# On the module structure of the center of hyperelliptic Krichever-Novikov algebras

## TL;DR

This paper studies the structure of the center of hyperelliptic Krichever-Novikov algebras, providing explicit generators, relations, and decompositions of the center into irreducible representations based on automorphism groups.

## Contribution

It offers a generator and relations description of the universal central extension and describes the decomposition of the center into irreducible representations for specific automorphism groups.

## Key findings

- Explicit generators and relations for the universal central extension.
- Decomposition of the center into irreducible representations.
- Analysis based on automorphism groups C_{2k} and D_{2k}.

## Abstract

We consider the coordinate ring of a hyperelliptic curve and let $\mathfrak{g}\otimes R$ be the corresponding current Lie algebra where $\mathfrak g$ is a finite dimensional simple Lie algebra defined over $\mathbb C$. We give a generator and relations description of the universal central extension of $\mathfrak{g}\otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and describe how the center $\Omega_R/dR$ decomposes into a direct sum of irreducible representations when the automorphism group is $C_{2k}$ or $D_{2k}$.