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
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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