Introduction - Fermilablss.fnal.gov/archive/other/itp-sb-92-04.pdf · ITP-S8-92-04 . New Solutions...

ITP-S8-92-04

New Solutions of the Yang-Baxter Equation Without Additivity

and Birman-Wenzl Algebra with Colour

Mo-Lin Gel and Kang Xue2

~

Abstract

New solutions of the Yang-Baxter equation associated with the fundamental repshy

resentations of A(n gt 1) B e and D without the additivity of spectral parameshy

ters are found in terms of the weight conservation The Birman-Wenzl algebra with

colour is constructed and the explicitmiddot examples are given in the cases of Bn en and

D_

~~~ ) lI~1t

--ijlt~I ~~

-- X fmiddot--middotmiddot- tt~

trn ~lt o~

t= ~ - v

bull --

trltt-- middot~middotII1 ~ ~ ~~ ~ ~---

ITheoretical Physics Division Nankai Institute of Mathematics Tianjin 300071 Peoples Rep~b~middotmiddot lic of China

2Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook New York 11794-3840 USA

1

bull

Introduction

The Yang-Baxter equation (YBE) plays a fundamental role in the integrable systems

in 1+1 dimensions and the exactly soluble models in statistical mechanics [1-4] Nlany

solutions of the YBE

(11)

x( = eta) and y(= etl ) are spectral parameters with additivity have been found

Among them the trigonometric solutions associated with the fundamental represenshy

tations of An En On and Dn were constructed based on the quantum group (QG)

[5J By taking the limit of R(x) the YBE (11) is reduced to the braid relation [6J

(12)

It is well kno~n that the solutions of the braid relation associated with the represenmiddot

tation of the Lie algebra can be constructed from theory of QG [781 and calculated

in terms of the weight conservation and the extended Kauffman diagrammetic techshy

niques [910] It is shown that there exist two types of the braid group representations

(BGRs) associated with the fundamental representations of An En On and Dn stanshy

dard and exotic [1112] and the structure of the Birman-Wenzl (BW) algebra [131 exists for the BGRs not only for the standard ones [8] but also for exotic ones [14J in the cases of En On and Dn The corresponding explicit solutions of the YBE(11 )

can be obtained by the Yang-Baxterization prescription [1516]

On the other hand the solutions of the YBE without additivity have been found

[17] As is known the complicated calculatins were made for deriving the nonmiddot

additivity of spectral parameters for YBE It is not clear what the relation is between

these solutions and the structure of Lie algebra Recently Murakami [18] has found a simple solution of YBE without additivity which can be expressed as a 4x4 matrix

tgt

o 1 (13)

tgt tilgt t1(tl - 1) -tSJ -1

where A and Jl are interpreted in colours and tgt tSJ are coloured parameters In Ref

[19] we solved directly the YBE

R12 ( A Jl )R23 ( A 1 )R12 (Jl 1) = R23(Jl 1 )R12 ( A 1 )R23( A Jl) (14)

2

2

and obtained two solutions

q o l1X(A)

(15)Rr(Ap) =

and q

o l1X(A) (16)Rll(Ap) =

1-1Y(I-) W( A 1-)

_q-l X(1)Y(P)

where X(A)g(A) and Y(A) are arbitrary functions of A and W(AI-) satisfies the

relation

W(A I-)WI- v) = (q - q-l X(I-)Y(I-))W(A v) (I) It is shown in Ref [18] that the solution (13) is a special case of solution (16)

When A = 1- the solutions (15) and (16) will be reduced to the standard and exotic bull

solutions of the braid group respectively In Ref [20] the new hierarchy of solutions

for SU(2) called colored-braid-group representations have been constructed As a

natural extension we will seek other solutions of the YBE(14) associated with the

representations of Lie algebra

