On the Commuting variety of a reductive Lie algebra

TL;DR

This paper proves that the commuting variety of a reductive Lie algebra is a normal scheme, ensuring its ideal of definition is prime, which advances understanding of its geometric and algebraic structure.

Contribution

It establishes the normality of the commuting variety scheme for reductive Lie algebras, a key property previously unconfirmed.

Findings

The commuting variety scheme is normal.

Its ideal of definition is prime.

Provides foundational results for Lie algebra geometry.

Abstract

The commuting variety of a reductive Lie algebra ${\goth g}$ is the underlying variety of a well defined subscheme of $\gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.