John Stillwell - Naive Lie Theory.pdf - Index of

6.1 Normal subgroups and ideals 117
6.1 Normal subgroups and ideals
In Chapter 5 we found the tangent spaces of the classical Lie groups: the
classical Lie algebras. In this chapter we use the tangent spaces to find
candidates for simplicity among the classical Lie groups G. Wedosoby
finding substructures of the tangent space g that are tangent spaces of the
normal subgroups of G. These are the ideals, 4 defined as follows.
Definition. An ideal h of a Lie algebra g is a subspace of g closed under
Lie brackets with arbitrary members of g. Thatis,ifY ∈ h and X ∈ g then
[X,Y ] ∈ h.
Then the relationship between normal subgroups and ideals is given by
the following theorem.
Tangent space of a normal subgroup. If H is a normal subgroup of a
matrix Lie group G, then T 1 (H) is an ideal of the Lie algebra T 1 (G).
Proof. T 1 (H) is a vector space, like any tangent space, and it is a subspace
of T 1 (G) because any tangent to H at 1 is a tangent to G at 1. Thus it
remains to show that T 1 (H) is closed under Lie brackets with members of
T 1 (G). To do this we use the property of a normal subgroup that B ∈ H and
A ∈ G implies ABA −1 ∈ H.
It follows that A(s)B(t)A(s) −1 isasmoothpathinH for any smooth
paths A(s) in G and B(t) in H. As usual, we suppose A(0)=1 = B(0), so
A ′ (0)=X ∈ T 1 (G) and B ′ (0)=Y ∈ T 1 (H). Ifwelet
then it follows as in Section 5.4 that
C s (t)=A(s)B(t)A(s) −1 ,
D(s)=C ′ s(0)=A(s)YA(s) −1
4 This terminology comes from algebraic number theory, via ring theory. In the 1840s,
Kummer introduced some objects he called “ideal numbers” and “ideal primes” in order to
restore unique prime factorization in certain systems of algebraic numbers where ordinary
prime factorization is not unique. Kummer’s “ideal numbers” did not have a clear meaning
at first, but in 1871 Dedekind gave them a concrete interpretation as certain sets of numbers
closed under sums, and closed under products with all numbers in the system. In the 1920s,
Emmy Noether carried the concept of ideal to general ring theory. Roughly speaking, a
ring is a set of objects with sum and product operations. The sum operation satisfies the
usual properties of sum (commutative, associative, etc.) but the product is required only
to “distribute” over sum: a(b + c) =ab + ac. A Lie algebra is a ring in this general sense
(with the Lie bracket as the “product” operation), so Lie algebra ideals are included in the
general concept of ideal.