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Roots of Unity: Representations of Quantum

arXiv:hep-th/9305180v1 31 May 1993Roots of Unity: Representations of QuantumGroupsW. A. Schnizer∗RIMS, Kyoto University, Kyoto 606, JapanandInstitute for Nuclear Physics, TU-Wien, 1040 Wien, AustriaFebruary 19, 1993AbstractRepresentations of Quantum Groups Uε(gn), gn any semi simpleLie algebra of rank n, are constructed from arbitrary representationsof rank n −1 quantum groups for ε a root of unity.

Representationswhich have the maximal dimension and number of free parameters forirreducible representations arise as special cases.1. )IntroductionDeformations of semi simple Lie algebras [18, 19] appear as a common alge-braic structure in the field of low dimensional integrable systems.

In manycases the deformation parameter is an N-th root of unity, where N can cor-respond e.g. to the number of states per site or to the lattice size in a twodimensional model.

We will denote the deformation parameter by ε, if the∗Supported by the Japan Foundation for the Promotion of Science1

parameter is an N-th root of unity (N the smallest integer such that εN = 1)and by q in the general case.The theories of chiral Potts [4, 5] type models, which saw dramatic devel-opments in recent years [6, 7, 8, 12, 24], are closely tied to the representationtheory of the quantum group Uε(sl(n, C)) in the case of ε being an N-throot of unity. The progress in the theories of chiral Potts models was partlystimulated by the better understanding of its deep connection to the repre-sentation theory of quantum groups.The representation theory in the case of ε an N-th root of unity is muchricher than for generic q, and several deep results by De Concini, Kac, Procesi[15, 17, 16] and Lusztig [20, 21, 22, 23] exist, laying the foundations of thegeneral representation theory in the roots of unity case.

Also considerableprogress has been made in directly constructing representations of quantumgroups Uε(gn). Accelerated by the development of chiral Potts type mod-els, much interest was devoted to find non-highest weight representations ofUε(gn) [25, 1, 2, 3, 9, 10, 11].

Finite dimensional non-highest weight repre-sentations, which do not exist in the representations theory of Uq(gn), are anew interesting feature of the representation theory in the roots of unity case.Non-highest representations of minimal dimension play a role similar to thatplayed by fundamental representations of Lie algebras [10]. The free param-eters which are characteristic of non-highest weight representations appearin the form of spectral parameters in chiral Potts type models.In this article we will show, that starting from an arbitrary representationof Uε(gn−1) one can construct a representation of Uε(gn).

The Lie algebrasgn and gn−1 will usually, but not always lie in the same series of Lie algebras.De Concini, Kac [15] showed that the maximal dimension and numberof parameters for representations of Uε(gn) for odd N is given by N∆+(gn)2

and dimgn, respectively. In this expression ∆+(gn) (= 1/2(dimgn −n)) isthe number of positive roots of the Lie algebra gn.

We use here and in thefollowing the term dimension of a representation to denote the dimension ofits representation space. The representations which will be constructed inthis article, are of dimension greater than or equal toN∆+(gn)−∆+(gn−1).For ε-deformations in the case of the An, Bn, Cn and Dn series such rep-resentations will be given in Section 3.

The exceptional E6, E7, E8, F4 andG2 cases of quantum groups are discussed in Section 4.The number offree parameters in all constructed representations is greater than or equal to2(∆+(gn) −∆+(gn−1)). Representations of maximal dimension and numberof free parameters arise as special cases for odd N and are discussed in section5.

These representations coincide with the maximal cyclic representations ofDate et al. [13] in the An case, and with the representations of [26] in theBn, Cn and Dn cases.

Conclusions are given in section 6 and in the appendixtwo relations which are important for the construction of representations inthe case of Uq(E8) and Uq(F4) are written down explicitly.2. )Definition of Uq(gn)In this article we will use the definition of quantum groups given by therelations among its Chevalley generators.

The quantized universal envelop-ing algebra Uq(gn) of a semi simple Lie algebra gn of rank n is generated by3

4n Chevalley generators {ei, fi, t±1i } which satisfy the commutation relationstitj=tjti,tit−1i= t−1i ti = 1tiejt−1i=qdiaijej,tifjt−1i= q−diaijfj[ei, fi]=δij {ti}qdi(1)and the Serre relations1−aijPv=0 (−1)v"1 −aijv#qdie1−aij−viejevi=01−aijPv=0 (−1)v"1 −aijv#qdif1−aij−vifjf vi=0(i ̸= j). (2)The matrix aij is the Cartan matrix of the Lie algebra gn, di non-zero integerssatisfying diaij = djaji.

The q-Gaussian is given as"mn#q=[m]q! [m −n]q![n]q!,[m]q!

