Realizations of the three point algebra $\mathfrak{sl}(2, \mathcal R) \oplus\left(\Omega_{\mathcal R}/d{\mathcal R}\right)$

TL;DR

This paper describes the universal central extension of a specific three point current algebra related to rak{sl}(2,\,\mathcal R) and constructs explicit realizations using partial differential operators.

Contribution

It provides the first detailed description of the universal central extension of the three point algebra rak{sl}(2,\,\mathcal R) and offers explicit realizations in differential operator form.

Findings

Explicit description of the universal central extension.

Constructed realizations using partial differential operators.

Enhanced understanding of three point current algebras.

Abstract

We describe the universal central extension of the three point current algebra $\mathfrak{sl}(2,\mathcal R)$ where $\mathcal R=\mathbb C[t,t^{-1},u\,|\,u^2=t^2+4t ]$ and construct realizations of it in terms of sums of partial differential operators.