On some unsolved problems in quantum group theory

Abstract

The general problem of the quantization of Poisson brackets onphase space is an ``old'' one. In the theory of quantum groups,which was created circa 1985, the objects to be quantized arePoisson Lie groups (Lie groups equipped with a Poisson bracketcompatible with the group structure) or Lie bialgebras (Liealgebras equipped with a compatible Lie coalgebra structure). Inthe first approach (Faddeev, Reshetikhin, Takhtajan) oneconstructs noncommutative deformations of the Hopf algebra offunctions on the group, with the infinitesimal given by thePoisson brackets, while in the second approach (the author,Jimbo) one constructs ``quantum universal enveloping algebras''or noncocommutative deformations of the universal envelopingalgebra, with the infinitesimal given by the Lie coalgebrastructure.\par The author's contribution to this conference onquantum groups is a list of 12 open problems of varyinggenerality. The author first poses the general problem of thequantization of Lie bialgebras, and more specifically that of theexistence of a ``universal'' quantization of a Lie bialgebra. Inparticular, he asks whether the Campbell Hausdorff series of aLie algebra can be quantized, i.e., replaced by a formal seriesin noncommuting variables. Some progress on these problems hasbeen made by N. Yu. Reshetikhin [Internat. Math. Res. Notices1992, no. 7, 143 151; MR\Cite{Reshetikhin92:Quantization:143--151}[93h:17041]]. The nextquestion concerns the quantization of the solutions of theclassical Yang Baxter equation. This problem had already beensolved in the special case of the triangular r matrices by theauthor himself, and by C. Moreno and L. Valero, who have sinceobtained new results on the uniqueness of that deformation.Further progress has been made by M. Gerstenhaber, A. Giaquintoand S. D. Schack [in Quantum groups (Leningrad, 1990), 9 46,Lecture Notes in Math., 1510, Springer, Berlin, 1992; MR\Cite{Gerstenhaber92:Quantum:9--46}[93j:17028]; ``Construction ofquantum groups from Belavin Drinfeld infinitesimals'', in Quantumdeformations of algebras and their representations, edited by A.Joseph and S. Shnider, Amer. Math. Soc., Providence, RI, toappear], tending to prove that deformations exist in all cases,and that the parameter space of the deformations is notconnected, and by L. Freidel and J. M. Maillet, using currentalgebras [see Phys. Lett. B 296 (1992), no. 3 4, 353 360; MR93k:81095].\par The author's work on quasi Hopf algebras hasshown the existence of a formal power series, Φ, in twononcommuting variables which permits the construction of aquasitriangular quasi Hopf algebra with a given ad invariantsymmetric infinitesimal t. If the twisting of the quantizedinfinitesimal \exp(ht/2), which is a solution of the quantumquasi Yang Baxter equation, into a solution of the quantum YangBaxter equation with a given infinitesimal r, whose symmetricpart is t, were possible in all cases, it would yield apositive answer to the question of the existence of the``sophisticated'' quantization of the solutions of the classicalYang Baxter equation which is the next question raised andtherefore to that of the ``naive'' quantization. The author alsomentions the analogous problem for the quantization of Lie quasibialgebras.\par Another problem concerns the quantization of theinfinite dimensional Lie algebra of the pseudo differentialoperators on the circle which can be equipped with an additionalstructure close to but not identical to that of a Lie bialgebra,and to which therefore the usual methods do not apply.\par Theaction of a finite group of a Lie bialgebra gives rise to a Liebialgebra structure on the subalgebra of fixed points. Such aconstruction yields the usual Lie bialgebra structure of thesimple Lie algebras of types B, C, F or G from that ofthe simple Lie algebras of types A, D or E, and that of thetwisted affine Kac Moody algebras from the nontwisted ones.Whether this construction can be carried out at the level of thequantum groups is another question which the author raisestogether with a related question concerning the Poisson algebraof functions on the associated Lie group.\par Other questionsconcern the representation theory of the quantized Kac Moodyalgebras and the quantum analogue of the Lie algebra associatedwith an arbitrary symmetrizable matrix.\par In the last section,the author considers the problem of the set theoretic solutionsof the quantum Yang Baxter equation. Little was known about it in1990, but the work of A. Weinstein and P. Xu [Comm. Math. Phys.148 (1992), no. 2, 309 343; MR\Cite{Weinstein92:Classical:309--343}[93k:58102]] has shown thatsuch solutions can be obtained from Lagrangian submanifolds inthe square of the symplectic groupoid of a Poisson Liegroup.\par Despite some notable progress the advances mentionedabove most of the preceding problems remain unsolved more thantwo years later.\par {For the entire collection see MR\Cite{Kulish92:Quantum:Springer-Verlag}[93d:00038].}