=mYj=1qj −q−jq −q−1and the curly bracket is defined by {x}q = (x −x−1)/(q −q−1). Further, weextend the algebra by adding the elements t1/ki, k ∈Z.

In case we specializeq to be a primitive N-th root of unity the letter ε is used instead (εN = 1).We will not make use of the Hopf algebra structure of Uq(gn).Some of the results in this article are most conveniently given in terms ofWeyl algebra generators. The Weyl algebra W is defined by the commutationrelation among its two generators x, zxz = qzx.

(3)We denote by ˜W a copy of this algebra with generators ˜x, ˜z.In the roots of unity case one can define a N dimensional representation4

σgh : W →End(CN) of W, depending on the two parameters g, h, byσgh(x) = g010· · ·0001· · ·0.........0· · ·011· · ·0σgh(z) = h100· · ·00ε0· · ·000ε2· · ·0...0· · ·εN−1.We will retain the notation σgh also for representations of tensor products ofWeyl algebras W. In this cases the letters g, h denote the set of parametersg := {g1, . .

. , gl} and h := {h1, .

. .

, hl}, with l the number of W algebras in⊗1≤i≤lWi.3. )The An, Bn, Cn and Dn seriesIn this section we want to demonstrate how one can construct, starting froman arbitrary representation of a quantum group Uq(gn−1) a representationof Uq(gn), in the cases of the An, Bn, Cn and Dn series of quantum groups.The next section will be devoted to the more complicated case of exceptionalquantum groups.

As a first step we investigate the algebra ¯Uq(gn−1) whichis defined as the algebra given by Uq(gn) generators Fi, (1 ≤i ≤n −1)and Ei, T ±1i, (1 ≤i ≤n) which satisfy the defining relations (1,2). Rep-resentations of algebras ¯Uq(gn−1) arise immedently from representations ofUq(gn−1), and in turn representations of Uq(gn) itself will be defined in termsof ¯Uq(gn−1) representations.

This will be discussed in the following.The Cartan matrices in the Bn and Cn correspond to Dynkin diagramswith the first root being the shortest or longest root, respectively. For theDn case we fix the notation in a way such that the Dynkin diagram nodes 1and 2 are both connected with node 3.

The integers di = 1 for 2 ≤i ≤n in5

all cases and d1 = 1, 1/2, 2, 1 in the An, Bn, Cn and Dn series, respectively.Lemma 0.1 Let {fi, ei, ti, t−1i } (i = 1, . .

. , n−1) be the generators of Uq(gn−1)and {Ei, Ti, T −1i} (i = 1, .

. .

, n), {Fi} (i = 1, . .

. , n −1) the generators of¯Uq(gn−1).

Then one obtains an algebra homomorphism ¯ρ from ¯Uq(gn−1) toUq(gn−1) by taking ¯ρ(Fi) = fi, ¯ρ(Ei) = ei, ¯ρ(T ±1i) = t±1i(i = 1, . .

. , n −1)and ¯ρ(En) = 0.

The action of the algebra homomorphism ¯ρ on the generatorsT ±1ndepends on the complex parameters λn and is defined by¯ρ(Tn) =qλnn−1Qi=1 t−inifor¯Uq(An−1)qλnn−1Qi=1 t−1ifor¯Uq(Bn−1)qλnt−121n−1Qi=2 t−1ifor¯Uq(Cn−1)qλnt−121 t−122n−1Qi=3 t−1ifor¯Uq(Dn−1)The above algebra homomorphism ¯ρ can be extended to define represen-tations of the quantum group Uq(gn). The above lemma, together with thetheorem below shows how any representation of Uq(gn−1) gives rise to a rep-resentation of Uq(gn) in the An, Bn, Cn and Dn series.Theorem 0.2 The following formulas define algebra homomorphisms ρ fromthe quantum groups Uq(gn) to (⊗iW) ⊗¯Uq(gn−1), for gn the An, Bn, Cn andDn series of semi simple Lie algebras.

The composition π := (σgh ⊗id) · ρdefines an algebra homomorphism in the roots of unity case.a. )ρ : Uq(An) →(n⊗i=1 Wi) ⊗¯Uq(An−1)ρ(fi)={zi−1z−1i }xi + x−1i−1xiFi−1,ρ(ei) = {ziz−1i+1Ti}x−1i+ Eiρ(ti)=z−1i−1z2i z−1i+1Ti6

b. )ρ : Uq(Bn) →(n⊗i=1 Wi) ⊗(n−1⊗i=1˜Wi) ⊗¯Uq(Bn−1)ρ(fi)={zi+1z−1i }xi + {˜zi˜z−1i+1}x−1i+1xi˜xi+1 + x−1i+1xi˜xi+1˜x−1i Fi,(i > 1)ρ(f1)={z2z−1/21}q1/2x1 + {˜z−12 z1/21}q1/2x−12 x1˜x2 + x−12 ˜x2F1ρ(ei)={ziz−1i−1˜z−1i−1˜z2i ˜z−1i+1Ti}x−1i+ {˜zi˜z−1i+1Ti}˜x−1i+ Ei,(i > 1)ρ(e1)={z1/21˜z−12 T1}q1/2x−11+ E1ρ(ti)=z−1i+1z2i z−1i−1˜z−1i−1˜z2i ˜z−1i+1Ti,(i > 1),ρ(t1) = z−12 z1˜z−12 T1c.

)ρ : Uq(Cn) →(n⊗i=1 Wi) ⊗(n−1⊗i=1˜Wi) ⊗¯Uq(Cn−1)ρ(fi), ρ(ei), ρ(ti) as in the Bn case (i > 1)ρ(f1)={z21˜z−22 }q2x−22 x1˜x22 + {z2˜z−12 }x−12 x1˜x2 + {z22z−21 }q2x1 + x−22 ˜x22F1ρ(e1)={z21˜z−22 T1}q2x−11+ E1,ρ(t1) = z−22 z41˜z−22 T1d. )ρ : Uq(Dn) →(n⊗i=1 Wi) ⊗(n−2⊗i=1˜Wi) ⊗¯Uq(Dn−1)ρ(fi), ρ(ei), ρ(ti) as in the Bn case (i > 2)ρ(f1)={z3z−11 }x1 + {z2˜z−13 }˜x3x−13 x1 + x−13 x1x−12 ˜x3F2ρ(f2)={z3z−12 }x2 + {z1˜z−13 }˜x3x−13 x2 + x−13 x2x−11 ˜x3F1ρ(e1)={z1˜z−13 T1}x−11+ E1,ρ(e2) = {z2˜z−13 T2}x−12+ E2ρ(t1)=z−13 z21˜z−13 T1,ρ(t2) = z−13 z22 ˜z−13 T2.In these formulas expressions of type xi ⊗Fj were abbreviated writing xiFj.Further, we set F0 = 0 and x−1i= ˜x−1i= 0, zi = ˜zi = 1 if the index i is outof range.Taking ¯Uε(gn−1) in an arbitrary representation ¯ρ′, then the algebra ho-momorphisms π becomes a representation π′ of dimension N∆+(gn)−∆+(gn−1)7

times the dimension of ¯ρ′.The number of free parameters in π′ becomes2(∆+(gn) −∆+(gn−1)) plus the number of free parameters in ¯ρ′.This theorem can be proven by directly verifying the defining relations ofthe algebras. We omit the details of these calculations.Remark 0.3 Representations with dimensions equal to N∆+(gn)−∆+(gn−1) and2(∆+(gn) −∆+(gn−1) + 12) free parameters are obtained by taking the trivialrepresentation ρ′0 of Uε(gn−1) (ρ′0(ei) = 0, ρ′0(fi) = 0, ρ′0(ti) = 1, 1 ≤i ≤n−1)to define ¯ρ in lemma 0.1.

In the case of Uε(An) the resulting representationsπ′ are minimal cyclic representations [1, 2, 9, 10, 12]. Further representationsof dimension N∆+(gn)−∆+(gn−1) were discussed in the quantum SO(5) case in[2] and for quantum SO(8) in [11].Remark 0.4 The algebra homomorphism π defined above for the An seriesis cyclic in the sense that the Chevalley operators in the algebra homomor-phism π to the power of N are non-vanishing scalars for generic values ofparameters.

The explicit expressions are given in the following formulas.π(fi)N=−hNi−1h−Ni(−1)N + h−Ni−1hNi(q −q−1)NgNi + g−Ni−1gNi F Ni−1π(ei)N=−hNi h−Ni+1T Ni (−1)N + h−NihNi+1T −Ni(q −q−1)Ng−Ni+1 + ENiπ(ti)=h−Ni−1h2Ni h−Ni+1T Ni.These formulas also show that restricted representations (π′(fi)N = 0, π′(ei)N =0, π′(ki)N = 1), as well as semi-cyclic representations (either π′(fi)N = 0 orπ′(ei)N = 0) can be derived from the cyclic representation by specializingsome or all of the free parameters. Cyclic and semi-cyclic representations ofUε(An) in dimensions higher than the minimal dimensions were discussed in[1, 14].8

Theorem 0.2 allows to construct a representation of Uq(gn) from represen-tations of smaller rank quantum groups Uq(gn−1) with gn and gn−1 being Liealgebras of the same series. To show that they need not to be necessarilyfrom the same series we present as an example how a representation of Uq(C3)follows from a representation of Uq(A2).

