noncommutative string theory, the r-matrix, and hopf algebras

17 February 2000

Ž .Physics Letters B 474 2000 295–302

Noncommutative string theory, the R-matrix, and Hopf algebras

Paul Watts 1

Dublin Institute for AdÕanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland

Received 12 November 1999; received in revised form 16 December 1999; accepted 21 December 1999Editor: P.V. Landshoff

Abstract

Motivated by the form of the noncommutative )-product in a system of open strings and Dp-branes with constantnonzero Neveu–Schwarz 2-form, we define a deformed multiplication operation on a quasitriangular Hopf algebra in termsof its R-matrix, and comment on some of its properties. We show that the noncommutative string theory )-product is aparticular example of this multiplication, and comment on other possible Hopf algebraic properties which may underlie thetheory. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 11.25.-w; 11.15.-qKeywords: Noncommutative geometry; R-matrix; Hopf algebras

1. Introduction

Although the subject of gauge theories in non-w xcommutative geometry is not a new one 1 , recently

it has enjoyed something of a resurgence. It reap-peared in the context of matrix models compactified

w xon tori 2 , where it was shown that such models mayŽ .be reformulated as super Yang–Mills SYM theo-

ries on noncommutative spacetimes when theŽ .Neveu–Schwarz NS 2-form B is constant andi j

nonzero. This is accomplished by replacing the usualcommutative multiplication of functions on the space

Žby a noncommutative one, denoted by ) which hadappeared previously in the context of the Moyal

w x.bracket 3–6 . Subsequent studies dealt with variousw xaspects of noncommutative spaces 7–17 . It has also

been realised that one gets a noncommutative SYM

1 E-mail: [email protected]; tel: q353-1-6140148; fax: q353-1-6680561

theory for the case of a system of open strings andŽDp-branes in a flat spacetime with the same condi-

.tion on B , provided that one not only changes alli j

ordinary products to )-products, but also deformsw xthe gauge fields and their transformations 18 . This

seems to suggest that noncommutative geometry maybe an underlying aspect of a large class of theories.

If this is so, then a mathematically consistent wayof dealing with this noncommutativity is needed, andone possibility might be to use the language of Hopf

Ž .algebras HAs . A HA extends the notion of analgebra by including information about its represen-

Ž .tations through the coproduct , and when it is quasi-triangular also gives the algebraic structure of themodules it is represented on through its associatedR-matrix, i.e. commutation relations between moduleelements. Perhaps the best-known cases where non-trivial quasitriangular HAs play key roles are quan-

w xtum groups 19,20 , which may be thought of con-sisting of matrices whose entries are noncommuting.

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 01485-9

( )P. WattsrPhysics Letters B 474 2000 295–302296

Ž .It is the intent of this paper to show in Section 3that the )-product mentioned above is in fact aparticular case of a more general multiplication whichmay be defined in terms of the R-matrix of a particu-lar quasitriangular HA. This may be a clue that thereis indeed a quasitriangular HA structure to thesenoncommutative theories, and might serve as a tenta-tive first step toward finding that structure. Thiscould in turn lead toward a way of formulatinggauge theories on a large class of spaces, not justcommutative ones. Some of the implications of sucha structure are commented upon in Section 4.

Parts of this work are somewhat pedagogical, butthis is because it is intended for an audience forwhom the language of HAs may not be too familiar;for those with some knowledge of the subject, Sec-tion 2 can be skimmed just to determine the notationwe use herein. Others who may be curious aboutHAs may find the short review useful.

2. Hopf algebras

This section is meant to be a review of both theformal aspects of what constitutes a Hopf algebraand the explicit example where we consider thealgebra of functions over a manifold and the partialderivatives acting on them.

2.1. Formal definitions

A Hopf algebra HH is an associative algebra withunit 1 over a field k which is also equipped with a

Ž .counit e :HH™k, a coproduct or comultiplicationD:HH™HHmHH and an antipode S:HH™HH; the firsttwo of these are defined to be homomorphisms, thethird an antihomomorphism, and all three satisfy therelations

Dm id D f s id mD D f ,Ž . Ž . Ž . Ž .em id D f s id me D f s f ,Ž . Ž . Ž . Ž .

m Sm id D f sm id mS D f se f 1,Ž . Ž . Ž . Ž . Ž .Ž . Ž .2.1Ž .

where fgHH, id is the identity map f¨ f and m isŽthe multiplication operation on HH which will usu-

.ally be suppressed . For future reference, the first ofŽ .2.1 is often called ‘coassociativity’.

w xFor clarity, we adopt Sweedler’s notation 21 , inŽ .which we write the coproduct as D f s f m f ,Ž1. Ž2.

where there is an implied summation. For example,Ž .using this convention, the third of 2.1 may be

. . Ž .written as S f f s f S f se f 1.Ž . Ž .Ž1 Ž2. Ž1. Ž2We can define the dually paired HA to HH, de-

noted HH ) , in the following way: As a vector spaceover k, HH ) is just the dual space to HH, so there is

) Žan inner product taking HH mHH which we assume. ² :to be nondegenerate to k, written as x , f for

xgHH ) and fgHH. HH ) may be given a Hopfalgebra structure by defining the multiplication, unit,coproduct, counit and antipode on HH ) via

² : ² :² :xy , f [ xmy ,D f , 1, f [e f ,Ž . Ž .

² :² :² :D x , fmg [ x , fg , e x [ x ,1 ,Ž . Ž .

² : ² :S x , f [ x ,S f . 2.2Ž . Ž . Ž .

It is straightforward to show that these operationssatisfy all the HA conditions.

HH ) may be thought of as an algebra of operatorsŽ . )on HH when we define the left action of xgHH

on fgHH, denoted xP f , as

² :xP f[ f x , f . 2.3Ž .Ž .Ž1. 2

Ž . Ž .Since it is easy to show that xy P fsxP yP f , thisis indeed an action of elements of HH ) on those ofHH, and thus gives a representation of HH ) with HH

as the module. Furthermore, there is a sort of LeibnizŽ . Ž .Ž . )rule: xP fg s x P f x Pg . To give HH anŽ1. Ž2.

Ž . Ž .interpretation, notice that P x [xye x 1 and0Ž . Ž . ) )P x [e x 1 are projections on HH , so HH s1

ker e[k1. Note that for xgker e , xP1s0, sothese may be thought of as derivatives on HH; ele-ments of k1 just multiply elements of HH by ele-ments of k. Finally, if we have a concept of deriva-tives on HH, we can define an integral on HH as alinear map H:HH™k such that H xP fs0 for anyxgker e .

A quasitriangular Hopf algebra is an HA HH forwhich there is a special invertible element RgHHmHH, called the R-matrix, which has the properties

t(D f sR D f Ry1 ,Ž . Ž . Ž .Dm id R sR R ,Ž . Ž . 13 23

id mD R sR R , 2.4Ž . Ž . Ž .13 12

( )P. WattsrPhysics Letters B 474 2000 295–302 297

where t : fmg¨gm f and the subscripts on R inthe latter two above tell in which pieces of HHm3 R

a Ž .lives, i.e. if Rsr mr sum implied , then R [a 13

r m1mr a. One consequence of these properties ona

the R-matrix is the fact that it must satisfy theŽ .Yang–Baxter equation YBE

R R R sR R R . 2.5Ž .12 13 23 23 13 12

ŽNote that if Dst(D, Rs1m1 satisfies all the.above, so all such HAs are trivially quasitriangular.

2.2. Hopf algebra of functions and deriÕatiÕes

Let FF be the space of functions mapping R pq1

into C. FF is made into a commutative associativeŽ .Ž . Ž . Ž .algebra in the usual way, e.g. fg x [ f x g x

for f , ggFF and xgR pq1. Furthermore, FF can beextended to an HA if we define the following on thecoordinate maps x i:

D x i [x i m1q1mx i ,Ž .e x i [0, S x i [yx i . 2.6Ž . Ž . Ž .

