Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices

Quantum algebra deforming maps, Clebsch–Gordan coefficients,coproducts, R and U matricesT. L. Curtright, G. I. Ghandour, and C. K. Zachos Citation: J. Math. Phys. 32, 676 (1991); doi: 10.1063/1.529410 View online: http://dx.doi.org/10.1063/1.529410 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v32/i3 Published by the American Institute of Physics. Related ArticlesCritical scaling dimension of D-module representations of N = 4,7,8 superconformal algebras and constraints onsuperconformal mechanics J. Math. Phys. 53, 103518 (2012) Levinson's theorem for graphs II J. Math. Phys. 53, 102207 (2012) Universal vertex-IRF transformation for quantum affine algebras J. Math. Phys. 53, 103515 (2012) Inversion operators in finite phase plane J. Math. Phys. 53, 103514 (2012) On left Hopf algebras within the framework of inhomogeneous quantum groups for particle algebras J. Math. Phys. 53, 102104 (2012) Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Quantum algebra deforming maps, Clebsch-Gordan coefficients, coproducts, R and U matrices

T. L. Curtright School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540 and Department of Physics, University of Miami, Box 248046, Coral Gables, Florida 33 I24

G. I. Ghandour Department of Mathematics, Ukversity of Miami, Box 249085, Coral Gables, Florida 33124

C. K. Zachos High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439

(Received 29 March 1990; accepted for publication 12 September 1990)

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch-Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang-Baxterization of U as well as R. All this is explicitly illustrated for the tensor product l/2 epj of SU( 2)4 using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.

I. INTRODUCTION ON QUANTUM ALGEBRA DEFORMING MAPS

Quantum algebras,‘-’ whose applications in physics are presently rather technical but nonetheless very promising, possess an important and useful representation theory4,5 virtually in complete correspondence to that of their classical limit algebras. In a recent study,’ a general method was de- veloped to find invertible quantum maps Q of classical Lie algebra generators g to their quantum-deformed correspon- dents G = Q(g). This construction codifies and system- atizes the representation theory of the q-algebra by direct reference to the representation theory of the underlying classical Lie algebra. This quantum mapping technique was illustrated in Ref. 6 for several deformations of SU(2). An earlier example for SU( 2)s was given by Jimbo’ (also see Ref. 7).

As pointed out in some of this previous work,6*7 invertible maps automatically induce a comultiplication A on the corresponding deformation Q by direct referral to the underlying classical Lie algebra. However, this comultiplication appears different than the usual simple comultiplication in- vented by the originators of the best-known SU (2) deformation.’ Nevertheless, the comultiplications are all related: they are equivalent via similarity transformations U. These transformations may be expressed either in terms of Clebsch-Gordan coefficients, or as “universal” solutions of operator-valued difference equations. In fact, a manifold of different comultiplication rules, including the “double,” are related in this manner, which we detail below in some gener- ality. The similarity transformations for this involve the universal R matrix that is shown below to be built out of U’s.

We choose to start our discussion with the quantum

maps for SU(2),, the Drinfeld-Jimbo deformation of SU(2). For pedagogical specificity, we will initially illustrate all our results with simple tensor products such as 2 o 2 and 2 o 3. Let us begin by briefly reviewing some properties of quantum maps.

Consider the classical algebra of SU (2 ) :

[h, j+ ] =.i+ f [j+ )L j- I =A, [j- 9 J-c-j =j- t (1.1)

whose Casimir operator is

I=V+j- +jo(io - 1) = 2j- j, -t-j,& + l)zjcj+ 1). (1.2)

Consequently, 2j+j- = o’+j,)(l +j-j,) and Y-j, = 0’ - j0 ) ( 1 + j + j, ) . The positive operator j is straightforward to solve for in terms of I: j= ( - 1 + jr)/2.

Now consider the SU( 2)s quantum deformation of Drinfeld and Jimbo:’

[JWJ, ] =J+ , [J, ,J- ] = 3[%]9, [J- Jo] =J-, (1.3)

t~l,=b-=-q-“Mq-q-‘) (1.4) is the “q-deformation”” of X. Note that q-l/q is an auto- morphism of this algebra, and that q+ 1 yields SU(2), of course, The invariant of this deformed algebra is

rq=w+ J- + [Jc]q[Jo - I]@ = 2J- J, + [Jo]q[Jc + l]q =fjlJj+ llq. (1.5)

676 J. Math. Phys. 32 (3). March 1991 0022-2488/91/030676-i 3$03.00 0 1991 American Institute of Physics 676


It is often useful in the following to write this as the pair of identities W, J- = [j+J,],[l +j-J,], and its antipodal form 2J- J, = [i-Jo]q[l +j+Jo],.

For most of our discussion, we take for simplicity q real (in one exceptional case below, q is a phase) which allows Hermitean conjugation 8 _ = St+ for the generator operators under discussion, and obviates checking the third commutation relation in ( 1.3 ). However, as the reader may ver- ify, if one draws the proper distinction between d _ and Fi + , then many of our stated results are valid for almost all complex q. [Seee.g., Eqs. (1.7), (3.17), (3.30)-(3.31) below. ]

The deformation functionals exhibited in Ref. 6 convert SU( 2) generators g into deformed operators G = Q(g) that obey the quantum algebra ( 1.3). For example:

Jo = QU,,, =j,,

J, =Qu+ )=dwj+. (1.6)

Note that the map‘ Q is actually a functional of all three SU ( 2) generators j, , j + , andj _ , since J + depends explicitly on the operator j as well as j,, and j + . This deformation map is readily invertible, except for special cases when the parameters are roots of unity,5 which can be analyzed as limits of the above results, as illustrated below.

If the map is not forced to preserve the Hermiticity relation J ‘+ = J- , then it admits a simple generalization6 to

J, (A) =Q,ci+ 1

J- (1) =Q,U- 1

-( “,~~~I~)““̂ ( “~=:=‘;,I~)“‘-“j-,

(1.7) for arbitrary 2. Of course, we also still require Jo = j,,. We remark that J, (il ) differ from J, = J * (2 = 0) by a similarity transformation

Jt (A) = V(q,A) -‘J, Uq,R), where

(1.8)

V(qJ) = ( F(l +j+je) F(l +j-j,)

) ’

r,(l +j+j,) r,(* +j-j,) ’ (1.9)

In ( 1.9) we have used the q analogX rq of the usual gamma function F, which obeys the relation Fq (X + 1) = [xlsFs (x). Note that the normalization of V is irrele-

vant, We have simply chosen it so that V( q,A = 0) = 1. Also note that V( q,R ) - ’ # V( q,R ) ‘, as expected, since Qn (j + ) # Q, (j _ ) + for generic /z # 0. Other such similarity transformations may be used to reduce either J, , or J- (but not both), to be just the undeformed chargej + , or j _ . (See Jimbo in Ref. 1.)