We define ¯U′q(A2) to be the algebragiven by the generators Ei, T ±1i, (i = 1, 2, 3) and F2, F3 of a Uq(C3) algebra.Similarly to 0.1 one can give an algebra homomorphism ¯ρ from ¯U′q(A2) toUq(A2) by setting ¯ρ(E1) = 0, ¯ρ(T1) = qλ1t−2/33t−4/32and the other generatorsequal to Uq(A2) generators.Example 0.5 The map ρ : Uq(C3) →(⊗6i=1 Wi) ⊗¯U′q(A2) given below, de-fines an algebra homomorphism. The composition π := (σgh ⊗id) · ρ givesan algebra homomorphism in the roots of unity case.

Taking ¯U′ε(A2) in anarbitrary representation ¯ρ′ one obtains a representation π′ of Uε(C3) withdimension N6 times the dimension of ¯ρ′. Twelve free parameters arise fromtaking representation σgh of (⊗6i=1 Wi) and further free parameters can arisein the representation ¯ρ′.ρ(f1)={z−21 }q2x1ρ(f2)={z2z−24 }x−11 x2x4 + {z3z−15 }x−11 x4x5 + {z21z−12 }x2 + x−11 x−13 x4x5F3ρ(f3)={z5z−26 }x−12 x3x−14 x5x6 + {z24z−15 }x−12 x3x5 + {z2z−13 }x3 + x−12 x3x−14 x6F2ρ(e1)={z21z−22 z44z−25 z46T1}q2x−11+ {z24z−25 z46T1}q2x−14+ {z26T1}q2x−16+ E1ρ(e2)={z2z−13 z−24 z25z−26 T2}x−12+ {z5z−26 T2}x−15+ E2ρ(e3)={z3z−15 T3}x−13+ E3ρ(t1)=z41z−22 z44z−25 z46T1ρ(t2)=z−21 z22z−13 z−24 z25z−26 T2ρ(t3)=z−12 z23z−15 T39

4.) The exceptional E6, E7, E8, F4 and G2 quantum groupsSo far we did not describe the actual construction method which leads tothe algebra homomorphisms which were given in the previous section.

Theprocedure will be outlined in the following in the cases of the exceptionalquantum Lie algebras and the An case will appear as an example.The construction method is based on a set of relations in Uq(gn). Therelations needed the derive algebra homomorphisms of quantum Lie algebrasin the E6, E7, E8 and F4 cases are given in the following.

We denote dividedpowers of fi generators f ji /[j]! by f (j)i .Relations 0.6 In Uq(gn) we have the commutation relations (compare [13,26])(i)fif (j)i= [j + 1]qdif (j+1)i(ii)fif (j1)kf (j2)i= f (j1)kf (j2+1)i[−j1 + j2 + 1]qdi + f (j1−1)kf (j2+1)ifk,if aik = aki = −1(iii)fif (j1)kf (j2)if (j3)k= f (j1−2)kf (j2+1)if (j3+2)k[−j2 + j3 + 1]q2 +f (j1−1)kf (j2+1)if (j3+1)k[−j1 + j2 + 2] + f (j1)kf (j2+1)if (j3)k[−j1 + j2 + 1]q2 +f (j1−2)kf (j2)if (j3+2)kfi, if aki = −2, aik = −1(iv)eif (j)i= f (j)i ei + f (j−1)i{qdi(1−j)ti}qdi(v)tif (j)k= q−jaikdif (j)k ti(vi)fif (j)k= f (j)k fi and f (j1)if (j2)k= f (j2)kf (j1)iif aik = 0,eif (j)k= f (j)k ei, if i ̸= k(vii)relation in lemma 0.14, appendix (used only in the E8 case)(viii)relation in lemma 0.15, appendix (used only in the F4 case)10

We will use the above relations to commute single generators fi, ei and tithrough monomials of l factors of typef (j1)i1· · · f (jl)il(4)(l = ∆+(gn)−∆+(gn−1)). We say a generator fi, ei or ti commutes with sucha monomial if the multiplication of the generator on the monomial from theleft gives a sum over monomials of the same type multiplied by a single ornone generator from the right.

In this sense we say that fi commutes withthe monomial f (j1)kf (j2)iaccording to relation (iii) in 0.6, if aik = aki = −1.One can find monomials with l factors for all quantum Lie algebras Uq(gn)which commute with all of its generators. Moreover, the relations 0.6 will besufficient to commute the generators of Uq(E6), Uq(E7), Uq(E8) and Uq(F4)with monomials in the corresponding algebras.The Uq(G2) case will betreated separately.Commuting a generator s ∈{fi, ei, ti} with a monomial of type (4) usingexclusively the relations 0.6 we denote by rel(sf (j1)i1· · · f (jl)il ).

