On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators

Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus ---the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Gao, Li

Contributors dc:contributor

• Junge, Marius
• Ruan, Zhong-Jin
• Boca, Florin P.
• Oikhberg, Timur

• Noncommutative Euclidean spaces
• Moyal Deformation
• Pseudo-differential operators

Statement dc:rights

• Copyright 2018 by Li Gao

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/101546

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/101546