From smooth dynamical twists to twistors of quantum groupoids

TL;DR

This paper links smooth dynamical twists on certain Poisson manifolds to the construction of twistors for quantum groupoids, advancing the understanding of equivariant star products in the context of Lie theory and deformation quantization.

Contribution

It demonstrates that smooth dynamical twists induce twistors on quantum groupoids, providing a new method to construct equivariant star products on Poisson homogeneous spaces.

Findings

Establishes a correspondence between dynamical twists and quantum groupoid twistors.

Provides a framework for constructing equivariant star products.

Connects the structure of twistors with nondegenerate polarized Lie algebras.

Abstract

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 $U(\mathfrak{g}) \otimes U(\mathfrak{g})\llbracket \hbar \rrbracket$, 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.