University of Illinois Urbana-Champaign

Local symplectic groupoids and the SGA equation

Zhang, Yuxuan

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ZHANG-DISSERTATION-2022.pdf

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https://hdl.handle.net/2142/116162

Description

Title
Local symplectic groupoids and the SGA equation
Author(s)
Zhang, Yuxuan
Issue Date
2022-06-27
Director of Research (if dissertation) or Advisor (if thesis)
Fernandes, Rui Loja
Doctoral Committee Chair(s)
Kerman, Ely
Committee Member(s)
  • Lerman, Eugene
  • Pascaleff, James
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • local symplectic groupoids
  • integrability of Poisson manifolds
  • SGA equation
Abstract
This thesis discusses several problems related to local symplectic groupoids. In Chapter 1, we prove that if a local symplectic groupoid has uniformly discrete associators, then its associative completion is a symplectic groupoid. It follows that a Poisson manifold is integrable if and only if any of its local integrations has uniformly discrete associators. In Chapter 2, we construct a local symplectic groupoid integrating the Heisenberg-Poisson manifold which is not 6-associative. In Chapter 3, we give the conditions for a function to be the generating function for some local symplectic groupoid structure on the cotangent bundle, both for a coordinate space and for an abstract manifold. We also compare different notions of generating functions and analyze the role of the SGA equation. In Chapter 4, we show that the algebraic equation in the SGA equation is equivalent to a groupoid 2-cocycle condition. Under mild assumptions on the local symplectic groupoid, we find a groupoid 2-cocycle which under the van Est map yields the underlying Poisson bivector.
Graduation Semester
2022-08
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/116162
Copyright and License Information
Copyright 2022 Yuxuan Zhang

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