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Naive Lie Theory

John Stillwell
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Allen Stenger
, on

This is a beautifully clear exposition of the main points of Lie theory, aimed at undergraduates who have studied calculus and linear algebra. The book is modeled after (and named in homage to) Halmos's Naive Set Theory. The key simplification is that it deals only with matrix groups.

The book is well equipped with examples, and it always ties the matrix groups back to concrete examples, especially the complex numbers and the quaternions. The book has a very strong geometric flavor, both in the use of rotation groups and in the connection between Lie algebras and Lie groups.

The book's most conspicuous weakness is that it treats Lie theory in isolation. It doesn't give any clue where the subject came from or what it is used for today. Each chapter ends with a very informative "Discussion" section, but what is discussed is the portions of Lie theory that we couldn't get to in this book. There's no mention of differential equations, Klein's Erlangen Program, or representation theory.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

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 Discussion

2 Groups
2.1 Crash course on groups
2.2 Crash course on homomorphisms
2.3 The groups SU(2) and SO(3)
2.4 lsometries of Rn and reflections
2.5 Rotations of R4 and pairs of quaternions
2.6 Direct products of groups
2.7 The map from SU(2) x SU(2) to SO(4)
2.8 Discussion

3 Generalized rotation groups
3.1 Rotations as orthogonal transformations
3.2 The orthogonal 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 Connectedness and discreteness
3.9 Discussion

4 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 Discussion

5 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 Discussion

6 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 sl(n, C) and su(n)
6.5 Simplicity of so(n) for n > 4
6.6 Simplicity of sp(n)
6.7 Discussion

7 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 tangent space
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 Discussion

8 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 Discussion

9 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 Discussion