On Casimir Elements of q-algebras U q (so n) and Their Eigenvalues in Representations

Transcription

1 Proceedigs of Istitute of Mathematics of NAS of Ukraie 2000, Vol. 30, Part 2, O Casimir Elemets of q-algebras U q so ad Their Eigevalues i Represetatios A.M. GAVRILIK ad N.Z. IORGOV Bogolyubov Istitute for Theoretical Physics, Metrologicha Street 14b, Kyiv 143, Ukraie The ostadard q-deformed algebras U qso are kow to possess q-aalogues of Gel fad Tsetli type represetatios. For these q-algebras, all the Casimir elemets correspodig to basis set of Casimir elemets of so are foud, ad their eigevalues withi irreducible represetatios are give explicitly. 1 Itroductio The ostadard deformatio U qso, see [1], of the Lie algebra so admits, i cotrast to stadard deformatio [2] of Drifeld ad Jimbo, a explicit costructio of irreducible represetatios [1, 3] correspodig to those of Lie algebra so i Gel fad Tsetli formalism. Besides, as it was show i [4], U qso is the proper dual for the stadard q-algebra U q sl 2 i the q-aalogue of dual pair so, sl 2. Let us metio that the algebra U qso 3 appeared earlier i the papers [5]. As a matter of iterest, this algebra arose aturally as the algebra of observables [6] i quatum gravity with 2D space fixed as torus. At >3, the algebras U qso are o less importat, servig as itermediate algebras i derivig the algebra of observables i 2+1 quatum gravity with 2D space of geus g>1, so that depeds o g, =2g + 2 [7, 8]. I order to obtai the algebra of observables, the q-deformed algebra U qso 2g+2 should be quotieted by some ideal geerated by combiatios of Casimir elemets of this algebra. This fact, alog with others, motivates the study of Casimir elemets of U qso. 2 The q-deformed agebras U q so Accordig to [1], the ostadard q-deformatio U qso of the Lie algebra so is give as a complex associative algebra with 1 geeratig elemets I 21,I 32,...,I, 1 obeyig the defiig relatios deote q + q 1 [2] q I 2 j,j 1 I j 1,j 2 + I j 1,j 2 I 2 j,j 1 [2] q I j,j 1 I j 1,j 2 I j,j 1 = I j 1,j 2, I 2 j 1,j 2 I j,j 1 + I j,j 1 I 2 j 1,j 2 [2] q I j 1,j 2 I j,j 1 I j 1,j 2 = I j,j 1, [I i,i 1,I j,j 1 ]=0 if i j > 1. 1 Alog with defiitio i terms of triliear relatios, we also give a biliear presetatio. To this ed, oe itroduces the geerators here k>l+1,1 k, l I ± k,l [I l+1,l,i ± k,l+1 ] q ±1 q±1/2 I l+1,l I ± k,l+1 q 1/2 I ± k,l+1 I l+1,l

2 O Casimir Elemets of q-algebras U qso ad Their Eigevalues i Represetatios 311 together with I k+1,k I + k+1,k I k+1,k. The 1 imply [I + lm,i+ kl ] q = I + km, [I+ kl,i+ km ] q = I + lm, [I+ km,i+ lm ] q = I + kl if k>l>m, [I + kl,i+ mp] =0 if k>l>m>p or k>m>p>l; 2 [I + kl,i+ mp] =q q 1 I + lp I+ km I+ kp I+ ml if k>m>l>p. Aalogous set of relatios exists ivolvig I kl alog with q q 1 let us deote this dual set by 2. If q 1 classical limit, both 2 ad 2 reduce to those of so. Let us give explicitly the two examples, amely = 3 the Odesskii Fairlie algebra [5] ad = 4, usig the defiitio [X, Y ] q q 1/2 XY q 1/2 YX: U qso 3 : [I 21,I 32 ] q = I + 31, [I 32,I + 31 ] q = I 21, [I + 31,I 21] q = I U qso 4 : [I 32,I 43 ] q = I 42 +, [I+ 31,I 43] q = I 41 +, [I 21,I 42 + ] q = I 41 +, [I 43,I 42 + ] q = I 32, [I 43,I 41 + ] q = I 31 +, [I+ 42,I+ 41 ] q = I 21, 4 [I 42 +,I 32] q = I 43, [I 41 +,I+ 31 ] q = I 43, [I 41 +,I 21] q = I 42 +, [I 43,I 21 ]=0, [I 32,I 41 + ]=0, [I+ 42,I+ 31 ]=q q 1 I 21 I 43 I 32 I The first relatio i 3 ca be viewed as the defiitio for third geerator eeded to give the algebra i terms of q-commutators. Dual copy of the algebra U qso 3 ivolves the geerator I31 =[I 21,I 32 ] q 1 ad the other two relatios similar to 3, but with q q 1. I order to describe the basis of U qso we itroduce a lexicographical orderig for the elemets I + k,l of U qso with respect to their idices, i.e., we suppose that I + k,l I+ m, if either k<m, or both k = m ad l<. We defie a ordered moomial as the product of odecreasig sequece of elemets I + k,l with differet k, l such that 1 l<k. The followig propositio describes the Poicaré Birkhoff Witt basis for the algebra U qso. Propositio. The set of all ordered moomials is a basis of U qso. 3Casimir elemets of U q so As it is well-kow, tesor operators of Lie algebras so are very useful i costructio of ivariats of these algebras. With this i mid, let us itroduce q-aalogues of tesor operators for the algebras U qso as follows: J ± k 1,k 2,...,k 2r = q rr 1 2 ε q ±1sI ± k s2,k s1 I ± k s4,k s3 I ± k s2r,k s2r 1. 6 s S 2r Here 1 k 1 <k 2 < <k 2r, ad the summatio rus over all the permutatios s of idices k 1,k 2,...,k 2r such that k s2 >k s1, k s4 >k s3,..., k s2r >k s2r 1, k s2 <k s4 < <k s2r the last chai of iequalities meas that the sum icludes oly ordered moomials. Symbol ε q ±1s q ±1 ls stads for a q-aalogue of the Levi Chivita atisymmetrictesor, ls meas the legth of permutatio s. Ifq 1, both sets i 6 reduce to the set of compoets of rak 2r atisymmetrictesor operator of the Lie algebra so. Usig q-tesor operators give by 6 we obtai the Casimir elemets of U qso.

