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Solutions of the Yang–Baxter Equation and Temperley–Lieb Algebraic Structure

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1996 Commun. Theor. Phys. 25 451

(http://iopscience.iop.org/0253-6102/25/4/451)

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Commun. Theor. Phys. 25 (1996) pp. 451-456 © International Academic Publishers Vol. 25, No. 4, June 15, 1996

Solutions of the Yang-Baxter Equation and Temperley-Lieb Algebraic Structure1

Lu-Yu WANG 2 and Yu CAI2

Department of Physics, Xinjiang University, Urumqi 830046, China

Han-Ying GUO and Shao-Ming FEI 2

Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China

(Received May 18, 1994)

A b s t r a c t The necessary and sufficient condition for the representations of Temperley-Lieb algebra. arising from representations of Heck algebra is presented. Using the condition, the represen­tations of braid group with a quadratic reduction relation are divided into two kinds. One has a representation of Temperley-Lieb algebra behind it and another has not. Some concrete examples are discussed with the condition and their Baxterizations are also mentioned.

I. Introduction T h e Temperley-Lieb algebras were discovered by Temperley and Lieb'1! who used them to

relate the spectrum of the six-vertex model to that of the Pot t s models. Their relevance to two-dimensional integrable models in statistical mechanics was further clarified by BaxterP>

Nowadays they are also of interest due to the fact tha t any representation of them is auto­matically given a representation of the braid group (BGR). Now the braid groups have yet a broader range of physical appl ica t ions .^ The Temperley-Lieb (TL) algebra is an associative algebra generated by the unit element 1 and e,-, i = 1,2, • • •, n — 1, subject to the following constraints

ef=y/qei q€C, eiei+iei = et, e^i = e^e,-, i f | i - j | > 2 . (1)

The first detailed account of their mathemat ical structure and their dependence on the value of the parameter q was given by J o n e s . ^

This paper focuses on connections between representation of braid group and tha t of T L algebra. After restricting to BGRs with a quadratic reduction relation, we derive a constraint for constructing the representations of the T L algebra from BGRs. Since the T L algebra is closely related to integrable models in statistical mechanics, we investigate a more general construction of the solutions of the Yang-Baxter equation (YBE) from representations of the TL algebra.

II. Temperley-Lieb Algebraic Structure for BGRs We first recall a few facts needed below about the representation theory of braid group.

Let an N2 x N2 mat r ix S € End (V <g> V) satisfy the special Yang-Baxter equation or braid

relation

S12S23S12 = S23S12S23 > (2)

where S12 = S ® I, S23 — I ® S, and I is the unit mat r ix in JV-dimensional vector space V.

Let Si € End (V®"" 1 )

Si = / W ® • • • ® j C - 1 * ® S ® J ( , + 2 ) ® • • • ® / ( " - 1 ) . (3)

Each solution of Eq. (3) leads to a representation of n-braid group, i.e., 5,- satisfies

SiSi+1Si = Si+lSiSi+i, SiS^SjSi, if | i - j | > 2 , (4)

1The project supported in part by National Natural Science Foundation of China and by Local Natural Science Foundation of Xinjiang Province

2CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China

452 Lu-Yu WANG, Yu CAI, Han-Ying GUO and Shao-Ming FEI Vol. 25

therefore we simply call the solution S of Eq. (3) BGR. Recently, the work [7] shows that there exist many new BGRs for the 4 x 4 case. In order

to classify these BGRs, we first reveal the algebraic structure of BGR. Let S be a BGR with two distinct eigenvalues, i.e., S satisfies a quadratic reduction relation

( 5 - . A 1 ) ( 5 - A 2 ) = 0, (5)

then S leads to a representation of the Heck algebra.^ The Heck algebra is an associative algebra generated by 1 and </,-, i = 1,2, • • •, n — 1, satisfying

