Simple superelliptic Lie algebras

TL;DR

This paper investigates the structure of superelliptic Lie algebras derived from specific algebraic curves, providing criteria for their simplicity, and exploring their central extensions, derivations, and automorphisms.

Contribution

It offers the first comprehensive criteria for simplicity and characterizes universal central extensions and automorphisms of superelliptic Lie algebras.

Findings

Criteria for simplicity of superelliptic Lie algebras

Explicit descriptions of universal central extensions

Classification of automorphisms and isomorphisms

Abstract

Let $m\in N$, $P(t)\in C[t]$. Then we have the Riemann surfaces (commutative algebras) $R_m(P)=C[t^{\pm1},u | u^m=P(t)]$ and $S_m(P)=C[t , u| u^m=P(t)].$ The Lie algebras $\mathcal{R}_m(P)=Der(R_m(P))$ and $\mathcal{S}_m(P)=Der(S_m(P))$ are called the $m$-th superelliptic Lie algebras associated to $P(t)$. In this paper we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras.