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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abelian action acts associated basis called Chapter closed subgroup commutative compact connected complex components conjugation consider contains continuous corresponding curve defined Definition derivative determine differentiable dimension discussed easy eigenvalues elements Equation Example exercise fact finite dimensional function given gives GLn(k GLn(R hence homogeneous space homomorphism identify identity implies important inner product inverse isomorphism k-algebra Let G Lie algebra Lie group linear manifold matrix group matrix subgroup maximal torus Mn(k Mn(R multiplication n x n norm Notice null vector obtain orthogonal orthogonal matrix particular path connected Pin(n positive Proof properties Proposition provides quaternions quotient R-linear transformation Recall respect result root system satisfies sequence simple skew smooth SO(n solution space standard subspace Suppose surjective symmetric symmetric matrix Theorem theory unique unit vector space write