9i9j = 9j9i, I* ~ il > 2 ,

9i9i+\9i = gi+i9i9i+i , 9i = (1 - t)9i + t, where t £ C. The equivalence between the BGRs with quadratic reduction relation and the representations of the Heck algebra is easily proved by defining gi = AiS,-, t = — A2/A1. Therefore in the following discussion, we do not distinguish the BGRs with quadratic reduction relation from the representations of the Heck algebra. Now we consider the connection between S and the TL algebra. Assume that we have a representation 7r of the TL algebra such that

w(ei) = Ui = I^®I^®---®I^-^®U®I(-i+^®---I(-n-1\ ' (7)

where U G End (V ® V) satisfies

U2 = U, (U®I)(I®U)(U®I) = (U®I), {I®U)(U®I)(I®U) = (I®U). (8)

Since S satisfies the quadratic relation

S2 = (Ai + A2)S - AiAa , (9)

we can assume that P € (V ® V)

P = aI + bS, P2 = P, (10)

where a, b EC. By solving Eq. (10) with Eq. (9), we get two solutions

a = -A 2 / (A! - A2), b = l/(Ax - A2) , (11)

a = - A 1 / ( A 2 - A 1 ) , 6 = 1 / ( A 2 - A i ) . (12)

It is obvious that the solutions (11) and (12) can be written in a united form

a = - A 2 / ( A ! - A 2 ) , 6 = l / ( A i - A 2 ) , (13)

where (Ai, A2) is permutation of (Ai, A2). Using the fact that 5,-5,±i5,- = 5,±i5,-5,±i, we can easily show that the matrix P defined by Eq. (10) satisfies

PiPi±iPi ~ a(l - a)Pi = Pi±iPiPi±i - a(l - a)Pi±1 (14)

PiPj^PjPi, | » - j |>2 . (15)

Comparing Eqs (8), (10) and (14) we immediately see that the necessary and sufficient con­dition for Eq. (10) qualifying as a {/-matrix is PiPi±\P{ — a(l — a)Pi — 0. Substituting P by Eqs (10) and (13), we find that for the BGR generating a representation of the TL algebra S has to satisfy the following equation

SiSi+1Si - X(SiSi+1 + Si+lSi) + X2(Si + Si+i) - A3/ = 0 , (16)

where A is an eigenvalue of S. [/-matrix is U — (i/\/AAi)(A — S), q = l / a ( l — a), where Ai is another eigenvalue of S. Since the parameter q depends on the eigenvalues Ai and A2, and A will rely on the parameters appearing in BGRs, q will depend on the multiparameter of BGRs. This topic will be discussed in the following concrete examples again.

According to the condition (16), the BGRs S with quadratic reduction relation can be divided into two kinds. One is that BGRs 5 satisfy the relation (16), and then possess a TL algebraic structure. Another is that BGRs S do not satisfy the relation (16), so they have no TL algebraic structure. In the following we discuss concrete BGRs in Ref. [7] by this classification.

No. 4 Solutions of the Yang-Baxter Equation and Temperley-Lieb Algebraic Structure 453

1) Solutions of BGR with Temperley-Lieb Algebras The first solution of BGRs we examine with condition (16) is

5(i) =

f Xn 0 0

0 0 X1 3

0 X12 (Xn-X13X12)/Xn

\ 0 . 0 0

0

0

0

Xn

There are two distinct eigenvalues for 5(i), Xn is a 3-fold eigenvalue and —(Xi2Xi3)/Xn is a 1-fold eigenvalue, satisfying (5(i) — Xn • -0(5(1) + (Xi2X 1 3 /Xu) • / ) = 0, where I is a 4 x 4 unit matrix. This solution 5 includes the universal i?-matrixof slg(2) Rq by taking X n = q, Xi2 = X13 = 1, 5(i) and the two-parameter solution Rpiq (see Ref. [8]) by Xn — q, X±2 — 1, X13 = q/p. It is obvious that if one takes A = Xn, 5(i) satisfies

