On the Universal Central Extension of Hyperelliptic Current Algebras

TL;DR

This paper describes the universal central extension of hyperelliptic current Lie algebras using combinatorial formulas, providing explicit generators and relations for these algebraic structures.

Contribution

It introduces a new explicit description of the universal central extension for hyperelliptic current Lie algebras using Faá de Bruno's formula and Bell polynomials.

Findings

Explicit generators and relations for the universal central extension

Application of Faá de Bruno's formula and Bell polynomials

Enhanced understanding of hyperelliptic current Lie algebras

Abstract

Let $p(t)\in\mathbb C[t]$ be a polynomial with distinct roots and nonzero constant term. We describe, using Fa\'a de Bruno's formula and Bell polynomials, the universal central extension in terms of generators and relations for the hyperelliptic current Lie algebras $\mathfrak g\otimes R$ whose coordinate ring is of the form $R=\mathbb C[t,t^{-1},u\,|\, u^2=p(t)]$.