foundimensional quantum space. OCR Output

Only two of the three R-matrices allow a differential structure on the reduciblethree R-matrices which can be constructed using four different SUq(2) doublets.the previously known R-matrices for SOq(3) and SOq(3,1). Also they are thetion. There are three solutions to the Yang-Baxter equation. They coincide with

'We consider S Uq(2) covariant R—matrices for the reducible 3 GB 1 representa

Abstract

Theresienstr. 37, D — 80333 Munich, Germany

Sektion Physik, Universitat Munchen

Fohringer Ring 6 , D - 80805 Munich, Germany\Verner-Heisenberg-Institut fiir Physik

Max—Planck—lnstitut fiir Physik

A. Lorek1,VV`.B. Schrnidkel and J. \xVess1’

for Reducible Representations

S U,,(2) Covariant R—rnatrices

October, 1993Paeaeeama MPI·Ph/ 93-89

|||l||||||Ill!|I||l||I||lll||l||l|||||II||||l|||QERN LIBRHRIES» GENEVH

> (xc) O) L·.O‘)+

4, 1, A `J s · if `J

a coproduct A. It essentially describes how the generators act on a tensor product of OCR Outputwhere q is the deformation parameter. Z»{,,(SU.,(2)) is a Hopf algebra and as such has

-(¢1 + q`1)T`"2T3T` — a2T‘T3(2-1)(q + <1‘1)T“’2T3T+ _ 0-2T+T3

"T+T` — aT`T+

T` and T3 which obey the relationsVVe begin by describing the algebra Z/{q(SUq(2)). lt is generated by the generators T1`,

2 Z/{q(SUq(2)) and Covariance

a differential calculus.

q—spinors as well. ln the final section we discuss the suitability of these R-matrices for

approach and we find that the three classes of R—matrices can be constructed from theexactly three classes of solutions. In the third section is then compared to the q—spinorsentations we solve the Yang-Baxter equation for the 3 ® 1 representation and findSUq(2) covariance and R-matrices. After two simple examples of irreducible repre

This paper is organized as follows. ln the next section we discuss the notion ofa representation with one triplet and one singlet under SUq(2).is a search for all possible R—matrices for a given reducible representation — in our casemethod that could be generalized to higher spin. The emphasis of this paper however

In this note we give an example of such R-matrices and construct them by aR-matrices are not known in general.there are candidates for a differential structure. The problem is that these reducibletimes, the generators and the R-matrix mix them differently. Among these R—matricesIf in the reduction of the product the same irreducible representation occurs severalare SUq(2) covariant but do not decompose the same way as the SUq(2) generators do.

There are however R—matrices defined on reducible representations of S U,,(2) that

the universal R-matrix.

momentum two or larger. Thus it is not possible to construct a Fock space based onsame if one tries to define creation and annihilation operators for particles with angularthat would satisfy the Poincaré -Birkhoff—\Vitt theorem. The situation is exactly thevalues. It is then not possible to build a differential structure based on this R-matrixon irreducible subspaces of the product space all projectors enter with different eigentwo or higher. The reason is that in the decomposition of the R-matrix into projectorsnot allow a differential structure on the representation spaces with angular momentumproduct of any two irreducible representations of S Uq(2). However, these R—rnatrices doFrom the universal R—ma.trix of the SUq(2) algebra. there follows an R-matrix for the

1 Introduction

or more spaces is independent of the order in which adjacent spaces are swapped. OCR Outputthe R-matrix. A consequence of this equation is that reordering of a product of threeHere the subscripts indicate which of three spaces in a tensor product is acted on by

(2-6)Rirasaiz Z Rsészézs.

posed. First it should satisfy the quantum Yang—Baxter equation:where repeated indices are summed. On such an R—matrix several properties are im

iumih ® |i2»m2)2 = R(j""")U°‘m’) <j;,m;>(j;,m;) lJ£»m§)2 ® |ii»mi)i (2-5)

product can be reversed. It is a set of numerical factors defining this action:

The R-matrix defines how the order of two different sets of states in a tensorSeveral explicit examples will be given in the following.(2.2) acting on the tensor product to be compatible with the representationform of the Clebsch-Gordan coefficients Cq is determined by requiring the coproductThe subscripts on the product state indicate the ordering of the underlying spaces. The

·]=lj1·j2

I

j1"l"j2

product can be written as a sum of states with different total angular momentum:two different sets of states \j1,m1}1 and ljz, m2)2. As in the classical case their tensor

VVe will be interested in tensor products of these representation spaces. Consider[rz], is the q—number defined by [nl, =

q‘1[2mlq-2 lj, m)T3 Um)

T‘ ILM) qw + mt-zu — m + itz nm — 1> <2-3)

T+ lim) (rk/[1+ m +11q—=U— mw IJ, m +1)

the generators on a state \j,m} isStates within each representation have eigenvalues of T3 labeled by m. The action ofthere is a Casimir operator with eigenvalues labeled by j, the total angular momentum.

