Biquantization of Lie bialgebras

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 $U(g^{\ast })$, and the symmetric bialgebra $S(g)$. We call $A_{u,v}(g)$ a biquantization of $S(g)$. We show that the bialgebra $A_{u,v}(g^{\ast })$ quantizing $U(g^{\ast })$, $U(g{)}^{\ast }$, and $S(g^{\ast })$ 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 $U(g)$.

67 pages, no figures

References in corpus (1)

• Quantization of Lie bialgebras, V

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