SiSi+iSi — Xn(SiSt+i + 5,+i5i) + Xn(S% + 5,+i) — X n = 0,

where 5 is 5(i). The relevant 17-matrix of the TL algebra reads

+ ~ j . t/(i) = - 7 = = = = ( A n - 5 ) , V5=-(- (17)

*kv = \

(18)

k V^12^13 ^11 ' \ /X l2Xl3

where U depends on two complex parameters Xn and X12X13, and q is determined by the parameters Xn,Xu and X13. Another example for solution with the TL algebra is

7 i n 0 0 0 \ 0 0 x12 0 0 -Xn X12 + Xn 0 0 0 0 Xn J

which has (5 — X"n)(5 - X i 2 ) = 0. Suppose that A = Xn, then one may show that

5,-5,.)-i5,- — XLI(5,-5,-+I + 5,+i5i) + X11(5,- + 5,+i) — X l t = 0,

where 5 is 5(2). As we discussed above this BGR gives a representation of the TL algebra. In this case the [/-matrixis given by U(2) = (l/VXnXi2)(Xn— 5(2))- Now y/q = i(y/Xi2/Xn —

y/Xn/Xn).

2) Solutions of BGR Without Temperley-Lieb Algebras We start with the following BGR 5

/ Xn 0 0 0

0 X12

0 0

5(3) =

V

0

- X 1 1 X 3

X12X3 + Xn

0 ^ 1 2 - ^ 3 /

This S corresponds to the two-parameter .R-matrix of super GL(1/1),[9J and the eigenvalues of 5(3) are still two, Xn and X12X13. But both of them now are 2-fold eigenvalues, and 5(3) satisfies equation (5(3) — X12X13 • -0(5(3) — Xn • I) — 0. A straightforward calculation shows that 5(3) does not satisfy the condition (16) for any eigenvalues A of 5(3). So there is no TL algebraic structure in 5(3).

More examples are in order

(Xn 0 0 0 \ (Xn 0 0 Xl7 \

5 ( 4 ) = 0 0 X u

0 Xn 0

0

0 > 5 ( 5 ) =

0 X u — X12

0 x 1 2

X u

0

0

0

\ 0 x6 x6 -Xn) \ 0 0 0 -X12)

454 Lu-Yu WANG, Yu CAI, Han-Ying GUO and Shao-Ming FEI Vol. 25

/ Xn

X3Xi

3(6)

Xi X3X?

S(7) =

Xu

X3Xi —Xn

\ 0 0

/ X$X2X3 — 2

x3 x6

x6

\ x2x3

x, -Xu

x3xl Xn

0

X2

X§X2X3

—XQX2X3

x6

0

0

0

-XnJ

X2

—X$X2X3

XsX2X3

Xe

X2X3

x2

x2 X$X2X3 -

x7~ 7 All of them satisfy quadratic reduction relation and the eigenvalues of them are 2-fold. But they are in no possession of the TL algebraic structures since they do not satisfy the condition (16) by taking A as their eigenvalues respectively.

I I I . The Yang-Baxter izat ion of the B G R We first consider the BGRs possessing a TL algebraic structure. Let U be a representation

of the TL algebra, we put

R{u) = / + f(u)U , (19)

where / is a function of u and u € C is a spectral parameter. Comparing

Ri2(u)R23(u + v)R12{v) = R23(v)Ri2(u + v)R23(u)

with Eq. (8) we see that the necessary and sufficient condition for Eq. (19) being a solution o fYBEis

/ («) + /(«) + y/qf(u)f(v) + / ( « ) / (« + v)f(v) -f(u + v) = 0. (20)

We see that y/q appears in Eq. (20). Therefore the connection between solutions of Eq. (20) and the representations of the TL algebra is given by y/q.