fn this paper we will show that there exist the solutions of the YBE(14) associated

with the fundamental representations of Ann gt 1) Bn Cn and Dn When A = 1-

these solutions will be reduced to the corresponding BGRS We construct the BW

algebra with colour which will give the usual BW algebra when all colours are the

same The explicit examples are calculated in the cases of Bn Cn and Dn In sec

II the solutions of the YBE (14) associated with the fundamental representations of

An(n gt 1) are found in terms of the weight conservation There exist two types of

variable-separation solutions that correspond to the standard and exotic ones In sec

III the similar method appearing in sec II is used to discuss the cases of Bn Cn

and Dn In sec IV the BW algebra with colour is constructed It is shown that this

structure exist for Bn Cn and Dn

Solutions of R(A 11-) Associated with An(n gt 1)

The structure of the BGR associated with the fundamental representations of An has

been given in terms of the weight conservation in Ref [12] We need to put the

3

spectral parameters without additivity or colours in this structure only Under the

circumstances we assume that R(x p) has the following form

R(x p) = L U4(X p)e44 rege44 +E W(4b)(X p)e44 regebb +LP(4b)(X p)e4bregeb4 (21) 4 4ltb 4~b

where (e4b)cci = 6acDbd a b c df[_N _Nl +1 Nl] N =n+l Ua() W(4b) (x p)

and p(4b)(X p) are determined parameters by solving the YBE (14)

Substituting (21) into (14) we obtain the relations

p(bc) (x V)p(4C)(p v) p(bc)(p V)p(4C) (x v) (a lt b)-p(4b)(X p )p(4C) (x v) - p(4b) (x V)p(4C)(X p) (b lt c) p(4b) (X ~)U4(x v) - p(4b)(X v)U4(X p) (a b) Ub (x v)p(4b)(p v) - Ub(p V)p(4b) (x v) (a b) W(4b) (x p )Ub(x v)W(4b) (p~ v) + p(4b) (A p) W(4b) (x v )p(b4) (p v)

- Ub(p v)W(4b)(X v)Ub(X p) (a lt b) U4(X p)W(4b)(X v)U4(p v) W(4b)(p v )U4(X v)W(4b)(X p)-

p(4b)(p V)W(4b)(X V)p(b4)(X p) (a lt b)+ (22)

and

p(4b) (x p)W(4C)(X V)W(b4)(p v) _ W(bc) (p V)p(4b) (x v)W(4C) (x p) (b lt a lt c) W(4b) (x p)W(bc) (x v )p(4b)(-l v) _ W(bc)(-l v)p(4b)( v)lV(4c) (a lt b lt c) W(4b) (x p )p(bc) (x v)W(4C) (p v) _ W(bc) (p v)W(4b) (x v )p(bc) (x p) (a lt blt c) W(4b) (x p )p(bc) (x v)W(4C)(p v) _ p(bc)(p V)W(4C)(X v)W(cb)(X p) (a lt c lt b) W(4b) (x p) W(bc) (x v)W(4b)(p v) + p(4b) (x p)W(4b)(X V)p(b4)(p v)

=W(bc)(p v)W(4b)(X v)nl(bc) (x p) + p(bc)(p v)W(4C) (x v)pcb) (x p) (altbltc) (23)

The relations (23) exist for n gt 1 only When n = 1 the solutions of the YBE (14)

have been discussed in Ref [19] So we consider the cases of n gt 1 here

Let us consider variable-separation solutions of Eqs (22) and (23) Suppose

U4( p) - g14(x)f14(p )U4

p(4b)JC) _ g2(4b)(x) f2(4b)p) (24)

Eq (22) leads to the following constraints

9i(x) = 9[4b)(x) = 9b() f(p) = fl4b)(P) = f4(P)

(U4 + Ub-1)lo4(p)4(JC) - 9b(p)lb(p)) =0

~V(4b) (x p) vV(4b)(pv) = (U494(P )4 (p) - Ul9b(p )fb(p))W(4b) (x v)

(25)

4

3

By solving Eqs (23) and (25) up to a common factor 1 (p )g1 (p) we obtain the

solutions

(i) Ua(Ap) = qga(A)gI(p)p(ab)(Ap) = gb(A)gI(p)

w(ab)(A p) = (q - q-l)gb(A)gI(P)ha(A)hb1(A)hI(p)hb(p)

(26)

and

(ii)Ua(Ap) - Uaga(A)gfj4Uafq_q-l

p(ab)(Ap) _ gb(A)gaa- 1(p)

w(ab)(Ap) _ (q - q-l)gb(A)gb1(P)ha(A)hb 1(A)hI(p)hb(p)

(27)

where q is an arbitrary parameter ga(A) and ha(A) are arbitrary f~nctions of A When

A= p the solutions (26) and (27) are reduced to the standard and exotic solutions

of the braid relation

Ua(A A) = Ua Ua =q or Uafq _q-l

p(ab)(A A) = gb(A)g(l) w(ab)(A A) = q _ q-l

p(ab)(A A)p(ba)(A A) = 1 (28)

Solutions of R(A Jl) Associated with Bn en and