ŽOnce we have the above as well as the relationsŽ . Ž . Ž . .D 1 s1m1, e 1 s1 and S 1 s1, of course , we

can use the fact that D and e are homomorphismsand S is an antihomomorphism to extend them to all

Žmonomials of the coordinate functions and ignoring.questions of completeness thus to all of FF.

We define FF ) , the dually paired HA to FF, to be� < 4spanned by elements E is0, . . . , p , where the in-i

ner product between E gFF ) and a monomial in FFi

is

N Nn pjpE , x s d d d , 2.7Ž . Ž .Ł Ý Łi i j n 1 n 0¦ ; p p q

q/pps1 ps1

² :as well as E ,1 s0. Once we have this, the HAi

structure of FF ) is immediate: The form of thecoproduct on FF tells us that FF ) is commutative,the commutativity of FF gives D E sE m1q1mŽ .i i

E , and e E s0 and S E syE .Ž . Ž .i i i i

The action of E on a monomial in FF can bei

computed as well; it will perhaps be no surprise tothe reader that the result is

N Nn nn y1p qpj j jp p qE P x s n x d x .Ž . Ž . Ž .Ł Ý Łi p i jp

q/pps1 ps1

2.8Ž .

Ž .Furthermore, for two arbitrary functions f x andŽ .g x ,

E P f x g xŽ . Ž .Ž .i

s E P f x g x q f x E Pg x , 2.9Ž . Ž . Ž . Ž . Ž .Ž . Ž .i i

so FF ) is indeed the space of partial derivatives onpq1 Žfunctions over R . Note: For brevity’s sake, from

now on we will omit the P when speaking of the.action of a partial derivative on a function of x.

3. The )-product

Motivated by the previously-mentioned literatureon noncommuting strings, we now use the formalismjust presented to introduce a new )-product on theHopf algebra HH. We examine its properties, andthen go on to show that the noncommutative productin string theory is a particular case of this multiplica-tion.

3.1. Formal definition of the )-product

Suppose we have two dually paired HAs HH andHH ) , and further suppose that there is an R-matrix onHH ). We can thus define a new operation ):HHmHH

™HH in terms of R and the usual multiplication onHH via

f ) g[ f g R , f mg . 3.1² : Ž .Ž1. Ž1. 21 Ž2. Ž2.

Ž ) .If HH is trivially quasitriangular, then f ) gs fg.This operation is actually associative as a conse-quence of the YBE. To see this explicitly, we pick

Ž .f , g,hgHH, and first compute f ) g ) h:

f ) g ) hs f g R , f mg ) h² :Ž . ž /Ž . Ž .1 1 21 Ž2. Ž2.

s f g h R , f g mh² :Ž1. Ž1. Ž1. 21 Ž2. Ž2. Ž2.

= R , f mg² :21 Ž3. Ž3.

s f g h R R R ,²Ž1. Ž1. Ž1. 32 31 54

f mg mh m f mg :Ž2. Ž2. Ž2. Ž3. Ž3.

s f g h R R R ,²Ž1. Ž1. Ž1. 32 31 21

f mg mh . 3.2: Ž .Ž2. Ž2. Ž2.


In going fron the first line to the second, we used theŽ .coassociativity of D; next, the third of 2.2 , the

Ž .second of 2.4 and a relabelling of the indices so asto stick everything into one inner product; and the

Ž .last step used the first of 2.2 . If we now calculateŽ .f ) g ) h , an exactly analogous compution gives the

result above with the left argument of the innerproduct replaced by R R R , and by using the21 31 32

Ž .YBE with the first and third tensor spaces swapped ,we find the two quantities are exactly the same.

ˆŽ . Ž .Hence, f ) g ) hs f ) g ) h and HH, the algebraŽ .constructed from the vector space HH and the )-

product is associative. Since it turns out that 1) fsˆf )1s f , HH is unital as well.

