Commuting varieties and the rank filtration of topological K-theory

TL;DR

This paper explores the topology of commuting elements in Lie algebras and connects their compactifications to rank filtrations in connective complex and real K-theory, revealing new structural insights.

Contribution

It establishes a novel relationship between commuting varieties in Lie algebras and subquotients of rank filtrations in connective K-theory, including real and complex cases.

Findings

Connected commuting varieties to K-theory filtrations

Described the topology of commuting elements in Lie algebras

Extended results to real K-theory variants

Abstract

We consider the space of $n$-tuples of pairwise commuting elements in the Lie algebra of $U(m)$. We relate its one-point compactification to the subquotients of certain rank filtrations of connective complex $K$-theory. We also describe the variant for connective real $K$-theory.