Covering Algebras II: Isomorphism of Loop Algebras

TL;DR

This paper characterizes when loop algebras derived from symmetrizable Kac-Moody Lie algebras and finite order automorphisms are isomorphic, providing precise criteria for their equivalence.

Contribution

It establishes necessary and sufficient conditions for isomorphism between loop algebras constructed from pairs of symmetrizable Kac-Moody algebras and automorphisms.

Findings

Derived explicit isomorphism criteria for loop algebras

Identified key invariants determining algebra isomorphism

Extended understanding of automorphism effects on algebra structure

Abstract

This paper studies the loop algebras that arise from pairs consisting of a symmetrizable Kac-Moody Lie algebra $\g$ and a finite order automorphism $\sigma$ of $\g$. We obtain necessary and sufficient conditions for two such loop algebras to be isomorphic.