A 2-categorical extension of Etingof–Kazhdan quantisation

Abstract

Let $\mathsf {k}$ be a field of characteristic zero. Etingof and Kazhdan (Sel. Math. (N.S.) 2:1–41, 1996) construct a quantisation $U_\hbar \mathfrak b$ of any Lie bialgebra $\mathfrak b$ over $\mathsf {k}$, which depends on the choice of an associator $\Phi$. They prove moreover that this quantisation is functorial in $\mathfrak b$ (Etingof and Kazhdan in Sel. Math. (N.S.) 4:213–231, 1998). Remarkably, the quantum group $U_\hbar \mathfrak b$ is endowed with a Tannakian equivalence $F_{\mathfrak b}$ from the braided tensor category of Drinfeld–Yetter modules over $\mathfrak b$, with deformed associativity constraints given by $\Phi$, to that of Drinfeld–Yetter modules over $U_\hbar \mathfrak b$ (Etingof and Kazhdan in Transform. Groups 13:527–539, 2008). In this paper, we prove that the equivalence $F_{\mathfrak b}$ is functorial in $\mathfrak b$.