Letters in Math. Phys. 28 (1993), 251–255

arXiv:math/9212209v1 [math.QA] 1 Dec 1992Letters in Math. Phys.

28 (1993), 251–255A QUANTUM GROUP LIKE STRUCTUREON NON COMMUTATIVE 2–TORIAndreas CapPeter W. MichorHermann SchichlInstitut f¨ur Mathematik, Universit¨at Wien,Strudlhofgasse 4, A-1090 Wien, Austria.IntroductionThis paper grew out of a project which aims at a general theory of bundles,and in particular principal bundles in non commutative differential geometry. Thefirst important question in this direction is: What should replace the cartesianproduct, expressed in terms of the algebras representing the factors.

Usually onetakes the (topological) tensor product of the two algebras. The tensor product isthe categorical coproduct in the category of commutative algebras, but does nothave a good categorical interpretation in the non-commutative case: (coordinateson) the two factors commute; why should they?One can as well consider algebras which have the same underlying vector space asthe tensor product but are equipped with a deformation of the usual multiplication.Clearly the question of cartesian products is also relevant for studying analogsof Lie groups in non commutative geometry, since the comultiplication mappingson the corresponding algebras represent a map from the cartesian product of the‘group’ with itself to the ‘group’.In this paper we show that in the case of non commutative two tori one gets ina natural way simple structures which have analogous formal properties as Hopfalgebra structures but with a deformed multiplication on the tensor product.1991 Mathematics Subject Classification.

16W30, 46L87, 58B30, 81R50.Supported by Project P 7724 PHY of ‘Fonds zur F¨orderung der wissenschaftlichen Forschung’Typeset by AMS-TEX1

2CAP, MICHOR, SCHICHL1. Non commutative 2–toriThe non commutative 2–tori are probably the simplest example of non commu-tative algebras which are commonly thought to describe non commutative spaces.They are quite well studied and arise in several applications of non commutativegeometry to physics, e.g.

in J. Bellisard’s interpretation of the quantum hall effect.We deal here with the smooth version of this algebras, so we consider the spaceS(Z2, C) of all complex valued Schwartz sequences (i.e. sequences which decay fasterthan any polynomial) (am,n) on Z2.

Now for a complex number q of modulus 1 wedefine a multiplication on this space as follows: Let δm,n be the sequence which isone on (m, n) and zero on all other points. Then any element of S(Z2, C) can bewritten as P am,nδm,n (this is a convergent sum in the natural Fr´echet topologyon S(Z2, C)), and we define the multiplication by δm,nδk,ℓ= q−knδm+k,n+ℓ.

Nowwe write U := δ1,0 and V = δ0,1, then by definition δm,n = U mV n, and the onlyrelevant relation is UV = qV U. Let us write Tq for the resulting algebra.2.

The comultiplication ∆q : C∞(S1, C) →TqFirst it should be noted that the non commutative 2–tori themselves are defor-mations of the topological tensor product of the algebra C∞(S1, C) ∼= S(Z, C) withitself, which corresponds to the case q = 1. In fact the natural comultiplicationon S(Z, C) induced by the group structure on S1 given by (an) 7→P an(UV )nmakes sense as an algebra homomorphism ∆q : S(Z, C) →Tq for any q, so thereis some kind of ‘multiplication mapping’ from any non commutative 2–torus to S1.It can be even shown that this comultiplication is coassociative in a similar senseas we will use below, but there are no counit and antipode mappings fitting to thiscomultiplication.3.

The quantum group like structure on Tq3.1. By the general principles of non commutative geometry the algebra Tq shouldbe a description of the orbit space S1/Z, where the action of Z on S1 is induced bymultiplication by q.

But this space clearly is a group so this should be reflected bysome kind of coalgebra structure on the algebra Tq.First we construct the algebra which we consider as a representative of the carte-sian product of the non commutative torus described by Tq with itself. As indicatedin the introduction we take as the underlying vector space of this algebra the tensorproduct, i.e.

the space S(Z4, C) of Schwartz sequences on Z4. As above we writeδk,ℓ,m,n for the sequence which is one on (k, ℓ, m, n) and zero everywhere else.

Thenwe define a multiplication on S(Z4, C) by:δk1,ℓ1,m1,n1δk2,ℓ2,m2,n2 = qk2n12−k2ℓ1−m1ℓ22−m1n2δk1+k2,ℓ1+ℓ2,m1+m2,n1+n2

NON COMMUTATIVE TWO TORI3Writing U1 := δ1,0,0,0, V1 := δ0,1,0,0, U2 := δ0,0,1,0 and V2 := δ0,0,0,1 we see thatδk,ℓ,m,n = U k1 V ℓ1 U m2 V n2 and the relations between these generators are:U1V1 = qV1U1U2V2 = qV2U2U1V2 = q−1/2V2U1V1U2 = q1/2U2V1U1U2 = U2U1V1V2 = V2V1Let us remark a little on these relations. Clearly we should have two canonicalcopies of the original algebra Tq contained, so the relation between U1 and V1 aswell as the one between U2 and V2 is clear.

Then it turns out that all other relationsfall out of the further development if one assumes that any product of two of thegenerators is a scalar multiple of the opposite product. We write P 2q for the resultingalgebra in the sequel.3.2.

Now we define the comultiplication ∆: Tq →P 2q by ∆(U) = U1U2 and∆(V ) = V1V2. Then this induces an algebra homomorphism, which is obviouslycontinuous, sinceU1U2V1V2 = q−1/2U1V1U2V2 = q3/2V1U1V2U2 = qV1V2U1U2.To formulate coassociativity we need a representative of the three fold product ofa non commutative torus with itself.

