Faithful representations of infinite-dimensional nilpotent Lie algebras

TL;DR

This paper constructs faithful representations of infinite-dimensional nilpotent Lie algebras using nilpotent operators on locally convex spaces, and bounded operators on Banach spaces for Banach-Lie algebras, advancing understanding of their structure.

Contribution

It introduces a method to represent infinite-dimensional nilpotent Lie algebras faithfully on locally convex spaces, including Banach spaces, with norm continuity for Banach-Lie cases.

Findings

Faithful representations by nilpotent operators on locally convex spaces.

Norm continuous representations by bounded operators on Banach spaces.

Extension of representation theory to infinite-dimensional nilpotent Lie algebras.

Abstract

For locally convex, nilpotent Lie algebras we construct faithful representations by nilpotent operators on a suitable locally convex space. In the special case of nilpotent Banach-Lie algebras we get norm continuous representations by bounded operators on Banach spaces.