The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry

Eberhard Zeidler
Additional contact information
Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 5 in Quantum Field Theory III: Gauge Theory, 2011, pp 355-370 from Springer

Abstract: Abstract The theory of Lie groups and Lie algebras is nothing else than a far-reaching generalization of Euler’s exponential function. The simplest case is the Lie group U(1) defined by $$U(1): = \{z \in \mathbb{C}: \;|z|=1\}$$ equipped with the usual multiplication of complex numbers. Equivalently, $$U(1)= \{\textrm{e}^{\textrm{i} \varphi}: \; \varphi \in \mathbb{R}\}.$$ The set U(1) is a real one-dimensional manifold, namely, the unit circle. This manifold is called the group manifold of the Lie group U(1). In particular, a Lie group $\mathcal{G}$ is called compact iff $\mathcal{G}$ is a compact manifold. For example, the Lie group U(1) is compact. In fact, the unit circle is a compact manifold.

Keywords: Discrete Fourier Transform; Haar Measure; Euclidean Plane; Dual Group; Addition Theorem (search for similar items in EconPapers)
Date: 2011
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-22421-8_6

Ordering information: This item can be ordered from
http://www.springer.com/9783642224218

DOI: 10.1007/978-3-642-22421-8_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().