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Quantization of Lie bialgebras, 3

Profile image of david kazhdandavid kazhdan

1996

Abstract

In this paper we construct explicitly the quantization of Lie bialgebras of a finite dimensional simple Lie algebra. by reducing the problem of quantization of the algebra of $\g$-valued functions on a curve with many punctures to the case of one puncture

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References (15)

  1. Alekseev, A., Grosse, H., and Schomerus, V., Combinatorial quantization of the Hamil- tonian Chern-Simons theory I, CMP 172 (1995).
  2. Alekseev, A., Grosse, H., and Schomerus, V., Combinatorial quantization of the Hamil- tonian Chern-Simons theory II, CMP 174 (1995).
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  4. A.A.Belavin, Dynamical symmetry of integrable quantum systems, Nucl. Phys. B180(FS2) (1981), no. 2, 189-200.
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  6. I.V.Cherednik, On R-matrix quantization of formal loop groups, Group Theoretical Methods in Physics, VNU Publ., (1986).
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  10. V.G.Drinfeld, On almost cocommutative Hopf algebras, Len. Math.J. 1 (1990), 321-342.
  11. P.Etingof and D. Kazhdan, Quantization of Lie bialgebras, I, q-alg 9506005, Selecta math. 2 (1996), no. 1, 1-41.
  12. P.Etingof and D. Kazhdan, Quantization of Lie bialgebras, II,, q-alg 9701038 (1996).
  13. N.Yu.Reshetikhin, L.A.Takhtajan, and L.D.Faddeev,, Quantization of Lie groups and Lie algebras, Len. Math. J. . 1 (1990), 193-225.
  14. N.Yu.Reshetikhin and M.A.Semenov-Tian-Shansky, Central extensions of quantum current groups, Lett. Math. Phys. 19 (1990), 133-142.
  15. L.Takhtajan, Solutions of the triangle equations with Z n × Z n -symmetry and matrix analogs of the Weierstrass zeta and sigma-functions, Zapiski Nauch. Sem. LOMI 133 (1984), 258-276.

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This paper is a continuation of "Quantization of Lie bialgebras, I" (q-alg/9606005). We show that the quantization procedure defined in "Quantization of Lie bialgebras, I" is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k[[h]] and the category quantized universal enveloping (QUE) algebras

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This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid...

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