3 312 A.M. Gavrilik ad N.Z. Iorgov Theorem 1. The elemets C 2r = 1 k 1 <k 2 <...<k 2r q k 1+k 2 + +k 2r r+1 J + k 1,k 2,...,k 2r J k 1,k 2,...,k 2r, 7 where r =1, 2,..., 2 x meas the iteger part of x, are Casimir elemets of U qso, i.e., they belog to the ceter of this algebra. I fact, for eve, ot oly the product which costitutes C of elemets C + J 1,2,..., + ad C J1,2,..., belogs to the ceter, but also each of them. We cojecture that, i the case of q beigotarootof1,thesetofcasimirelemetsc 2r, r =1, 2,..., 1 2, ad the Casimir elemet C+ for eve geerates the ceter of U qso, i.e., ay elemet of the algebra U qso which commutes with all other elemets ca be preseted as a polyomial of elemets from this set of Casimir elemets. Let us give explicitly some of Casimir elemets. For U qso 3 adu qso 4 wehave C 2 3 = q 1 I I+ 31 I 31 + qi2 32 = qi I 31 I q 1 I32 2, C 2 4 = q 2 I I q2 I q 1 I 31 + I 31 + qi+ 42 I 42 + I+ 41 I 41, C 4+ 4 = q 1 I 21 I 43 I 31 + I qi 32I 41 + = qi 21I 43 I31 I 42 + q 1 I 32 I41 = C4 4. For U qso 5 the fourth order Casimir elemet is C 4 5 = q 2 J + 1,2,3,4 J 1,2,3,4 + q 1 J + 1,2,3,5 J 1,2,3,5 + J + 1,2,4,5 J 1,2,4,5 + qj+ 1,3,4,5 J 1,3,4,5 + q2 J + 2,3,4,5 J 2,3,4,5, where J + i,j,k,l = q 1 I + ji I+ lk I+ ki I+ lj + qi+ kj I+ li ad J i,j,k,l = qi ji I lk I ki I lj + q 1 I kj I li.foru qso 6, we preset oly the highest order Casimir elemet: C 6+ 6 = q 3 I 21 I 43 I 65 q 2 I + 31 I+ 42 I 65 + q 1 I 32 I + 41 I 65 q 2 I 21 I + 53 I q 1 I + 31 I+ 52 I+ 64 I 32 I + 51 I q 1 I 21 I 54 I + 63 I+ 41 I+ 52 I qi+ 42 I+ 51 I+ 63 I+ 31 I 54I I+ 41 I+ 53 I+ 62 q 2 I 43 I + 51 I qi 32I 54 I + 61 q2 I + 42 I+ 53 I q3 I 43 I + 52 I+ 61. Fially, let us give explicitly the quadratic Casimir elemet of U qso, C 2 = q i+j 1 I ji + I ji. 1 i<j This formula coicides with that give i [4], ad is a particular case of 7. 4 Irreducible represetatios of U q so Let us give a brief descriptio of irreducible represetatios irreps of U qso. More detailed descriptio of these irreps ca be foud i [1, 3]. As i the case of Lie algebra so, fiite-dimesioal irreps T of the algebra U qso are characterized by the set m m 1,,m 2,,...,m 2, here x meas the iteger part of x of

