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The existence of these little—known elementary proofs convinced me that a naive approach to Lie theory is possible and desirable． The aim of this book is to carry it outdeveloping the central concepts and results of Lie theory by the simplest possible methods， mainly from single—variable calculus and linear algebra． Familiarity with elementary group theory is also desirable， but I provide a crash course on the basics of group theory in Sections 2．1 and 2．2．

1 Geometry of complex numbers and quaternions 1.1 Rotations of the plane 1.2 Matrix representation of complex numbers 1.3 Quaternions 1.4 Consequences of multiplicative absolute value 1.5 Quaternion representation of space rotations 1.6 Discussion2 Groups 2.1 Crash course on groups 2.2 Crash course on homomorphisms 2.3 The groups SU(2) and SO(3) 2.4 Isometrics of R'' and reflections 2.5 Rotations of R4 and pairs of quaternions 2.6 Direct products of groups 2.7 The map from SU(2)SU(2) to SO(4) 2.8 Discussion3 Generalized rotation groups 3.1 Rotations as orthogonal transformations 3.2 The orthogonai and special orthogonal groups 3.3 The unitary groups 3.4 The symplectic groups 3.5 Maximal tori and centers 3.6 Maximal tori in SO(n), U(n), SU(n), Sp(n) 3.7 Centers of SO(n), U(n), SU(n), Sp(n) 3.8 Connectcdness and discreteness 3.9 Discussion4 The exponential map 4.1 The exponential map onto SO(2) 4.2 The exponential map onto SU(2) 4.3 The tangent space of SU(2) 4.4 The Lie algebra su(2) of SU(2) 4.5 The exponential of a square matrix 4.6 The affine group of the line 4.7 Discussion5 The tangent space 5.1 Tangent vectors of O(n), U(n), Sp(n) 5.2 The tangent space of SO(n) 5.3 The tangent space of U(n), SU(n), Sp(n) 5.4 Algebraic properties of the tangent space 5.5 Dimension of Lie algebras 5.6 Complexification 5.7 Quaternion Lie algebras 5.8 Discussion6 Structure of Lie algebras 6.1 Normal subgroups and ideals 6.2 Ideals and homomorphisms 6.3 Classical non-simple Lie algebras 6.4 Simplicity of (n,C) and su(n) 6.5 Simplicity of o(n) for n > 4 6.6 Simplicity of p(n) 6.7 Discussion7 The matrix logarithm 7.1 Logarithm and exponential 7.2 The exp function on the tangent space 7.3 Limit properties of log and exp 7.4 The log function into the tangentspace 7.5 SO(n), SU(n), and Sp(n) revisited 7.6 The Campbell-Baker-Hausdorff theorem 7.7 Eichler's proof of Campbell-Baker-Hausdorff 7,8 Discussion8 Topology 8.1 Open and closed sets in Euclidean space 8.2 Closed matrix groups 8.3 Continuous functions 8.4 Compact sets 8.5 Continuous functions and compactness 8.6 Paths and path-connectedness 8.7 Simple connectedness 8.8 Discussion9 Simply connected Lie groups 9.1 Three groups with tangent space R 9.2 Three groups with the cross-product Lie algebra 9.3 Lie homomorphisms 9.4 Uniform continuity of paths and deformations 9.5 Deforming a path in a sequence of small steps 9.6 Lifting a Lie algebra homomorphism 9.7 DiscussionBibliographyIndex
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