To apply rela-tions 0.6 in this way will be the first step in a construction procedure whichleads to representations of Uq(gn) in the exceptional cases.The second step introduces the Weyl algebra generators. We shall give arule Ωwhich applies on expressions of the following typeqr1j1+···+rljlf (j1+α1)i1· · · f (jl+αl)ils,(5)wherein the integer quantities j1, · · ·, jl are regarded as “free variables” andrk, αk ∈N, 1 ≤k ≤l.

The factor s is either 1 or a single Uq(gn) generatorfm, em, t±1m , (1 ≤m ≤n).If a, b are two expressions of type (5) and βa rational function in q which does not depend on the jk’s, then ΩsatisfiesΩ(a+b) = Ω(a)+Ω(b) and Ω(βa) = βΩ(a). The rule Ωapplies on expressions11

(5) according to the following assignmentΩ: qr1j1+···+rljlf (j1+α1)i1· · · f (jl+αl)ils 7→xα11 z−r11· · · xαll z−rllS. (6)If s = 1, fm, em, t±1m then S = 1, Fm, Em, T ±1m , respectively.Capital Ei, Fiand T ±1iare generators of a second Uq(gn) algebra, and will generate in thecoming examples the algebras ¯Uq(gn−1).To illustrate the working of the above two steps in the construction proce-dure of an algebra homomorphism we consider the An case of section 3.Example 0.7 The expressions for the algebra homomorphism ρ from Uq(An)to W1 ⊗· · · ⊗Wn ⊗¯Uq(An−1) in theorem 0.2 can be constructed in a uniqueway using relations 0.6 and Ω.

If one defines y to bey = f (j1)1f (j2)2· · · f (jn)nthen the formula for ρ(fi) in theorem 0.2, a.) is given by Ω(rel(fiy)), usingrelations (vi, ii) in 0.6 and (6).

Similarly, one obtains ρ(ei) = Ω(rel(eiy))and ρ(ti) = Ω(rel(tiy)), using the relations (iv, v, vi) in 0.6 and (6). Themonomial y which is used to construct the algebra homomorphism ρ in theUq(Bn) and Uq(Cn) case (theorem 0.2, b.

), c.)) isy = f (j1)nf (j2)n−1 · · · f (jn−1)2f (jn)1f (jn+1)2· · · f (j2n−1)n,and a similar monomialy = f (j1)nf (j2)n−1 · · · f (jn−1)2f (jn)1f (jn+1)3· · · f (j2n−2)n,is used to derive the algebra homomorphism in the Uq(Dn) case.We start the investigation of the exceptional quantum Lie algebras withthe Uq(E6), Uq(E7), Uq(E8) cases and define the following numbering of thenodes in the corresponding Dynkin diagrams12

❜❜❜❜❜❜❜❜16782345The integers di are equal to 1 for all i. Let us introduce the algebra ¯U′q(D5) be-ing generated by Fi̸=2, Ei, T ±1i∈Uq(E6).

Similarly, one can define ¯Uq(E6) asFi̸=7, Ei, T ±1i∈Uq(E7) and ¯Uq(E7) as given by the generators Fi̸=8, Ei, T ±1i∈Uq(E8). Algebra homomorphism ¯ρ in case of these algebras arise analogouslyas in lemma 0.1 by taking¯ρ(T2)=qλ2t−341 t−543 t−324 t−15 t−126 ,¯ρ(T7)=qλ7t−11 t−232 t−433 t−24 t−535 t−436 ,¯ρ(T8)=qλ8t−321 t−12 t−23 t−34 t−525 t−26 t−327 ,respectively.Let us abbreviate monomials of generators in Uq(gn) of typef (jk)if (jk+1)i±1· · · f (jk+l−1)i±l∓1f (jk+l)i±lby f (jk)ii±l.

Using this notation we define the following monomials in Uq(E6),Uq(E7) and Uq(E8)y6=f (j1)63 f (j5)1f (j6)46 f (j9)25 f (j13)1f (j14)42y7=f (j1)73 f (j6)1f (j7)47 f (j11)26f (j16)1f (j17)45f (j19)34f (j21)23f (j23)1f (j24)47y8=f (j1)83 f (j7)1f (j8)48 f (j13)27f (j19)1f (j20)46f (j23)35f (j26)24f (j29)1f (j30)42f (j33)53f (j36)64f (j39)1f (j40)72f (j46)84f (j51)1f (j52)38(7)Similar to example 0.7 one can obtain the expressions for algebra homo-morphisms in the case of Uε(E6), Uε(E7) and Uε(E8) by using relations 0.6and Ω.13

Theorem 0.8 The following expressions define an algebra homomorphismρ in the case of the quantum groups Uq(E6), Uq(E7) and Uq(E8). The com-position π := (σgh ⊗id) · ρ defines an algebra homomorphisms in the roots ofunity case.

