John Stillwell - Naive Lie Theory.pdf - Index of

5
The tangent space
PREVIEW
The miracle of Lie theory is that a curved object, a Lie group G, can be
almost completely captured by a flat one, the tangent space T 1 (G) of G at
the identity. The tangent space of G at the identity consists of the tangent
vectors to smooth paths in G where they pass through 1. ApathA(t) in G
is called smooth if its derivative A ′ (t) exists, and if A(0)=1 we call A ′ (0)
the tangent or velocity vector of A(t) at 1. T 1 (G) consists of the velocity
vectors of all smooth paths through 1.
It is quite easy to determine the form of the matrix A ′ (0) for a smooth
path A(t) through 1 in any of the classical groups, that is, the generalized
rotation groups of Chapter 3 and the general and special linear groups,
GL(n,C) and SL(n,C), we will meet in Section 5.6. For example, any
tangent vector of SO(n) at 1 is an n × n real skew-symmetric matrix—a
matrix X such that X + X T = 0. The problem is to find smooth paths in the
first place. It is here that the exponential function comes to our rescue.
As we saw in Section 4.5, e X is defined for any n × n matrix X by the
infinite series used to define e x for any real or complex number x. Thismatrix
exponential function provides a smooth path with prescribed tangent
vector at 1, namely the path A(t)=e tX ,forwhichA ′ (0)=X. In particular,
it turns out that if X is skew-symmetric then e tX ∈ SO(n) for any real t, so
the potential tangent vectors to SO(n) are the actual tangent vectors.
In this way we find that T 1 (SO(n)) = {X ∈ M n (R) : X +X T = 0},where
M n (R) is the space of n × n real matrices. The exponential function similarly
enables us to find the tangent spaces of all the classical groups: O(n),
SO(n), U(n), SU(n), Sp(n), GL(n,C), andSL(n,C).
J. Stillwell, Naive Lie Theory, DOI: 10.1007/978-0-387-78214-0 5, 93
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