Now we consider some concrete solutions of Eq. (20). 1) One obvious class of solutions of Eq. (20) is

/i(w) = sinh u/ sinh (77 — u), (21)

where 77 depends on q by y/q = 2 cosh rj. As is well known, R(u)

sinh u R(i)(u) = psinh (rj — u) \l + "} (22)

sinh (T? — u)

connects with g-state Potts model,t2,1°] where p is an overall normalization factor. 2) Another class of solutions of Eq. (20) is the one by setting y/q = 2, the corresponding

f(u) reads f2 = —u/(l + u). When we take the BGR S as the permutation P, i.e., P e End(V<g) V) and P : / j ® r = rigi/j, from the above discussion, we can obtain a {/-matrix Uf — 2Ui, UiUi±iUi = Ui, UiUj = UjUi, \i - j \ > 2. The relevant J?-matrix is R(2)(u) = I + fo)U. This .R-matrix is a rational solution for sl(iV). For N = 2, it reads

/ 1 0 0 0 \

0 1/(1+ «) « / ( ! + «) 0 0 u / ( l + u) 1/(1+ «) 0

V 0 0 0 1 /

It is first given by McGuire^ and YangJ6!

R\2)(u)

No. 4 Solutions of the Yang-Baxter Equation and Temperley-Lieb Algebraic Structure 455

3) A third class of solutions of Eq. (20) is obtained by setting y/q •• relevant / (u) is

i /ATA e u - l h{u) =

(Ai-A)/iA/AAT. The

(23) A e« + (A!/A)'

where A and Ai are arbitrary parameters. If [/-matrix is constructed from BGR 5, these A are related eigenvalues. The corresponding R(u) is R(u) oc Ai - e u 5 - 1 + (l/X^S, which is the trigonometric-type solution of YBE getting from Yang-BaxterizationJ11^

Secondly we consider the Yang-Baxterization of BGRs without TL algebraic structure. In this case, the BGRs 5,- satisfy a quadratic equation 5 2 = (Xi+X2)S — XiX2-I- Then we simply put R(u) = (1/A2)5 + Ai e - ° 5 _ 1 , YBE follows immediately. Corresponding to 5(4), 5(5), 5(6) and S(7) which we mentioned above-, the solutions of the Yang-Baxter equation are

/ e - " - l 0 0 0 \

ft(4)(") = 0

0

0

e - u - l

e - " - l

0

V 0 X6(e-« - \)/Xn X6(e-« - 1 ) /X U 1 - e " " /

fi(5)(u) =

Xi2e" X l i

Xl2

0

0

V

X12

0

R(6)(u)

,(X2X3-Xn)e-U+X2l

X2X3 + X2,

X1X1 1X3(1 - e~») X2X3 + X2

n

X1XnX3(l - e~»)

XfXs + X^ \ 0

where a-= [(X2X3 + Xu)e~u - X2n)/{XlX3 + X2

n).

0 0

X12 - Xn X n ( e - » - 1)

Xl2 e-»(X12 - Xn)

X\2

0

XiXn{l-e-») X2X3+X2

n

X2X3

X2X3+X2n

X2n(e-"-l) X2X3 + X2

n

0

X1 7(e-» - 1) ^

Xl2

0

0

X12 - e - « I „ Xl2

X i X u ( l - e - " ) X2X3 + X2

n

X2n(e-»-l)

x2x3 + x2n

xfx3 x2x3 + x2,

0

o\

0

0

a /

fl(7)(u) =

/ an(u) a12(u) ai3(u) ai4(u) \

02l(u) «22(«) a23(u) ^(ll)

a3i(«) 032(11) a33(u) a3i(u)

\ a4i(w) a42(u) a43(u) a44(u) /

where

an(u

Ol2(«

ai4(u

o2i(u

a22(w

fl23(«

= a44 = -(X2X6X2e-« + X2X6Xi - 2)/X3 ,

= a13(«) = a24(«) = a34(«) = X2{e~u - l)X2(e~u - 1),

= X22X3(e-u - 1), a41(u) = X3X

2(e-» - 1),

= a3 i(") = a42(") = a43(u) = X6(e~u - 1),

= a33(u) = [(2 - X2AVY32)e-u + X2X6X

2]/X3 ,

= a32(u) = X2X3X6(l - e~u).

The physical application of these solutions will be discussed in our forthcoming papers.