Substituting specific SU( 2) matrix representations in the above connection formulas produces the corresponding representations of SU( 2), in a direct, mechanical fashion. The map from SU (2) to SU( 2), is effected for finite matri-

ces essentially just by the replacement of integers n by [n 1, as has been widely observed. For example, the 2 representation (j = l/2, Pauli matrices) of SU (2):

j. =+(k T,), j, =$-(i L) (1.10)

maps under ( 1.6) to

Jo =j,,, J, =j+. The3 (j= 1):

(1.11)

(1.12)

maps to

J, =j,, J, =dF.j+ =fij+;, 13)

The4 (j= 3/2):

3/2 0 0

0~0 0

j+ 0

= ( 0 0 Jz 1 0 0 O&3 0 0 0 0

maps to

(1.14)

, (1.15)

and so forth. Now observe that [3], = 0 for q = exp(2ri/3), and

hence the 4 representation J + now has only one nontrivial entry and J ‘+ = 0; the middle commutator in ( 1.3) breaks up, so the representation reduces: 4 = 1 B 2 @ 1. This reduction obtains for roots of unity with period smaller than the dimensionality of the representation.’

Having tied the representation theory of the SU(2), algebra to the representation theory of SU( 2), it follows that the composition laws for representations are likewise linked. Specifically, in the addition of angular momenta, two parallel operators of SU( 2)) e.g., realized in two spaces by two independent representations, tensor multiply to give an operator satisfying the same SU( 2) commutation relations. In our matrix example, this operator is a reducible representation of SU(2), the reduction effected by the Clebsch-Gor- dan coefficient matrix ie (the standard orthogonal matrices for specific matrix representations) :

A(g)=lOg+gel=ce(g’~g,eg,e...)ce-’. (1.16)

Now it is evident that the invertible map Q from SU(2)

677 J. Math. Phys.. Vol. 32, No. 3, March 1991 Curtright, Ghandour, and Zachos 677


generators g to SU( 2)s generators G = Q(g) induces a tensor coproduct of G’s:

e(A(g))=ecl~Q-‘(G)+Q-‘(G)sl), (1.17)

which obeys the SU (2) 4 commutations, since its argument obeys SU(2). The map-induced coproduct simply classi- cizes the SU(2), representations through the inverse map Q - ‘. It composes them at the classical level, and then it quantizes the answer through Q.

II. CLEBSCH-GORDAN COEFFICIENTS AND COPRODUCT RELATIONS

The coproduct ( 1.17) is a representative of a broad class of equivalent coproducts that obey SU( 2) 4. Any similarity transformations U - ‘QU on the coproduct ( 1.17) will also serve to produce an admissible coproduct (isomorphic comultiplication). In particular, the same Clebsch-Gordan matrix ‘% appearing in ( 1.16) will automatically also reduce the coproduct ( 1.17) :

%‘-‘~A(g,)‘t’=G,caG,ceG,~.... (2.1)

For instance, for the 2 QP 2 case,

Ati,) = diag( l,O,O, - l),

AV+)=$

QUO’, ))=,/mA(i+ 1, (2.2)

reduce through

to 3 &t 1 blocks. Likewise, for the 2 8 3 case,

A&) =diag $,$-,--$,$-, ,- 1 3 -?--- , 2 2 >

(2.3)

reduce by

1 0 0 0 0 0

0 $73 0 -l/Jj 0 0

cf= 0 0 ( l/8 0 -y/m 0

0 l/Jj 0 $73 0 0

0 0 $273 0 l/J3 0 00 0 0 0 1

(2.5) to 4 CB 2 blocks. The same % also reduces e(A (i +. )).

Recall, however, the well-known’ comultiplication rule for (1.3)

At,(J,, =&el+ 16sJ,,

A,CJ, 1 = J, @q%-q-JO@JJ, (2.6) and its q -+ 1 /q “double”

&(Jo) =J,ol + l@J,,

A,,,(J, 1 = J, eq-%-q%+J,. (2.7) These appear different than the induced coproduct ( 1.17)) and from each other (by a tensor permutation). They are reduced to direct sums by the q-Clebsch-Gordan coeficients (unitary matrices for specific representations’ -in fact orthogonal for real q) C, and Cljp, respectively.

For instance, for the 2 Q 2 case,

A,(J*)=$ t ; 0 0 */sq o o & (2.8) 0 0 0 0

reduce through

/’ O 0 O\

e, = I

0 ‘qrqi -&my 0 0 J;17i-27J l/JD& 0

(2.9)

\o I

0 0 */ to 3 cg 1 blocks, as specified in ( 1.13 ) .

Likewise, for the 2 69 3 case,

0

0 \ l/q 1 0 0 0 0

o 0 blmLjz 0

0 0 0 0 Jmi 0 0 0 0 0

(2.10)

reduces through

67% J. Math. Phys., Vol. 32, No. 3, March 1991 Curtright, Ghandour, and Zachos 678


1 0 0 0 0 0 Jm 0 - 4&miy 0

O 0 0 $f = L wl~ -JIqmr7

9 0 4qrq 0 Jm 0

0 0 Jm 0 mm 0 0 0 0 0

to4@2blocks,identifiablein (l.lO)-(1.15):

om 0 00 0

PI, 0 0 0

gq-‘A,(J+ )gq ’ 001’

0

0

1 0

0 1

(2.11)

(2.12)

A central theme of our discussion now clearly emerges from the context of the above examples: All comultiplication rules relate among themselves through conversion of the appropriate Clebsch-Gordan coejicients. In our examples, the map-induced coproduct ( 1.17) converts to (2.6) and (2.7) through

MA(g))= Y%‘,‘A,(G)%,%-’

= ‘Z%&A,,,(G)%,,,g -‘. (2.13)

Thus the map-defined coproduct is related to the conventional coproduct, and its “double,” by similarity transformations amounting to U, = Ceq Ce - ’ and U,,, = Ce ,,q Ce - ‘. These are nothing but the unitary transformations that convert V: to Ceq and 4fl ,,q, respectively. Moreover, it is immediately evident from (2.13) that A, transforms to its “double” A,,, through the combination U,U,&‘= (e,Ce-‘%%‘,,j = ‘%,%‘;b,whichisthetrans- formation that converts 9 Ce I,q to %‘,. Conventionally, this latter transformation is called the R matrix, and is known to

I

provide a realization of the braid group. Here we have shown that

A, = R,A,,,R q- ‘,

with

(2.14)

u, =vqwl, R, = U,U,/; = V,%,:;. (2.15)

From this last result and the unitarity of the ieq’s, it follows that

RI = R,,, = R q- ‘. (2.16) The above relation between R and q-Clebsch-Gordan coefficients has appeared before in the literature,“” although the discussion of this relation involving the map-defined coproduct and the U matrix has not. As we discuss below, further combinations of such similarity transformations connect virtually all coproducts among themselves.