This is in general a noncommutative multiplica-ˆtion: Even if HH itself is commutative, HH may not

be if R /R. To examine this a bit more, it can be21

shown that the defining relations for R imply thatŽ .Ž . Ž .Ž .em id R s id me R s1m1, so that we maydefine a new quantity Qgker e mker e as Q[Ry1m1. When written in terms of Q , the )-productbecomes f ) gs fgq f g Q , f mg . Sup-² :Ž1. Ž1. 21 Ž2. Ž2.pose we now define Q'u mu a and computea

u a P f u Pg :Ž .a

u a P f u Pg su a P fg u , g² :Ž . Ž .Ž .a 1 a Ž2.

s f g u a , f g u , g² :² :Ž1. Ž1. Ž2. Ž2. a Ž3.

s f g D u a mu ,² Ž .Ž1. Ž1. a

f mD g , 3.3:Ž . Ž .Ž .Ž2. 2

where in the last step we have split up f g to getŽ2. Ž2.a Ž .the coproduct of u . The third of 2.4 in terms of Q

Ž .Ž .is id mD Q sQ qQ qQ Q ; if we use13 12 13 12.this, and then get rid of D g by exchanging it forŽ .Ž2

a multiplication in the HH ) argument of the innerproduct, we find

u a P f u PgŽ .a

s f g Q q 1mk R , f mg , 3.4² :Ž . Ž .Ž1. Ž1. 21 21 Ž2. Ž2.

where k[u au . Let us now suppose that there is aa

kind of ‘tracelessness’ condition on Q , and k van-ishes. We must stress that this is purely an assump-tion, but if it does in fact hold, then we conclude that

f ) gs fgqu a P f u Pg , 3.5Ž . Ž .a

in other words, f ) g and fg differ by a ‘totalderivative’, since u a gker e . It follows that if thereis an integral H defined over HH, H f ) gsH fg.

A comment on the above: The above can beeasily extended to the case where we have a N=N

Ž .matrix-valued Hopf algebra, i.e. HHmM k , inN

which case the )-product includes matrix multiplica-Ž . i i ktion as well: f ) g s f ) g . Note that this is notj k j

the same as a quantum group, since HH is not beingthought of as the set of functions over the groupmanifold. Thus, when we take coproducts et al., the‘matrix part’ is unaffected.

Also, this )-product may not be unique: If weŽ .assume that ) is defined by 3.1 without assuming

Ž .that R satisfies 2.4 , then the requirement that ) isassociative means R must satisfy

Dm id R R s id mD R R . 3.6Ž . Ž . Ž . Ž . Ž .21 21 21 32

As shown above, R satisfies this, but so does Ry1,21

and unless the HA is triangular, i.e. R sRy1, there21

may be more than one associative )-product.

3.2. NoncommutatiÕe String Theory

w xRecently 18 , it has been shown that when oneconsiders open strings and Dp-branes in a space withmetric g and constant nonvanishing Neveu–i j

Ž .Schwarz 2-form B i, js0, . . . , p , the theory cani j

be reformulated as a SYM theory on a space wherethe coordinates no longer commute, but instead sat-isfy the deformed relation

x i) x j yx j

) x i s iu i j , 3.7Ž .where

i j1 12Xi ju [y 2pa B .Ž . X Xž /gq2pa B gy2pa B

3.8Ž .Ž . Ž .More generally, it is shown that if f x and g x

Ž .are matrix-valued functions, the noncommutativeŽ . Ž .product between two functions f x and g x is

f x ) g xŽ . Ž .i E E

i ju2 Ej i Ez j[e f xqj g xqz . 3.9Ž . Ž . Ž .

jszs0

The action then can be expressed in SYM formprovided that all ordinary multiplications are re-


placed by this )-product, the ‘closed string metric’g is replaced by the ‘open string metric’ G [gi j i j i j

y 2paX 2 Bgy1B and the gauge field A isŽ . Ž . ii j

ˆ Žreplaced by A , which depends on both A and itsi i. i jderivatives and u .

