Twisted divided powers and applications

TL;DR

This paper introduces twisted divided powers to extend the theory of differential and q-difference equations in positive characteristic, enabling new algebraic structures and correspondences.

Contribution

It develops the concept of twisted divided powers relative to q, leading to the twisted Weyl algebra, twisted p-curvature, and twisted Simpson correspondence.

Findings

Constructed twisted divided powers and the twisted Weyl algebra.

Established a twisted p-curvature map describing the algebra's center.

Built a divided p-Frobenius for Azumaya splitting and duality applications.

Abstract

In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We introduce here a notion of twisted divided powers (relative to q) and show that one can recover the twisted Weyl algebra and obtain a twisted p-curvature map that describes the center of the twisted Weyl algebra. We also build a divided p-Frobenius that will give, by duality, a formal Azumaya splitting of the twisted Weyl algebra as well as a twisted Simpson correspondence.