The representations of Z/{q(SUq(‘2)) are well known As in the classical case

and is coassociative.

Here we define A = q — q‘1. This coproduct is a homomorphism 0f the algebra (2.1)

MTB) T3®1+(1-AT3)®T3

A(T‘) T·®1+ (1—)\T3)f ®T— (22)

A(T+> T+®1+ 1-AT3®T+()?

representation spaces. For Z/{q(SUq(2)) we have

a = ——q ' or a = —q‘. (2.11) OCR Output

b = 1 we find two solutions to the Yang—Baxter equation;ln the classical limit q —-> 1, a : -1 and b = 1. Fixing the normalization by setting

OWU Z ¤ |0»0>2i ILM).? =b|1»m}2i (2-10)

parameterize the R-matrix:(The overall normalizations have been chosen for convenience.) Following (2.7) we

1¤]·>12 ·1%%ql». .·>®I%7%}

1aO>l2 (<1’+1)`i <1a—a>.®1aa>.+ <r1l%»%>.® la-%>.><2—9>

17 _1>12 q`1l%»—%}.®l%=‘i>2

and for the triplet we have

(28)0»0>r2 = q`1l%»·%1 2 . ® If ·%}2)® if %>— 1%%)

for the singlet are found to betriplet. Using the coproduct and representation rules the Clebsch-Gordan coefficientsdoublets. According to the rule 2 ® 2 = 3 ® 1 their product will have a singlet and a

The first and simplest example we will consider is the case of two Z/{q(SUq(2))be analyzed to determine its projector decomposition.purely cubic in R the overall normalization of R is not fixed.) The R-matrix may thendetermined by solving the Yang—Baxter equation. (Since the Yang-Baxter equation istions the R-matrix can be parameterized according to (2.7). The parameters are then

VVe may now construct covariant R-matrices. For any two underlying representaThis is the most general form of a covariant R—matrix.

(2-7)J» Mliz = Ri. |J» M};1 ·

may be rewritten in the product state basis:multiplets may mix. Labeling different multiplets with the same J as |J,]VI)’, (2.5)reducible there may be different multiplets with the same J , and in reordering theseall have different total angular momentum J. However when an underlying space isare preserved. When the underlying spaces are both irreducible the product multipletsnumbers are conserved and the relative normalizations within multiplets of a given Jwhen the corresponding product states are reordered according to (2.5) their quantumdefinite quantum numbers J and NI by inverting (2.4). Covariance of R means thatOne way of stating this is as follows. Product states may be arranged into states of

A final property of our R—matrices is that they should be Z/{q(SUq(2)) covariant.

as will be discussed in section 4.

decomposition. This will be especially important for constructing a differential calculusIl = ZiPi. It is useful to characterize R-matrices by their eigenvalues and projector

jectors: R = Zi /\;Pi. The projectors obey PiPj = 6UPi and sum to the identity matrix:Another feature of R—matrices is that they can be decomposed into a sum of pro

R Z P5 ·I· q_PT — q}),4 OCR Output6-4

q"6 with multiplicity 1. The projector decomposition isFor the former the eigenvalues are 1 with multiplicity 5, -q'4 with multiplicity 3 andThese solutions correspond to the usual R-matrix for Z/{q(SOq(3)) and its inverse

(2.17); ; {qa, Op Z ; iq,_ 6_ -6 ‘

to the Yang-Baxter equation with parameter valuesFor q —> 1, cz = 1, b = -1 and c = 1. Normalizing so that c = 1 there are two solutions

0,0),, : a 10,0),, I1,m)12 : b |1,m)21 I2,m)12 : c|2,m)21. (2.16)

simple formAs in the previous case the multiplets all have different J and the R-matrix has the

(9 ) ...152,2),2 oc q`2|1,1)1®|1,1)2.

