AP08_2.vp 1 Definition of quantum groups, Lie bialgebras and twists In mathematics and theoretical physics, quantum groups are certain non-commutative algebras that first appeared in the theory of quantum integrable models, and which were then formalized by Drinfeld and Jimbo. Nowadays quantum groups are one of the most popular objects in modern mathematical physics. They also play a sig- nificant role in other areas of mathematics. It turned out that quantum groups provide invariants of knots, and that they can be used in quantization of Poisson brackets on manifolds. Quantum groups gave birth to quantum geometry, quantum calculus, quantum special functions and many other “quan- tum” areas. At first, quantum groups appeared to be a useful tool for the following program: Let R satisfy the quantum Yang- -Baxter equation (QYBE) and let R be decomposable in a series in formal parameter �. Then r, which is the coefficient at the first order term, satisfies the so-called classical Yang- -Baxter equation (CYBE). In many cases CYBE can be solved and the problem is to extend the solutions of CYBE to solu- tions of QYBE. However, the most important applications of quantum groups relate to the theory of integrable models in mathemat- ical physics. The presence of quantum group symmetries (or so-called hidden symmetries) was the crucial point in ex- plicit solutions of many sophisticated non-linear equations, such as Korteweg-de Vries or Sine-Gordon. Quantum groups changed and enriched representation theory and algebraic topology. Quantum groups were defined by V. Drinfeld as Hopf al- gebra deformations of the universal enveloping algebras (and also their dual Hopf algebras). More exactly, we say that a Hopf algebra A over � �C [ ]� is a quantum group (or maybe a quasi-classical quantum group) if the following conditions are satisfied: i. The Hopf algebra A A� is isomorphic as a Hopf algebra to the universal enveloping algebra of some Lie algebra L. ii. As a topological � �C [ ]� -module, A is isomorphic to � �V [ ]� for some vector space V over C. The first examples of quantum groups were quantum uni- versal enveloping algebras Uq( )g , quantum affine Kac-Moody algebras Uq( �)g , and Yangians Y( )g . Further, it is well-known that the Lie algebra L such that A A� � U( )L is unique: � �L � � � � �a a a aA A� : ( )� 1 1 . If A is a quantum group with A A� � U L( ) , then L pos- sesses a new structure, which is called a Lie bialgebra structure �, and it is given by: �( ) ( ( ) ( )) modx a aop� � �1 � � , where a is an inverse image of x in A. In particular, the classi- cal limit of Uq( )g is g, the classical limit of Uq( �)g is the affine Kac-Moody algebra, which is a central extension of g[ , ]u u 1 , and the classical limit of Y( )g is g[ ]u (the corresponding Lie bialgebra structures will be described later). The Lie bialgebra ( , )L � is called the classical limit of A, and A is a quantization of ( , )L � . So, any quantum group in our sense has its classical limit, which is a Lie bialgebra, and the following natural problem arises: Given a Lie bialgebra, is there any quantum group whose classi- cal limit is the given Lie bialgebra? Or in other words: Can any Lie bialgebra be quantized? In the mid 1990’s P. Etingof and D. Kazhdan gave a posi- tive answer to this problem. Now, let g be a simple complex finite-dimensional Lie algebra and let g[ ]u be the corresponding polynomial Lie algebra. If you ask a physicist which quantum group is a quantization of g[ ]u , you will almost certainly hear that the quantization of g[ ]u is the Yangian Y( )g . Let us explain this in greater detail. Set � � �� I I� �, where � �I � is an orthonormal basis of g with respect to the Killing form. Then it is well-known that the function r( , ) ( , )u v u v u v � � � � 2 (1) is a rational skew-symmetric solution of the classical Yang- -Baxter equation that is [ ( , ), ( , )] [ ( , ), ( , )] [ r r r r r 12 1 2 13 1 3 12 1 2 23 2 3u u u u u u u u 13 1 3 23 2 3 0( , ), ( , )]u u u ur � (2) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 25 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. More generally, we call r( , )u v a rational r-matrix if it is skew-symmetric, satisfies (2) and r( , ) ( , )u v u v� �2 polynomial ( , )u v . Sometimes �2( , )u v is called Yang’s classical r-matrix. Further, the formula � ��� ( ( )) ( , ), ( ) ( )p u u v p u p v� � ��2 1 1 (3) defines a Lie bialgebra structure on g[ ]u , since � is a g-invari- ant element of g g� and this implies that �2( ( ))p u is a poly- nomial in u,v. The Yangian Y( )g is precisely the quantization of this Lie bialgebra. On the other hand, it is easy to see that all rational r-matrices introduced in [13] define Lie bialgebra structures on g[ ]u by the formula � ��( ( )) ( , ), ( ) ( )p u u v p u p v� � �r 1 1 (4) Of course, the next question is: Which quantum groups quantize the other “rational” Lie bialgebra structures on g[ ]u ? Before we give an answer to this question we need the fol- lowing two definitions: Definition: Let �1 be a bialgebra structure on L. Suppose s ��2L satisfies [ , ] [ , ] [ , ] ( )( )s s s s s s Alt id s12 13 12 23 13 23 1 � �� , where Alt x x x x( ): � 123 231 312 for any x � �L 3. Then � �2 1 1 1( ): ( ) [ , ]a a a a s� � � defines a Lie bialgebra structure on L. We say that: �2 is obtained from �1 by twisting via s, and s is called a classical twist. At this point we note that any rational r-matrix is a twist of Yang’s r-matrix. Let H be a Hopf algebra and F H H� � be an invertible element. Let F satisfy F id F F id F12 23( ) ( )� �� � � . (5) Such F is called a quantum twist. The formula � �F a F a F( ) ( )� 1 (6) defines a new co-multiplication on H. 2 Conjecture and scheme Although it is clear that any quantum twist on a quantum group induces uniquely a classical twist on its classical limit, the converse statement remained unknown for a long time. It was formulated in [9] in 2004. Conjecture 1. Any classical twist can be extended to a quantum twist. The conjecture was proved by G. Halbout in [3]. In partic- ular, we can now give an answer to the question posed in the previous section: A quantum group which quantizes any ratio- nal Lie bialgebra structure on L u� g[ ] is isomorphic to the Yangian Y( )g as an algebra and it has a twisted co-algebra structure defined by the corresponding rational solution of CYBE. However, there might exist Lie bialgebra structures on g[ ]u of a different nature, and for classification purposes one has to find all of them. Therefore, in order to classify quantum groups which have a given Lie algebra L as the classical limit one has to solve the following four problems: 1. Describe all basic Lie bialgebra structures on L (in other words all Lie bialgebra structures up to classical twisting). 2. Find quantum groups corresponding to the basic struc- tures. 3. Describe all the corresponding classical twists. 4. Quantize all these classical twists. 3 Lie bialgebra structures on the polynomial Lie algebras and their quantization According to the results of an unpublished paper by Montaner and Zelmanov [11], there are four basic Lie bi- algebra structures on the polynomial Lie algebra P u� g[ ]. Let us describe them (and, hence, we do the first step in the classification of Lie bialgebra structures on g[ ]u ). When it is possible we make further steps. Case 1. Here the one-cocycle �1 0� In this case it is not difficult to show that there is a one-to-one correspondence between Lie bialgebra structures of the first type and finite-dimensional quasi-Frobenius Lie subalgebras of g[ ]u . The corresponding quantum group is U u( [ ])g . Classi- cal twists can be quantized following Drinfeld’s quantization of skew-symmetric constant r-matrices. Case 2. In this case the Lie bialgebra structure is described by �2 2 1 1( ( )) [ ( , ), ( ) ( )]p u u v p u p v� � �� , where �2( , )u v is Yang’s rational r-matrix. The corresponding Lie bialgebra structures are in a one-to-one correspondence with the rational solutions of CYBE described in [13]. The corresponding quantum group is a Yangian Y( )g . Quantum twists were found for g � sln for n � 2 3, in [9]. Case 3. Here the basic Lie bialgebra structure is given by �3 3 1 1( ( )) [ ( , ), ( ) ( )]p u u v p u p v� � �� with � � 3( , )u v v v u rDJ� , where rDJ is the classical Drinfeld-Jimbo modified r-matrix. This Lie bialgebra is the classical limit of the quantum group U g uq( [ ]), which is a certain parabolic subalgebra of a non- -twisted quantum affine algebra Uq( �)g (see [15] or [10]). There is a natural one-to-one correspondence between Lie bialgebra structures of the third type and the so-called quasi- 26 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 -trigonometric solutions of CYBE. A complete classification of the classical twists for sln was presented in [9] and for the arbi- trary g in [12]. Details on quantization of the classical twists for sln can be found in [9]. Case 4. Finally, the 4-th basic Lie bialgebra structure on g[ ]u is de- fined as follows: �4 4 1 1( ( )) [ ( , ), ( ) ( )]p u u v p u p v� � �� , with � � � 4 1 1( , )u v u v uv v u � � . It was proved in [14] that there is a natural one-to-one corre- spondence between Lie bialgebra structures of this kind and the so-called quasi-rational solutions of CYBE. The quasi-ratio- nal solutions of CYBE for g � sln were classified in [14]. Some ideas indicate that quasi-rational r-matrices do not exist for g � g f e2 4 8, , . The corresponding quantum group remains unknown, but, it is rather clear that it is related to the dual Hopf algebra of Y( )g . 4 Some open questions 1. Following steps 1–4 in the classification scheme, it is natural to classify quantum groups related to affine Kac- -Moody algebras. A conjecture is that in this case we have only two basic types of Lie bialgebra structures and the corresponding quantum groups are quantum affine alge- bras and so-called doubles of Y( )g . 2. More generally: Let L be a Lie algebra. Is it possible to de- scribe the moduli space of Double(L), all Lie bialgebra structures on L modulo action of classical twists? 3. It is well-known that there is a 1-1 correspondence be- tween Lie bialgebras and simply connected Poisson-Lie groups. Let us consider a Lie bialgebra L and let G be the corresponding Poisson-Lie group. Let M be a Poisson homogeneous space that is a homogeneous space with a Poisson structure such that the multiplication m G M M: � is a Poisson map. Now, let H be the quantum group which quantizes L. Conjecture 2. Let Fun( )M be the Poisson algebra of smooth functions on M. Then it can be quantized in an equivariant way, i.e., there ex- ists an associative algebra A, which is a deformation of the Poisson algebra Fun( )M , and which is a Hopf module algebra over H. This conjecture is based on some results proved in [6, 7, 8]. A special case of this conjecture has been proved in [2]. More exactly, in this paper the conjecture was proved for Lie bialgebras of a special type L D K� ( ), where K is another Lie bialgebra and L D K� ( ) is the corresponding classical double. Recent progress in related questions was achieved in [4]. Here, a result similar to the conjecture above was proved un- der the following assumptions: L is the so-called coboundary Lie bialgebra and L acts freely on M (in other words M is far from being a homogeneous space). 5 Appendix: Solutions for sl(2) and deformed Hamiltonians The aim of this section is to present concrete examples of quantum twists and the corresponding Hamiltonians follow- ing the results obtained in [5]. We consider the case sl(2). Let � � � e , � � e21 and � z e e� 11 22. Recall that in sl(2) we have two quasi-trigono- metric solutions, modulo gauge equivalence. The non-trivial solution is X z z X z z z z1 1 2 0 1 2 1 2( , ) ( , ) ( )( )� � � � . This so- lution is gauge equivalent to the following: X z z z z z a z z a b z z z z , ( , ) ( 1 2 2 1 2 1 2 1 4 � � � � � � � � � � � � � � � � � � � �) ( ).b z z (7) The above quasi-trigonometric solution was quantized in [5]. Let 1 2( )z be the two-dimensional vector representation of U slq( )2 . In this representation, the generator e � acts as a matrix unit e21, e� � as ze21 and h� as e e11 22 . The quan- tum R-matrix of U slq( )2 in the tensor product 1 2 1 1 2 2( ) ( )z z� is the following: R z z e e e e z z q z qz e e e 0 1 2 11 11 22 22 1 2 1 1 2 11 22 ( , ) ( � � � � 22 11 1 1 1 2 2 12 21 1 21 12 � � � e q q q z qz z e e z e e ) ( ). (8) Proposition 1. The R-matrix given by R R z z z z q z qz b az q az q z: ( , ) (( ) ( � � 0 1 2 1 2 1 1 2 2 1 1 � � b b az q az qb z) ( )( ) ) � � � � � �2 1 1 (9) is a quantization of the quasi-trigonometric solution Xa b, . Corollary 1. The rational degeneration R u u u u u u P u u u u F z( , ) ( ( 1 2 1 2 1 2 12 1 2 2 1 1� � � � � � ��� � ��� � � �z u u2 2 1( ). (10) where P12 denotes the permutation of factors in C C 2 2� , is a quantization of the following rational solution of CYBE: © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 27 Acta Polytechnica Vol. 48 No. 2/2008 � � r