25 Years of Quantum Groups: from Definition to Classification A

Acta Polytechnica Vol. 48 No. 2/2008 25 Years of Quantum Groups: from Definition to Classification A. Stolin

In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.

Keywords: Lie bialgebra, quantum group, Yang-Baxter equation, quantization, classical twist, quantum twist, integrable model.

1 Definition of quantum groups, Lie Further, it is well-known that the Lie algebra L such that bialgebras and twists AAh = U()L is unique: In mathematics and theoretical physics, quantum groups L =Î{}aaaaAAh :()D =Ä+Ä11 . are certain non-commutative algebras that first appeared in h = the theory of quantum integrable models, and which were If A is a quantum group with AAUL(),thenL pos- then formalized by Drinfeld and Jimbo. sesses a new structure, which is called a Lie bialgebra structure Nowadays quantum groups are one of the most popular d, and it is given by: objects in modern mathematical physics. They also play a sig- - d()xaa=-hh1 (DD ()op ())mod , nificant role in other areas of mathematics. It turned out that quantum groups provide invariants of knots, and that they where a is an inverse image of x in A. In particular, the classi- can be used in quantization of Poisson brackets on manifolds. $ cal limit of Uq()g is g, the classical limit of Uq(g) is the affine Quantum groups gave birth to quantum geometry, quantum - calculus, quantum special functions and many other “quan- Kac-Moody algebra, which is a central extension of g[,uu 1 ], tum” areas. and the classical limit of Y()g is g[]u (the corresponding Lie At first, quantum groups appeared to be a useful tool for bialgebra structures will be described later). the following program: Let R satisfy the quantum Yang- -Baxter equation (QYBE) and let R be decomposable in a The Lie bialgebra (,)L d is called the classical limit of A, series in formal parameter ÿ.Thenr, which is the coefficient and A is a quantization of (,)L d . at the first order term, satisfies the so-called classical Yang- So, any quantum group in our sense has its classical limit, -Baxter equation (CYBE). In many cases CYBE can be solved which is a Lie bialgebra, and the following natural problem and the problem is to extend the solutions of CYBE to solu- arises: tions of QYBE. Given a Lie bialgebra, is there any quantum group whose classi- However, the most important applications of quantum cal limit is the given Lie bialgebra? Or in other words: Can any Lie groups relate to the theory of integrable models in mathemat- bialgebra be quantized? ical physics. The presence of quantum group symmetries In the mid 1990’s P. Etingof and D. Kazhdan gave a posi- (or so-called hidden symmetries) was the crucial point in ex- tive answer to this problem. plicit solutions of many sophisticated non-linear equations, such as Korteweg-de Vries or Sine-Gordon. Quantum groups Now, let g be a simple complex finite-dimensional Lie changed and enriched representation theory and algebraic algebra and let g[]u be the corresponding polynomial Lie topology. algebra. If you ask a physicist which quantum group is a Quantum groups were defined by V. Drinfeld as Hopf al- quantization of g[]u , you will almost certainly hear that the gebra deformations of the universal enveloping algebras (and also their dual Hopf algebras). More exactly, we say that a quantization of g[]u is the Yangian Y()g . h Hopf algebra A over C[][]is a quantum group (or maybe a Let us explain this in greater detail. Set W =Äå IIll, quasi-classical quantum group) if the following conditions are where {}I l is an orthonormal basis of g with respect to the satisfied: Killing form. Then it is well-known that the function i. The Hopf algebra AAh is isomorphic as a Hopf algebra W ==G to the universal enveloping algebra of some Lie algebra L. r(,)uv2 (,) uv (1) uv- ii. As a topological C[][]h -module, A is isomorphic to V[][]h for some vector space V over C. is a rational skew-symmetric solution of the classical Yang- -Baxter equation that is The first examples of quantum groups were quantum uni- [rr12 (uu , ),13 ( uu , )]+ [ rr12 ( uu , ),23 ( uu , )] versal enveloping algebras Uq()g , quantum affine Kac-Moody 12 13 12 23 (2) $ + 13 23 = algebras Uq(g), and Yangians Y()g . [r (uu13 , ),r ( uu23 , )] 0