Dn

According to the similar consideration in sec II the R(A p) associated with the

fundamental representations of Bn en and Dn has the following structure

R(A Jl) = L Ua(A p)eaa reg eaa +L w(ab)(A p)eaa reg ebb + L p(ab)(A p)eab reg eba a altb acentb

+ (ab) to ~ q eab ICJ e-a-b (31) ab

where a bf[- N-l N-l] iV = 2n + 1 and 2n for Bn and Cn(Dn) respectively In

(31) we assume that

q(ab) (A p) la=plusmnb = q~~~ labgto = q~~~) la~Obgtlal = q(ab) (A p) Ib~Oagt Ibl = 0 (32)

5

Substituting (31) and (32) into (14) we still obtain Eqs (22) and 23) in the cases

of a + b i= 0 a + c i= 0 and b + c i= O by solving these equations we determine the

parameters Ua()J-l)(a i= O)p(ab)(AJ-l)(a + b i= 0) and lV(ab)()J-l)(a + b i= 0) given

by

Ua()p) - qga())gl(p)(a i= 0)

p(ab)()p) _ go())gl (p)(a +bF 0)

WCob)()p) _ (q - q-l)gb())gbl(p)(a +bF 0) (33)

where for simplicity we have taken Uo = q and h()) = 1 The other constraints

given by the YBE (14) are different for Bn en and Dn In the following let us first

consider the case of Bn

(i) The case of Bn Under the case besides Eqs (22) and (23) the YBE (14)

gives riseto the constraints

UO(A Il)Uo() 1I) = pCo-o)() lI)p(Oo)()p)(a gt 0)

p(ao)(AlI)p(-aO)(plI) = Uo(p lI)Uo(A lI)(a gt 0)

p(Oa) (A Il )q(o-a) (A 1I )p(a-a)(Il 1I) + w(Oa) (A Il )p(aO) (A 1I )q(o-a) (J-l 1I )

= p(oO)(p lI)q(o-a)(A lI)Ua(A Il)(a gt 0)

p(o-a)(A p)q(O-a)(A p)U-o(p 1I) = WC-aO)(p lI)p(O-o)(A lI)q(O-o)(A Il)

+p(-ao) (p 1I )q(o-a)() Il )p(a-o) (A p)( a gt 0)

q(-aO)(A p)q(O-a)(A lI)p(O-a)(A 1I) + W(Oa)() p)p(aO)(A lI)w(-aO)(p 1I)

=p(aO) (p lI)w(-ao) (A lI)W(Oa)(A p)(a gt 0)

p(O-a) (A p)w(Oa)(A lI)WC-aO)(p 1I) = q(-aO)(p lI)q(O-o)(A lI)p(amiddotO) (A p)

+w(Oa)(p lI)p(O-a) (A lI)w(Oa)(A p)(a gt 0)

q(o-a)(A p)q(9-b)(A lI)p(-o-b)(p v) +q(O-b)() p)p(b-a)() v)w(-b-a)(p 1I) = 0(0 lt a lt b)

q(O-a)(p v)q(a-b)(A v)p(ba)(Ap) + q(O-b)(IllI)w(ab)(AIl) = 0(0 lt a lt b)

q(-ba)(Ap)q(-aO)(AlI)p(oO)(IllI) +q(-bO(AIl)p(Oa)(AlI)W(Oa)(plI) = 0(0 lt a lt b)

q(-ba)(p lI)q(-ao)(A lI)p(o-a)(Ap) +q(-bO)(p lI)p(-aO) (A lI)wlt-aO)(Ap) = 0(0 lt a lt b)

q(-aO)(Ap)q(O-b)(A lI)p(O-b)(p II) +q(-a-b)(Ap)p(bO)(A lI)W(-bO)(p II) = O(a b gt 0)

q(-ao)(p lI)q(O-b)(A lI)p(bO)(Ap) + q(-a-b)(p lI)p(O-b)(A lI)W(Ob)() p) = O(a b gt 0)

q(O-a) (A J-l )Ua(A 1I) W( -aa) (p II) +L q(O-b) (A p)W(bo) () II )q(-b-a) (Il II) blta

+Uo(Ap)w(Oa)(AII)q(O-a)(plI) = l-v(oa)(plI)q(o-a)(AlI)Ua(Il)(a gt 0)

~V( -aO) () J-l )q(O-a)() II)U-a (p II) = q(O-a) (Il lI)U -a () II) lV(-aa) () Il)

6

+I q(O-b)(Jl v)w(-e-b)(A v)q(-b-e)(A Jl) + Uo(Jl v)w(-eO)(A v)q(O-e)(A Jl)(a gt0) blte

(3 f)and

p(e-e)(AV)U_o(JlV) = p(-be)(Jlv)p(b-e)(AV)(O lt ab lt a)

p(eb) (A p)p(ete-b)(A v) =p(ot-o)(A V)Uo(A p)(O lt a b lt a)

q(-b-o)(A Jl )q(e-6) (A v)p(-o-b)(p v) +w(-b6)(A p )p(b-o) (A v) w(-b-e)(Jl v)

= p(b-o)(Jl v)w(-bt-e)(A v)w(-eb)(Ap)(O lt a lt b)

p(e-b)(Jl)w(-bte)(Jl v)w(eb)(A v) = q(-be)(Jl v)q(e-b)(A V)p(be) (A Jl)

+W(-btb)(Jl v)p(o-b)(A v)w(eb)(A Jl )(0 lt a lt b)

q(-eb)(A Jl )qlt-b-e) (A v)p(b-o)(Jl v) + w(-oe) (A Jl )p(ob) (A v) w(-eb) (Jl v)

pC-b-e)(A Jl)wC-be)(A v)w(-e-b)(Jl v) = q(-eb)(Jl v)q(-b-e)(A v)p(e-b)(A Jl)

+W( ~eo) (jamp v )p(-b-e)(A v)W(-be) (A Jl )(0 lt b lt a)

q(e-b)(A Jl)Ub(A v)q(-be)(Jl v) + I q(e-c)(A Jl)q(-ce)(Jl ~)W(cb)(A v) eltcltb

+p(e-e)(A Jl )w(eb)( v )p(-ee)(Jl v) = p(-eb(Jl v) w(eb)(A v )p(b-e)(A Jl )(0 lt a lt b)

r(tl~V) p(-be(A Jl) w(eb) (A v) = p(e-b(Jl v)p(e-e) (Jl v) w(-b-e)( A v )p(-ee)( A Jl)

+q(e-b)(p V)U_b(AV)q(-bo)(AJl) + I q(e-c)(Jl v)W(-b-c)(A v)q(-ce)(AJl)(O ltqltb) eltcltb

