On central extensions and simply laced Lie algebras

TL;DR

This paper presents a construction linking central extensions of coinvariant groups of simply laced root lattices to semisimple Lie algebras with automorphisms, generalizing previous methods used in algebraic curve studies.

Contribution

It introduces a new construction that produces semisimple Lie algebras with automorphisms from central extensions of coinvariant groups, extending prior approaches.

Findings

Provides a method to construct Lie algebras from algebraic curve families.

Generalizes Thorne's construction for plane quartics.

Connects lattice automorphisms with algebraic geometry applications.

Abstract

Let $\Lambda$ be a simply laced root lattice and $w$ an elliptic automorphism of $\Lambda$ of order $d$. This paper gives a construction that begins with a central extension of the group of coinvariants $\Lambda_w$ and produces a semisimple Lie algebra of Dykin type $\Lambda$ with an automorphism of order $d$ lifting $w$. The input for this construction naturally arises when considering certain families of algebraic curves. The construction generalizes one used by Thorne to study plane quartics and one used by the author and Thorne to study a family of genus-2 curves.