456 Lu-Yu WANG, Yu CAI, Han-Ying GUO and Shao-Ming FEI Vol. 25

IV. Conclusions and Remarks We conclude with several remarks.

Remark 1. The investigation of the connection between the representation of the Heck algebras (RHA) and that of the TL algebras shows that there exists an equivalence between the algebra generated by Si satisfying Eqs (4), (5) and the algebra generated by Pt- satisfying Eqs (14), (15). Such an equivalence leads a necessary and sufficient condition (16) to judge when the given BGR will have a representation of the TL algebra. Restricting the BGR S to the 4 x 4 matrix, all BGRs (which satisfy quadratic reduction relation) in Ref. [7] show us a very interesting phenomenon. Whether a representation of the TL algebra can be arisen from a given RHA may depend on the multiplicity of the eigenvalues of RHA. Since all examples show that the representation of the TL algebra will arise from those RHA in which one eigenvalue is 1-fold and another is 3-fold. The RHA in which the two eigenvalues are both 2-fold do not have representations of the TL algebra. The reason for such a phenomenon remains to be explored.

Remark 2. From the discussion we see that y/q in the TL algebra depends on the eigenvalues of BGRs. Since the eigenvalues will depend on the parameters which appear in BGRs, the multiparameter BGRs will lead to multiparameter representations of the TL algebra.

Remark 3. Equation (20) shows us that after restricting to the representations of the TL algebra, to solve YBE becomes to solve a function equation (20). The property of concrete representation of the TL algebra will be given by ^/q. Several types of solutions of Eq. (20) are discussed in this paper. It is obviously that to get more types of solution of Eq. (20) will be of significance.

Acknowledgements We are thankful to Profs Z.M. QIU, K. WU, Drs J.H. DAI, Y.Q. LI and J. ZHANG for

helpful discussions. The authors (S.M. FEI and L.Y. WANG) would like to thank CCAST (World Laboratory) for hospitality and support. The work is finished during the Summer Institute sponsored by CCAST and co-sponsored by ITP.

References [1] H.V. Temperley and E.H. Lieb, Proc. R. Soc. A322 (1971) 251. [2] R.J. Baxter, Exactly Solved Models in Statistical Physics, Academic Press, New York (1982). [3] V.G. Drinfeld, ICM Proceedings, Berkeley (1986) 798; M. Jimbo, Commun. Math. Phys. 102

(1986) 537; T. Kohno, Ann. Inst. Fourier. 37 (1987) 139; J. Frohlich, Statistics of Fields, The Yang-Baxter Equation, and Theory of Knots and Links, Cargese Lectures (1987); M. Wadati, T. Deguchi and Y. Akutsu, Phys. Rep. 180 (1989) 248.

[4] V.F.R. Jones, Invent. Math. 72 (1983) 1; Ann. Math. 126 (1987) 335; H. Wenzl, Invent. Math. 92 (1988) 349; R.B. ZHANG, J. Phys. A24 (1991) L535.

[5] C.N. YANG, Phys. Rev. Lett. 19 (1967) 1312. [6] J.B. McGuire, J. Math. Phys. 5 (1964) 622. [7] S.M. FEI, H.Y. GUO and H. SHI, preprint, ASTTP-91-48; CCAST-91-49. [8] A. Schirrmacher, J. Wess and B. Zumino, Z. Phys. C49 (1991) 317. [9] L. Dabrowski and L.Y. WANG, preprint, INFN/AE-91/06.

[10] E.H. Lieb and F.Y. Wu, Phase Transitions and Critical Phenomena, ed. C. Domb and M.S Green, Vol. 1, Academic Press, London (1972) p. 321.

[11] V.F.R. Jones, Commun. Math. Phys. 136 (1989) 195; M.L. GE, Y.S. WU and K. XUE, preprint, ITP-SB-90-02; L.Y. WANG, Braid Group Representations and Their Trigonometric Yang-Baxterization, Ph.D. Thesis, Lanzhou University (1990).