For our specific examples, these similarity transformations take the form

/l 0 0 O\ 0

14mq WI, wq4w21,/l/miy 0

uq = 1 0 (q - l/q) [1/2l,/J~ 1/JWr l/21, 0 0 0 0 1

1 ’

0

Rq = i

1 0 0 0 (l/q - 0

0 0 2421, 4V[21, 1

(q- l/q)/[21, 2/[21, 0 0 0 1

= 2”;;1- q, (J, czaJ_ -J- sJ+ ) + 2 - Y

( &)Jo@Jo+(++&)‘“1.

(2.17)

(2.18)

for 2 o 2 [which is a special case of Eq. (3.3 1) below]. Note this is Eq. (3.5) in the last paper in Ref. 1 for the special parameter x= -1.

Similarly,

679 J. Math. Phys., Vol. 32, No. 3, March 1991 Curtright, Ghandour, and Zachos 679


1

( 0

0 0 0 0 0 0 c(q) 0 s(q) 0 0

uq 0 c(l/q) 0 -&l/q) =

0 -s(q) 0 c(q) 0 0 0 0 s(l/q) 0 c(l/q) 0 0 0 0 0 0 0 I 1

4s) = (jzqz + qvj-qq-, s(q) = Cj-Tqz - fiqvjq-q, ‘1 0 0 0

0

R, = 0

( > q-L dTT[3/21

4 [31

0 0

[21 + 1 [31

0

( > q-L d%[3/21 9 [31

0 0 0 0

= (J, eJ- -J- @J+ )+2 l-

+ [21,+1 1~1

131, '

( > --q JrzTc3/21 1

4 [31

0 0

[21+ 1 [31

0

0 1,

(2.20)

for 2 % 3. [Again, this is a special case of eq. (3.3 1) below.] To illustrate the general situation when q is a root of unity, we now describe in some detail what happens in the particular

case of 2 % 3 when q = exp( 2rrV3). Under this condition, ?Yq diverges, and so does U,, but not R,. This is as it should be, since the rhs of (2.12) decomposes completely, whereas A, of (2.10) is not fully reducible. As discussed by Pasquier and Saleur in Ref. 11, the nonunitary 6 of (2.10) is reducible, but not decomposable to a 4 and a 2, as their collective q-dimension vanishes: [4] + [2] = 0. Specifically, since J- = Jt~nsposr, the norm is u*v = u’v. The six states a’= (0 0 0 0 0 i), % , , 3 , d’= ( - l,O,O,O,O,O), beJ+ a, csJ._ d, b”= q$(O,O,l,O, - i,O), c”=q2~(0,i,0,0, - l,O), contain the doublet ofzero- norm states b and c, which only transform to each other: J + b = c/G, J _ c = b /@, J _ b = 0, J + c = 0. However, as evident above, a and d are not singlets, and may, in turn, be reached from elsewhere: J_ b ’ = a, J, c’ = d; b-b ’ = c-c’ = I, and J _ c’ = (b + b ’ )/vB, J + b ’ = (c + c’)/$. Full reduction fails by dint of the divergence of 5’,. (This is also implicit in Ref. 12.)

We may extend the simple examples above to the direct product ofspin I/2 with spinj, in a semiabstract setting, treating the variables for spinj as operators, but using the explicit 2 x 2 matrix realization for spin l/2. The extension is provided by a Clebsch-Gordan operator whose matrix elements between states which realize an integer value for 2j are just the usual Clebsch-Gordan coefficients. This operator, which generalizes (2.9) and (2.11)) is

4

i

(43 -

J, d’ +j+ J*)l‘dG

--4 (I+j+J&Z

[l +j:Jo]q J-

4 (Jo - ’ - i)‘2 jm + jmq 8, + J{, /J ‘(2*21)

Using (1.6), we may express ‘~5’~ in terms of the undeformed charges. Also, as q- 1, (2.21) reduces to the usual SU(2) Clebsch-Gordan operator for l/2 %j,

jm+W Sl+j+,t,

g=1 ~~

i

2 -

1 +j+jo j-

j, 2

II-

Al- ,

1 +j+j, JFOi + OTT 4-k],,

1

(2.22)



Note that Ceq is a special orthogonal matrix (det %‘, = + 1) when evaluated for any integer 2j matrix representation for which j _ is the transpose ofj + . This may in fact be used to derive the form of gq up to the overall phases of the four Jo-dependent blocks in the matrix, and up to a single function ofq, j, and Jo. The latter function may be determined by requiring that gq have well-ordered entries that avoid “O/O” ambiguities for - jq; ~1 and that Ce4 diagonalize the co-Casimir defined by A,. (Actually, there are two equivalent, antipodal solutions which are interchanged by the J, c+J- , Jo -, - Jo isomorphism of the algebra. For simplicity, we only display one of these solutions.) We remind the reader that this is the essential defining property of the Clebsch-Gordan coefficients for specific matrix representations.

Also note that Ceq contains Kronecker delta terms Sk, with So = 1 and zero otherwise, which are present even as q+ 1. These are absolutely necessary for consistency and may be understood as fulfilling certain boundary conditions in the tensor product space. One of these, the Sj+ j0 term, is crucial to achieve agreement with the specific examples above [cf. (2.9) and (2.11)].Theothers,involvingS,+j+j,,, are irrelevant for those particular examples, and have in fact only been displayed in the upper left-hand block in the above expressions. Other 6, + j + j0 terms are present in the off-diagonal blocks, and are required to validate certain formal operator properties, but they do not contribute in the examples and so they have been harmlessly suppressed in (2.21)- (2.22).

Given %‘, , we may proceed to construct U and R as in (2.13)-( 2.15). The construction is straightforward, and the results are given in the next section [cf. (3.1), (3.30), and (3.31) ] where they are obtained using operator difference methods. These same methods may be used to fill in the details of the derivation of %’ q.