q(e-b)(A Jl)Ub(A v)W(-bb)(p v) + p(e-e)(A Jl)w(eb)(A v)q(-e-b)(p v)

= yenv(~eb)(Jlv)q(e-b)(AV)Ub(AJl)(O lt a lt b)

w( -be)(A Jl )q(e-b)(A v)U-b(Jl v) = p(e-o) (Jl v)Ivlt-b-e)(A~ v )q(-c-b) (Jl v)

+q(e-b)(Jl v)U-b(A v )W(-bb)(A Jl )(0 lt a lt b) (3middot5)

Substituting (33) into (34) and (35) we obtain the solutions as follows

UO(AJl) =go (A)gO-teJl) p(e-e)(AJl) =q-lg_o(A)gl(p)

w(-ee)(AJl) = (q - q-l)(l - q-2e+l)ge(A)g1(Jl)

q(O-e)(AJl) = -(q - q-l) q-e+l2go(A)g1(Jl)(a gt 0)

q(-eO)(AJl) = -(q - q-l)q-e+l2ge (A)gol(Jl)(a gt 0)

q(o-b)(AJl) = -(q - q-l)qo-bg_e(A)gbl(Jl)(O lt a lt b)

q(-be)(AJl) = -(q - q-l)qe-bgb(A)g(Jl)(O lt a lt b)

q(-e-b)(A Jl) = -(q - q-l)q-e-b+lge(A)gbl(Jl)(a b lt 0)

(36)

7

4

where g() should be satisfied the relation

gCl()g-Cl() is independent of index a (37)

(ii) The cases of en and Dn Under the case th~ YBE (14) leads to the algebraic

equations (33) and (35) The distinction between Cn and Dn is that the element

W(-1212)( p) of R( p) - matrix is equal to nonzero and zero for en and Dn respectively By solving Eqs (35) directly we obtain the solutions

UCl(p) - qgCl()gl(p) (CIb)(p) = gb()gl(p) (a +bf 0)

Wltcab) (A p) _ (q - q-l)gb()gl(p)(a +bf 0) (ca-ca) (A p) = q-lg_ca()gl(p)

q(ca-b)(A p) _ -(q - q-l)qCl-bg_Cl (A)gb1(Jl) (0 lt a lt b)

q(-bca)(p) _ _ (q _ q-l) qca-bgb(A)g(p) (0 lt a lt b)

wlt-aa)( Jl) _ (q- q-l)(1 +q-2ca-l)gca(A)gl(p) for en (q - q-l )(1 - q-2C1+1 )gCl( A)g1 (Jl) -for Dn

(q - q-l )q-a-b-lga(A)gb 1(Jl) for Cnq(-a-b)( Jl) = (3 8) -(q - q-l)q-ca-b+lga (A)gb1(Jl) for D1l bull

where a bt( -n + 12 -n +32 n - 12) and ga(A) are also constrained by the

relation (37)

We emphasize that the solutions (33) (36) and (38) are one type of the solutions

associated with the fundamental representations of Bn en and Dn only which will - ~

be reduced to the standard BGRs when = p It is worth mentioning that there

also exists another type of the solutions corresponding to the exotic BGRs which

can be obtained by changing the coefficients depending on and Jl of the elements of

R( Jl )-matrix Considering that the calculation is lengthy and complicated we will

not write it here

BW Algebra With Colour

In the R(AJl) satisfied the YBE (14) the parameters and Jl can be understood

as the colours According to this consideration we try to construct BV algeba

with colour As is known the usual BW algebra is generated by the unit I the

braid operators Gi and the monoid operators Ei and depends on two independent

parameters m and e [13] In order to colour the usual BW algebra we introduce

8

the operators li(Ap)RGi(Ap) and Ei(Ap) instead of IGi and Ei Gi(Ap) and

li(A p) satisfy the relations

G( A p )Gi +1(A II )G(p II) - Gi+l CP II )G(A II )Gi+1(A p)

G(Ap)Gj (lIp) - Gj (lIp)G(Ap) Ii - il ~ 2 (41)

and

Ii(A P)1+1 (A II )I(p II) - 1+1 (p II )1(A II )+1 (A p)

Ii(Ap)Ij(lIp) - Ii(Ap) Ii - il ~ 2

1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)

where I is unit We define the following algebraic relations

Ei(Ap) - m-1(G(Ap) +Gi1(pA)) - Ii(Ap)

E(Ap)Gi(pA) - G(Ap)Ei(pA) = t-1E(Ap)Ii(pA) = t-1I i(Ap)Ei(pA)

Ei(A Jl )Ei+1(A II )Ei(p II) - li( A p )1+1 (A II )E(p II)

- E(A P )1+1(A II )(p 1I) = 1+1 (p 1I )Ei(A 1I )+1(A p)

E+1 (p II )Ei(A 1I )E+1(A Jt) - 1+1 (Jt II )1(A II )E+1(A p)

- Ei+l(Jt 1I )(A 1I )1+1 = 1(A Jt )E+1(A II )I(Jt II)

Gat A Jt )Gi+1(A II )Ei(Jt 1I) - Ei+1(Jt 1I )G(A 1I )G+1 (A Jt)

- Ei+l (Jt 1I )Ei(A II )Ii+l (A Jt) = Ii( A Jt )E+1 (A II )Ei(Jt 1I)

Ei(A Jt)G+l(A lI)G(Jt 1I) - Gi+1(Jt lI)G(A lI)Ei+l(A p)

- Ei(A Jt )E+l (A 1I )li(Jt 1I) = 1+1 (Jt 1I )Ei(A 1I )Ei+1 (A Jt)

