From smooth dynamical twists to twistors of quantum groupoids

Consider a Lie subalgebra $\mathfrak{l}\subset \mathfrak{g}$ and an $\mathfrak{l}$-invariant open submanifold $V\subset {\mathfrak{l}}^{\ast }$. We demonstrate that any smooth dynamical twist on $V$, valued in $U(\mathfrak{g})\otimes U(\mathfrak{g})\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$, establishes a twistor on the associated quantum groupoid when combined with the Gutt star product on the cotangent bundle $T^{\ast }L$ of a Lie group $L$ that integrates $\mathfrak{l}$. This result provides a framework for constructing equivariant star products from smooth dynamical twists on those Poisson homogeneous spaces arising from nondegenerate polarized Lie algebras, leveraging the structure of twistors of quantum groupoids.