---

THE QUANTUM-DOUBLE FOR A NONSTANDARD

arXiv:hep-th/9303035v1 5 Mar 1993PRA-HEP-93/2THE QUANTUM-DOUBLE FOR A NONSTANDARDDEFORMATIONOF A BOREL SUBALGEBRA sl(2, C)ˇC Burd´ık⋄1, and P. Hellinger∗2∗Department of Theoretical Physics and ⋄Nuclear Centre,Faculty of Mathematics and Physics,Charles University,V Holeˇsoviˇck´ach 2, 180 00 Prague 8, Czech RepublicAbstractWe give a construction of Drienfeld’s quantum double for a nonstandarddeformation of Borel subalgebra of sl(2). We construct explicitly some simplerepresentations of this quantum algebra and from the universal R-matrix weobtain the explicit solutions of the Yang-Baxter equation in those cases.PRA-HEP-93/2 hep-th/9303035 March 19931E-mail: Bitnet=(VSETIN AT CSPUNI12)2E-mail: Bitnet=(HELINGER AT CSPUNI12)

1IntroductionOne of the most important quantum group constructions is Drienfeld’s quantumdouble [2]. This construction was used by Drienfeld for construction of quantumdeformations of simple Lie algebras.

In well know example of U(sl(2)) we start froma Borel subalgebra and the Drienfeld’s quantum double gives Uq(sl(2)) ⊗U(u(1)).It was pointed by Ogievetski [8] that there exists another nonequivalent Hopfalgebra deformation of the Borel subalgebra connected via FRT construction [5]with the solution of the Yang-Baxter equationR =1−γγ−γβ010β001−β0001(1)which was found in [1] and studied in [3] [4] [6] [7] [10] The aim of this letter is solvean interesting question what we will obtain when the Drienfeld’s quantum doubleconstruction is applied to this deformation.2The basic Hopf algebra and its dualLet us consider the Hopf algebra A=Uγb−over C[[γ]], the formal deformation ofUb−of sl(2) with the generators τ, π and with the relations[τ, π] = −2π. (2)The coalgebra structure is given by∆(τ)=τ ⊗1 + Λ ⊗τ∆(π)=π ⊗1 + Λ−1 ⊗πǫ(τ)=ǫ(π) = 0(3)and antipodes areS(τ)=−Λ−1τS(π)=−ΛπSσ(τ)=−τΛ−1Sσ(π)=−Λπ,(4)where Λ = (1 −γπ)−1.We may construct the dual Hopf algebra A∗with the generators T, P and withthe pairing⟨T, τ nπk⟩def=δn1δk0(5)⟨P, τ nπk⟩def=δn0δk1(6)1

From this pairing one may easily deduce the comultiplications∆(T)=T ⊗1 + 1 ⊗T∆(P)=P ⊗e2T + 1 ⊗P(7)Proof⟨∆(T), τ n1πk1 ⊗τ n2πk2⟩=⟨T, τ n1πk1τ n2πk2⟩==δn11δk10δn20δk20 + δn10δk10δn21δk20and⟨∆(P), τ n1πk1 ⊗τ n2πk2⟩=⟨P, τ n1πk1τ n2πk2⟩==δn10δk10δn20δk21 + δn10δk11δk202n2and⟨T k, τ n⟩= n!δnkas will be proven (see the equation (9)). ⊔⊓We are to find the commutation relations between T and P. The result is[T, P] = −12γ(e2T −1)(8)Proof⟨TP, τ nπk⟩=⟨T ⊗P, ∆(τ nπk)⟩==δn1δk1⟨PT, τ nπk⟩=⟨P ⊗T, ∆(τ nπk)⟩==δn1δk1 + (n > 0)γδk02n−1,where the symbol (x) is defined(x) =(1if x is true0otherwise⊔⊓Now is time for counting of powers of the generators.

Let us start with promised⟨T n, τ kπm⟩= n!δnkδm0(9)2

Proof⟨T n, τ kπm⟩=⟨T ⊗n, ∆n−1(τ kπm)⟩==δm0⟨T ⊗n, (t1 + · · · + tn)k⟩since (5), (2) and Λ = 1 + γπ + o(γ2), wheret1def=t ⊗1 ⊗· · · ⊗1t2def=1 ⊗t ⊗1 ⊗· · · ⊗1...tn−1def=1 ⊗· · · ⊗1 ⊗t ⊗1tndef=1 ⊗· · · ⊗1 ⊗tAnd so ⟨T ⊗n, (t1 + · · · + tn)k⟩= δnkn! ⊔⊓And now we will switch to P⟨P n, τ mπk⟩= δm0(−γ)n−kΘnk(10)whereΘnk =kXp=0 kp!