G( A Jt )1+1 (A 1I )I(Jt 1I) - 1+1 (Jt 1I )G(A II )+1 (A Jt) = I i(A Jt )+1(A 1I )Gi(Jt 1I)

G+I(Jt II)I(A lI)li+l(~ Jt) - I(A Jt)G+l I (Jt 1I) = I+l(Jt lI)l(A lI)G+l(A Jt) (43)

where m l are arbitrary parameters independing on the colours Gil (p A) is the

inverse of G(p A) In terms of the algebraic relations (41)-(43) it is easy to derive

other algebraic relations given by

G( A p )(p A) - li(A p )G(p A)

Ei(~ p) Ii (Jt A) - li(A P)Ei(P A)

Ei(A Jt )Ei(Jt A) - (m-l(pound +pound-1) - 1)E(A p )Ii(p A)

G(A Jt )Gi(Jt A) - m(G(A p) +pound-1 E(A p )(p A) + I

Gi ( A p )Gi (p A)Gi ( A Jt) - (m + pound-1)Gi ( A p )G(p A)Ii (A Jl )

9

-(i-1m + 1)Gi(A Jl) + i-I li(A Jl)

Gi(AJl)Ei+l (AII)Ei(JlII) - Gi+l(IIJl)Ei(AII)li+l(AJl) = li(AJl)G~I(IIA)Ei(JlII) Gi+l (Jl II )Ei(A II )Ei+1(A Jl) - Gil (Jl II)Ei+1 (A II )li(Jl II) = 1+1 (Jl II)Gl( II A )Ei+l (A Jl)

Ei(A Jl )Ei+1 (A II )Gi(Jl II) - E(A Jl )G-Jl (II A)li(Jl II) = li+l (p II )Ei(A II )G1 (p II)

Ei+l(PII)Ei(AII)Gi+l(Ap) - Ei+l(PII)Gl(lIp) = l i(Ap)Ei+1(AII)GI(IIp)

Ei(Ap)Gi+1(A1I)Ei(plI) - lIi+1(pII)Ei(AII)I+1(Ap)

Ei+1 (p II )Gi(A II )Ei+l (A p) - t1i(A p )Ei+l (A II )li(p II)

Gi(A p)Ei+1(A II )Gi(p II) - G-1 (II P )Ei(A II)G-Jl (p A)

Gi+1(p II )Ei(A II )Gi+1(A p) - G-I(p A)Ei+1(A II )G-l(lI p)

Ei(A p)li+l(A II)Ei(p II) - (m-1(1 + 1-1) - I)Ei(Ap)li+l(A 1I)li(p II)

Ei+l (p v)li(A II )Ei+1(A p) - (m- 1(1 + i-I) - 1)li+l (p II )li(A II )Ei+1 ( A p)

Gi(A p)li+l (A II )Gi(p II) - (m(Gi(A p) + i-IEi(A praquo + li(A praquoli+l (A II )li(p II)

Gi+l (p II )li(A II )Gi+l (A p) - (m(Gi+l (p II) + i-IEi+l (p IIraquo) + li+l (p II) )li (A II )li+l (A Jl)

(44)

For reason that the algebraic relations (41 )-(44) will be reduced to the BV

algebra when A = p = II [13] we shall call the algebra generated by the operators

li(A p) Gi(A p) and E(A p) the BW algebra with colour In the following we will

show that the R(Ap) associated with the fundamental representations of Bnen and

Dn admits this structure

Before doing so we express these generators as the tensor product

Ai(A p) = 1(1)(Al) reg reg l(i-l)(Ai_l) reg A(A p) reg l(i+2)(Ai+l) reg reg l(n)(An) (45)

where A(Ap) stands for I(Ap)G(Ap) and E(Ap)I(j)(Aj) is the iInit denoting

by color Aj In terms of the method appearing in Ref [69] we introduce the

diagrammatic symbols defined by

) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10

the algebraic relations (41)-(44) can be represented by Figs 1-4 (the coefficients are

omitted) respectively

--)

FIG 1 Yang-Baxter algebra with colour

A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap

FIG 2 The operator I(A p) satisfies the relations

11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I

) shyI -~~~ I I bull

I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot

_Imiddot middotmiddot

) P bull- I f middot ( ~

A P III middotmiddotmiddotmiddotmiddotmiddotmiddot1

I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A

AIIllI II lmiddotmiddotmiddotmiddot I

FIG 3 Some algebraic relations given by (43)

A P II

XJ - (-t 1

i

~ po J I )(-middotmiddotl

J ~

~

~ p II A J II l) I t

t

-- J _ 1middotmiddotll

I f iI P ~

) 1-4 II x j4 Ii A ampI

I r -1

middotbullbullmiddotmiddotshyXI f ~ ~ I middotmiddot 1

=

I _ -I J

I ~ I

v)A A II )4 -


As is known the linear representation of the generator R(A p)(I(A p) and E(A p))

is the R(A p)-matrix on the space V(A)regV(p)(R(Ap) V(A)regV(p) ~ V(JL)reg V(A))

The R(A p)-matrix associated with the fundamental representations of Bn en and

Dn has been given in Sec III The corresponding R-1 (JL A) has the form

