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).

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).

For any finite-dimensional Lie bialgebra $\mathfrak{g}$, we construct a bialgebra $A_{u,v}\left(\mathfrak{g}\right)$ over the ring $C\left[u\right]\left[\left[v\right]\right]$, which quantizes simultaneously the universal enveloping bialgebra $U\left(\mathfrak{g}\right)$, the bialgebra dual to $U\left({\mathfrak{g}}^{\ast }\right)$, and the symmetric bialgebra $S\left(\mathfrak{g}\right)$. Following Turaev, we call $A_{u,v}\left(\mathfrak{g}\right)$ a biquantization of $S\left(\mathfrak{g}\right)$. We show that the bialgebra $A_{u,v}\left({\mathfrak{g}}^{\ast }\right)$ quantizing $U\left({\mathfrak{g}}^{\ast }\right)$, $U{\left(\mathfrak{g}\right)}^{\ast }$, and $S\left({\mathfrak{g}}^{\ast }\right)$ is essentially dual to the bialgebra obtained from $A_{u,v}\left(\mathfrak{g}\right)$ by exchanging $u$ and $v$. Thus, $A_{u,v}\left(\mathfrak{g}\right)$ contains all information about the quantization of $\mathfrak{g}$. Our construction extends Etingof and Kazhdan’s one-variable quantization of $U\left(\mathfrak{g}\right)$.

For any finite-dimensional Lie bialgebra $\mathfrak{g}$, we construct a bialgebra $A_{u,v}\left(\mathfrak{g}\right)$ over the ring $C\left[u\right]\left[\left[v\right]\right]$, which quantizes simultaneously the universal enveloping bialgebra $U\left(\mathfrak{g}\right)$, the bialgebra dual to $U\left({\mathfrak{g}}^{\ast }\right)$, and the symmetric bialgebra $S\left(\mathfrak{g}\right)$. Following Turaev, we call $A_{u,v}\left(\mathfrak{g}\right)$ a biquantization of $S\left(\mathfrak{g}\right)$. We show that the bialgebra $A_{u,v}\left({\mathfrak{g}}^{\ast }\right)$ quantizing $U\left({\mathfrak{g}}^{\ast }\right)$, $U{\left(\mathfrak{g}\right)}^{\ast }$, and $S\left({\mathfrak{g}}^{\ast }\right)$ is essentially dual to the bialgebra obtained from $A_{u,v}\left(\mathfrak{g}\right)$ by exchanging $u$ and $v$. Thus, $A_{u,v}\left(\mathfrak{g}\right)$ contains all information about the quantization of $\mathfrak{g}$. Our construction extends Etingof and Kazhdan’s one-variable quantization of $U\left(\mathfrak{g}\right)$.

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