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Some new solutions of the Yang-Baxter equation
.__ ii3
13 February 1995
&b&I-& $1
PHYSICS
LETTERS
A
4._
ELXVIER
Physics Letters A 198 ( 1995) 3942
Some new solutions of the Yang-Baxter equation Susumu Okubo Department of Physics and Astronomy, University of Rochestec Rochestec NY 14627, USA
Received 11 October 1994; revised manuscript received 19 December 1994; accepted for publication 20 December 1994 Communicated by A.R. Bishop
Abstract
We have found some new solutions of both rational and trigonometric types by rewriting the Yang-Baxter equation as a triple product equation in a vector space of matrices.
We introduce [ 51 two O-dependent triple products by
The Yang-Baxter equation (YBE)
N
N
c
d=l N
N
=
c
R~,,(e”)R~:(B’)R~(,,,
[ed,eb,eali
(1)
j,k.l=l 8+
8” = e’,
(2)
appears in many fields ranging from statistical physics
[ 11, exactly solvable two-dimensional field theories [ 1I, and braid-groups [ 2,3], as well as quantum groups [ 1,4]. Let V be an N-dimensional vector space with a symmetric bilinear non-degenerate form (xly) = (y(x). For a fixed basis el,e2,. . . ,t?N of V, we set gjk = gkj
= (ejlek),
N c8’
.4=l
kek.
(4)
= c R$XO)e, C=l
(5b)
so that we have @b(e) =
(ed~[ec~ea~eblf?)
= (ecl[ed9eb7eali). (6)
Then as we noted in Ref. [ 51, the YBE can be rewritten as a triple product equation, N c
tu,
tkej,Zle~,
[ej,x,yltil~N
j=l N
(3)
with its inverse tik. We raise and lower indices as usual in terms of these metric tensors as ej =
(54
[eC,ea,eblo =~ed$(Q,
Rik ah (8)R’a2(B’)RCZb2(8”) kc1 I[
j,k.l=l
=
c
[u,
[~,ej,Xl~~,
[ej,z,yl,*ttls.
(7)
j=l
As a matter of fact, if we identify x = e,, , y = eb, , z =e,,,u=e “*,ando=eQ inEq. (7),andifwenote Eqs. (5), then we can readily verify that Eq. (7) will reproduce Eq. ( 1) . Similarly, we have the validity of
0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(94)01024-2
S. Okubo / Physics Letters A 198 (I 99.5) 39-42
40
(4[u7X,Yle)
=
(4[u,Y,xli3 j=l
for any u, u, x, y EV in view of Eq. (6). In Ref. [ 51, some solutions of Eq. (7) for the case of [x, y, z ] $ = [x, y, z ] e have been found for some triple systems including the case of the octonionic solution of de Vega and Nicolai [ 61. The purpose of this note is to present other simpler solutions in terms of n x FZmatrices. Let V now be a vector space consisting of all II x n matrices with N = n2, i.e.
-
[U, [u,ej,Xl~~,
[ej,z,y18*tf18}
=
iyo(uxyzu - uzyxu) + Kl (yzuxu - uxuzy)
-
R,tyxuzu
- uzuxy) + K*{ (2[u)yxu - (ulx)uzy}
A
-
x;{(&J)YZ~
-
(+$Jxy)
+ K3{ (XlY)UZV
-
(ylxuz)u}
-
&{
(z
ly)uxu - (y~zux)~}
+ K4{(YIX)UZU - (ylzvx)u)
V = {x] x = n x n matrix}
(9)
- %{(ylz)uxu
- (y/xuz)v}
+Kd”dY)(zI+
- $(ZlY)(XIU)~,
(13)
and set where we have set for simplicity (X/Y) =Tr(xy).
