Chapter 1

Lie Groups

1. An example of a Lie group

A Lie group is a set that has both a manifold and a group struc-

ture, which are compatible. So, we will begin this discussion with an

example that exhibits these two properties.

Let MnR be the set of all n x n real matrices. We associate to

the matrix A = (a^) the point in the Euclidean space Rn whose

coordinates are an , ai2,..., a,nn- Hence, topologically MnR is simply

the Euclidean n2 space. Next we define the general linear group GLnR

to be the group (under usual matrix multiplication) of all n x n real

matrices A — (a^) with determinant del A ^ 0. Since detA is a

polynomial of degree n in the coordinates, it is a smooth function on

MnR. Furthermore, since the set R \ {0} forms an open set in R,

and since the inverse image of an open set under a continuous map is

open, the set GLnR is an open subset of MnR. Hence, topologically

GLnR is an open subset of a Euclidean space, and as such is an

n2-

dimensional manifold, as will be seen later on. This takes care of the

manifold and the group structure structure of GLnR. Let us now see

how they interact.

Since (ab)ij = ^ a ^ f r ^ , the product matrix AB has coordinates

that are smooth functions of the coordinates of A and B. Also, from

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http://dx.doi.org/10.1090/stml/022/01