More generally, we call r(,)uv arationalr-matrix if it is g[]u of a different nature, and for classification purposes one =+G skew-symmetric, satisfies (2) and r(,)uv2 (,) uv polynomial G has to find all of them. Therefore, in order to classify quantum (,)uv. Sometimes 2(,)uv is called Yang’s classical r-matrix. Further, the formula groups which have a given Lie algebra L as the classical limit d =Ä+ÄG one has to solve the following four problems: 2(())pu[]2 (,),() uv pu11 pv () (3) W 1. Describe all basic Lie bialgebra structures on L (in other definesaLiebialgebrastructureong[]u , since is a g-invari- words all Lie bialgebra structures up to classical twisting). Ä d ant element of ggand this implies that 2(())pu is a poly- 2. Find quantum groups corresponding to the basic struc- nomial in u,v. tures. The Yangian Y()g is precisely the quantization of this Lie 3. Describe all the corresponding classical twists. bialgebra. On the other hand, it is easy to see that all rational 4. Quantize all these classical twists. r-matrices introduced in [13] define Lie bialgebra structures on g[]u by the formula 3 Lie bialgebra structures on the d(())pu=Ä+Ä[]r (,),() uv pu11 pv () (4) polynomial Lie algebras and their Of course, the next question is: Which quantum groups quantization quantize the other “rational” Lie bialgebra structures ong[]u ? According to the results of an unpublished paper by Montaner and Zelmanov [11], there are four basic Lie bi- Before we give an answer to this question we need the fol- algebra structures on the polynomial Lie algebra Pu= g[]. lowing two definitions: Let us describe them (and, hence, we do the first step in the d classification of Lie bialgebra structures on g[]u ). When it is Definition: Let 1 be a bialgebra structure on L. Suppose possible we make further steps. s ÎÙ2L satisfies 12 13++=Ä 12 23 13 23 d [,ss ][, ss ][,] ss Altids (1 )(), Case 1. 123 231 312 Ä3 d = where Alt(): x=++ x x x for any x ÎL . Here the one-cocycle 1 0 Then In this case it is not difficult to show that there is a one-to-one dd=+Ä+Ä 21():aaa () [11 as ,] correspondence between Lie bialgebra structures of the first defines a Lie bialgebra structure on L. type and finite-dimensional quasi-Frobenius Lie subalgebras d d We say that: 2 is obtained from 1 by twisting via s,ands is ofg[]u . The corresponding quantum group isUu([g ]). Classi- called a classical twist. cal twists can be quantized following Drinfeld’s quantization At this point we note that any rational r-matrix is a twist of of skew-symmetric constant r-matrices. ÎÄ Ya n g ’ s r-matrix. Let H be a Hopf algebra and FH Hbe Case 2. an invertible element. Let F satisfy In this case the Lie bialgebra structure is described by FidFFidF12()DDÄ= 23 () Ä.(5) d =Ä+ÄG 22(())[pu (,),() uv pu11 pv ()], Such F is called a quantum twist.Theformula where G (,)uv is Yang’s rational r-matrix. DD= -1 2 F()aFaF () (6) The corresponding Lie bialgebra structures are in a one-to-one defines a new co-multiplication on H. correspondence with the rational solutions of CYBE described in [13]. The corresponding quantum group is a Yangian Y()g . = = 2 Conjecture and scheme Quantum twists were found for g sln for n 23, in [9]. Although it is clear that any quantum twist on a quantum group induces uniquely a classical twist on its classical limit, Case 3. the converse statement remained unknown for a long time. It Here the basic Lie bialgebra structure is given by was formulated in [9] in 2004. d =Ä+ÄG 33(())[pu (,),() uv pu11 pv ()] Conjecture 1. W G = v + Any classical twist can be extended to a quantum twist. with 3(,)uv rDJ, vu- The conjecture was proved by G. Halbout in [3]. In partic- where r is the classical Drinfeld-Jimbo modified r-matrix. ular,wecannowgiveananswertothequestionposedinthe DJ previous section: A quantum group which quantizes any ratio- This Lie bialgebra is the classical limit of the quantum group nal Lie bialgebra structure on Lu= g[]is isomorphic to the Uguq([]), which is a certain parabolic subalgebra of a non- $ Ya n g i a n Y()g as an algebra and it has a twisted co-algebra -twisted quantum affine algebra Uq(g) (see [15] or [10]). structure defined by the corresponding rational solution of There is a natural one-to-one correspondence between Lie CYBE. However, there might exist Lie bialgebra structures on bialgebra structures of the third type and the so-called quasi-

