Matrix Groups for UndergraduatesMatrix groups are a beautiful subject and are central to many fields in mathematics and physics. They touch upon an enormous spectrum within the mathematical arena. This textbook brings them into the undergraduate curriculum. It is excellent for a one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups. Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, and maximal tori. The volume is suitable for graduate students and researchers interested in group theory. |
Contents
| 1 | |
| 5 | |
Chapter 2 All matrix groups are real matrix groups | 23 |
Chapter 3 The orthogonal groups | 33 |
Chapter 4 The topology of matrix groups | 51 |
Chapter 5 Lie algebras | 67 |
Chapter 6 Matrix exponentiation | 79 |
Chapter 7 Matrix groups are manifolds | 93 |
Chapter 8 The Lie bracket | 113 |
Chapter 9 Maximal tori | 135 |
| 163 | |
| 165 | |
Back Cover | 169 |
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Common terms and phrases
abelian angle bijective boundary points called Chapter clopen closed compact matrix groups conjugation contains converges defined Definition denoted derivative describe det(A dfp(v diagonal diffeomorphism differentiable path dimension element of SO(3 elements of G equals Equation Euclidean space example exponential finite function f G₁ geometric GLn(K identity inner product integer integral curve inverse function theorem isometry isomorphic Lemma Let G C Lie algebra Lie bracket Lie group linear groups linear map linear transformation manifold matrix groups means Mn(C Mn(K multiplication neighborhood norm Notice On(K orthonormal basis path y(t path-connected power series Proof Prove Proposition quaternionic real numbers restriction rotation scalar sequence skew-field SLn(K smoothly isomorphic SO(n Sp(n standard maximal torus SU(n subgroup subset subspace symmetry group tangent space Theorem 7.1 topology unit-length v₁ vector field vector space verify