(10)
K. = P;A’P,
- PI’A’P2,
K1 = P,“P[C - P:‘P$z,
The completeness condition of the space V can then be expressed as
K2 = Pz”P;P, f Ppz
R, = ctr,;q
- C”P{P2,
-I- ctfcfq
- CtiP2C i- nPTC’P2, & = P:‘PiPI i- C”C’P, i- PpX
N c
ejxej
=
(Trx) 1
(11)
- CitP{C i- nP:C’PI,
j=i
K3 = P~P~P,~P~AtA~C’tP~A-CttA’P2~nP~P~A,
for any x EV, where 1 stands for the unit n x n matrix. Following Ref. f5], we seek a solution with the ansatz
k3 = P,“P;P, + AttArP, + AttP(C - P,“A’C f nA”P[Pl, K4 = P:‘P:P~~P:‘A’A~C”P~‘A--C’/A’PI
+nP/P(A,
ri, = P:‘P,‘P2 +- A”A’P2 f A”P;C
In, y, Zle = PI (@yZX: -t ~2(@I~ZY
- P,“A’C -I- nAttPiP2, K5 = P,“P(A + P,“P;A i- AttP;P2 + A”P;P, i- P,“C’P, +- PpC’P2 i- n(P,/‘CtA
+ PPC’A
-tAftP~A+A”P,‘A~A~‘C’P~+AttCtP2}+CttCtA - CItA’C + AffC’C +- AttAtA + n2A”CtA,
which satisfy constr~nt Rq. (8). Here, PI(@), P2(@), A (0)) and C (8) are some functions of 8 to be determined. Also, the products yzx and xzy in Eqs. (12) represent the standard associative matrix products in V. We insert expression (12) into both sides of Eq. (7) and note the validity of Rq. ( 1I ) . This yields the following equation,
& = A”P;P, + PtC’Pl
$- A”PfP2 -+ P:‘P;A + P,“P,‘A + PrC’P2 + n{A”CtP,
+ A”C’P2
+ A”P;A -I- A”P;A + P,“CtA + P:‘C!A) +A”C’C-CttAtC+CttCtA+A”A’A+n2Af’C’A. (14)
S. Okubo /Physics
Letters A 198 (199.5) 39-42
41
Here, P”, P’, and P for example stand for k(8 + ~9)= logp P = P(e),
P” = P(B”).
P’ = P(B’),
(15)
We note that Kj ( j = 1,2,3,4,5) is the same function as Kj except for the interchanges of 8 c-t 0” and PI H P2. The YBE can be satisfied if we have
for (I),
= log(n2 - 1)
N
(16)
We can solve these eleven coupled function equations as in Refs. [5,6] to find the following trigonometric solutions, assuming that at least one of the PI (19) and P2 (6) is not identically zero: Solution (I). We have PI (0) = P2 (6). Setting A=;(nf&z-4),
@ ek,
(23b)
&( 0)) R*(O),
and triple
N c
j=l
ek0 -AZ
Pz(e) = 0,
A(@
(18b)
’
(n2 - 1) -ekO’
= -
Now, the unitarity relation for all solutions is expressed in the form
(20)
1) -ekB’
$)
[X?Y,ZlS
(21)
1
for example for solutions related to 0 by
=J?(e)&*(-e)
(I) -( III)
=c(e)c(-e)Id, (25)
In Eqs. ( 19) and (20)) k is again an arbitrary constant including k = ha We remark that these solutions satisfy the so-called crossing relation [ 2,7 ] which can be expressed as [y,x,zlg=
(24b)
’
n ekO (n2-
[e’,y,xl8 63ej.
n
eke -1
P2(W
[ej,x,yllf
N
g-B)l?(B)
c(e) -
=o,
PW
j=l
j=l
(19)
[ej,y,xl$ @eej,
&*(O)XX_Y=Fej@
C'
c(e) n = &O-l’ PI(e) neke
Solution (III).
&
Rc(e)ej
@ eb = c
The relationship between products is then given by
=
A(@) p2(8)=
(234
j,k=l
=
where /? can assume two possible values, A2 or -A4, and k is an arbitrary constant including k = fem. Solution (II).
PI(e)
@ ek,
N
&*(e)e,
(18a)
’
A(ekO -1)
9(e)=
@,(O)ej
j&l
j=l
A(eke -p)
C(6) - _
@ et, = c
R(B)*~y=~ej~[ej,x,rln
A(6) - _ A2ekO-p PI (6)
R(f))e,
(17)
the solution is given by
Pl(@
(22)
They also satisfy the unitarity relation. To see this, we introducel?(8) andg(8): V@V-+V@Vby
K. = K1 = 8, = K2 = K2 = K3 = I& =K4=&=K5=&=0.
for (II).
(I) and (II),
where ?? is
where Id is the identity map in V@V. Note especially that we have r(8) = R(0) and [x,y,z]; = [x,y,z]~ for solution (I). We have also found another solution of the YBE when we replaced Eqs. (12) by
[x,y,zls = [x,y,zl; = PI(e)zxy+ +W@(.+)z
+C(@(z
IX)Y,
P2(~)~XZ (26)
which is consistent with Eq. (8). Repeating the same procedure, the solutions are now found to be of rational type given by
42
S. Okubo / Physics Letters A 198 (1995) 3942
Solution (I). S(e)
B(e) = PI(@)
= d,(e),
-
c(e)
ke
’
ff
9(e>=is VW
Solution (II).
