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

Miaskiwskyi, L.T.

doi:10.4233/60c65f23-8c3c-4050-85e4-39ef675e2168

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

Doctoral thesis (2022)

Authors

L.T. Miaskiwskyi Analysis -

Research Group

Analysis () (TU Delft)

DOI: 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

To reference this document use:

http://resolver.tudelft.nl/uuid:60c65f23-8c3c-4050-85e4-39ef675e2168

More Info

expand_more

Published Date

2022

Language

English

Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

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.

Files

Miaskiwskyi_dissertation.pdf

(pdf | 1.58 Mb)

Unknown license