Biquantization of Lie bialgebras

Christian Kassel (1) , Vladimir Turaev (1)

1 IRMA - Institut de Recherche Mathématique Avancée

For any finite-dimensional Lie bialgebra g, we construct a bialgebra Auv(g) over the ring C[u][[v]], which quantizes simultaneously the universal enveloping bialgebra U(g), the bialgebra dual to U(g*), and the symmetric bialgebra S(g). We call Auv(g) a biquantization of S(g). We show that the bialgebra Auv(g*) quantizing U(g*), U(g)*, and S(g*) is essentially dual to the bialgebra obtained from Auv(g) by exchanging u and v. Thus, Auv(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan's one-variable quantization of U(g).

Mots clés: quantization; Hopf algebra; Poisson algebra

Domaines: Algèbres quantiques [math.QA]

Christian Kassel, Vladimir Turaev. Biquantization of Lie bialgebras. Pacific Journal of Mathematics, 2000, 195, pp.297-369. ⟨hal-00124670⟩