The mappings ρ are defined bye6. )ρ : Uq(E6) →(16⊗k=1 Wk) ⊗¯U′q(D5),e7.

)ρ : Uq(E7) →(27⊗k=1 Wk) ⊗¯Uq(E6),e8. )ρ : Uq(E8) →(57⊗k=1 Wk) ⊗¯Uq(E7),ρ(fi) = Ω(rel(fiyn)),ρ(ei) = Ω(rel(eiyn)),ρ(ti) = Ω(rel(tiyn)),∀i, and n = 6, 7, 8Taking ¯U′ε(D5), ¯Uε(E6), ¯Uε(E7) in an arbitrary representation ¯ρ′, one obtainsrepresentations π′ which dimensions are given respectively by N16, N27 andN57 times the dimensions of ¯ρ′.

The number of free parameters of the repre-sentations π′ is 32, 54 and 114, respectively plus the number of free parame-ters in a representation ¯ρ′.The remaining two cases of exceptional Lie algebras are discussed in thefollowing theorems. We fix the numbering of nodes in the Dynkin diagramfor the F4 case as4321❝❝❝❝>The integers di are defined as d1 = d2 = 2, d3 = d4 = 1.

We write ¯U′q(B3)for the algebra generated by Fi̸=4, Ei, T ±1i∈Uq(F4).Similarly to lemma0.1 one can give an algebra homomorphism ¯ρ in the case of ¯U′q(B3) start-ing from Uq(B3) and defining ¯ρ(T4) = qλ4t−1/21t−12 t−3/23.Using of the re-lations 0.6 and the operation Ωin the Uε(F4) case, one can again define14

the algebra homomorphism in a short way.Let y4 denote the monomialy4 = f (j1)14 f (j5)2f (j6)31 f (j9)24 f (j12)2f (j13)31.Theorem 0.9 The following expressions define an algebra homomorphismρ from Uq(F4) to W1 ⊗· · · W15 ⊗¯U′q(B3) in a unique way and by compositionwith σgh an algebra homomorphism in the roots of unity case.f. )ρ : Uq(F4) →(15⊗i=1 Wi) ⊗¯U′q(B3)ρ(fi) = Ω(rel(fiy4)),ρ(ei) = Ω(rel(eiy4)),ρ(ti) = Ω(rel(tiy4)),∀iTaking ¯U′q(B3) in an arbitrary representation ¯ρ′ one obtains a representa-tion π′ of dimension N15 times the dimension in ¯ρ′.

The number of freeparameters in π′ is 30 plus the number of free parameters in ¯ρ′.Let G2 be defined by the Cartan matrix with a11 = 2, a12 = −3, a21 = −1and a22 = 2. The integers di are given as d1 = 1, d2 = 3.

In the following¯U′q(A1) is defined by the Uq(G2) generators F1, E1, E2, T ±11, T ±12. In analogyto lemma 0.1 one can easily give a corresponding algebra homomorphism ¯ρ,starting from U′q(A1) and taking ¯ρ(T2) = qλ2t−3/21.Theorem 0.10 The following formulas define an algebra homomorphism ρfrom Uq(G2) to W1 ⊗· · · ⊗W5 ⊗¯U′q(A1) and by composition with σgh an al-gebra homomorphism π := (σgh ⊗id) · ρ in the roots of unity case.g.

)ρ : Uq(G2) →(5⊗i=1 Wi) ⊗¯U′q(A1)ρ(f1)={z33z−24 }x−11 x−12 x3x24 + {z4z−35 }x−11 x−12 x24x5 + {x22x−33 }x−11 x2x3 +{z2z−14 }{q2}x−11 x3x4 + {z31z−12 }x2 + x−11 x−12 x4x5F1ρ(f2)={z−31 }q3x1ρ(e1)={z2z−33 z−35 z24T1}x−12+ {z4z−35 T1}x−14+ E115

ρ(e2)={z31z63z65z−32 z−34 T2}q3x−11+ {z33z65z−34 T2}q3x−13+ {z35T2}q3x−15+ E2ρ(t1)=z−31 z−33 z−35 z22z24T1ρ(t2)=z61z63z65z−32 z−34 T2Taking ¯U′ε(A1) in an arbitrary representation ¯ρ′ one obtains a representationπ′ which dimension (number of free parameters) is N5 (10) times (plus) thedimension (number of free parameters) of ¯ρ′, respectively.5. )Representations of maximal dimensionsThe representations constructed in section 3 and 4 depend on the arbitraryrepresentations ¯ρ′ of the corresponding ¯Uε(gn−1) algebras.

