Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics

1. Front Matter

   Pages i-ix

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2. Lie Groups and Algebras

   1. Front Matter

      Pages 1-1

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   2. Lie Groups

      • D. H. Sattinger, O. L. Weaver

      Pages 3-20

   3. Lie Algebras

      • D. H. Sattinger, O. L. Weaver

      Pages 21-31

   4. Lie Groups and Algebras: Matrix Approach

      • D. H. Sattinger, O. L. Weaver

      Pages 32-40

   5. Applications to Physics and Vice Versa

      • D. H. Sattinger, O. L. Weaver

      Pages 41-54

3. Differential Geometry and Lie Groups

   1. Front Matter

      Pages 55-55

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   2. Calculus on Manifolds

      • D. H. Sattinger, O. L. Weaver

      Pages 57-75

   3. Symmetry Groups of Differential Equations

      • D. H. Sattinger, O. L. Weaver

      Pages 76-88

   4. Invariant Forms on Lie Groups

      • D. H. Sattinger, O. L. Weaver

      Pages 89-108

   5. Lie Groups and Algebras: Differential Geometric Approach

      • D. H. Sattinger, O. L. Weaver

      Pages 109-115

4. Algebraic Theory

   1. Front Matter

      Pages 117-117

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   2. General Structure of Lie Algebras

      • D. H. Sattinger, O. L. Weaver

      Pages 119-129

   3. Structure of Semi-Simple Lie Algebras

      • D. H. Sattinger, O. L. Weaver

      Pages 130-152

   4. Real Forms

      • D. H. Sattinger, O. L. Weaver

      Pages 153-159

5. Representation Theory

   1. Front Matter

      Pages 161-161

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   2. Representation Theory

      • D. H. Sattinger, O. L. Weaver

      Pages 163-180

   3. Spinor Representations

      • D. H. Sattinger, O. L. Weaver

      Pages 181-186

6. Applications

   1. Front Matter

      Pages 187-187

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   2. Applications

      • D. H. Sattinger, O. L. Weaver

      Pages 189-207

This book is intended as an introductory text on the subject of Lie groups and algebras and their role in various fields of mathematics and physics. It is written by and for researchers who are primarily analysts or physicists, not algebraists or geometers. Not that we have eschewed the algebraic and geo­ metric developments. But we wanted to present them in a concrete way and to show how the subject interacted with physics, geometry, and mechanics. These interactions are, of course, manifold; we have discussed many of them here-in particular, Riemannian geometry, elementary particle physics, sym­ metries of differential equations, completely integrable Hamiltonian systems, and spontaneous symmetry breaking. Much ofthe material we have treated is standard and widely available; but we have tried to steer a course between the descriptive approach such as found in Gilmore and Wybourne, and the abstract mathematical approach of Helgason or Jacobson. Gilmore and Wybourne address themselvesto the physics community whereas Helgason and Jacobson address themselves to the mathematical community. This book is an attempt to synthesize the two points of view and address both audiences simultaneously. We wanted to present the subject in a way which is at once intuitive, geometric, applications­ oriented, mathematically rigorous, and accessible to students and researchers without an extensive background in physics, algebra, or geometry.

• Algebra
• Algebras
• Applications
• Geometry
• Groups
• Mechanics
• Physics
• Representation theory
• Symmetry group

• School of Mathematics, University of Minnesota, Minneapolis, USA

  D. H. Sattinger

• Department of Physics, Kansas State University, Manhattan, USA

  O. L. Weaver

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