R(p A) - L U~(p A)eaa reg eQ4 + L w(ab)(JL A) reg ebb + L p(ab)(JL A)eab reg eba a agtb aeb

+ L qab)(p A)eab ~ e-a-b) (47) ab

12

and

q(IIb)(p ) latb = q(IIb)(p )Iablto = q(ab) I lt0 = q(ab) I bSO = 0 (48) Olaquoa) 1Ilaquob)

The nonzero elements of the fl- 1(p) are given by

U~(p A) = q-lga(A)gl(p)(a =F 0) U~(p Al =go(A)gol(p)

w(a6)(pA) = -(q - q-l)gb(A)gbl(p)(a + b =F 0)

w(a-a)(p A) = -(q - q-l )(1 - q2a-lg_a(A)g(p)(a gt 0)

p(ab)(pA) = gb(A)gl(p)(a+ b =F O)pt(-aa)(pA) = qga()g(J)

q(aO)(pA) = (q_q-l)qa-l2g_a(A)gol(p)(a gt 0)

q(Oa)(p A) = (q - q-l)qA-l2gb(A)g_A -l(p)(a gt 0)

q(-ba)(p A) = (q - q-l)q-b+lg(A)g_a-1(0 lt b lt a)

qt(a-b)(JA) = (q - q-l)q-b+2g_a(A)g-1(p)(0 lt b lt a)

q(a)(J~) = (q_q-l)qa+-lg_a(A)g-I( m)(abgt 0) (49)

for En and

U~(p) = q-lgll(A)ga-1(J) w(ab)(J) = -(q - q-l)g()gbl(p)(a + b =F 0)

p(ab)(p A) =g(A)gl(p)(a +b =F O)pt(-aqga()g_a -1(J )

a++l pound r C q(a) (p A) = (q - q-l )g-a(A)g-b-l(p) ~~b-l r degD n (a b gt 0)

q lor n

w(a-a) ( ) = ~( _ -1) () -1 ( ) (1 +q2A+l) for Cn (410)J q q g-a g-a J (1 _ q2a-l) for Dn

for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)

from 5th equation in the relation (44) we obtain

(412) a

Substituting (411) and (412) into the first equation in (43) we derive to E( J) in

the following form

(413) ab

13

where EtJ = 1 for Bn and Dn ~ = l(a lt 0) or -l(a gt 0) for Cn and

(414)

a+12(altO)

_ 0) ~ B d D _ a - 12 (a lt 0)( - lor nann a - (415)a a a= a +12 (a gt 0) for en

a -12(a gt 0)

By the direct calculation we can prove that the matrices 1(gt p) G(gt p) and E(gt p)

given by (411)-(415) satisfy the algebraic relations (41)-(43) It is shown that the

R(gt p )-matrix associated with the fundamental representations of Bn Cn and Dn admits the structure of the BW algebra with colour

14

Discussion In this paper we have derived new solutions of the YBE (14) associated

with the fundamental representations of An(n gt 1)BC and D and constructed

the BW algebra with colour It has been shown that the R( A Jl) for B C and Dn

admits this algebraic structure

It is worth notice that there also exist solutions of the YBE (14) associated with

other representations of Lie algebra but which can not give rise to the BW algebra

with colour This situation will appear in the high dimensions

We would like to point out that starting from the R(A Jl )-matrix we can construct

second Yang-Baxterization prescription to generate the solutions of YBE (11) if

we regard the parameters A and Jl as the colours In this prescription the solutions of

YBE (11) will satisfy the initial copndition the unitarity condition etc The details

will ~e discussed in a following paper

Acknowledgements

We wish to express our sincere thanks to professor CN Yang for drawing attenshy

tion to this subject and much encouragement

ML Ge is partly supported by the National Science Foundation of China K

Xue is supported by the C M Cha Fellowship through the Committee for Educational

Exchange with China

15

References

1 CN Yang Phys Rev Lett 19 (1967) 1312 Phys Rev 168 (1968) 1902

2 RJ Baxter Exactly Solved Models in Statistical 11echanics (Academic Lonshy

don 1982)

3 LD Faddeev Integrable Models in I + I)-dimensional Quantum Field Theory

(Les Houches Sesio XXXIX 1982) p 536

4 M Jimbo Braid Group Knot Theory and Statistical Mechanics edited by CN

Yang and ML Ge (World Scientific Singapore 1989) p 111

5 M Jimbo Commun Math Phys 102 (1986) 537

6 Y Akutsu and M Wadati J Phys Soc Japan 56 (1987) 3039 M Wadati T

Deguchi and Y Akutgu Phys Rep 180 (1989) 247