P2(e) = c(e) = 0,
$g =p+ke.
(27b)
z =
(27~)
Solution (III).
4(e)
=c(e)
=o,
This paper is supported in part by the U.S. Department of Energy Grant DE-FG-02-91ER40685.
P + k8,
for arbitrary constants (Y (# 0) , j3, and k. However, we will not go into the details of the calculations. To end this note, we remark the following. Currently, there exist [ 8-121 many physically motivated solutions of the YBE, which are related to quantum field theory and statistical mechanics. The present trigonometric solutions ( 18) and ( 19) have forms quite similar to some solutions given in this literature (especially Ref. [ 12] ), although they are not identical. If we wish, we can construct the corresponding triple systems for these solutions by Eqs. (5). Whether the present model may correspond to some physically relevant theory is not clear at the moment.
References [ 11M. Jimbo, Yang-Baxter equations in integrable systems (World Scientific, Singapore, 1989). [2] C.N. Yang and M.L. Ge, Braid group, knot theory, and statistical mechanics (World Scientific, Singapore, 1989). [3] L.H. Kauffman, Knots and physics (World Scientific, Singapore, 1991) [ 41 Y.1.Martin, Quantum groups and non-commutative geometry (Univ. of Montreal Press, Montreal, 1988). [S] S. Okubo, J. Math. Phys. 34 (1993) 3273, 3292. [6] H.J. de Vega and H. Nicolai, Phys. Lett. B 244 (1990) 295. [7] A.B. Zamolodchikov and A1.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. [8] C.L. Schultz, Phys. Rev. Lett. 46 (1981) 629. [9] V.V. Bazhanov and Y.G. Stroganov, Nucl. Phys. B 205 (1982) 505; 230 (1984) 435. [lo] H. Au-Yang et al., Phys. Lett. A 123 (1987) 219. [ 111 H. Sauler and J.-B. Zuber, Integrable lattice models and quantum groups, in: String theory and quantum gravity (World Scientific, Singapore, 1991) p. 1. [ 121 H.J. de Vega and G. Giavarini, Nucl. Phys. B 410 (1993) 550.
13 February 1995
&b&I-& $1
PHYSICS
LETTERS
A
4._
ELXVIER
Physics Letters A 198 ( 1995) 3942
Some new solutions of the Yang-Baxter equation Susumu Okubo Department of Physics and Astronomy, University of Rochestec Rochestec NY 14627, USA
Received 11 October 1994; revised manuscript received 19 December 1994; accepted for publication 20 December 1994 Communicated by A.R. Bishop
Abstract
We have found some new solutions of both rational and trigonometric types by rewriting the Yang-Baxter equation as a triple product equation in a vector space of matrices.
We introduce [ 51 two O-dependent triple products by
The Yang-Baxter equation (YBE)
N
N
c
d=l N
N
=
c
R~,,(e”)R~:(B’)R~(,,,
[ed,eb,eali
(1)
j,k.l=l 8+
8” = e’,
(2)
appears in many fields ranging from statistical physics
[ 11, exactly solvable two-dimensional field theories [ 1I, and braid-groups [ 2,3], as well as quantum groups [ 1,4]. Let V be an N-dimensional vector space with a symmetric bilinear non-degenerate form (xly) = (y(x). For a fixed basis el,e2,. . . ,t?N of V, we set gjk = gkj
= (ejlek),
N c8’
.4=l
kek.
(4)
= c R$XO)e, C=l
(5b)
so that we have @b(e) =
(ed~[ec~ea~eblf?)
= (ecl[ed9eb7eali). (6)
Then as we noted in Ref. [ 51, the YBE can be rewritten as a triple product equation, N c
tu,
tkej,Zle~,
[ej,x,yltil~N
j=l N
(3)
with its inverse tik. We raise and lower indices as usual in terms of these metric tensors as ej =
(54
[eC,ea,eblo =~ed$(Q,
Rik ah (8)R’a2(B’)RCZb2(8”) kc1 I[
j,k.l=l
=
c
[u,
[~,ej,Xl~~,
[ej,z,yl,*ttls.
