Lie Groups: An Introduction Through Linear Groups
This book is an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level. It covers the essentials of the subject starting from basic undergraduate mathematics. The correspondence between linear Lie groups and Lie algebras is developed in its local and global aspects. The classical groups are analyzed in detail, first with elementary matrix methods, then with the help of the structural tools typical of the theory of semisimple groups, such as Cartan subgroups, root, weights and reflections. The fundamental groups of the classical groups are worked out as an application of these methods. Manifolds are introduced when needed, in connection with homogeneous spaces, and the elements of differential and integral calculus on manifolds are presented, with special emphasis on integration on groups and homogeneous spaces. Representation theory starts from first principles, such as Schur's lemma and its consequences, and proceeds from there to the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups.
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action acts analytic assume basis called character classical groups closed coefficients compact complex components condition conjugate connected consider consists contains continuous converges coordinate system Corollary corresponding covering defined definition denoted determinant diagonal differential domain eigenvalues element entries equal equation equivalent Example exp(TX Explain exponential fact finite fixed follows formula function given gives GL(n group G hence Hermitian homomorphism identified identity implies integral invariant inverse irreducible isomorphic Lemma Let G Lie algebra linear group linear transformation manifold matrix means multiplicity neighborhood normal notation operation orthogonal polynomial positive power series problem Proof Proposition prove relation remains Remark representation represented respect root satisfies says scalar sense Show subgroup subset subspace Suggestion Suppose Table tangent vector Theorem unique vector space weight write written