Ž . Ž .If we compare 3.9 to 3.1 , it suggests that acandidate for the R-matrix is

ii jy u E mEi jRse , 3.10Ž .2

but we must make sure it satisfies all the appropriaterelations. Since FF ) is commutative, and D E isŽ .i

Ž .symmetric, it is evident that the first of 2.4 isŽ .satisfied. To check the second of 2.4 , note that

k 1D E sŁ Ý Ýi llž / ll ! ky ll !Ž .lls1 s llFk

=E . . . E mE . . . E ,i i i is Ž1. s Ž ll . s Ž llq1 . s Ž k .

3.11Ž .

Ž .where s is a permutation of 1, . . . ,k . Therefore,

Dm id RŽ . Ž .k` yiŽ .

i jp ps u D E m EÝ Ł Ł Łi jž /k ž /ž / ll q2 k! pFk qFkllFkks0

k` yiŽ .i jp ps uÝ Ý Łk ž /2 k! ll ! ky ll !Ž . pFkks0 llFk

=ll k

E m E m EÝ Ł Ł Łi i jŽ . Ž .m n ž /s s qž /ms1 qFknsllq1s

ll` yiŽ .i jp ps u EÝ Ł iž /ll p2 ll ! pFlllls0

ky ll kyiŽ .i jn nm u EÝ Ł ikyll nž /2 ky ll !Ž . nsllq1kGll

m E , 3.12Ž .Ł jž /qqFk

where in the last step we have switched the sumsover k and ll , and also used the commutativity ofthe partial derivatives in the third tensor space to get

Ž .rid of s picking up a k! in the process . If we nowsplit the product in the third space into two, one from

1 to ll and the other from llq1 to k, and letky llsr, then we get

llqr` yiŽ .i jp pDm id R s u EŽ . Ž . Ý Ł iž /llqr p2 ll !r ! pFllll ,rs0

m u iXn jX

nE XŁ iž /nnFr

m E E XŁ Ł Xj jž /q qXq FrqFll

ll` yiŽ . lli js u E m1mEŽ .Ý i jll2 ll !lls0

=

r` yiŽ . rX Xi j

X X1mu E mEŽ .Ý i jr2 r !rs0

sR R . 3.13Ž .13 23

A very similar calculation confirms that the last ofŽ . Ž .2.4 holds as well, so 3.10 is in fact an R-matrix,and FF a quasitriangular HA. The YBE is thereforesatisfied by this R, and thus ) is associative, asproven above. Actually, any element of FF ) mFF )

Ž . Ž .satisfying 3.6 must have the form 3.10 , whichmeans for this particular HA, the )-product is unique.

To check that this R-matrix gives us the correctŽ .commutation relation 3.7 , we simply compute

x i) x j:

x i) x j

s x i x j R , x i m x j² :Ž . Ž . Ž . Ž .Ž . Ž . Ž . Ž .1 1 2 221

i j² : i jsx x R ,1m1 qx R ,1mx² :21 21

qx j R , x i m1 q R , x i mx j² : ² :21 21

ii j i jsx x q u . 3.14Ž .

2

Therefore, by switching i and j and subtracting, weŽ . Ž .recover 3.7 . 3.9 holds almost by definition, since,

Ž .from 3.1 , f ) g is simply the product between theaction of the first tensor space of R on f and the21

action of the second on g.This R also satisfies the ‘tracelessness’ condition:

If we subtract 1m1 from R and multiply the two21

tensor product spaces together, we immediately getk , which is evidently zero due to the antisymmetryŽ . i jand constancy of u and commutativity of thepartial derivatives, so f ) g and fg differ by a total


derivative. And since the original multiplication wasŽ . Ž .commutative to begin with, Htr f ) g sHtr g ) f .

4. Noncommutative gauge theories

We now make some comments on noncommuta-tive gauge theories and how they may or may notrelate to HAs.