(7)
j=l
As a matter of fact, if we identify x = e,, , y = eb, , z =e,,,u=e “*,ando=eQ inEq. (7),andifwenote Eqs. (5), then we can readily verify that Eq. (7) will reproduce Eq. ( 1) . Similarly, we have the validity of
0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(94)01024-2
S. Okubo / Physics Letters A 198 (I 99.5) 39-42
40
(4[u7X,Yle)
=
(4[u,Y,xli3 j=l
for any u, u, x, y EV in view of Eq. (6). In Ref. [ 51, some solutions of Eq. (7) for the case of [x, y, z ] $ = [x, y, z ] e have been found for some triple systems including the case of the octonionic solution of de Vega and Nicolai [ 61. The purpose of this note is to present other simpler solutions in terms of n x FZmatrices. Let V now be a vector space consisting of all II x n matrices with N = n2, i.e.
-
[U, [u,ej,Xl~~,
[ej,z,y18*tf18}
=
iyo(uxyzu - uzyxu) + Kl (yzuxu - uxuzy)
-
R,tyxuzu
- uzuxy) + K*{ (2[u)yxu - (ulx)uzy}
A
-
x;{(&J)YZ~
-
(+$Jxy)
+ K3{ (XlY)UZV
-
(ylxuz)u}
-
&{
(z
ly)uxu - (y~zux)~}
+ K4{(YIX)UZU - (ylzvx)u)
V = {x] x = n x n matrix}
(9)
- %{(ylz)uxu
- (y/xuz)v}
+Kd”dY)(zI+
- $(ZlY)(XIU)~,
(13)
and set where we have set for simplicity (X/Y) =Tr(xy).
(10)
K. = P;A’P,
- PI’A’P2,
K1 = P,“P[C - P:‘P$z,
The completeness condition of the space V can then be expressed as
K2 = Pz”P;P, f Ppz
R, = ctr,;q
- C”P{P2,
-I- ctfcfq
- CtiP2C i- nPTC’P2, & = P:‘PiPI i- C”C’P, i- PpX
N c
ejxej
=
(Trx) 1
(11)
- CitP{C i- nP:C’PI,
j=i
K3 = P~P~P,~P~AtA~C’tP~A-CttA’P2~nP~P~A,
for any x EV, where 1 stands for the unit n x n matrix. Following Ref. f5], we seek a solution with the ansatz
k3 = P,“P;P, + AttArP, + AttP(C - P,“A’C f nA”P[Pl, K4 = P:‘P:P~~P:‘A’A~C”P~‘A--C’/A’PI
+nP/P(A,
ri, = P:‘P,‘P2 +- A”A’P2 f A”P;C
In, y, Zle = PI (@yZX: -t ~2(@I~ZY
- P,“A’C -I- nAttPiP2, K5 = P,“P(A + P,“P;A i- AttP;P2 + A”P;P, i- P,“C’P, +- PpC’P2 i- n(P,/‘CtA
+ PPC’A
-tAftP~A+A”P,‘A~A~‘C’P~+AttCtP2}+CttCtA - CItA’C + AffC’C +- AttAtA + n2A”CtA,
which satisfy constr~nt Rq. (8). Here, PI(@), P2(@), A (0)) and C (8) are some functions of 8 to be determined. Also, the products yzx and xzy in Eqs. (12) represent the standard associative matrix products in V. We insert expression (12) into both sides of Eq. (7) and note the validity of Rq. ( 1I ) . This yields the following equation,
& = A”P;P, + PtC’Pl
$- A”PfP2 -+ P:‘P;A + P,“P,‘A + PrC’P2 + n{A”CtP,
+ A”C’P2
+ A”P;A -I- A”P;A + P,“CtA + P:‘C!A) +A”C’C-CttAtC+CttCtA+A”A’A+n2Af’C’A. (14)
S. Okubo /Physics
Letters A 198 (199.5) 39-42
41
Here, P”, P’, and P for example stand for k(8 + ~9)= logp P = P(e),
P” = P(B”).
P’ = P(B’),
(15)
We note that Kj ( j = 1,2,3,4,5) is the same function as Kj except for the interchanges of 8 c-t 0” and PI H P2. The YBE can be satisfied if we have
for (I),
= log(n2 - 1)
N
(16)
We can solve these eleven coupled function equations as in Refs. [5,6] to find the following trigonometric solutions, assuming that at least one of the PI (19) and P2 (6) is not identically zero: Solution (I). We have PI (0) = P2 (6). Setting A=;(nf&z-4),
@ ek,
(23b)
&( 0)) R*(O),
and triple
N c
j=l
ek0 -AZ
Pz(e) = 0,
A(@
(18b)
’
(n2 - 1) -ekO’
= -
Now, the unitarity relation for all solutions is expressed in the form
(20)
1) -ekB’
$)
[X?Y,ZlS
(21)
1
for example for solutions related to 0 by
=J?(e)&*(-e)
(I) -( III)
=c(e)c(-e)Id, (25)
In Eqs. ( 19) and (20)) k is again an arbitrary constant including k = ha We remark that these solutions satisfy the so-called crossing relation [ 2,7 ] which can be expressed as [y,x,zlg=
(24b)
’
n ekO (n2-
[e’,y,xl8 63ej.