Even without such details, the form of the Clebsch- Gordan operator in (2.21) for general SU( 2) realizations j underscores a central point of our discussion. All similarity transformations connecting the various coproducts may be viewed in the abstract, say as operators within the universal

I

l/det,,,(j,, + 1) 0 R(q) = 0 Wet,,, ci, 1

enveloping algebra. They are defined by their formal algebraic properties quite precisely, as we will see further in the next two sections. The specific cases we have considered in this section are simply the images of the general objects pro- jected onto specific matrix realizations.

III. DIFFERENCE ANALYSIS OF 1/2@j

For the case of spin l/2 combined with spin j, i.e., 2 @ (2j + l), we employ the standard 2 X 2 representation matrices [cf. ( 1.10) ] to express the general U matrix in the following form:

U(q) = c, tie 1 aqtio)j- /Jz

bqUoio)i+ /G > dq(io) * (3.1)

Below, we will determine the functions a4 u. ), 6, (i. ), cq (j. ), and d, (i. ) by solving the four first-order difference equations implicit in

A, u(q) = Wq)Q,, (A). (3.2)

Note that it is trivial to replace ordinary SU( 2) operators in (3.1) by their quantum counterparts, for generic q, simply by inverting ( 1.7).

The inverse of the general U(q) in (3.1) is

l/det, (i. + 1) 0 U(q) - ’ = 0 l/det, (i,)

X d, Uo + 1) -aq(io).i- /Jz

- bqCio)j+ I$ > c,uo-1) ’ (3.3)

where

detq(.jo)=cq(j, - l)dqtioi,) -&u+j,)(l +j-j,)

xa,(j, - l)b,U,). (3.4) Appropriately, the diagonal l/det, matrix in (3.3) commutes with the matrix to its right, and also with U(q).

Making useof (3.1) and (3.3), an R matrix defined as in (2.15),R(q)-U(q)U(l/q)-‘,becomes

cq(io)dl,qCio + 1) -tti-j, )a,Cio)b,,,tio + 1) (a,(iO)c,,,UO) -a,,,(iO)c,CiO)) i- /\lz X

(b, U. Ml,, cio ) - h,, cio Id, ci, 1) j, 14 1 cl,,Uo - l)d,(i,) -$O’+j- h,,(io - l)bqUo) ’ (3.5)

The four difference equations that determine a4 (i, ), 6, (j. ), cq (jO ), and d, (i, ) may be made explicit by right- multiplying (3.2) with projection operators for the j -+ l/2

I

irreducible subspaces of l/2 o j:

Taking the general form of the map in ( 1.7)) we obtain

WqX!,(AO’+ ))p,+ i/2

(1 +j-jo)/Jz

’ (3.7)

(2 +j-ioIl+ where D is a diagonal matrix consisting of the two blocks

(3.6)

D I.’

X(1 +j+jo)(cq(jo) -t/O’-jo>aqUo)), (3.8)



D,,, = ( woolq)“““( ‘;~;~~,I~)“‘-~

X(dqW +g./++jl-J)bq~~)). Similarly,

UJ.t )Wq)p,+ ,,2

(3.9)

i .,i+ =- 1 + 2j (

(1 +j-jo)/@ Jzjz, > (2+j-j,)j, ’

(3.10)

where D ’ is also a diagonal matrix consisting of the blocks

D;.’ = ( “,=“,a’q)“~+~( y;~$‘lq)“‘-^ by’

X(c,G - 1) +$(l +j-jo)aq(io - 1)) +dY~qUo) ++t+jo)b,Cio)~, (3.11)

D;,2 = ( [;~~olq)“~+~( ‘wl’u,‘q)“‘”

X&(d,(i, - 1) +jC.j+j, - l)b,(j, - 1)). (3.12)

Therefore, WqX?d~*O:+ ))P,+ l,2 = Aq(J+ )U(q)P,+ l,2 if and only if D = D ‘. [We assume at this point that the solutions for a4 (&, ), b, (jO ), cq (j,, ), and d, r& ) are analytic inj. A priori this need not be the case. It might be possible for the solutions to involve Kronecker delta terms of generic form Sj - n f j,, for various non-negative n, as does the operator expression for the Clebsch-Gordan coefficients in Sec- tion II. Since U(q) goes smoothly into the identity operator as q+ 1, however, we believe it is a reasonable Ansatz to exclude such Kronecker deltas from Uitself, and suppose all such nonanalytic structure in the q-Clebsch-Gordan coefficients is the result of an analytic Uacting on the usual SIJ( 2) Clebsch-Gordan coefficients. 1 This yields the pair of difference equations

( [l +j-j,], 2+j-j, "'-' =

1 +j-,A [2 +.i--joI, )

x,&‘&‘, - 1) + $o’- 1 +.&lb,& - 1)) (3.13)

and

[l +j+j,], “2+d 1 +j+j, (1 +j+h)X cqUo) ++- U-jobq(jo) - ) ([:=iloly+*

x o’+h) -h ( CPU0 - 1) ++(I +j-jo)aq(io - 1) = ) (

1 t-j-j, [l +j-jolq )“lid.‘(dqC.jo) +t (i+jO)b,(j,,)),(3.14)

Similar considerations on U(q) Q, (A tj+ ))4 _ ,,* = A, (J + ) U(q) P, _ l,2 give the second pair of difference equations

&kio) -+(l +j-jo)bp(jO) = [j+j,], j- 1 +j,

j-t.& [j- 1 fjo]q d,(io - 1) --$2+j-j,,)b,(j, - I)>

and

_ (“,;y?)“‘̂ ~-io+qW --) (1 +j+j,)a,&) + ) (

[* +j-.hlq )

“*-’ (1 +j-j,) 1 +j-j, &

These two pairs of difference equations are interchanged, (3.13)-( 3.14)~( 3.15)-( 3.16), under il-+ - /i and j. General solutions of the difference equations (3.13)-( 3.16) are found to be

a,(&) = --& V;4’+3~lJ2+f3q-jf+2) * +;+jo ( [;;;;+lq)“2+d + --& cf,s”“J’-f‘K?+( [;;;lq)‘/2-d,

b,Cj,,) = 2q’“/’ f, (:y;+-$” [(

o )“‘” +-( [;,~olq)“2+*],

CqtiQ) = ~cr,q’+**2+1;9-*~2~([~~~~~~lq)“z+d+~

x ( _ f2ql + W* +fhq -/h/2 I( [;:;Jv’-^,

d,(j,) =dd2 [ 1 +j-j,], 1’2-A

1 i-j-h The four constants of integrationf; are of course independent ofje but may depend on q andj.

-+--

(3.15)

(3.16)

.I -j.

