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

Gao, Li

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https://hdl.handle.net/2142/101546 Copy

Description

Title

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

Author(s)

Gao, Li

Issue Date

2018-07-10

Director of Research (if dissertation) or Advisor (if thesis)

Junge, Marius

Doctoral Committee Chair(s)

Ruan, Zhong-Jin

Committee Member(s)

• Boca, Florin P.
• Oikhberg, Timur

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2018-08

Type of Resource

text

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

Copyright and License Information

Copyright 2018 by Li Gao

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