Representations for three-point Lie algebras of genus zero

TL;DR

This paper develops new representation theories for three-point Lie algebras of genus zero, including functorial constructions and vertex representations, expanding the understanding of their module structures.

Contribution

It introduces functors transforming modules from affine algebras to three-point affine algebras and constructs vertex and Fock modules for these structures.

Findings

Constructed functors from affine modules to three-point affine modules.

Developed vertex representations for three-point affine algebra.

Built Fock modules for three-point Virasoro algebra.

Abstract

In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox-Jurisich's presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into simple modules over the three-point affine algebras of genus zero. As a corollary, vertex representations are constructed for the three-point affine algebra of genus zero using vertex operators. Moreover, we construct a Fock module for certain quotient of three-point Virasoro algebra of genus zero.