In this section wewant to show how a natural choice of this representation of ¯Uε(gn−1) leadsto representations of Uε(gn), which have maximal dimensions and number ofparameters for odd N.Irreducible representations of quantum groups Uε(gn) in the roots of unitycase exist only in dimensions smaller than or equal to N∆+(gn) and the numberof free parameters is maximally dim(gn), for odd N [15]. The simplest rep-resentation of this type is the non-highest weight representation π′ := σgh · ρof Uε(A1) which is defined in terms of Weyl algebra generators byρ(f) = {z−1}x,ρ(e) = {ελ1z}x−1,ρ(t) = ελ1z2 .Representation π′ has dimension N and the map σgh together with the com-plex parameter λ1 gives rise to the 3 free parameters of the representation.Starting from this representation one can step by step construct represen-tations of higher rank Lie algebras, using inductively representations ¯ρ of16

¯Uε(gn−1) and representations π′ of Uε(gn). A representation of Uε(gn) ob-tained in this way will have 2∆+(gn) number of free parameters induced bythe free parameters of the mappings σgh and in addition n parameters λicoming from the definition of ¯ρ (see e.g.

lemma 0.1). Also, the dimensionof such a representation is induced by the map σgh and adds up to N∆+(gn).This gives rise to the following theorem.Theorem 0.11 Using inductively the representations of theorems 0.2-0.10to define representations of ¯Uε(A1), · · · , ¯Uε(gn−2), ¯Uε(gn−1) one obtains rep-resentations of Uε(gn) which have maximal dimensions and number of freeparameters for odd N, in all cases of semi simple Lie algebras gn.Remark 0.12 In the case of Uε(An) one obtains the maximal cyclic rep-resentations of [13].In [13] it is also proven, that this representation isgenerically irreducible.

In the case of quantum SO(5) irreducible representa-tions of maximal dimension were obtained in [3]. For the general Bn, Cn andDn series the representations of maximal dimension of theorem 0.11 coincidewith those in [26], for which the irreducibility for odd N was established inthe simplest examples.Remark 0.13 The construction of algebra homomorphisms and represen-tations for Uq(gn) was based on the action of the generators ei, fi, t±1ionmonomials of length ∆+(gn) −∆+(gn−1).

Such monomials were defined e.g.in example 0.7 and in (7). Adjoining these monomials according to the in-ductive procedure of theorem 0.11 gives monomials f (j1)i1· · · f (jl)ilof lengthl = ∆+(gn).

In all cases of quantum Lie algebras discussed above the order-ing in these monomials corresponds to the ordering of simple reflexions in alongest element in the Weyl group of gn.17

6.) ConclusionsStarting from an arbitrary representations of Uq(A1) one can construct rep-resentations for all higher rank semi simple Lie algebras by “adding” the ad-ditional generators which arise with the adding of a new node to the Dynkindiagram.

The dependence of the constructed representations of Uε(gn) onthe algebra Uε(gn−1) results in representations of quantum groups in dimen-sions greater than or equal to N∆+(gn)−∆+(gn−1). Only comparable very fewrepresentations in such dimensions were previously known.

Even more repre-sentations, especially highest weight representations of Uε(gn) can be foundby specializing some or all of the free parameters. Although, the constructedrepresentations do not yet lead to representations of quantum groups at ε anN-th root of unity in all possible dimensions for irreducible representations,one could hope that this problem might be settled in the future.The minimal and maximal cyclic representations of Uε(An), which areboth special cases of the above described representations are basic alge-braic structures related to the generalizations of the chiral Potts model in[7, 13, 24].

The next step is to find statistical models related to representa-tions of the other ε-deformed Lie algebras gn discussed above. In analogy tothe Uq(sln) case one would expect that starting from the affine extension ofan ε-deformation of an arbitrary semi simple Lie algebra gn one could findalgebraic varieties which determine relations among the spectral parametersto allow the existence and the construction of the intertwining R-matrix oftwo representations.

This article is meant to be a small contribution to theresearch going in this direction.But applications of roots of unity representations are not restricted to chi-ral Potts type models.Many further applications are found to lie in the18

development of other integrable models, conformal field theory (e.g. fusionrules), and areas in mathematics such as the representation theory of affineLie algebras or the theory of semi-simple groups over fields of positive char-acteristic.Acknowledgement I am greatly indebted to M. Jimbo and T. Miwa formany discussions and their steady interest in this work, and to the ResearchInstitute of Mathematical Sciences, Kyoto University for its kind hospitality.7.