(3.17a)

(3.17b)

(3.17c)

(3.17d)



When the four constants of integration are unrelated, the above solution is singular: aq ci, ) has poles at j = j, and - 1 -j =je. Nonsingular solutions are obtained by eliminating the residues at these poles, which leads to two relations

between the&: f3 = -q-‘-2% (3.18a)

eliminates the pole at - 1 -j = j,,, and f4 zq’f2if2 (3.18b)

eliminates the pole at j = j,, , A solution completely free of singularities is therefore parametrized by only two&. As complicated as the results in (3.17a)-( 3.17d) may appear to be, they produce a simple result for det, in (3.4)) which is

actually independent ofj, :

det, = I(1 + W(q - l/q) l&K (qI.6 (q) -I-f2 (q).fi (4)). The other components in the R matrix of (3.5) may also be expressed in terms of theA. We find

~,U~)d,,~(i, + 1) -tU-h)(l +j+jO)a,UO%,,Ci, + 1)

(3.19)

=z [(A (4% ($)-f’ (++ (’ +2h)‘2 + (A <l)h (4) t-h (i)& w)d - ’ - 2jo)12]1 (3.20)

aqUo)cl,q(iO) --al,,(io)cqG3jO) 2(1 + Y)

( 1 +j+j,

= (q- l/4)*(1 +j+jo)o’-jo) [l +j+h], )“2+*( [;E;],)“-

x[f’(4x (+’ (+?) -h(q)h (+)+A ($)Jm + (fi (J-)& ((I) +fi (+)A (q)), - ’ - 2io - (A Cc?).& ($) +.h (4)fi (-$4’ + “]y

b, (io M,, (io 1 - b,,, (io Id, ci, 1

=2(1+2j) (

[l +.i-joI, 11-j-h

)I”-*( [-!~olq)“2+d(A (4% (d) -fi &m).

(3.21)

(3.22)

So we have obtained the most general solution for U(q), and its associated R matrix, for the case of l/2 8 j. For completeness, we also give the simplifications of (3.17a), (3.17~) and (3.19)~( 3.21)) which result from imposing the

pole-eliminating conditions in (3.18 ) :

(3.17a’)

(3.17c’)

"2-d aqUo)l~3.18) =2dd2 fiq--’

[ (

[l +j+j,], 1 +j+jo >

-fzql+j( “,-_j$oIq)“2+A],

cqtio)l~3.18~ =dd2[fiq-‘[i +j+j,],( ’ +f++ )“2mA+~q’+j[j-jo~q( LlfIjhl )““̂ I, [I +I+Jolq 0 Q

detqI,3.18, = (1 + 2)) 11 + ?d,ti (4% (41, cqUoM,q(io + 1) -to’-h)(l +j+h)~q(io)h,q~o + 1)lc3.~~,

= (1 + 2j)V; (q)f,(l/q)q’-‘-2~“2[ 1 +j+jo] +fi (l/q)f,(q)q”+2i”2[j-j~]), a, (i. hq (i. 1 - al,q U. )cq U0 )Iw~)

(3.19’)

(3.20’)

=2(1+2j) (

[ 1 +j+h], 1 +j +jo

)“2-d(y..-!01q)“2+d(h(9)h (J-)4-‘-*wl (-+2wq’+*~), (3.21’)

Other considerations may fix the A, either partially or completely. For example, it follows from (3.5) that the R matrix is upper triangular only if b,,,(j,)d,(i,) - b, (jr, )d,,, ci, ) = 0. Or from (3.22), in terms of theh this

requirement is

fl (4X ( l/q) =.fl (l/qv2 (4). (3.23)

Similarly, R is lower triangular only if aI,q (j. )c4 (j. ) - a, (i. )c,,~ (j. ) = 0. In general, this re- quires two conditions on the constants of integration, as is clear from (3.2 1 ), but given the constraints for a nonsingular U in ( 3.18)) these reduce to the single requirement

I

fi (qv2 (l/q) =fl ( l/q)f, (q)q2 + *. (3.24) For il = 0 the fourA are constrained by unitarity, i.e.,

U - ’ = Ut. It turns out that only for the Hermiticity-pre- serving case of the map, i.e., il = 0, can this constraint be satisfied. In this case, unitarity applied to the form of U in (3.1) leads to three equations that we consider for real a,, b,, c,, and d,:

O=aqCio)dq(io + 1) +cqUo)bqtio + 11, (3.25)

l=Cici0) +tO’-j,)(l +j+jo)ai(io), (3.26) l=dih) +$o’+j,)(l t-j-j,)bi(j,,). (3.27)



From these it follows that the four constants of integration are completely determined up to some choices of phases. For example, starting with the expressions containing all fourA, (3.17b), (3.17d), and (3.27) give (for realfl)

f: = s’

(l+2jHl+2jlq , f;= 4-l-j

(1+2jNl+2jl, ’ (3.28)

while(3.17a), (3.17c),and (3.26) giveinaddition

f3 = -q-‘-25, f, =q’+*s. (3.29)

Note that ( 3.29) is precisely (3.18) again, so requiring U to be unitary automatically eliminates the pole singularities. Given (3.28) and (3.29), it is straightforward to check that (3.25) is satisfied and gives no new conditions.

Finally, the q- 1 limit selects the appropriate roots for fi and& in (3.28). Requiring U(q = 1) = I selects theposi- tive roots for both. The pieces a,, b,, cq, and d, of V, when /z = 0, are then

I

aqUa) = 2q’J~-i’/2 (J3 J(l -!-?i)[l -I-Z&

- q(l +*lk/*J [hI$o)), (3.30a)

b,Cio) =Gq( -ji) = -a,(jo - 11, (3.30b)

cqo?o) = Q

(I;, - W2

J(1 +Zj)[l +Zjl,

X(J[l +j-t-j,],(l +j+j,)

$9 (1 + *j)/Z J[j--jOl,(i-jo)L (3.3&I d,(j,) =c,,~( -jo) =cstjo - 1). (3.30d)

The reader may check that (3.30a)-( 3.30d) yield a rotation matrix for U and reduce to the specific cases (2.17) and (2.19) above. Also, U(q)Q(A.ci- )) = A,(L )U(q) issat- isfied automatically for unitary U(q), given U(q)Q@O’+ 1) = A4(J+ )Wq).