271)12 K

2»0I12 O< 94 IL -1Ii ® IL U2 + <1(<12 +1)|L0>1 ® ILO); + IL III ® IL —lI2

2v _2>12 OC (12IL—lI1 ® Ilv _l>2

we give only the appropriate linear combinations:Since the normalizations for the 5 are complicated and not important for our purposes,

1,1l12 1 q-1I]·70)]®I]‘7]‘>2-qI]‘7]·>]_®I]‘70>2'

1, 0)12 17 _l)]_ ® I]`7 ]`>2 _ I1? 1)] ® I]`7 -—1>2 U- A I]'70>1 ® Il’0>2

1,—1l12 q-1 I1? _]‘>1 ® I17O>2 —® Il"? _]‘)2

and the triplet states are

(L0}12 = <12IL1I1 ® IL —1}2 ·<1|L0I1 ® U.0)2 + IL —1}1 ® IL 1}; (2-13)

will have three multiplets: 3 ® 3 = 5 @ 3 ® 1. The singlet isFor our next example we will take two Z/(q(SUq(2)) doublets. The product space

Here PS is the q—deformed symmetrizer and PA is the q—deformed antisymmetrizer.

R = Pg — q"PA. (2.12)

and —q‘2 with multiplicity 1 and the projector decomposition isL{q(SUq(2)) and its inverse For the former the eigenvalues are 1 with multiplicity 3the basis of (2.5) it is seen that these solutions correspond to the usual R-matrix for(There is also a. solution for a which has the wrong classical limit.) When iewritten in

correct limit for q —> 1. OCR Outputthese terms (al, dg, and eg) are undetermined and may be any functions of q with theinvolves product states with an underlying singlet [O, 0), or IO, 0),. The parameters forprecisely the R-matrix for SOq(3) from the previous example. The rest of the R-matrix(2.5) the 16 ><_16 matrix is found to be block diagonal. There is a 9 >< 9 block which issically are nonzero, and there is no mixing of multiplets. When written in the basis of

The first solution is quite simple. Only the parameters which do not vanish clasin the appendix. Here we will discuss the general features of the solutions.Z/{q(SUq(2)) covariant R-matrices. The parameter values for these solutions are listedto a known R-matrix. So we learn that the known R-matrices are the only possibleclass having an R-matrix and its inverse. Each of the three solutions corresponds

Solving the Yang-Baxter equation we find exactly three classes of solutions, each

vanish.

In the limit q —-> 1, al = by == dg = eg : f = 1, cl = —1 and the rest of the parameters

fllmlgflm)?

l»m>T2 sS 61 I1.m>?+ €2l1»m>§f+ ealhmlgi

idl |1,m)§{+d2|1,m)§{+d3|1,m)fflim}?(221) i

r Scrl1,m>$f+<¤¤I1.m>€Y+¤3I1.m>£hm)?

"fbi |0,0)§f + bg |0,0)§?OM

?Sai jU,O)§j+Cl2jO,O)g¤.¤>?§

parameterize the R—matrix as follows:Now there can be mixing among the two singlets and among the three triplets. We

0,u>1®|i,i>2 i,i>{f Z |1,1>1®|o,0>2.1,1>f?

0,0)1®j1,0)2 1,0){§ : j1,0>1®|0,0)2 (2.20)1,0>f}

rSO>0>1®l]‘7_1>2 i17_1>12:1,—1>fZ

and two triplets:

(2.19)0,0)f§ Z |0,0)1 ® |0,0)2

1

and we write them as \J, 1W)§. There are three additional multiplets, a singlet:superscript to the state kets. The TT multiplets have the same form as (2.13)—(2.15)in the product are triplets or singlets. In the following we will append a corresponding511 ® 311 ® 111 @ 351 @ 315 ® 155. The subscripts indicate whether the two spacesZ/{q(SUq(2)). Their product has six rnuitiplets according to the rule (3 ® 1) ® (3 ® 1)

We now consider the interesting case of two 3 ® 1 reducible representations ofPS and the antisymmetrizer PA.where the q—dcf0rmed projectors are the trace projector PT, the traceless symmetrizer

1The second solution corresponds to R; and the third solution corresponds to Ru in [3, 4]. OCR Output

xl = Ié, -2;) and x2 = q"1 [1,, With a second copy y°‘ the appropriate R—matrixspinors. The spinors x°‘, cr = 1,2 are a Z2{q(SUq(2)) doublet with the assignment

The simplest example of such coordinates is the quantum plane, or quantumin which adjacent coordinates are swapped.product of the coordinates may be reordered and the result is independent of the orderwhere k is an arbitrary normalization. Since R satisfies the Yang—Baxter equation any

(3.1)xixf Z JQRU Hxkxl(2.5) may be written ascopy X' corresponding to a second space m],. Dropping the tensor product notation

Cpnsider coordinates xi corresponding to some representation m], and a seconddoublet of L{q(SU,,(2)).3 @ 1 representation can be constructed from spinor coordinates corresponding to thetion spaces discussed in the previous section. We will show how the R—matrices for theln this section we will consider q-deformed coordinates as examples of the representa