4.1. Algebraic structure

In Section 2.2, we showed that for the commuta-tive case, there is a HA structure to both functionsand derivatives on R pq1. However, even though the

ˆnoncommutative FF is an associative unital algebra,it is not a HA, as can be seen from the following: As

ˆwe proved, the unit in FF is the unit in FF, so that ifFF is a HA and has a counit e , then the fact that thisˆ

ˆcounit is a homomorphism from FF to C meansŽ . i j j ie 1 s1. Now, consider x ) x yx ) x ; the counitˆ

will map this to zero. But this commutator is iu i j1,ˆso we have a contradiction. Therefore, FF cannot be

a HA.This is no real surprise. In the first place, the

requirements necessary for a space to be a HA arevery restrictive, and in general there is no reason toexpect an arbitrary algebra to also be a HA. Further-

ˆ ) Žmore, although FF has a coalgebra structure aˆ.counit and a coproduct due to the fact that FF is

unital and associative, there is no ‘deformed deriva-tive’. We can certainly define an inner product be-tween the two spaces, but due to the lack of a

ˆ ˆ )coproduct on FF, there is no action of FF on it, andthus no concept of derivative. This is borne out bythe fact that we must use the ordinary partial deriva-tive to define the noncommutative SYM field

ˆ ˆ ˆ ˆ w xstrength, via F sE A y iA ) A 18 ; if a de-i j w i j x w i j xformed derivative were available, it would be themore natural choice, but this is not the case.

However, although there is no interpretation ofˆ )elements of FF as objects with a local action on

FF, it might still be possible to interpret them asnonlocal operators. As an illustration of this, con-sider the case of the 2-dimensional quantum hyper-

Žplane: The coordinates x, y generate an algebra the.functions on the plane modulo the commutation

relation xysqyx, and the ‘derivatives’ E ,E act onx y

Ž . Ža function f x, y ordered so that all xs appear to.the left of all ys as

f qy2 x , y y f x , yŽ .Ž .E f x , y s ,Ž .x y2q y1 xŽ .

f qy2 x ,qy2 y y f qy2 x , yŽ . Ž .E f x , y s , 4.1Ž . Ž .y y2q y1 yŽ .

w xwhere qgR 22 . Note that as q™1, these becomeordinary derivatives, but otherwise they are nonlocaldifference operators. The string case could be simi-lar, with u i j playing the role of lnq. This granulari-sation of the spacetime might explain the absence of

w xsmall instanton singularities 23,18 in the noncom-muting theory, by smearing out such objects overmore than one point.

4.2. Gauge fields and Hopf algebras

The fact that the )-product can be defined interms of an abstract HA and includes the noncommu-tative string theory case hints at the possibility ofdescribing the entire theory using a quasitriangularHA, where the R-matrix depends on u i j. However,this is certainly not sufficient, since we have not

ˆconsidered the gauge field A . We have also notiˆaddressed the matter of the map A ¨A whichi i

allow us to cast the action in SYM form. We nowaddress both of these issues.

u i j is inherently a HA parameter; it appears in theR-matrix and therefore describes the HA structure ofFF and FF ). This can be seen either explicitly, as in

Ž .Ž . Ž . y1 )the relation t(D x sRD x R on any HH , orvia the commutation relations of elements of HH,which may be expressed as

² : y1g fs R , f mg f g R , f mg . 4.2² : Ž .Ž . Ž .1 1 Ž2. Ž2. Ž3. Ž3.

So, motivated by these facts, it therefore seemsreasonable to conjecture that all u-dependence in thetheory is in the HA structure of FF and FF ).

