Biquantization of Lie bialgebras

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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). 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).