Ebook: Matrix Groups: An Introduction to Lie Group Theory

Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.
The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.
Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.

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Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.
The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.
Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.

---

Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.
The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.
Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.

Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Real and Complex Matrix Groups....Pages 3-43
Exponentials, Differential Equations and One-parameter Subgroups....Pages 45-65
Tangent Spaces and Lie Algebras....Pages 67-97
Algebras, Quaternions and Quaternionic Symplectic Groups....Pages 99-128
Clifford Algebras and Spinor Groups....Pages 129-156
Lorentz Groups....Pages 157-178
Front Matter....Pages 179-179
Lie Groups....Pages 181-209
Homogeneous Spaces....Pages 211-233
Connectivity of Matrix Groups....Pages 235-247
Front Matter....Pages 249-249
Maximal Tori in Compact Connected Lie Groups....Pages 251-265
Semi-simple Factorisation....Pages 267-288
Roots Systems, Weyl Groups and Dynkin Diagrams....Pages 289-302
Back Matter....Pages 303-330

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Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.
The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.
Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.

Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Real and Complex Matrix Groups....Pages 3-43
Exponentials, Differential Equations and One-parameter Subgroups....Pages 45-65
Tangent Spaces and Lie Algebras....Pages 67-97
Algebras, Quaternions and Quaternionic Symplectic Groups....Pages 99-128
Clifford Algebras and Spinor Groups....Pages 129-156
Lorentz Groups....Pages 157-178
Front Matter....Pages 179-179
Lie Groups....Pages 181-209
Homogeneous Spaces....Pages 211-233
Connectivity of Matrix Groups....Pages 235-247
Front Matter....Pages 249-249
Maximal Tori in Compact Connected Lie Groups....Pages 251-265
Semi-simple Factorisation....Pages 267-288
Roots Systems, Weyl Groups and Dynkin Diagrams....Pages 289-302
Back Matter....Pages 303-330
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