Noetherianity up to conjugation of locally diagonal inverse limits

TL;DR

This paper proves that certain inverse limits of Lie algebra duals are Noetherian under group actions and classifies conjugation-stable subsets of infinite matrices, advancing understanding of algebraic structures in infinite dimensions.

Contribution

It establishes Noetherianity up to conjugation for inverse limits of dual Lie algebras and classifies conjugation-stable subsets of infinite matrices, extending algebraic theory.

Findings

Inverse limits of dual Lie algebras are Noetherian under group action.

Classification of conjugation-stable closed subsets of infinite matrices.

Extension of algebraic properties to infinite-dimensional settings.

Abstract

We prove that the inverse limit of the sequence dual to a sequence of Lie algebras is Noetherian up to the action of the direct limit of the corresponding sequence of classical algebraic groups when the sequence of groups consists of diagonal embeddings. We also classify all conjugation-stable closed subsets of the space of $\mathbb{N}\times\mathbb{N}$ matrices.