n
eke -1
P2(W
[ej,x,yllf
N
g-B)l?(B)
c(e) -
=o,
PW
j=l
j=l
(19)
[ej,y,xl$ @eej,
&*(O)XX_Y=Fej@
C'
c(e) n = &O-l’ PI(e) neke
Solution (III).
&
Rc(e)ej
@ eb = c
The relationship between products is then given by
=
A(@) p2(8)=
(234
j,k=l
=
where /? can assume two possible values, A2 or -A4, and k is an arbitrary constant including k = fem. Solution (II).
PI(e)
@ ek,
N
&*(e)e,
(18a)
’
A(ekO -1)
9(e)=
@,(O)ej
j&l
j=l
A(eke -p)
C(6) - _
@ et, = c
R(B)*~y=~ej~[ej,x,rln
A(6) - _ A2ekO-p PI (6)
R(f))e,
(17)
the solution is given by
Pl(@
(22)
They also satisfy the unitarity relation. To see this, we introducel?(8) andg(8): V@V-+V@Vby
K. = K1 = 8, = K2 = K2 = K3 = I& =K4=&=K5=&=0.
for (II).
(I) and (II),
where ?? is
where Id is the identity map in V@V. Note especially that we have r(8) = R(0) and [x,y,z]; = [x,y,z]~ for solution (I). We have also found another solution of the YBE when we replaced Eqs. (12) by
[x,y,zls = [x,y,zl; = PI(e)zxy+ +W@(.+)z
+C(@(z
IX)Y,
P2(~)~XZ (26)
which is consistent with Eq. (8). Repeating the same procedure, the solutions are now found to be of rational type given by
42
S. Okubo / Physics Letters A 198 (1995) 3942
Solution (I). S(e)
B(e) = PI(@)
= d,(e),
-
c(e)
ke
’
ff
9(e>=is VW
Solution (II).
P2(e) = c(e) = 0,
$g =p+ke.
(27b)
z =
(27~)
Solution (III).
4(e)
=c(e)
=o,
This paper is supported in part by the U.S. Department of Energy Grant DE-FG-02-91ER40685.
P + k8,
for arbitrary constants (Y (# 0) , j3, and k. However, we will not go into the details of the calculations. To end this note, we remark the following. Currently, there exist [ 8-121 many physically motivated solutions of the YBE, which are related to quantum field theory and statistical mechanics. The present trigonometric solutions ( 18) and ( 19) have forms quite similar to some solutions given in this literature (especially Ref. [ 12] ), although they are not identical. If we wish, we can construct the corresponding triple systems for these solutions by Eqs. (5). Whether the present model may correspond to some physically relevant theory is not clear at the moment.
References [ 11M. Jimbo, Yang-Baxter equations in integrable systems (World Scientific, Singapore, 1989). [2] C.N. Yang and M.L. Ge, Braid group, knot theory, and statistical mechanics (World Scientific, Singapore, 1989). [3] L.H. Kauffman, Knots and physics (World Scientific, Singapore, 1991) [ 41 Y.1.Martin, Quantum groups and non-commutative geometry (Univ. of Montreal Press, Montreal, 1988). [S] S. Okubo, J. Math. Phys. 34 (1993) 3273, 3292. [6] H.J. de Vega and H. Nicolai, Phys. Lett. B 244 (1990) 295. [7] A.B. Zamolodchikov and A1.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. [8] C.L. Schultz, Phys. Rev. Lett. 46 (1981) 629. [9] V.V. Bazhanov and Y.G. Stroganov, Nucl. Phys. B 205 (1982) 505; 230 (1984) 435. [lo] H. Au-Yang et al., Phys. Lett. A 123 (1987) 219. [ 111 H. Sauler and J.-B. Zuber, Integrable lattice models and quantum groups, in: String theory and quantum gravity (World Scientific, Singapore, 1991) p. 1. [ 121 H.J. de Vega and G. Giavarini, Nucl. Phys. B 410 (1993) 550.