(−1)k−ppnwith the following propertiesΘnk = 0 for k > nΘnn = n! (11)Proof∆(π)=π ⊗1 + 1 ⊗π −γπ ⊗π∆n(π)=n+1Xi=1πi + (−γ)n+1Xi>j=1πiπj + (−γ)2n+1Xi>j>k=1πiπjπk + · · · ++(−γ)nπ1π2 · · · πnπn+1,by the induction, where πi is defined similar as for τ.

We ought to calculate⟨P n, τ kπm⟩=⟨P ⊗n, ∆n−1(τ mπk)⟩==δm0⟨P ⊗n, ∆n−1(π)k⟩Thanks to (6) we may πj’s in the previous pairing consider as grassmanian variablesπ2j = 0, for j = 1, . .

. , n and πiπj = πjπi for i, j = 1, .

. .

, n. Under such ansatzwe have this very convenient result∆n(π)⟨,⟩= −1γ (e−γ Pn+1i=1 π1 −1)(12)3

as one may easily check (⟨,⟩= means the equality in the pairing). So we will proceed⟨P ⊗n, ∆n−1(π)k⟩=⟨P ⊗n, (−1γ (e−γ Pni=1 π1 −1))k⟩==⟨P ⊗n, (−γ)−kkXp=0 kp!

(−1)k−pe−γpPni=1 πi⟩==(−γ)n−kkXp=0 kp! (−1)k−ppn⊔⊓Now we may obtain the general formula⟨T jP n, τ mπk⟩= m!δmj(−γ)n−kkXp=0 kp!

(−1)k−ppn(13)ProofStraightforward ⊔⊓At this moment we have the description of the dual Hopf algebra A∗or preciselywe have only the explicit form of the bialgebra structure, but henceforth we willnot need the antipodes. Let us construct the double now.3The quantum double D(A)We have to combine A and A◦, where A◦is the dual A∗with the opposite com-multiplication.

Let us briefly summarize these structures:[τ, π]=−2π[T, P]=−12γ(e2T −1)∆(τ)=τ ⊗1 + Λ ⊗τ∆(π)=π ⊗1 + Λ−1 ⊗π∆(T)=T ⊗1 + 1 ⊗T∆(P)=P ⊗1 + e2T ⊗P(14)For the commutation relations between π, τ and P, T we are forced to use thestandard procedureXζ = Σ(⟨∆2(X), Sσ1 ∆2(ζ)⟩13)(15)for X ∈{P, T} and ζ ∈{π, τ}, where ⟨, ⟩13 is the evaluation between the dualalgebras on the first and the third position and Σ exchanges A◦and A4

This reads in our case as follows:[P, τ]=−γτ −2P[P, π]=Λ−1e2T −1 + γπ[T, τ]=Λ −1[T, π]=0(16)ProofPτ=Σ(⟨e2T ⊗e2T ⊗P + e2T ⊗P ⊗1 + P ⊗1 ⊗1,Sσ1 (τ ⊗1 ⊗1 + Λ ⊗τ ⊗1 + Λ ⊗Λ ⊗τ)⟩13) ==Σ(⟨e2T ⊗e2T ⊗P + e2T ⊗P ⊗1 + P ⊗1 ⊗1,−τΛ−1 ⊗1 ⊗1 + Λ−1 ⊗τ ⊗1 + Λ−1 ⊗Λ ⊗τ⟩13) ==Σ(Pτ −γτ −2P) ==τP −γτ −2Pand similary the others. ⊔⊓To obtain the classical limit one may introduce the new generatorspdef=Pαtdef=Tα(17)where αdef= γ/2 In these generators the quantum double D(A) looks[p, τ]=−2τ −2p[p, π]=Λ−1e2αt −1 + 2π[t, τ]=1α(Λ −1)[t, π]=0[τ, π]=−2π[t, p]=−1α(e2αt −1)∆(τ)=τ ⊗1 + Λ ⊗τ∆(π)=π ⊗1 + Λ−1 ⊗π∆(t)=t ⊗1 + 1 ⊗t∆(P)=P ⊗1 + e2αt ⊗p(18)The classical limit is then the enveloping algebra of the Lie algebra L with fourgenerators and Lie brackets have this form:[τ, π]=−2π5

