Groups, Lie Groups, and Lie Algebras
 Nadir Jeevanjee
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Abstract
Chapter 4 introduces abstract groups and Lie groups, which are a formalization of the notion of a physical transformation. The chapter begins with the definition of an abstract group along with examples, then specializes to a discussion of the groups that arise most often in physics, particularly the rotation group O(3) and the Lorentz group SO(3,1)_{ o }. These groups are discussed in coordinates and in great detail, so that the reader gets a sense of what they look like in action. Then we discuss homomorphisms of groups, which allows us to make precise the relationship between the rotation group O(3) and its quantummechanical ‘doublecover’ SU(2). We then define matrix Lie groups and demonstrate how the socalled ‘infinitesimal’ elements of the group give rise to a Lie algebra, whose properties we then explore. We discuss many examples of Lie algebras in physics, and then show how homomorphisms of matrix Lie groups induce homomorphisms of their associated Lie algebras.
Inside
Within this Chapter
 Groups—Definition and Examples
 The Groups of Classical and Quantum Physics
 Homomorphism and Isomorphism
 From Lie Groups to Lie Algebras
 Lie Algebras—Definition, Properties, and Examples
 The Lie Algebras of Classical and Quantum Physics
 Abstract Lie Algebras
 Homomorphism and Isomorphism Revisited
 Problems
 References
 References
Other actions
 V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, Berlin, 1989
 A. Cannas Da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, Berlin, 2001
 T. Frankel, The Geometry of Physics, 1st ed., Cambridge University Press, Cambridge, 1997
 H. Goldstein, Classical Mechanics, 2nd ed., AddisonWesley, Reading, 1980
 B. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Springer, Berlin, 2003
 I. Herstein, Topics in Algebra, 2nd ed., Wiley, New York, 1975
 K. Hoffman and D. Kunze, Linear Algebra, 2nd ed., Prentice Hall, New York, 1971
 B. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, Cambridge, 1980
 V.S. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Springer, Berlin, 1984
 F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, Berlin, 1979
 Title
 Groups, Lie Groups, and Lie Algebras
 Book Title
 An Introduction to Tensors and Group Theory for Physicists
 Pages
 pp 87143
 Copyright
 2011
 DOI
 10.1007/9780817647155_4
 Print ISBN
 9780817647148
 Online ISBN
 9780817647155
 Publisher
 Birkhäuser Boston
 Copyright Holder
 Springer Science+Business Media, LLC
 Additional Links
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 eBook Packages
 Authors

 Nadir Jeevanjee ^{(1)}
 Author Affiliations

 1. Department of Physics, University of California, 366 LeConte Hall MC 7300, Berkeley, CA, 94720, USA
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