7 VG Drinfeld Sovit Math Dod 32 (1985) 254 Quantum Group IC1f

Proceedings Berkeley 1986 p 788

8 N Yu Reshetikhin LOMI E-4-87 E-14-87 (1988)

9 VG Turaev Invent Math 92 (1988) 527

10 ML Ge LY Wang K Xue and YS Wu Int J Mod Phys 4 (1989) 3351

ML Ge YQ Li and K Xue J Phys A 23 (1980) 605 619

11 HC Lee and M Couture Chalk River preprint Canada (1989)

12 ML Ge and K Xue J Math Phys 32 (1991) 1301

13 J Birman and H Wenzl Osaka J Math 24 (1987) 745

14 Y Cheng ML Ge and K Xue Commun Math Phys 136 91991) 195

15 ML Ge YS Wu and K Xue Int J Mod Phys A 6 91991) 3735

16 ML Ge GC Liu and I Xue J Phys A 24 (1991) 2679

17 H Au -Yang B~1 McCoy JHH Perk S Tang and ML Yan Phys Lett A

123 (1987) 1219 BM Mccoy JHH Perk and CH Sah Phys Lett A 125

(19887) 9 RJ Baxter JHH Perk and H Au-Yang Phys Lett A 128 (1988)

138

16

18 J Murakami A state model for the multi-varible Alexander polynomial preprint Osaka 1990 Talk at Int Workshop on Quantum Group (Euler International

Mathematical Institute Leningrad 2-18 December 1980)

19 ML Ge and K Xue J Phys A 24 (1991) L89S

20 Y Akusu and T Deguchi Phys Rev Lett 67 (1991) 777

17

1

bull

Introduction

The Yang-Baxter equation (YBE) plays a fundamental role in the integrable systems

in 1+1 dimensions and the exactly soluble models in statistical mechanics [1-4] Nlany

solutions of the YBE

(11)

x( = eta) and y(= etl ) are spectral parameters with additivity have been found

Among them the trigonometric solutions associated with the fundamental represenshy

tations of An En On and Dn were constructed based on the quantum group (QG)

[5J By taking the limit of R(x) the YBE (11) is reduced to the braid relation [6J

(12)

It is well kno~n that the solutions of the braid relation associated with the represenmiddot

tation of the Lie algebra can be constructed from theory of QG [781 and calculated

in terms of the weight conservation and the extended Kauffman diagrammetic techshy

niques [910] It is shown that there exist two types of the braid group representations

(BGRs) associated with the fundamental representations of An En On and Dn stanshy

dard and exotic [1112] and the structure of the Birman-Wenzl (BW) algebra [131 exists for the BGRs not only for the standard ones [8] but also for exotic ones [14J in the cases of En On and Dn The corresponding explicit solutions of the YBE(11 )

can be obtained by the Yang-Baxterization prescription [1516]

On the other hand the solutions of the YBE without additivity have been found

[17] As is known the complicated calculatins were made for deriving the nonmiddot

additivity of spectral parameters for YBE It is not clear what the relation is between

these solutions and the structure of Lie algebra Recently Murakami [18] has found a simple solution of YBE without additivity which can be expressed as a 4x4 matrix

tgt

o 1 (13)

tgt tilgt t1(tl - 1) -tSJ -1

where A and Jl are interpreted in colours and tgt tSJ are coloured parameters In Ref

[19] we solved directly the YBE

R12 ( A Jl )R23 ( A 1 )R12 (Jl 1) = R23(Jl 1 )R12 ( A 1 )R23( A Jl) (14)

2

2


q o l1X(A)

(15)Rr(Ap) =

and q


1-1Y(I-) W( A 1-)

_q-l X(1)Y(P)


relation





















3










and






U4( p) - g14(x)f14(p )U4

p(4b)JC) _ g2(4b)(x) f2(4b)p) (24)


9i(x) = 9[4b)(x) = 9b() f(p) = fl4b)(P) = f4(P)

(U4 + Ub-1)lo4(p)4(JC) - 9b(p)lb(p)) =0


(25)

4

3


solutions



(26)

and




(27)






p(ab)(A A)p(ba)(A A) = 1 (28)


Dn






(31) we assume that


5




by

Ua()p) - qga())gl(p)(a i= 0)

p(ab)()p) _ go())gl (p)(a +bF 0)


























6


