Representation of small conformal algebra

TL;DR

This paper clarifies the mathematical structure of the small conformal algebra by demonstrating that the operator L_0 is a difference operator and providing representations for L_1 and L_{-1} within this framework.

Contribution

It introduces a new representation of the small conformal algebra operators as difference operators, resolving previous issues with their definitions.

Findings

L_0 is a difference operator

Representations of L_1 and L_{-1} as difference operators

Clarification of the operator structure in conformal algebra

Abstract

In hep-th/0202087 it was argued that the operator L_0 is bad defined in kappa-basis as a kernel operator. Indeed, we show that L_0 is a difference operator. We also find a representation of L_1 and L_{-1} in a class of difference operators.