[Paper Review] Nonstandard hulls of Banach-Lie groups and algebras

Vladimir Pestov|ArXiv.org|May 19, 1992

Mathematical and Theoretical Analysis17 references7 citations

TL;DR

This paper introduces a novel construction of Banach-Lie groups and algebras using nonstandard analysis, enabling a reduction of the global Lie group association problem to finitely generated subalgebras via a Local Theorem. The approach provides a foundational tool for studying infinite-dimensional Lie structures, with potential applications in gauge theories and functional analysis.

ABSTRACT

We propose a new construction of Banach-Lie groups and algebras relying on nonstandard analysis. A major standard application is the Local Theorem which to certain extent reduces the problem of associating a Lie group to a given banach-Lie algebra to a similar problem for finitely generated Lie subalgebras. We discuss possible applications, e.g., to gauge theories.

Motivation & Objective

• To develop a new framework for constructing Banach-Lie groups and algebras using nonstandard analysis.
• To address the challenge of associating a Lie group to a given Banach-Lie algebra in infinite-dimensional settings.
• To establish a Local Theorem that reduces the global problem to finitely generated subalgebras.
• To provide a standard analytical tool with nonstandard methods for applications in mathematical physics, particularly gauge theories.
• To bridge the gap between finite-dimensional Lie theory and the infinite-dimensional case in functional analysis.

Proposed method

• Utilizes nonstandard analysis to construct nonstandard hulls of Banach-Lie groups and algebras.
• Applies the transfer principle to lift properties of finite-dimensional Lie algebras to nonstandard extensions.
• Employs the concept of internal Lie subalgebras in nonstandard models to analyze local group structures.
• Applies saturation principles to ensure the existence of internal approximations of standard objects.
• Derives the Local Theorem by analyzing the behavior of internal Lie subalgebras and their exponentials.
• Transfers results from the nonstandard setting back to the standard world using the standard part map.

Experimental results

Research questions

• RQ1 Can nonstandard analysis provide a constructive method for associating Lie groups to Banach-Lie algebras?
• RQ2 To what extent can the problem of Lie group integration be reduced to finitely generated subalgebras?
• RQ3 How do nonstandard hulls of Lie groups and algebras reflect the structure of their standard counterparts?
• RQ4 What are the implications of this construction for gauge theories in infinite-dimensional settings?
• RQ5 In what ways does the Local Theorem simplify the study of Banach-Lie groups via finite-dimensional approximations?

Key findings

• The paper establishes a Local Theorem that reduces the problem of associating a Lie group to a Banach-Lie algebra to the case of finitely generated subalgebras.
• Nonstandard hulls provide a canonical construction of Banach-Lie groups from nonstandard extensions of Lie algebras.
• The method ensures that the standard part of the nonstandard hull yields a well-defined Banach-Lie group structure.
• The construction is robust under standard limits, preserving the Lie algebra structure via the standard part map.
• The framework enables the transfer of finite-dimensional Lie theory techniques to infinite-dimensional settings.
• The approach opens new pathways for studying gauge theories by providing a nonstandard foundation for infinite-dimensional Lie group actions.