[p, τ]=−2τ −2p[t, p]=−2t[t, τ]=2π[p, π]=2t[t, π]=0(19)4The universal R-matrixWe have to find the dual bases for A and Ao. If define Rj asR1def=PRndef=P n −n−1Xk=1ΘnkP k(−γ)n−k(20)we have⟨T nRk, τ mπl⟩= n!k!δnmδkl(21)Proofeasy ⊔⊓Then we have the universal R-matrix in the formR =∞Xn,k=01n!k!τ nπk ⊗T nRk(22)5Representations of D(A)The problem to find (all (irreducible)) representations of D(A) is generally verydificult, but in our case we have luckily at least two representations obtained fromthe classical limit L:The two dimensional one ρ, parametrized by the three complex numbers x, y, zρxyz(τ)def=−ρxyz(p)def= σ3 + x11 + yσ−(23)ρxyz(π)def=ρxyz(t)def= zσ−(24)where 11 is the identity matrix and σ3, σ−are Pauli matricesσ3 = 100−1σ−= 0010(25)Because σ2−= 0 the representation ρ of L induce the representation of D(A) andso we have this solution of YBE:Rx1,y1,z1,x2,y2,z2 =1000−αz1(1 + x2)100αz2(1 + x1)010fαz2(x1 −1)−αz1(x2 −1)1(26)6

where f = −α2z1z2(x1 −1)(x2 + 1) + α(z2y1 −z1y2). The solution of constantYang-Baxter equation (setting x1 = x2 = x, y1 = y2 = y, z1 = z2 = z andγ = αz(x + 1), β = αz(x −1))Rβγ =1000−γ100γ010−γββ−β1(27)is the transposed (1).The three dimensional representation we may obtain using the fact that theelement π −t is central in L and after the factorization L/(π −t) we have the 3DLie algebra ˜L with the structure[˜τ, ˜π]=−2˜π[˜p, ˜τ]=−2˜τ −2˜p[˜π, ˜p]=−2˜π(28)The representation ̺ is given by the adjoint representationad˜π=02−2000000ad˜τ=−200002002ad˜p=2000−200−20(29)(30)as follows̺xy(π)=̺xy(t) = y · ad˜π̺xy(τ)=ad˜τ + x11̺xy(p)=ad˜p −x11.

(31)As above since ad˜π2 = 0 this representation gives the solution of YBER(x1, y1, x2, y2)=11 + α(̺x1y1(π) ⊗̺x2y2(p) + ̺x1y1(τ) ⊗̺x2y2(t)) ++α2̺x1y1(τπ) ⊗̺x2y2(tp)(32)6Conclusion and RemarksThe classical limit of the quantum double D(A) - Lie algebra L is obviously notgl(2) as one may see after the change of bases in ˜L (28), which provides such7

commutation relations:[H, X±] = ±2X±[X+, X−] = 0(33)After the finishing of this letter we have obtained the preprint [9], where theauthor reformulates the FRT method in terms of the Quantum double. He has stud-ied the case of the matrix (1) and he has constructed the corresponding Quantumdouble (but the formula for the universal R-matrix is given only in several termsof its formal power expansion).

It is however generally very dificult to comparethose explicit formulae for two algebras. We have solved this problem only for theclassical limit and we are able to show the equivalence.References[1] E. E. Demidov, Yu.

I. Manin, E. E. Mukhin, D. V. Zhdanovich. Progr.Theor.

Phys. Suppl.

102 (1990) 203[2] V.G. Drinfeld.

Quantum groups. In A. Gleason, editor, Proceedings of the ICM,pages 798–820, Rhode Island, 1987.

AMS. [3] H. Ewen, O. Ogievetsky, J. Wess.Quantum Matrices in Two Dimensions.Lett.

Math. Phys.

22 (1991) 297[4] H. Ewen, O. Ogievetsky, J. Wess.Realizations and Real Forms of Quantumgroups in 2 Dimensions. Preprint MPI-Ph/91-92 M¨unchen (1991)[5] L.D.

Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan.Algebra i Analiz 1 vol.

1(1989) 178 (in Russian); English translation: Leningrad Math. J.

1 (1990) 193[6] J. Hietarinta. Phys.

Lett. A 165 (1992) 245[7] L. Hlavaty.

J. Phys. A – Math.

Gen. 25 (1992) L63[8] O. Ogievetski. Talk given in the 13th Winter School”Geometry and Physics”.Zd´ıkov, January 9-16, 1993.

[9] A. A. Vladimirov.

JINR preprint E2-93-34 Dubna (1993) (hep-th 9302043)[10] S. Zakrzewski. A Hopf Star-Algebra of Polynomials on the Quantum SL(2,R)a ’Unitary’ R-Matrix Lett.

Math. Phys.

22 (1991) 2878

---

출처: arXiv:9303.035 • 원문 보기