John Stillwell - Naive Lie Theory.pdf - Index of

7.3 Limit properties of log and exp 145
7.2.4 Suppose that, for each A in some neighborhood N of 1 in G, thereisa
smooth function A(t), with values in G, such that A(1/n)=A 1/n for n =
1,2,3, .... ShowthatA ′ (0)=logA,sologA ∈ T 1 (G).
7.2.5 Suppose, conversely, that log maps some neighborhood N of 1 in G into
T 1 (G). Explain why we can assume that N is mapped by log onto an
ε-ball N ε (0) in T 1 (G).
7.2.6 Taking N as in Exercise 7.2.4, and A ∈ N , show that t logA ∈ T 1 (G) for
all t ∈ [0,1], and deduce that A 1/n exists for n = 1,2,3, ....
7.3 Limit properties of log and exp
In 1929, von Neumann created a new approach to Lie theory by confining
attention to matrix Lie groups. Even though the most familiar Lie
groups are matrix groups (and, in fact, the first nonmatrix examples were
not discovered until the 1930s), Lie theory began as the study of general
“continuous” groups and von Neumann’s approach was a radical simplification.
In particular, von Neumann defined “tangents” prior to the concept
of differentiability—going back to the idea that a tangent vector is the limit
of a sequence of “chord” vectors—as one sees tangents in a first calculus
course (Figure 7.1).
A 1
A 2
A 3
P
Figure 7.1: The tangent as the limit of a sequence.
Definition. X is a sequential tangent vector to G at 1 if there is a sequence
〈A m 〉 of members of G, and a sequence 〈α m 〉 of real numbers, such that
A m → 1 and (A m − 1)/α m → X as m → ∞.