4 O Casimir Elemets of q-algebras U qso ad Their Eigevalues i Represetatios 313 umbers, which are either all itegers or all half-itegers, ad satisfy the well-kow domiace coditios m 1, m 2, m 2 1, m 2, if is eve, m 1, m 2, m 1, 0 if is odd. 2 To give the represetatios i Gel fad Tsetli basis we deote, as i the case of Lie algebra so, the basis vectors α of represetatio spaces by Gel fad Tsetli patters α. The represetatio operators T m I 2p+1,2p adt m I 2p,2p 1 acto α by the formulae p T m I 2p+1,2p α = A r 2pα m +r 2p Ar 2pm r 2p m r 2p, r=1 p 1 T m I 2p,2p 1 α = B2p 1α m r +r 2p 1 Br 2p 1m r 2p 1 m r 2p 1 +ic 2p 1 α. r=1 Here the matrix elemets A r 2p,Br 2p 1, C 2p 1 are obtaied from the classical o-deformed oes by replacig each factor x withitsrespectiveq-umber [x] q x q x /q q 1 ; besides, the coefficiet 1 2 i the classical Ar 2p is replaced with the l r,2p-depedet expressio [l r,2p ][l r,2p +1]/[2l r,2p ][2l r,2p +2] 1/2, where l r,2p = m r,2p + p r. 5 Casimir operators ad their eigevalues The Casimir operators the operators which correspod to the Casimir elemets, withi irreducible fiite-dimesioal represetatios of U qso take diagoal form. To give them explicitly, we employ the so-called geeralized factorial elemetary symmetric polyomials see [9]. Fix a arbitrary sequece of complex umbers a =a 1,a 2,... The, for each r =0, 1, 2,...,N, itroduce the polyomials of N variables z 1,z 2,..., z N as follows: e r z 1,z 2,...,z N a = z p1 a p1 z p2 a p2 1...z pr a pr r p 1 <p 2 < <p r N The Casimir operators i the irreducible fiite-dimesioal represetatios characterized by the set m 1,,m 2,,..., m N,, N = 2, by the Schur Lemma, are presetable as here 1 deotes the uit operator: T m C 2r =χ m 2r 1. Theorem 2. The eigevalue of the operator T m C 2r is χ 2r m = 1 r e r [l 1, ] 2, [l 2, ] 2,...,[l N, ] 2 a where a =[ɛ] 2, [ɛ +1] 2, [ɛ +2] 2,..., l k, = m k, + N k + ɛ. Hereɛ =0for =2N ad ɛ = 1 2 for =2N +1. I the case of eve, i.e., =2N, T m C + = T m C = 1 N [l1, ][l 2, ]...[l N, ]1. The eigevalues of Casimir operators are importat for physical applicatios. Let us quote some of Casimir operators together with their eigevalues. For U qso 3, = [m 13 ][m 13 +1]1. T m13 C 2 3

5 314 A.M. Gavrilik ad N.Z. Iorgov For U qso 4 wehave T m14,m 24 C 2 4 = [m 14 +1] 2 +[m 24 ] 2 1 1, T m14,m 24 = T m14,m 24 = [m 14 +1][m 24 ]1. C 4+ 4 C 4 4 Fially, for U qso 5 the Casimir operators are T m15,m 25 C 2 5 = [m 15 +3/2] 2 +[m 25 +1/2] 2 [1/2] 2 [3/2] 2 1, = [m 15 +3/2] 2 [1/2] 2 [m 25 +1/2] 2 [1/2] 2 1. T m15,m 25 C Cocludig remarks I this ote, for the ostadard q-algebras U qso we have preseted explicit formulae for all the Casimir operators correspodig to basis set of Casimirs of so. Their eigevalues i irreducible fiite-dimesioal represetatios are also give. We believe that the described Casimir elemets geerate the whole ceter of the algebra U qso of course, for q beig ot a root of uity. As metioed, the algebras U qso for>4 are of importace i the costructio of algebra of observables for 2+1 quatum gravity with 2D space of geus g > 1 servig as certai itermediate algebras. For that reaso, the results cocerig Casimir operators ad their eigevalues will be useful i the process of costructio of the desired algebra of idepedet quatum observables for the case of higher geus surfaces, i the importat ad iterestig case of ati-de Sitter gravity correspodig to egative cosmological costat. Ackowledgemets The authors express their gratitude to J. Nelso ad A. Klimyk for iterestig discussios. This research was supported i part by the CRDF Award No. UP1-309, ad by the DFFD Grat 1.4/206. Refereces [1] Gavrilik A.M. ad Klimyk A.U., Lett. Math. Phys., 1991, V.21, 215. [2] Drifeld V.G., Soviet Math. Dokl., 1985, V.32, 254; Jimbo M., Lett. Math. Phys., 1985, V.10, 63. [3] Gavrilik A.M. ad Iorgov N.Z., Methods Fuc. Aal. ad Topology, 1997, V.3, N 4, 51. [4] Noumi M., Umeda T. ad Wakayama M., Compositio Math., 1996, V.104, 227. [5] Odesskii A.V., Fuc. Aal. Appl., 1986, V.20, 152; Fairlie D., J. Phys. A, 1990, V.23, 183. [6] Nelso J., Regge T. ad Zertuche F., Nucl. Phys. B, 1990, V.339, 516. [7] Nelso J. ad Regge T., Phys. Lett. B, 1991, V.272, 213; Comm. Math. Phys., 1993, V.155, 561. [8] Gavrilik A.M., Talk give at It. Cof. Symmetry i Noliear Mathematical Physics 99, Kyiv, July 1999, these Proceedigs. [9] Molev A. ad Nazarov M., Math. A., 1999, V.313, 315, q-alg/