John Stillwell - Naive Lie Theory.pdf - Index of

1.1 Rotations of the plane 3
Thus we can represent the geometric operation of combining successive
rotations by the algebraic operation of multiplying matrices. The main
aim of this book is to generalize this idea, that is, to study groups of linear
transformations by representing them as matrix groups. For the moment
one can view a matrix group as a set of matrices that includes, along with
any two members A and B, the matrices AB, A −1 ,andB −1 . Later (in Section
7.2) we impose an extra condition that ensures “smoothness” of matrix
groups, but the precise meaning of smoothness need not be considered yet.
For those who cannot wait to see a definition, we give one in the subsection
below—but be warned that its meaning will not become completely clear
until Chapters 7 and 8.
The matrices R θ , for all angles θ, form a group called the special orthogonal
group SO(2). The reason for calling rotations “orthogonal transformations”
will emerge in Chapter 3, where we generalize the idea of
rotation to the n-dimensional space R n and define a group SO(n) for each
dimension n. In this chapter we are concerned mainly with the groups
SO(2) and SO(3), which are typical in some ways, but also exceptional
in having an alternative description in terms of higher-dimensional “numbers.”
Each rotation R θ of R 2 can be represented by the complex number
z θ = cosθ + isinθ
because if we multiply an arbitrary point (x,y)=x + iy by z θ we get
z θ (x + iy)=(cosθ + isinθ)(x + iy)
= xcos θ − ysinθ + i(xsin θ + ycosθ)
=(xcos θ − ysinθ, xsinθ + ycosθ),
which is the result of rotating (x,y) through angle θ. Moreover, the ordinary
product z θ z ϕ represents the result of combining R θ and R ϕ .
Rotations of R 3 and R 4 can be represented, in a slightly more complicated
way, by four-dimensional “numbers” called quaternions. We introduce
quaternions in Section 1.3 via certain 2 × 2 complex matrices, and to
pave the way for them we first investigate the relation between complex
numbers and 2 × 2 real matrices in Section 1.2.
What is a Lie group?
The most general definition of a Lie group G is a group that is also a smooth
manifold. That is, the group “product” and “inverse” operations are smooth