Formal confluence of quantum differential operators

TL;DR

This paper demonstrates that classical differential operators can be viewed as limits of quantum differential operators, introducing twisted operators of infinite level and providing explicit formulas to establish this connection.

Contribution

It introduces the concept of twisted differential operators of infinite level and proves their independence from the twist, linking quantum and classical differential operators.

Findings

Classical differential operators are limits of quantum differential operators.

Explicit formulas for the limit process are provided.

Twisted differential operators of infinite level are independent of the twist.

Abstract

We prove that a usual differential operator is formally the limit of quantum differential operators. For this purpose, we introduce the notion of twisted differential operator of infinite level and prove that, formally, such an object is independent of the choice of the twist. Our method provides explicit formulas.