The R matrix of (2.15) is now obtainable from the above unitary U, We find

R, 1 ( [l +j+J01, + [j-Jolq - Jz(q - l/q) u-t 1/2l,J-

[l +$I, Jz(q- l/q)V+ 1/2lJ+ [l +j-JOlg + [j+JOlq 1,

This result agrees with previous determinations’ of R. Multiplying R with the projectors Pjk ,,* as defined in

(3.6) gives a resolution on the invariant subspaces of l/2 ej which conforms with general expectations.‘*” Nevertheless, it is perhaps more instructive to think ofR as simply a sum of two q-deformed versions of the projection operators onto the j & l/2 subspaces,

R(q)=Rj+,/*(q) +Rj-,/2(q), where

(3.32)

Rj+ 1/2 (4) = 1

[1 + 2j1,

and

( [l +j+J,], @q’-1-2i’/2J-

x 4% ( ’ + =jtf2J + I1 +j--JoI, > (3.33)

Rj- I/2 (4) 1

= [1+2jl,

( [j- Jolq 4% (I + 2j>/*J

X _ Jzq’ - I - 2N2J+

[j+Jolq - > ’

(3.34)

As q-+ 1, Rj, 112 (9) +Pj;t 112. However, Rjr, I,2 (4) as just defined are not projection operators for q# 1. Rather, they are true deformations of projection operators in the same sense that SU (2) 4 is a deformation of SU( 2) : For q# 1 the algebra obeyed by R, * ,,* is rtot that of projections. This is evident in

684 J. Math. Phys., Vol. 32, No. 3, March 1991

(3.31)

I Rjk I/Z (4) = Nj, 112 (q)Rj* l/2 (4)

= Rj, t/t (q)&* 112 (4) (3.35) and

R~, I/Z (q)RJF I/Z (4) = Jfi, I/Z (q)RJ, l/2 (4) = 4, i/2 (q)M,, ,,2 (l/q),

(3.36)

where

Nj * l/2 (4)

( 00 r!x l/2), = &2), 0

0 CJ, T l/2), ’ (3*37) )

Mj+ l/2 (4) = 1 (4 - 14)

G’-l- m, (q + l/q)

( 4 X

(-I-*j)~*[l+j+JO]q

0 _ q( 1 + 2j)/2 yl +j-J,], ’ )

MJ- t/Z (4) =

1 (4 - l/q) ci + m, (4 f l/q)

0 X

( --4 ('+*i)/*[j-Jo],

0 Q > t-~-2j)/2[j+JO]q '

(3.38)

In the above, we have also defined

Cuttright, Ghandour, and Zachos 684


{x} ,@+q+ 9 q+l/q *

(3.39)

Each of RJ + I/2 commutes with its own “normalization factor,”

[R,,,,,(qLN,~,,2(q)l =Q (3.40)

which inappropriately suggests that one need only divide by these factors to obtain bona fide projectors. However, (3.36) shows that this is not a sufficient modification. Even after such a change in normalization, the product R, + ,,2(q)Rj- ,,2(q) doesnot vanishforq#l.

Nonetheless, it is true that each of R,, 1,2 is aZmost a projection operator. To make this precise, define the similarity transformed projections using the previous Umatrix:

‘J i ,/2 (4) = ‘,‘j, 1/2 ‘, ‘- (3.41)

Ofcourse, P , + ,,* (q) depend on q, but they are not deformed in the sense that they obey the usual algebraic properties of projection operators: pi+ I/2 (q12 = pj* I/2 (4),

', f l/2 (d”, 1/2 (q) = '*

After changing variables to the deformed charges through the use of the inverted deforming map (1.6), we explicitly have

JO ‘J-1 ,/2(q) =

Ll Jzjl,

X (

[ 1 +j+J,],q-j

$4 - “*J+

JO ‘J-f,*(q) = u1 ;2jI,

y([j-Jolq4’+j

,bq”*J-

> [l +.F Jol,d ’ (3.42)

- fiq”*J- \ “\-fiq-“‘J, [j+ Jolsq-‘-jj f

(3.43) For matrix representations where J, = J t~n~Wse, it follows

that P, i ,,2 (9) = P,, ,,* (q)tra”nrxr5e. From (3.42)-(3.43), and (3.33)-(3.34), we then ob-

tain the precise relation between R,, ,,2 (q) and P, + L,2 (q):

RJ + 1/2 (9) = p,+ ,,2 (4)

(3.44)

r(2 +i+j,) r(l +j-j,) 1

Uq,~)P, + ,,* = i ( r,(2+j+jo) r,(l +j-j,) )

0

/( r(l+j+j,) IYj-jo) A

1

- 1

’ -j-J<, 0

Rj- 112 (4) = Pj- I/2 (9) 0 1 4

tj-4,

4 J,+l+j

O = )p,-,,* (t). (3.45) 0 qJo-‘-’

Once again, for matrix representations where J, = JtyPOse, it follows that Rj+ 1/2 (4) = Rj f ,,2 ( ~/q)‘ra”sp“ce.

Given the sometimes subtle effects of similarity transformations, it is not immediately evident that the diagonal matrices that appear in (3.44)-( 3.45) actually deform the algebra of projection operators. However, they do indeed, as is clear from the results in (3.35)-( 3.36).

While for generic A # 0 the similarity transformation defined by the solutions (3.17a)-( 3.17d) is not unitary, and therefore largely unconstrained, there is still a preferred choice of the integration constants in those results. Namely, we may construct the transformation for /z #0 through a combination of U(q), given by (3.30a)-(3.30d), and V(q,A), given by (1.9). Since Vis not unitary, neither is the combination. For clarity, we will denote by U(q,il) this special combined similarity transformation that converts the coproduct induced by the general map J, (A.) into A,. So we have

A4 = WsJ)Q, (A) WqJ) - ‘, where

(3.46)

WqJ) = U(q) UqJ). (3.47) The latter factor V(q,A) is to be constructed on the tensor product space exactly as in ( 1.9)) only with j replaced by the cospin Aj, and j, replaced by A ci, ) = 1 o j, + j. o 1.

Now, inasmuch as il could be complex, QL (A) actually defines a two-dimensional manifold of coproducts, which is connected in its entirety by similarity transformations to the conventional coproducts, A, and A,,,. Of course, a product form as in (3.47) would suffice for any tensor product of representations, but in the case of l/2 8 j, an explicit form for U( q,A ) results from resorting to the 2 x 2 operator-valued matrices again, as in (3.1)) and projecting by P, i l,2 of (3.6) to reduce the cospin to j & l/2. The j, dependence is again tractable, in the sense that we obtain a diagonal matrix whose entries are functions of j,: just the ratios of r functions in (1.9):

0

( w +j+j,) r(2+j-jo)

>

1

k

j + l/2 5

r,(l+j+jo) r,(2+j-j,)

0 \

UqJ)P,- ,,* =

I

\r4(l +i+j,) r,O'-j,)/

0 (

W+j,) r(l+j-j,) A j- I/*-

r,u+io) r,(l +j-j,) )r

(3.48)

(3.49)



It is interesting that U(q,A) gives the same R as U(q), without any /z dependence,

R(q) = (I(q,il)U(l/q,;l) --I = U(q)U(l/q)-‘. (3.50)

This is understood from V( q,iE ) = V( l/q,/2 ), which, in turn, follows from the property of the q-analog gamma function, r4 (n) = l-,/q (xl.

