λ
1 2
u = −
12
1 λ λ
1 2
−
in terms of which the “semi-cyclic” intertwiner reads as
00 01 10 11
00 1 0 0 0
 
01 0 1 u u 0
R12 = − 12 12 (4.2)
 
10 0 u 1+u 0
 12 12 
 − 
11 0 0 0 1 
 
and the Yang–Baxter equation becomes
u+v u+v
R (u)R R (v) = R (v)R R (u) (4.3)
12 13 23 23 13 12
(cid:18)1+uv(cid:19) (cid:18)1+uv(cid:19)
Note that only when Ω = 1 can we parametrize the whole R–matrix in terms of a single
quantity u . In the usual trigonometric solutions to the Yang–Baxter equations, the rapidi-
12
ties u and v add up linearly, to u+v. Here the rapidities add up like relativistic velocities!
It thus turns out that the labels of the irreps under the quantum group may be identified
with kinematical parameters: the two–dimensional Poincar´e group is thus a manifestation of
an internal quantum symmetry. This situation at p = 2 is very similar to the limit p
→ ∞
of the general semi-cyclic intertwiner [5].
The braid group limit of the R–matrix (4.2) is obtained when u 1, i.e. in the
→ ±
extreme relativistic regime. Letting R = lim R(u), we find
u 1
± →±
1 0 0 1 0 0
0 0 1 0 0 2 1 0
R+ = 0 1 −2 0  R = 0 1 0 0  (4.4)
−
−
0 0 0 1  0 0 0 1 
   
Let us concentrate on one of them, say R (the analysis is identical for R ). It can be
+
−
viewed as a particular case of the more general
1 0 0
0 1 bc b 0
R(b,c) = 0 −c 0 0  (4.5)
0 0 0 1 
 
which satisfies Yang–Baxter without spectral parameter and is thus a good starting point
for the construction of an extended Yang–Baxter system and hence a link invariant [14].
8
Indeed, we find that µ = bc and µ =1 satisfy
0 1
kℓ
µ µ µ µ R = 0
i j k ℓ ij
−
(cid:0) kj (cid:1) k
R µ = ABδ
ij j i
Xj (4.6)
kj
1 1 k
R µ = A Bδ
− j − i
Xj (cid:16) (cid:17)ij
with A 1 = B = √bc. Accordingly, if α B is a word of the braid group, the link invariant
− n
∈
associated with its closure αˆ is
w(α) n n
T(αˆ) = A B tr ρ(α) µ (4.7)
− − ⊗
⊗
(cid:0) (cid:1)
where w(α)isthewraithnumber ofα andρistherepresentation ofthebraidgroupassigning
1 1
to each generator σ the matrix R acting on the i-th and (i+1)-th strands. The link
i± (±b,c)
invariant satsifies the skein rule
2 2 1
A P A P = A A P (4.8)
+ − − 0
− − (cid:16) − (cid:17)
and is so normalized that N disconnected unknots are assigned the polynomial (A+A 1)N.
−
This is just Jones polynomial in t = A2. This is a nice result, although the particular value
bc = 1 in (4.4) is in fact singular: T (αˆ) = 0 for all α B .
(1, 1) n
− ∈
−
5. Other cyclic solutions to Yang–Baxter
Have we found all the R–matrices which satisfy Yang–Baxter and intertwine among
cyclic representations of U (sℓ(2)) with q2 = 1? This is interesting because it would
q
−
represent a major, if not final, step towards the classification of all the 4 4 R–matrices
×
satisfying Yang–Baxter. We have not been able to prove that the family R(Ω) exhausts the
solutions, although we strongly suspect that this is indeed the case except for exceptional”
situations. For example, a notable curiosity of U (sℓ(2)) with q = 1 is that there exists a
q
−
parameter region 4µν = 1 for which the raising and lowering generators coincide, up to a
proportionality factor: F = 2νE. The quadratic Casimir is then zero. Fixing furthermore
ν = 1 (thus µ = 1/4) allows us to find a different R(λ ,λ ), namely
1 2
1 0 0 0
0 u 1 u 0
R(λ1,λ2) = R(u12) =  0 012 −1 12 0  (5.1)
1 u 0 0 u 
12 12
 − 
9