)AppendixIn the case of applying the construction procedure of section 4 to derive al-gebra homomorphisms for Uq(E6), Uq(E7) quantum Lie algebras only Uq(A2)type relations in 0.6 were necessary. For the construction of representationsin the case of E8 and F4 it is necessary to use two further relations in Uq(E8)and Uq(F4).Lemma 0.14 Let f1, .

. .

, f8 be generators in Uq(E8). Then the generatorf1 commutes with the monomial f (j1)4f (j2)5f (j3)3f (j4)4f (j5)1f (j6)4f (j7)5f (j8)3f (j9)4asfollowingf1f (j1)4f (j2)5f (j3)3f (j4)4f (j5)1f (j6)4f (j7)5f (j8)3f (j9)4=f (j1−1)4f (j2−1)5f (j3−1)3f (j4−1)4f (j5)1f (j6+1)4f (j7+1)5f (j8+1)3f (j9+1))4f1 +f (j1−1)4f (j2−1)5f (j3−1)3f (j4−1)4f (j5+1)1f (j6+1)4f (j7+1)5f (j8+1)3f (j9+1)4[−j5 + j6 + j9 + 1] +f (j1−1)4f (j2−1)5f (j3−1)3f (j4)4f (j5+1)1f (j6+1)4f (j7+1)5f (j8+1)3f (j9)4[−j4 −j6 + j8 + j7 + 1] +f (j1−1)4f (j2−1)5f (j3−1)3f (j4)4f (j5+1)1f (j6)4f (j7+1)5f (j8+1)3f (j9+1)4[−j4 + j9 + 1] +f (j1−1)4f (j2−1)5f (j3)3f (j4)4f (j5+1)1f (j6+1)4f (j7+1)5f (j8)3f (j9)4[−j3 + j7 + 1] +f (j1−1)4f (j2)5f (j3−1)3f (j4)4f (j5+1)1f (j6+1)4f (j7)5f (j8+1)3f (j9)4[−j2 + j8 + 1] +19

f (j1−1)4f (j2)5f (j3)3f (j4)4f (j5+1)1f (j6+1)4f (j7)5f (j8)3f (j9)4[−j2 −j3 + j4 + j6 + 1] +f (j1)4f (j2)5f (j3)3f (j4−1)4f (j5+1)1f (j6+1)4f (j7)5f (j8)3f (j9)4[−j1 + j6 + 1] +f (j1)4f (j2)5f (j3)3f (j4)4f (j5+1)1f (j6)4f (j7)5f (j8)3f (j9)4[−j1 −j4 + j5 + 1]The proof of this lemma is solely based on the A2-type Serre relations whichdefine the commutation relations among the fi generators.Lemma 0.15 Let f1, . .

. , f4 be generators of Uq(F4).

Then the action of f3on the monomial f (j1)4f (j2)3f (j3)2f (j4)1f (j5)3f (j6)2f (j7)3f (j8)4f (j9)3f (j10)2f (j11)3is given byf3f (j1)4f (j2)3f (j3)2f (j4)1f (j5)3f (j6)2f (j7)3f (j8)4f (j9)3f (j10)2f (j11)3=f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5−1)3f (j6−1)2f (j7−1)3f (j8)4f (j9+1)3f (j10+1)2f (j11+1)3f4 +f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5−1)3f (j6−1)2f (j7−1)3f (j8+1)4f (j9+1)3f (j10+1)2f (j11+1)3[j11 + j9 + 1 −j8] +f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5−1)3f (j6−1)2f (j7)3f (j8+1)4f (j9+1)3f (j10+1)2f (j11)3[2j10 + 1 −j7 −j9] +f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5−1)3f (j6−1)2f (j7)3f (j8+1)4f (j9)3f (j10+1)2f (j11+1)3[j11 + 1 −j7] +f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5−1)3f (j6)2f (j7)3f (j8+1)4f (j9+1)3f (j10)2f (j11)3[−2j6 + j7 + j9 + 1] +f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5)3f (j6)2f (j7−1)3f (j8+1)4f (j9+1)3f (j10)2f (j11)3[−j5 + j9 + 1] +f (j1−1)4f (j2+1)3f (j3)2f (j4)1f (j5)3f (j6)2f (j7)3f (j8+1)4f (j9)3f (j10)2f (j11)3[−j5 −j7 + j8 + 1] +f (j1)4f (j2+1)3f (j3)2f (j4)1f (j5)3f (j6)2f (j7)3f (j8)4f (j9)3f (j10)2f (j11)3[−j1 + j2 + 1]The identity of this lemma was obtained using the commutation relationsamong fi generators with terms f (j)kin the cases of Uq(A2), Uq(B2) andUq(C2).With the help of the above lemmas it is possible construct the representa-tion of Uε(E8) and Uq(F4) in section 4.References20

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