Cohomological local-to-global principles and integration in finite- and infinite-dimensional Lie theory

Doctoral thesis (2022)

Authors

L.T. Miaskiwskyi Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
Copyright: © 2022 L.T. Miaskiwskyi
DOI
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https://doi.org/10.4233/uuid:60c65f23-8c3c-4050-85e4-39ef675e2168
Brownian motion Lie theory Lie algebra cohomology Gelfand-Fuks cohomology Factorization algebras Loday-Quillen-Tsygan theorem Formal vector fields Diagonal cohomology Topological tensor products Fréchet algebras LF-spaces Slodowy slices Orbital limits Wilson loops Haar measure Cosheaf theory Hochschild-Serre spectral sequence
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Published Date

2022

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

The subject of this thesis is twofold: The first part is the study of local-to-global principles for the continuous Lie algebra (co-)homology of certain infinite-dimensional Lie algebras of geometric origin, specifically, Gelfand-Fuks cohomology and continuous cohomology of gauge algebras. It includes both an exposition to classical results of Gelfand and Fuks, and new methods to construct general local-to-global spectral sequences for the Lie algebra cohomology of section spaces of Lie algebroids. This includes a close functional-analytics study of the involved spaces and attention to complications within LF- and Fréchet spaces. The second part contains a study of two certain measure-theoretic problems on Lie groups. The first such problem is the study of Haar measures of certain identity-neighbourhoods relevant to de Leeuw inequalities in Harmonic Analysis. The second problem is the evaluation of expectation values of polynomials on compact Lie groups, motivated by the study of Weingarten functions and Wilson loops from lattice gauge theory.