Despite this simple example, it is important to emphasize that not all the U’s obtained by choosing the constants of integration in (3.17) are related to U(q) by simple right multiplication with V’s that satisfy V(q) = V( l/q). Hence, not all the R ‘s obtained from the U’s are identical, nor even equivalent via other similarity transformations. This can be seen explicitly by comparing the “triangular” and “unitary” examples discussed above and leads naturally to our next discussion.

IV. FURTHER GENERALIZATIONS, REMARKS ON UNIVERSAL UAND R, AND YANG-BAXTERIZATION

Most of the remarks we have illustrated in the previous sections naturally generalize to all QUE-algebras,2*6*7 all varieties of deforming maps, and, of course, all representations. In principle, a straightforward calculation of Clebsch- Gordan coefficients gives the ingredients necessary to construct a U and an R for particular tensor products of representations-a compact expression is ultimately a matter of

elegance if one is only interested in unitary U’s and the related R matrices.

However, it is useful to carry out the construction of U and R using operator methods similar to those of Sec. III, only generalized to any pair of representations, beyond l/2 @j. Such general operator constructions constitute “universal’” U and R matrices, and provide a framework to discuss inequivalent forms for the Uand R matrices, as pointed out in thecontext of l/2 epjin the last section. Such inequivalent R ‘s and, by logical extension, Ll’s are the basis of the Yang-Baxterization problem.

While this has been substantially discussed in the literature’*‘*“’ for R, we are not aware of any such discussion for U. Here we consider both universal matrices in parallel, to emphasize some similarities and contrast some differences between them.

A general universal operator form for U is

U(q) =djod3) “I- 2 a,(io&)(i+ @i- 1” n=I

+ ~,,cio~o)U- ej, 1” (4.1) and, as before, all operators are trivially converted to J’s by inverting the deforming map. The coefficients c, a,, and b, are determined by solving the hierarchy of coupled, first- order, partial difference equations implied by

.h,(J+ )Wq) = u(q)Q(Ao’, )I. (4.2) The explicit form of the map in (1.6) allows (4.2) to be written

I

= U(q) [Aj+ l@jO+jO~l],[lsl+Aj-- lepjO-jO~l]g (lej (Aj+ lepj, i-jo@l)(lel +Aj- laj, -jOal)

+ +j

* sl)

I (4.3)

where we have used subscripts to distinguish the independent Casimirs for the left and right representations in the coproduct (i.e., we are considering j, e3, j, ), and we have de- noted the ordinary SU( 2) cospin by Aj. The relation between the cospin and the standard SU( 2) co-Casimir is

Aj=$( - lsl+~lel +4AI). (4.4) In addition, in terms of the individual left and right operators, the co-Casimir is

dence in the map to produce A,, which has no such dependence. As a consequence of this intricacy, we will not give here a detailed discussion of the U hierarchy. The most general solution for U, extending that for the l/2 @j case, is an unsolved technical problem. Given the framework we have provided for its explicit construction, however, the existence of a universal U is evident.

Similarly, a general universal operator form for R is

AZ=Z’““@l + 1sZ”ght+2jO@j0 R(q) = CCJOJO) “I- 2 A,,(J,,J,)(J+ @J- )” +2j+ ej- +2j- 8j+,

where Z ‘rft~nght =j,., ( 1 + j,, ) .

(4.5) tl=l

t &?U,,Jo 1 (J- eJ+ 1”. (4.6)

Expanding both left- and right-hand sides of (4.3) in Imposing (2.14) in the form powers of j t 8 j _ and j _ @j + , and identifying the coeffi- A,(J+ )R(qI = R(q)A,,,(J+ ) alsoleadstoaninfiniteset cients of these independent monomials leads to a set of coupled, first-order, partial difference equations in the two oper-

of coupled, first-order, partial difference equations for the JO dependent coefficients. Only now the infinite set is much

ator variables 1 @j, and j, 69 1. The resulting equations have an intricate dependence on the two Casimirs. This is expect-

easier to display, since neither A, nor A,,, has any explicit

ed, since U must compensate for the explicit Casimir depen- Casimir dependence when expressed in terms of the SU( 2)9 operators. Explicitly, this hierarchy ofequations is



c(Jo,J,Hlxq-4’) +&4,(J,,Jo)q-‘fJo Q (,yht - MmJo + 11,) =CCJo - LJo)UeqJ”) +j‘4,,Jo,Jo - l)q-J”

8 (,;ght - [Jo 14 [Jo - l],), (4.7)

C(Jo,Jo)(q% I) +p, (J,,J,,

X(Zb”“- [Jo]q[Jo + l]q)eq’-Jo

=CCJo,Jo - l)(q+‘sl) +$B,(J, - l,J,)

X(Zb’“- [Jol,[Jcl - 11,) @d”, (4.8) and, for n> 1,

A,(Jo,J,ww-“-J~) +~,,+,(J,,J,)q-“-‘+J” a (z;gh’ - [Jo +n]q[Jo + 1 +n],)

=A,(J, - 1,JoH18qJ”) +&‘f”+,(Jo,Jo -up)

@ (py’- [Jo]q[Jo - l]qA

B,(Jo,JfJ)(q”+JJ”8 1) +;&+I (Jo,Jo)

(4.9)

X(Zb’“- [Jo +n],[J, + 1 +n],)sq”+‘-J”

=B,(Jo,Jo - lHq-J”el) +p,+,(Jo - 1,Jo)

X(Zb’“- [Jo]q[Jo - 11,) 4”. (4.10)

This set of linear equations for R has been discussed earlier by Jimbo,‘** who proved that there is at most a one parameter space of nonsingular solutions. He also constructed an explicit solution for the R matrix associated with the coproduct of an arbitrary representation with the fundamental representation of a general linear algebra. A similar proof and explicit construction should work for the U matrix as well.