3 q—Deformed Coordinates

tions of these projectors consistent with the algebra of q-Minkowski coordinates.As noted in [3, 4] these two R-matrices and their inverses are the only linear combina

R = PT + q·PS - q1>+ - q1>_. (2.22)‘*·2·2

and for the third solution is

R = PT + PS -— qP+ — qP_ (2.22]2'2

decomposition for the second solution isthe antisymmetrizer splits into selfdual and antiselfdual projectors Pi. The projectorAgain there is a trace projector PT and a traceless symmetrizer PS. Here howevermay be extracted from each of them. However taken together we have four projectors.9 and 6. Since each of these matrices has three distinct eigenvalues three projectorsrespectively. The third solution has eigenvalues 1, q"* and q‘2 with multiplicities 1,The second solution has eigenvalues 1, —q2 and —q`2 with multiplicities 10, 3 and 3q—Lorentz group, where a triplet and a singlet were combined into a four-vectorl [3, 4].in the basis of (2.5) it is found that these are the R—matrices as constructed for thein the relative normalization between the underlying singlet and triplet. When writtenwith the same J. For both solutions there is a free parameter, but it may be absorbedall parameters are nonvanishing and there is maximal mixing between all multiplets1,m}multiplet. The parameters ag and bl are always zero. For the third solution

TS

TTmultiplets, between the ]1,m]multiplet and either the ]1,m}ST multiplet or thelinear combinations of the same projectors. The second solution has some mixing of

The second and third solutions should be discussed together since they are different

A,XX = O. (4.1) OCR Outputindices, we write

product of two coordinates to be annihilated by the antisymmetrizers. SuppressingE, S; + Z, A,. Since the coordinates are classically commuting objects, we require thesymmetrizers S; and antisymmetrizers Ai, which sum to the identity matrix: 11from the same copy. Suppose there is a set of q—deformed projectors on the space,the quantum space. However we have not specified the algebra among coordinates

Equation (3.1) gives the algebra between coordinates from two different copies ofdecomposition of the R-matrix used to formulate the calculus.in the previous section. lt will be seen that there are constraints on the projectorln this section we will consider a differential calculus on the quantum spaces discussed

4 Differential Calculus

Similar results have been obtained in [5] from a different approach.construction followed here could be generalized to higher dimensional representations.generators of L{,,(SU,,(2)), as is the case for the universal R—matrix. The method ofNote that we have constructed R-matrices which cannot be expressed in terms of thewe showed that these R-matrices may be constructed from the basic R-matrixconstructed R—matrices are unique assuming Z/{q(SUq(2)) covariance. ln this section(3.3) gives the inverse solutions. From the previous section We know that these threethird solutions respectively from the previous section. Replacing q`1R by qR"1 inA direct calculation shows that these cases l, ll and lll lead to the first, second and

Case lll : X = mu, X =yv.

Case ll : X = xv, X = yu (3.4)Casel : X:-xy, Xzuv

spinors into two 3 GB 1’s:

where we have suppressed indices. Then there are three distinct ways to group the

xv = q‘1Rvx uv = q"1Rvu

mu = q‘1Ru;1: yv = q`1Rvycry : q`1Rya; yu = q'1Ruy

and ordering:spinor plane: a:°‘, y‘*, u" and v"‘. Following (3.1) we choose the following normalization

For two copies of the 3 EB- 1 X’ and X‘ we will need four copies of the quantumthese bispinors by X', 1 f i f 4.into a four-dimensional 3 EB 1 reducible representation of L{q(SUq )/Ve will denoteAs in the previous section these two copies of the quantum plane may be combined

0 O 0 q

) )(3.2)RGBW5 : i lg l\ fE

q O 0 0identification:

for (3.1) is the R-matrix constructed using (2.10), (2.11) and using this coordinate

taking the value 1 in the limit q —> 1. (For the R-matrix constructed in section 3 we OCR OutputThe parameters al, dg and eg are undetermined. They may be any functions of q

(A.1)bg = q`° cl = -q"

The first solution is quite simple, with no mixing of different multiplets. \Ve have

obtained for all of the constructed R-matrices.

not fix the normalization of R. \¢Ve fix the normalization by choosing f = 1, the valuefor the R-matrix parameterized in (2.21). Recall that the Yang-Baxter equation doesln this appendix we list the parameter values for solutions of the Yang—Baxter equation

A Appendix

cannot be used for a differential calculus.

