Abstract

For any finite-dimensional Lie bialgebra g, we construct a bialgebra A_u,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 Turaev, we call A_u,v(g) a biquantization of S(g). We show that the bialgebra A_u,v(g^*) quantizing U(g^*), U(g)^*, and S(g^*) is essentially dual to the bialgebra obtained from A_u,v(g) by exchanging u and v. Thus, A_u,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s one-variable quantization of U(g).

Authors