Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures

TL;DR

This paper explicitly constructs and computes Drinfeld twists for Hopf algebroids associated with linear Poisson structures, advancing the understanding of deformation quantization in noncommutative geometry.

Contribution

It provides an explicit description and computation of Drinfeld twists for Hopf algebroids related to linear Poisson structures, connecting previous theoretical frameworks.

Findings

Explicit formula for the Drinfeld twist as a product of exponentials

Confirmation that the twisted bialgebroid matches the constructed Hopf algebroid

Enhanced understanding of deformation quantization for linear Poisson structures

Abstract

In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every deformation quantization leads to a Drinfeld twist of the associative bialgebroid of h-adic series of differential operators on a fixed Poisson manifold. In the case of linear Poisson structures, the twisted bialgebroid essentially coincides with our construction. Using our explicit description of the Hopf algebroid, we compute the corresponding Drinfeld twist explicitly as a product of two exponential expressions.