Quantum differential operators on K[x]

TL;DR

This paper characterizes the structure of quantum differential operators on affine and projective spaces, extending classical differential operator theory with quantum analogs and explicit algebraic descriptions.

Contribution

It explicitly describes the ring of quantum differential operators on affine space and constructs the global quantum differential operators on the projective line.

Findings

The ring on the affine line is generated by classical operators plus two new operators.

The structure of the quantum differential operators ring is given by generators and relations.

Extension of the theory to affine n-space and the projective line is achieved.

Abstract

Following the definition of quantum differential operators given by Lunts and Rosenberg in (Sel. math., New ser. 3 (1997) 335--359), we show that the ring of quantum differential operators on the affine line is the ring generated by x and \del, the familiar differential operators on the line, along with two additional operators which we call \del^\beta^1 and \del^\beta^-1. We describe this ring both as a subring of the ring of graded endomorphisms and as a ring given by generators and relations. From this starting point, we are able to describe the ring of quantum differential operators on affine n-space and to construct the ring of global quantum differential operators on the projective line.