-trigonometric solutions of CYBE. A complete classification of bialgebras of a special type LDK= (),whereK is another Lie = the classical twists for sln was presented in [9] and for the arbi- bialgebra and LDK()is the corresponding classical double. trary g in [12]. Details on quantization of the classical twists Recent progress in related questions was achieved in [4]. for sln can be found in [9]. Here, a result similar to the conjecture above was proved un- der the following assumptions: L is the so-called coboundary Case 4. Lie bialgebra and L acts freely on M (in other words M is far Finally, the 4-th basic Lie bialgebra structure on g[]u is de- from being a homogeneous space). fined as follows: d =Ä+ÄG 44(())[pu (,),() uv pu11 pv ()], WW 5 Appendix: Solutions for sl(2) and G = = uv with 4(,)uv -- . uv11- vu- deformed Hamiltonians It was proved in [14] that there is a natural one-to-one corre- The aim of this section is to present concrete examples of spondence between Lie bialgebra structures of this kind and quantum twists and the corresponding Hamiltonians follow- the so-called quasi-rational solutions of CYBE. The quasi-ratio- ing the results obtained in [5]. nal solutions of CYBE for g = sl were classified in [14]. Some s + = s - = n We consider the case sl(2). Let e12, e21 and ideas indicate that quasi-rational r-matrices do not exist for s z =- ee11 22. Recall that in sl(2) we have two quasi-trigono- = g gfe248,,. The corresponding quantum group remains metric solutions, modulo gauge equivalence. The non-trivial unknown, but, it is rather clear that it is related to the dual =+-Äss++ solution is Xzz112(, ) X 012 (, zz ) ( z 1 z 2 )( ).Thisso- Hopf algebra of Y()g . lution is gauge equivalent to the following: z W -+1 Xzz(, )= 2 +Ä+ss sszz Ä ab, 12 - 4 Some open questions zz12 4 (7) +Ä-Äss- zz ss--+Ä-Ä sssszz - 1. Following steps 1–4 in the classification scheme, it is az( 12 z )(b ). natural to classify quantum groups related to affine Kac- The above quasi-trigonometric solution was quantized in -Moodyalgebras.Aconjectureisthatinthiscasewehave p [5]. Let 12()z be the two-dimensional vector representation only two basic types of Lie bialgebra structures and the ¶ of Uslq()2 . In this representation, the generator e-a acts as corresponding quantum groups are quantum affine alge- - amatrixunite21, eda- as ze21 and ha as ee11 22.Thequan- bras and so-called doubles of Y()g . ¶ tum R-matrix of Uslq()2 in the tensor product 2. More generally: Let L be a Lie algebra. Is it possible to de- ppÄ 12()zz 1 12 () 2 scribe the moduli space of Double(L), all Lie bialgebra is the following: structures on L modulo action of classical twists? =Ä+Ä Rzz012(, ) e 11 e 1122 e e 22 3. It is well-known that there is a 1-1 correspondence be- zz- + 12(eeeÄ+Ä e ) (8) tween Lie bialgebras and simply connected Poisson-Lie -1 - 11 22 22 11 qz12 qz groups. Let us consider a Lie bialgebra L and let G be the - qq1 - corresponding Poisson-Lie group. Let M be a Poisson + ().zeÄ+ e ze Ä e -1 - 2 12 21 1 21 12 homogeneous space that is a homogeneous space with qz12 qz a Poisson structure such that the multiplication Proposition 1.TheR-matrix given by ´® mG: M M zz- - RRzz:(,)=+12 (()baz+Ässz 012 -1 - 2 is a Poisson map. qz12 qz Now, let H be the quantum group which quantizes L. +--1 ss- Ä z (qazq1 b) (9) ++---1 +ss Ä Conjecture 2. ()()bazqazqb2 1 ) Let Fun()M be the Poisson algebra of smooth functions on M. is a quantization of the quasi-trigonometric solution Xab, . Then it can be quantized in an equivariant way, i.e., there ex- Corollary 1. The rational degeneration ists an associative algebra A, which is a deformation of the uu- P - RuuFz(, )= 12 (1 - hxss12 -Äu Poisson algebraFun()M ,andwhichisaHopfmodulealgebra 12 --h - 2 uu12 uu12 (10) over H. --- +-xh)ssx(u Ä+z 2uu(). - h)ss Ä This conjecture is based on some results proved in [6, 7, 8]. A 1 21 22Ä special case of this conjecture has been proved in [2]. More where P12 denotes the permutation of factors in CC,isa exactly, in this paper the conjecture was proved for Lie quantization of the following rational solution of CYBE:

