Birational Equivalences and Kac-Moody Algebras

TL;DR

This paper demonstrates that all Kac-Moody algebras and their quantized versions are birationally equivalent to specific algebraic structures involving Weyl algebras, polynomial algebras, and their quantum analogues.

Contribution

It establishes a unifying birational equivalence framework for Kac-Moody and quantized Kac-Moody algebras, linking them to Weyl and polynomial algebra structures.

Findings

Kac-Moody algebras are birationally equivalent to smash biproducts of Weyl algebras and polynomial algebras.

Quantized Kac-Moody algebras are similarly equivalent when Weyl algebras are replaced with quantum analogues.

The results provide a new perspective on the algebraic structure of Kac-Moody algebras.

Abstract

We show that every Kac-Moody algebra is birationally equivalent to a smash biproduct of two copies of a Weyl algebra together with a polynomial algebra. We also show that the same is true for quantized Kac-Moody algebras where one replaces Weyl algebras with their quantum analogues.