Local symplectic groupoids and the SGA equation

Zhang, Yuxuan

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

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

Copyright and License Information

Copyright 2022 Yuxuan Zhang

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