Biquantization of Lie bialgebras

For any finite-dimensional Lie bialgebra g, we construct a bialgebra Au,v(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). Following [Tur89], we call Au,v(g) a biquantization of S(g). We show that the bialgebra Au,v(g ) quantizing U(g), U(g), and S(g) is essentially dual to the bialgebra obtained from Au,v(g) by exchanging u and v. Thus, Au,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s one-variable quantization of U(g) [EK96]. Mathematics Subject Classification (1991): 17B37, 17B99, 16W30, 53C15, 81R50