If this is true, then the u-dependence in the non-commuting theory must arise from the underlyingHA describing the commutative theory. This gives usa bit of information about the gauge fields: We knowthat the change of variables between the gauge fields

w xof the two theories involves u 18 , which meanssome sort of HA-derived operation is involved in


going from one to the other. Since, as we proved inˆthe previous section, the noncommutative algebra FF

is not a HA, there must be some element WgFFŽ .with given HA properties coproduct, etc. which is

ˆrelated to both A and A . The assumption that alli i

u-dependence is in the HA structure and not in FFŽ .as a vector space itself leads to the conclusion thatW is independent of u , and thus must be relatedto A , since this is the gauge field for the u i j s0i

case. If we also assume that the only dependence onthe open string metric G is from constructingi j

Lorentz-invariant quantities in the integral, e.g.pq1 i j'Hd x G G a b , then W should also be G-inde-i j

pendent.At this writing, we do not know precisely what

this element might be, but a natural candidate wouldŽbe the Wilson line which explains why we call it

.W : Recall that the Wilson line W is given byŽ .C x , x0

i AHW [Pe , 4.3Ž .Ž .C x , xŽ .C x , x 00

where C x , x is a path going from x to x. ItŽ .0 0

depends on the gauge field of the commutative the-ory, and is independent of u and G, so it fits thecriteria we just outlined. We might also be able to

ôbtain A in the following way: Our proposed Wi

must relate both gauge fields in a HA-dependentway, but be independent of u . The relation of W toC

A is through the multiplication on FF, via the path-iôrdered exponent. We could therefore define A toi

be that function for which the same W can beC

written as a path-ordered exponential as well, butŽ .this time using the )-product instead denoted e . Inˆ

other words,

î AHW sPe Ž .C x , xŽ .C x , x 00

ˆs1q i AyH HŽ . Ž .C x , x C x , x0 0

= ˆ Â) A q . . . 4.4Ž .Hž /Ž .C x , x0 1

Then the condition

E î AHPe s0 4.5Ž .ˆ Ž .C x , x0i jEu

ˆ i jwould give a differential equation involving A , ui

and W , which we could presumably solve to findC

ˆ i j Ž .A as a function of u and derivatives of A . Ouri i

first calculations show enough similarities to the firstŽ . w xof Eq. 3.5 of Ref. 18 to be encouraging.

As for the gauge transformation, the sameidea applies: A finite transformation on W givenC

Ž . i lŽ x . Ž .by a unitary matrix U x s e gives U xW Uy1 x . This should be the same as if weŽ .Ž .C x , x 00 ˆstarted with W in terms of ) and A , and thenC i

transformed by U expressed in terms of ) and a newî lˆmatrix l via Use . Then by solvingˆ

E ˆ ˆ ˆŽ .i l x i A yi lŽ x .H 0e ) e ) e s0, 4.6Ž .ˆ ˆ ˆŽ .C x , x0i jEu

Ž . w xwe would obtain the second of 3.5 in Ref. 18 .So the sought-after WgFF could be related to the

Wilson loop W , even though we do not claim toC

have offered anything more than a vague justifica-tion for this feeling. We have not said anything about

Žthe HA properties of W although the HA enters in.going from e to e, via ) , nor have we said how theˆ

curve C x , x would be chosen, since we haveŽ .0

taken it to be completely arbitrary. And perhaps mostimportantly, we have not considered what the action

Ž .might be as a function of W and R such that weultimately end up with a SYM form when we go tothe noncommuting spacetime. Regardless, there isenough evidence to consider our Wilson line guessas reasonable.

Everything we have done above has been for thespecific case of a noncommutative string, where we

Ž pq1.started with a commutative space R ; formulat-ing it in the HA language as we propose would alsobe a way of coming up with gauge theories wherethe original space might be noncommutative. Stepsin this direction have been made when the gauge

Ž .group and possibly also the spacetime is deformedw x24 , and this may mean there is some hope ofsuccess for the present problem.

5. Conclusions

We have shown that the product appearing innoncommutative string theory is simply a specificcase of one which may be defined in terms of aquasitriangular HA. This fact has lead us to speculatethat it may be possible to relate an arbitrary noncom-


mutative gauge theory to a quasitriangular HA in thisway, and we have commented on some ways inwhich this may be done. We hope to address thispossibility in future work.

Acknowledgements

´I would like to thank David Fairlie, Lochlainn ORaifeartaigh, Jan Pawlowski and Jorg Teschner for¨their helpful comments, suggestions and advice.

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noncommutative string theory, the r-matrix, and hopf algebras

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