the antisymmetrizers Pd: enter the R—matrix with different eigenvalues. This R-matrixwas the matrix used for the q-Poincaré algebra in Finally we see in (2.22) thatthird solution can also be used to construct a differential calculus with C = q2R. ThisThen C = q"R is suitable for a differential calculus. Looking at (2.23) we see that theantisymmetrizers all have eigenvalue —q"‘, the value for the S Oq(3) part of the matrix.tation. For the first solution the undetermined parameters may be chosen so that the

We may now apply these considerations to the R-matrices for the 3 QB 1 represenantisymmetrizers having eigenvalue -1.Yang—Baxter equation. Thus for a differential calculus C must be an R—matrix with allobjects X ‘, 0, and §' are considered together consistency requires that C satisfy thed = §‘O;, where f' = dXf are the differentials of the coordinates. When all three

ln [6, 7] the differential calculus has been developed with an exterior derivativethe antisymmetrizers with eigenvalue -1.In order to have a consistent differential calculus the matrix C in (4.2) must have all

C = 20gS; —so we find

vanishes for consistency we see that ori = -1. This must be true for all of the projectors,where in the last step we used A.Aj : 6,jA; and A,Sj = O. Since the left hand side

(4 3)8A.XX : A,(1l + C)X + cubic terms

on the eigenvalues for the antisymmetrizers:a derivative to (4.1) and use (4.2) to move the derivative to the right we get constraintsHere C is some linear combination of the projectors C = E, 035} + Zi mA). If we apply

(4.2)kl0.X] = 6{ -1- Cj;;X0k.

O; acting on the coordinates. This action is\wVe now establish a differential calculus on this space by introducing derivatives

This must be true for all of the antisymmetrizers.

Again r is a free parameter and the constructed R—matrices are recovered for r : 1. OCR Output

-77qA.cz = n(<1"+1)= nq2Ar"d2 = —r]q»\= qq2/\r‘1 v(q"+)

= v(q%—2<1") C2 = -n(<1"+1W C3 v( 4 T qTm (A6)= w(<1’—<1)r2 be = n(q"+q’)= v(q"+1) az = vq“`>~r‘

The inverse solution has parameters

(A.5)nq/\.A7··1 n(<12 + <1`2)

maAr··1 n(<1“’ + W)_2q_2) C2 W12 + <1‘2)*r v(q*’ + q‘2)M‘

n — b2 n(q‘“‘ + 1)77(q2+ -2 q ) —nq‘2M’2

venience we define the common factor ry = (q2 + l)`1. Then we haveThe third solution is more complicated, with no parameters vanishing. For con

underlying singlet and triplet. The constructed R—matrices are recovered for r : 1.Here r is a free parameter which may be absorbed in the relative normalization of the

el = Ar°1 1+ A2dl Z 0

O (A.4)Cl Z -1 — A2 (q2 + q·2)iTbl Z 0

G1 Z 1

and for the inverse solution

C1 Z 0

dl Z A7"-1 1+/\2C] Z *1 — A2 (q2 + q‘2)M (A-3)bl = 0

(L1 = 1

For the second solution we find

rest of the parameters vanish.Afunctions of q (For the constructed R—matrix we had al = dg = cz = q2.) Again theAgain the parameters al, d3 and e2are undetermined and may be any appropriate

bz = q° cl : —q (A2)we Hnd

had al = dg = cg = q‘2.) The rest 0f the parameters vanish. For the inverse solution

10

based on work with B. Zumino, preprint KA—THEP·1990-22 (1990)at Third Centenary Celebrations of the Mathematische Gesellschaft, March 1990,J. Wess: Differential Calculus on Quantum Planes and Applications. Talk given{7]

Suppl.) 18 B (1990) 302J. Wess and B. Zumino: Covariant Differential Calculus Nucl. Phys. B (Proc.[6]

V. Jain and O. Ogievetsky, Mod. Phys. Lett. A 7 (1992) 2199[5]

First German-Polish Symposium on ‘Particles and F ields’, Rydzyna, Poland, 1992O. Ogievetsky, \rV.B. Schmidke, J. \Vess and B. Zumino, to appear in Proc. The[4]

150 (1992) 495O. Ogievetsky, \wV.B. Schmidke, J. Wiess and B. Zumino, Commun. Math. Phys.[3]

(1990) 193[2] L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, Leningrad Math. J. 1

Verlag, New York and Berlin, 1991), and references therein_Rapallo, Italy, 1990, ed. C.Bartocci, U.Bruzzo and R.Cia11ci, p. 61 (Springer

{1] L.C.Biedenham, Proc. Differential Geometric Methods in Theoretical Physics,

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