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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


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


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


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


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


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


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


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.


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


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


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


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


Mathematical methods

G. Barr, R. Devenish, R. Walczak, and T. Weidberg

in Particle Physics in the LHC Era

Published in print:
2016
Published Online:
March 2016
ISBN:
9780198748557
eISBN:
9780191811203
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198748557.003.0002
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter provides a resume of the mathematical tools that are used to explain and understand the Standard Model of particle physics and the experimental data that support it. Various aspects of ... More


Maps and vector 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/9780199605880.003.0005
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This discussion of maps and vector bundles covers the pull-back construction; pull-backs and Grassmannians; pull-back of differential forms and push-forward of vector fields; invariant forms and ... More


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