W -- ru(, u )= +Ä-Äxs (uu szz s s) . (11) [3] Halbout, G.: Formality Theorem for Lie Bialgebras and 12 - 12 uu12 Quantization of Twists and Coboundary r-Matrices. Adv. Math.Vol.207 (2006), p. 617–633. The Hamiltonians of the periodic chains related to the [4] Halbout, G.: Quantization of r–Z-quasi-Poisson Mani- twisted R-matrix were computed in [5]. We recall this result: folds and Related Modified Classical Dynamical r-Matri- We consider ces. [arXiv math. QA/0801.2789]. tz()= TrR (, zz ) R- (, zz )K R (, zz ) (12) 00NN 2 0 1 2 01 2 [5] Khoroshkin,S.,Stolin,A.,Tolstoy,V.:q -Power Function a family of commuting transfer matrices for the correspond- over q-Commuting Variables and Deformed XXX and ing homogeneous periodic chain, [(tz¢¢¢ ),( tz )]= 0,wherewe XXZ Chains. Phys. Atomic Nuclei,Vol.64 (2001), No.12, = p. 2173–2178. treat z2 as a parameter of the theory and zz1 as a spectral parameter. Then the Hamiltonian [6] Karolinsky, E., Stolin, A., Tarasov, V.: Dynamical Yang- -Baxter Equation and Quantization of Certain Poisson =--11d - Hqqzabz,, ()()()tz z= z t z2 (13) 22dz Brackets. Noncommutative Geometry and Representa- tion Theory in Mathematical Physics. 175–182, Contemp. can be computed by a standard procedure: Math., Vol. 391 (2005), Amer. Math. Soc., Providence, z --z - - HH=+å(( Css+ + ss+ ) + D ss+ ). (14) RI. a,, b z2 XXZ k kkk11kk 1 k [7] Karolinsky, E., Stolin, A., Tarasov, V.: Irreducible High- - =-q 1 -1 =+-1 + est Weight Modules and Equivariant Quantization. Adv. Here Cbazq2 ()2 , Dazbqazqb()()2 2 , s + = s- = s z =- Math.Vol.211 (2007), No. 1, p. 266–283. e12, e21, ee11 22 and [8] Karolinsky, E., Stolin, A., Tarasov, V.: Quantization of æ + -1 ö =++ç ss+ -- ss+ qq ssz z ÷ Poisson Homogeneous Spaces, Highest Weight Mod- HXXZåç kk+11 kk+ kk+1 ÷. (15) 2 k è ø ules, and Kostant’s Problem. In preparation. [9] Khoroshkin,S.,Pop,I.,Stolin,A.,Tolstoy,V.:OnSome We see that, by a suitable choice of parameters a, b and z , 2 Lie Bialgebra Structures on Polynomial Algebras and wecanaddtotheXXZHamiltoniananarbitrarylinearcom- Their Quantization. Preprint no. 21, 2003/2004, Mittag- bination of the terms -Leffler Institute, Sweden. ssz --+ ssz ss- - å k kkk+11+ and å kk+1 [10]Khoroshkin,S.,Pop,I.,Samsonov,M.,Stolin,A.,To- k k lstoy, V.: On some Lie Bialgebra Structures on Polyno- and the model will remain integrable. mial Algebras and Their Quantization. [ArXiv math. QA/0706.1651v1]. To appear in Comm. Math. Phys. Moreover,itwasprovedin[5]thattheHamiltonian =----11h d - 1 [11] Montaner, F., Zelmanov, E.: Bialgebra Structures on Hqquqhx,,uuu(( ) )tu ( )= t ( u2 ) (16) 22du Current Lie Algebras. Preprint, University of Wisconsin, Madison, 1993. for = K [12] Pop, I., Stolin, A.: Classification of Quasi-Trigonometric tu() TrR00NN (, uu 2 ) R 0- 1 (, uu 2 ) R 01 (, uu 2 ), (17) Solutions of the Classical Yang–Baxter Equation. Sub- is given by the same formula (13), where mitted. - -1 1xh =-x q -1 q =-xh2 -1 [13] Stolin, A.: On Rational Solutions of Yang-Baxter Cu2 and Duquq2()2 . 2 2 Equation for sl(n). Math. Scand., Vol. 69 (1991), No. 1, Now it also makes sense in the XXX limit q =1: p. 57–80. z --z - - [14] Stolin A., Yermolova–Magnusson, J.: The 4th Structure. HHhx =+å(( Css+ + ss+ ) + D ss+ ), (18) ,,u2 XXX k kkk11kk 1 k Czech. J. Phys., Vol. 56 (2006), No. 10/11, p. 1293–1297. xh [15] Tolstoy ,V.: From Quantum Affine Kac–Moody Algebras where C =- and Duu=-xh2 (). 2 22 to Drinfeldians and Yangians. Kac–Moody Lie algebras and Related topics. Contemp. Math.Vol.343 (2004), References Amer. Math. Soc., p. 349–370. [1] Belavin, A. A., Drinfeld, V. G.: On Classical Yang-Baxter Equation for Simple Lie Algebras. Funct. An. Appl., Alexander Stolin Vol. 16 (1982), No. 3, p. 1–29. e-mail: [email protected] [2] Etingof, P., Kazhdan, D.: Quantization of Poisson Alge- braic Groups and Poisson Homogeneous Spaces. Department of Mathematics Symétries quantiques (Les Houches, 1995), North-Hol- University of Göteborg land, Amsterdam 1998, p. 935–946. Göteborg, Sweden