In the R case, in contrast to U, the Casimir dependence in the hierarchy of Eqs. (4.7)-( 4.10) is actually quite tractable. In fact, it is easily seen that all Casimir dependence is eliminated from the R hierarchy if and only if

A” (JO,J” I(4 - n+4)@ 1) =A,(J,,J, - l)(q-%l) (4.11)

and

B, (Jo ,Jo ) ( 1 o q” - “1 = B, (Jo - l,J, ) ( 18 4”). (4.12)

It is a straightforward consequence of (4.1 l)-(4.12) that the resulting solution to the hierarchy is an arbitrary linear combination of the knowniV” upper-triangular R matrix, R tria”g”‘ar( q), and its lower-triangular conjugate “double” R ‘ria”g”‘ar( l/q) +. These are given by

R frl~fl&r(q) = q-*J”@JO 2 - 2”(1/4- qYqncn+ I)/2

i?=O

[nl

4!

x(q -“J”sq”4b)(J+ tsJ_ )“, (4.13)

and

R tW,,,gU’X( l/q)+ = q2J@Jc)

xcq- n4’.q”4’)(J- sJ+ )“, (4.14)

where En],!= [n], [n - I] ***[2],[1],. The single parameter in the space of nonsmgular solutions for R is easily understood as the ratio of coefficients in an arbitrary linear combination of the upper- and lower-triangular solutions.

Let us consider this last point in more detail for the l/2 8 j example of the preceding sections, to clarify how Ca- simir dependence can occur in the solution for R. In this regard, it is important to realize that (3.1) is (4.1) with the summation truncated to n = 1, since for the spin l/2 representation J *+ = 0. Now, define a one-parameter solution for the R hierarchy as

R(q) -a(q)R rriangular(q) + ptq)R triangular( l/qjt.

(4.15) This is a one-parameter (say a/p) solution since the overall normalization is irrelevant to the task of converting A, into A I/q.

Upon making a detailed comparison of (4.15) to (3.3 1 ), we see that the Casimir dependence of the latter is recovered by choosing

J&(q) = 4+ ,,* +14’- ,,* , P(4) = a (+) . (4.16)

This illustrates the general situation: Casimir dependence for a nonsingular solution of the R hierarchy is obtained by taking linear combinations of the Casimir independent triangular R matrices, with explicit Casimir dependent coefficients. Thus the two Casimir independent points in the solution manifold of the hierarchy may be joined by following an explicitly Casimir dependent path connecting the two points.

Of course, there should be a corresponding statement for the Umatrix, except that the Usolution space always has Casimir dependence. One may hope to minimize but never completely eliminate that dependence.

There is one other important property of the R matrix that warrants a discussion here, if only broadly. It should satisfy the Yang-Baxter relation (cf. the review by Jimbo” and references therein). We have not determined what if any restrictions this places directly on the solution manifold of either the U or R hierarchies. At least for the R matrix, however, this is a well-studied problem which has led to the idea of Yang-Baxterization.‘3 This is the precise determina- tion of a continuous family of R(x)‘s which also satisfy the relation

R,, (x)R,, (xY)&, (Y) = R23 (y)R,3 (yx)R,, (X).

(4.17)

Given a solution to this relation for, say, x = y = 1, the rest can be determined (for a useful guide to the literature on this problem, see Ref. 13 ) .

Now, as we have discussed, there is a manifold of solutions to the R hierarchy. This manifold is continuously parametrized by the initial data for the hierarchy of difference equations. The hierarchy solution manifold therefore pro- vides all the candidates for the Yang-Baxterization problem. The individual R(x) above are just points in that manifold where x corresponds to the initial data. Insofar as Yang- Baxterization may be defined as the explicit parametrization of dissimilar, inequivalent similarity transformations, or



“paths”, between A, and its “double,” which also obey the trilinear Yang-Baxter relation, then the general solution to the Yang-Baxterization problem is completely contained within the R hierarchy solution manifold described above.

By obvious logical extension, there should be a similar idea involving Yang-Baxterization for the U matrix, which may be viewed as providing a “factorization” of the corresponding problem for R. This idea remains to be fully inves- tigated.

Let us stress that, in general, R, and also V, for different data are not only unequal, but also inequivalent and are not related by similarity transformations, as the reader may ver- ify for the l/2 ej example of the preceding sections. In a strict sense, the manifold of such solutions is connected (since the solutions are continuous functions of the initial data, as in the l/2 % j case), but the manifold is not “similarity connected.”

V. CONCLUDING REMARKS

To summarize, we have explored the broad classes of similarity transformations that connect all equivalent realizations ofa given quantum algebra-SU (2) 4 in our illustra- tions. Such transformations connect the various alternative coproducts whose particular features (e.g., Casimir-inde- pendence for Ap ) may be of special utility in suitable phys- ical applications. To find these similarity transformations, we set up, and, in some cases, solve, the systems of operator difference equations that suffice to determine them. This procedure may be further extended and adapted to any particular physics situation as it arises. The basic building blocks of the similarity transformations discussed are the generic Clebsch-Gordan coefficients, the universal U operator introduced here, and the standard universal R matrix, all of which were shown to be simply related.

As an example of a physics application, we conclude by outlining the utility of the results in this paper to the physics of deformed spin chains. ” Evidently, a two step process should permit a direct solution of a deformed spin-chain model in terms of any known solution of the undeformed model, as constructed using, say, the Bethe Ansatz for energy eigenstates.

Given an undeformed solution for any representation of SU( 2), i.e., for any size spin in the chain, the deforming map serves to express all operators in the deformed model in terms of undeformed operators. This alone would seem to permit, in principle, the calculation of correlation functions for deformed operators. However, another step is required since the usual formulation of deformed spin chains makes use of the conventional coproduct A, to write the deformed Hamiltonian, so the energy eigenstates are linked to the conventional coproduct. Therefore, we need to convert between A, and the map-defined coproduct Q(A), to obtain the states most appropriate to the deforming map. This conversion is provided by the U matrix discussed above, and could be carried out for the simplest spin chains using the explicit results we have given for l/2 sj, or in more general cases using the universal Umatrix defined by the hierarchy of Sec. IV.

ACKNOWLEDGMENTS

We are indebted to D. Fairlie, P. G. 0. Freund, M.-L. Ge, N. Jing, P. Kulish, L. Mezincescu, R. Nepomechie, A. Polychronakos, V. Rittenberg, and S. Vokos for discussions.

The work of author T.L.C. was supported by the Monell Foundation and an NSF Grant PHY-87 03390. The work of author C.K.Z. was supported by the U.S. Dept. of Energy, Division of High Energy Physics, Contract W-31-109- ENG-38.

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Quantum algebra deforming maps, Clebsch–Gordan coefficients, coproducts, R and U matrices