Again we want this to be a deformation ofthe third tensor power of Tq, so the underlying vector space is S(Z6, C). We writedown this algebra only in terms of generators and relations, so we write a Schwartzsequence as P ai,j,k,ℓ,m,nU i1V j1 U k2 V ℓ2 U m3 V n3 .

The choice of relations in this algebra isdictated by requiring the same deformation which led to P 2q in the first and secondfactor as well as in the second and third factor and no additional deformations.This gives the following relations:U1V1 = qV1U1U2V2 = qV2U2U3V3 = qV3U3U1V2 = q−1/2V2U1U2V3 = q−1/2V3U2U3V1 = V1U3V1U2 = q1/2U2V1V2U3 = q1/2U3V2V3U1 = U1V3U1U2 = U2U1U2U3 = U3U2U3U1 = U1U3V1V2 = V2V1V2V3 = V3V2V3V1 = V1V3We write P 3q for the resulting algebra.3.3. Taking into account the canonical vector space isomorphisms P 2q ∼= Tq ˆ⊗Tq andP 3q ∼= Tq ˆ⊗Tq ˆ⊗Tq, where ˆ⊗denotes the projective tensor product, we get continuous

4CAP, MICHOR, SCHICHLlinear maps (∆, Id) and (Id, ∆) from P 2q to P 3q which are induced by ∆ˆ⊗Id andIdˆ⊗∆, respectively.Now it turns out that in this setting these maps are evenalgebra homomorphisms. We show this only for (∆, Id), the proof for (Id, ∆) iscompletely analogous.On the generators of P 2q the map (∆, Id) is given byU1 7→U1U2V1 7→V1V2U2 7→U3V2 7→V3,and this induces an algebra homomorphism sinceU1U2V3 = q−1/2U1V3U2 = q−1/2V3U1U2V1V2U3 = q1/2V1U3V2 = q1/2U3V1V2U1U2U3 = U3U1U2V1V2V3 = V3V1V2But then coassociativity is obvious since both (∆, Id) ◦∆and (Id, ∆) ◦∆arecontinuous algebra homomorphisms Tq →P 3q which map U to U1U2U3 and V toV1V2V3.3.4.

Let us next turn to the counit ε : Tq →C. We cannot expect to get a counitwhich is an algebra homomorphism since there are no nonzero homomorphismsfrom Tq to a commutative algebra.

This can be interpreted as the fact that thenon commutative torus has no classical points. So we define ε as a linear map byε(U kV ℓ) := qkℓ2 .

Clearly this defines a continuous linear mapping. Again using thecanonical vector space isomorphism P 2q ∼= Tq ˆ⊗Tq we get continuous linear mappings(ε, Id) and (Id, ε) from P 2q to Tq.

These maps are given by (ε, Id)(U k1 V ℓ1 U m2 V n2 ) =qkℓ2 U mV n and by (Id, ε)(U k1 V ℓ1 U m2 V n2 ) = qmn2 U kV ℓ. To prove that ε is in facta counit for the comultiplication ∆we have to show that both (ε, Id) ◦∆and(Id, ε) ◦∆are the identity map.

Since these are continuous linear maps it sufficesto check this on elements of the form U kV ℓ. But∆(U kV ℓ) = (U1U2)k(V1V2)ℓ= U k1 U k2 V ℓ1 V ℓ2 = q−kℓ2 U k1 V ℓ1 U k2 V ℓ2 ,so the result is obvious.3.5. Finally we define the antipode map S : Tq →Tq.The obvious choice isS(U) := U −1 and S(V ) := V −1.

Then this extends to a continuous algebra homo-morphism since U −1V −1 = qV −1U −1. So in contrast to the case of Hopf algebraswe get an antipode which is a homomorphism and not an anti homomorphism.

This

NON COMMUTATIVE TWO TORI5seems rather positive since algebra homomorphisms should correspond to mappingsof the underlying ‘spaces’ while the meaning of anti homomorphisms is rather un-clear.As before we get continuous linear mappings (S, Id) and (Id, S) from P 2q toitself using the vector space isomorphism with the tensor product. These mapsare given by U k1 V ℓ1 U m2 V n2 7→U −k1V −ℓ1U m2 V n2 and U k1 V ℓ1 U m2 V n2 7→U k1 V ℓ1 U −m2V −n2,respectively, and they are not algebra homomorphisms (since the multiplication onP 2q is twisted).

Moreover the same method leads to a continuous linear map µ :P 2q →Tq which represents the multiplication of Tq and is given by µ(U k1 V ℓ1 U m2 V n2 ) =U kV ℓU mV n = q−ℓmU k+mV ℓ+n.To prove that the antipode is really an analog of a group inversion we have toshow that (µ◦(S, Id)◦∆)(x) = ε(x)·1 and likewise with (S, Id) replaced by (Id, S).Since all these maps are linear and continuous it suffices to check this on elementsof the form U kV ℓ. But for these we get:(µ ◦(S, Id) ◦∆)(U kV ℓ) = (µ ◦(S, Id))(q−kℓ2 U k1 V ℓ1 U k2 V ℓ2 )= µ(q−kℓ2 U −k1V −ℓ1U k2 V ℓ2 )= q−kℓ2 qkℓ= qkℓ2 = ε(U kV ℓ) · 1and likewise with (Id, S).Institut f¨ur Mathematik, Universit¨at Wien, Strudlhofgasse 4, A-1090 Wien, Aus-tria.E-mail address: michor@pap.univie.ac.at

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