## Introduction

*Mark Green, Phillip Griffiths, and Matt Kerr*

### in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.003.0001
- Subject:
- Mathematics, Analysis

This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of ... More

## Higgs Bundles and Geometric Structures on Surfaces

*William M. Goldman*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0008
- Subject:
- Mathematics, Geometry / Topology

The theory of Higgs bundles, pioneered by Hitchin and Simpson, provides an analytic approach to studying surface group representations and their deformation space. This chapter describes the basic ... More

## Lie Groups and Lie Algebras: A Physicist's Perspective

*Adam M. Bincer*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199662920
- eISBN:
- 9780191745492
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199662920.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This book is based on lectures given to graduate students in physics at the University of Wisconsin-Madison. Group theory has been around for many years and the only thing new in this book is my ... More

## Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

*Mark Green, Phillip A. Griffiths, and Matt Kerr*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.001.0001
- Subject:
- Mathematics, Analysis

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive ... More

## RADON TRANSFORM ON GROUPS

*Leon Ehrenpreis*

### in The Universality of the Radon Transform

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.003.0007
- Subject:
- Mathematics, Mathematical Physics

This chapter applies the book's methods to various Lie groups. In particular, to the horocyclic and geodesic transforms on G/K.

## Breaking Symmetries

*Gian Francesco Giudice*

### in A Zeptospace Odyssey: A Journey into the Physics of the LHC

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199581917
- eISBN:
- 9780191723001
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199581917.003.0008
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Symmetries play a fundamental role in modern physics. This chapter introduces this concept by discussing the meaning of symmetries in mathematics and in physics, explaining the significance of Lie ... More

## Symmetries And Renormalization

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with linear continuous symmetries corresponding to compact Lie groups because they imply interesting formal properties; consequences of discrete symmetries can also be studied but ... More

## Lie groups and Lie algebras

*Adam M. Bincer*

### in Lie Groups and Lie Algebras: A Physicist's Perspective

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199662920
- eISBN:
- 9780191745492
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199662920.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Lie groups and Lie algebras are defined. SU(2), the group whose elements are 2x2 unitary unimodular matrices is described providing an example of a 3-dimensional Lie group. Infinitesimal generators ... More

## Fundamental Vector Fields

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0011
- Subject:
- Mathematics, Educational Mathematics

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, ... More

## Matrices and Lie groups

*Clifford Henry Taubes*

### in Differential Geometry: Bundles, Connections, Metrics and Curvature

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0002
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter discusses the following: the general linear group; Lie groups; examples of Lie groups; some complex Lie groups; the three most commonly seen subgroups of Gl(n; ℂ): Sl(n; ℂ), U(n) and ... More

## Introduction

*Ercüment H. Ortaçgil*

### in An Alternative Approach to Lie Groups and Geometric Structures

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198821656
- eISBN:
- 9780191860959
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821656.003.0001
- Subject:
- Mathematics, Pure Mathematics

This introductory chapter explain the unique feature of this book that makes it different from the classical books on the subject.

## Principal bundles

*Clifford Henry Taubes*

### in Differential Geometry: Bundles, Connections, Metrics and Curvature

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0010
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines principal bundles, which is defined as the Lie group analog of a vector bundle. It covers principal bundles constructed from vector bundles; examples of Lie group quotients; ... More

## Non-Abelian Gauge Theories: Introduction

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0019
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter constructs a field theory invariant under local, that is, space-dependent, transformations of a general compact Lie group G. Inspired by the abelian example of Chapter 18, it introduces ... More

## A Universal Bundle for a Compact Lie Group

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0008
- Subject:
- Mathematics, Educational Mathematics

This chapter looks at a universal bundle for a compact Lie group. By Milnor's construction, every topological group has a universal bundle. Independently of Milnor's result, the chapter constructs a ... More

## Integration on a Compact Connected Lie Group

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0013
- Subject:
- Mathematics, Educational Mathematics

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a ... More

## Topology, Manifolds, Dynamical Systems, Measure, and Bundles

*Xin Wei Sha*

### in Poiesis and Enchantment in Topological Matter

- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780262019514
- eISBN:
- 9780262318914
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262019514.003.0006
- Subject:
- Philosophy, Aesthetics

This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological ... More

## Curvature and Instability

*S. G. Rajeev*

### in Fluid Mechanics: A Geometrical Point of View

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805021
- eISBN:
- 9780191843136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805021.003.0011
- Subject:
- Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

The Euler equations of a rigid body can be understood as the geodesic equations for a metric on the rotation group. A rapid introduction to the Riemannian geometry of Lie groups (following Milnor) is ... More

## An Alternative Approach to Lie Groups and Geometric Structures

*Ercüment H. Ortaçgil*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198821656
- eISBN:
- 9780191860959
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821656.001.0001
- Subject:
- Mathematics, Pure Mathematics

This book is about the foundations of geometric symmetry, namely, Lie groups and differential geometry. Although this is a classical subject about which hundreds of books have been written, this book ... More

## Harish-Chandra Modules over ℤ

*Günter Harder*

### in Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions: (AMS-203)

- Published in print:
- 2019
- Published Online:
- September 2020
- ISBN:
- 9780691197890
- eISBN:
- 9780691197937
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691197890.003.0008
- Subject:
- Mathematics, Number Theory

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted ... More

## The Maurer–Cartan Form

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0015
- Subject:
- Mathematics, Educational Mathematics

This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The ... More