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
1
http://dx.doi.org/10.1090/stml/022/01
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