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 adjoint action Affn(k automorphism Chapter Clifford algebras closed subgroup closed subset commutative compact connected Lie conjugation connected Lie group defined diffeomorphism differentiable curve differential equation dimension division algebra Dynkin diagrams eigenvalues eigenvector elements Example exponential finite dimensional GLn(C GLn(k GLn(R group G group homomorphism hence homogeneous space inner product inverse isometry isomorphism k-algebra k-Lie k-vector Lemma Let G Lie algebra Lie homomorphism Lie subgroup linear Lor(n manifold matrix group matrix subgroup maximal torus metric space Mn(C Mn(k Mn(R multiplication n x n non-trivial non-zero norm Notice obtain open subset orthogonal orthonormal basis path connected Pin(n polynomial Proposition quaternions quotient R-Lie R-linear transformation real number result root system satisfies Section skew symmetric SLn(k SLn(R smooth SO(n Sp(l Sp(n spinor groups SU(n subalgebra subspace surjective tangent space Theorem topological space upper triangular vector space Weyl group