u u u u u uz z( , ) ( .1 2 1 2 1 2� � � � � � � � � (11) The Hamiltonians of the periodic chains related to the twisted R-matrix were computed in [5]. We recall this result: We consider t z Tr R z z R z z R z zN N( ) ( , ) ( , ) ( , )� 0 0 2 0 1 2 01 2� (12) a family of commuting transfer matrices for the correspond- ing homogeneous periodic chain, [ ( ), ( )]t z t z� �� � 0, where we treat z2 as a parameter of the theory and z z� 1 as a spectral parameter. Then the Hamiltonian H q q z z t z t za b z z z, , ( ) ( ) ( )2 2 1 1 2� � d d (13) can be computed by a standard procedure: H H C Da b z XXZ k z k k k z k k k , , ( ( ) )2 1 1 1� � � � � � � � . (14) Here C b az q q � 1 2 2 1( ), D az b q az qb� ( )( )2 1 2 , � � e12, � � � � e , � z e e� 11 22 and H q q XXZ k k k k k z k z k � � � � � � � � � � � � � � � �1 1 1 12 . (15) We see that, by a suitable choice of parameters a, b and z2, we can add to the XXZ Hamiltonian an arbitrary linear com- bination of the terms � � � �k z k k k z k � 1 1 and � �k k k � 1 and the model will remain integrable. Moreover, it was proved in [5] that the Hamiltonian H q q u q u t u t uu u u� �, , (( ) ) ( ) ( )2 2 1 1 1 2� � d d (16) for t u Tr R u u R u u R u uN N( ) ( , ) ( , ) ( , )� 0 0 2 0 1 2 01 2� , (17) is given by the same formula (13), where C u qq � �1 1 2 2 1 2 and D u q u q� �2 2 1 2( ). Now it also makes sense in the XXX limit q � 1: H H C Du XXX k z k k k z k k k � � � � � � �, , ( ( ) )2 1 1 1� � , (18) where C � �2 and D u u� � 2 2 2( ) . References [1] Belavin, A. A., Drinfeld, V. G.: On Classical Yang-Baxter Equation for Simple Lie Algebras. Funct. An. Appl., Vol. 16 (1982), No. 3, p. 1–29. [2] Etingof, P., Kazhdan, D.: Quantization of Poisson Alge- braic Groups and Poisson Homogeneous Spaces. Symétries quantiques (Les Houches, 1995), North-Hol- land, Amsterdam 1998, p. 935–946. [3] Halbout, G.: Formality Theorem for Lie Bialgebras and Quantization of Twists and Coboundary r-Matrices. Adv. Math. Vol. 207 (2006), p. 617–633. [4] Halbout, G.: Quantization of r–Z-quasi-Poisson Mani- folds and Related Modified Classical Dynamical r-Matri- ces. [arXiv math. QA/0801.2789]. [5] Khoroshkin, S., Stolin, A., Tolstoy, V.:q -Power Function over q-Commuting Variables and Deformed XXX and XXZ Chains. Phys. Atomic Nuclei, Vol. 64 (2001), No.12, p. 2173–2178. [6] Karolinsky, E., Stolin, A., Tarasov, V.: Dynamical Yang- -Baxter Equation and Quantization of Certain Poisson Brackets. Noncommutative Geometry and Representa- tion Theory in Mathematical Physics. 175–182, Contemp. Math., Vol. 391 (2005), Amer. Math. Soc., Providence, RI. [7] Karolinsky, E., Stolin, A., Tarasov, V.: Irreducible High- est Weight Modules and Equivariant Quantization. Adv. Math. Vol. 211 (2007), No. 1, p. 266–283. [8] Karolinsky, E., Stolin, A., Tarasov, V.: Quantization of Poisson Homogeneous Spaces, Highest Weight Mod- ules, and Kostant’s Problem. In preparation. [9] Khoroshkin, S., Pop, I., Stolin, A., Tolstoy, V.: On Some Lie Bialgebra Structures on Polynomial Algebras and Their Quantization. Preprint no. 21, 2003/2004, Mittag- -Leffler Institute, Sweden. [10] Khoroshkin, S., Pop, I., Samsonov, M., Stolin, A., To- lstoy, V.: On some Lie Bialgebra Structures on Polyno- mial Algebras and Their Quantization. [ArXiv math. QA/0706.1651v1]. To appear in Comm. Math. Phys. [11] Montaner, F., Zelmanov, E.: Bialgebra Structures on Current Lie Algebras. Preprint, University of Wisconsin, Madison, 1993. [12] Pop, I., Stolin, A.: Classification of Quasi-Trigonometric Solutions of the Classical Yang–Baxter Equation. Sub- mitted. [13] Stolin, A.: On Rational Solutions of Yang-Baxter Equation for sl(n). Math. Scand., Vol. 69 (1991), No. 1, p. 57–80. [14] Stolin A., Yermolova–Magnusson, J.: The 4th Structure. Czech. J. Phys., Vol. 56 (2006), No. 10/11, p. 1293–1297. [15] Tolstoy ,V.: From Quantum Affine Kac–Moody Algebras to Drinfeldians and Yangians. Kac–Moody Lie algebras and Related topics. Contemp. Math. Vol. 343 (2004), Amer. Math. Soc., p. 349–370. Alexander Stolin e-mail: astolin@chalmers.se Department of Mathematics University of Göteborg Göteborg, Sweden 28 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 2/2008 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends false /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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