Large-$ N $ Limits of Spaces and Structures

Irfan Alam; Ambar N. Sengupta

doi:10.3934/cpaa.2023019

Article Contents

2023, Volume 22, Issue 4: 1009-1042. Doi: 10.3934/cpaa.2023019

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Large-$ N $ Limits of Spaces and Structures

Irfan Alam^1, and
Ambar N. Sengupta^2, ,

1.
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
2.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

^*Corresponding author: Ambar N. Sengupta
^*Corresponding author: Ambar N. Sengupta

Received: October 2022

Revised: December 2022

Early access: January 2023

Published: April 2023

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Abstract

For a sequence of spaces $ X_N $, with topological, algebraic or measure-theoretic structures, we show how a large-$ N $ limit $ X_\infty $ with corresponding structures is obtained. For example, when each space is a topological group $ G_N $, such as $ G_N = U(N) $, a limiting group $ G_\infty $ with topology results. Using the Weil-Kodaira construction, for compact topological groups $ G_N $ equipped with normalized Haar measures, we obtain a topological structure on $ G_\infty $ that also makes the group operations continuous. When each $ G_N $ is a Lie group we describe a Lie algebra associated to $ G_\infty $.

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Mathematics Subject Classification: Primary: 28E04, 54J05; Secondary: 28C10, 28C20.

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