(3 f)and


























(36)

7

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17

2


q o l1X(A)

(15)Rr(Ap) =

and q


1-1Y(I-) W( A 1-)

_q-l X(1)Y(P)


relation





















3










and






U4( p) - g14(x)f14(p )U4

p(4b)JC) _ g2(4b)(x) f2(4b)p) (24)


9i(x) = 9[4b)(x) = 9b() f(p) = fl4b)(P) = f4(P)

(U4 + Ub-1)lo4(p)4(JC) - 9b(p)lb(p)) =0


(25)

4

3


solutions



(26)

and




(27)






p(ab)(A A)p(ba)(A A) = 1 (28)


Dn






(31) we assume that


5




by

Ua()p) - qga())gl(p)(a i= 0)

p(ab)()p) _ go())gl (p)(a +bF 0)


























6


(3 f)and


























(36)

7

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17










and






U4( p) - g14(x)f14(p )U4

p(4b)JC) _ g2(4b)(x) f2(4b)p) (24)


9i(x) = 9[4b)(x) = 9b() f(p) = fl4b)(P) = f4(P)

(U4 + Ub-1)lo4(p)4(JC) - 9b(p)lb(p)) =0


(25)

4

3


solutions



(26)

and




(27)






p(ab)(A A)p(ba)(A A) = 1 (28)


Dn






(31) we assume that


5




by

Ua()p) - qga())gl(p)(a i= 0)

p(ab)()p) _ go())gl (p)(a +bF 0)


























6


(3 f)and


























(36)

7

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17

3


solutions



(26)

and




(27)






p(ab)(A A)p(ba)(A A) = 1 (28)


Dn






(31) we assume that


5




by

Ua()p) - qga())gl(p)(a i= 0)

p(ab)()p) _ go())gl (p)(a +bF 0)


























6


(3 f)and


























(36)

7

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17




by

Ua()p) - qga())gl(p)(a i= 0)

p(ab)()p) _ go())gl (p)(a +bF 0)


























6


(3 f)and


























(36)

7

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17


(3 f)and


























(36)

7

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17

4













relation (37)







not write it here







8





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17





and



1(A p )I(p A) - Ii(A A) = Ii(p p) = I (42)






















9













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17













(44)











) P ) p ~

GCAgt = ~ ft A)A A

A )A

(j-I~lI) = X JA A

(46)

10



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17



--)


A fo I

0 J jJ

i I J

~

~

p V

) p ) P)4 ~ (I

- -shy - - I

h fo X ~ )Ap


11

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17

) JI ~

-I

~ P JJ - I

I f

I II A

-I

v P A

y) I

~ p

) P II I I


I

liP

~ P AI

I - ~ = ~~

v JA-

p vl () = aJ J4 A

--

11

- middot




I

J P

=

~

--

A v middot J 1shy

~ ~f 1 bull

)I )J- A



A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17


A P II

XJ - (-t 1

i


J ~

~


t


I f iI P ~


I r -1


=

I _ -I J

I ~ I

v)A A II )4 -








12

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17

and












for En and




q lor n


for Cn and Dn

Setting

G(Ap) = (-1)12fl(Ap)

(411)


(412) a


the following form

(413) ab

13


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17


(414)

a+12(altO)


a -12(a gt 0)




14













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17













Acknowledgements





Exchange with China

15

References



don 1982)























138

16





17

References



don 1982)























138

16





17





17

Introduction - Fermilablss.fnal.gov/archive/other/itp-sb-92-04.pdf · ITP-S8-92-04 . New Solutions...

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Transcript of Introduction - Fermilablss.fnal.gov/archive/other/itp-sb-92-04.pdf · ITP-S8-92-04 . New Solutions...