John Stillwell - Naive Lie Theory.pdf - Index of

120 6 Structure of Lie algebras
An example of a matrix Lie group with a nontrivial normal subgroup is U(n).
We determined the appropriate tangent spaces in Section 5.3.
6.1.3 Show that SU(n) is a normal subgroup of U(n) by describing it as the kernel
of a homomorphism.
6.1.4 Show that T 1 (SU(n)) is an ideal of T 1 (U(n)) by checking that it has the
required closure properties.
6.2 Ideals and homomorphisms
If we restrict attention to matrix Lie groups (as we generally do in this
book) then we cannot assume that every normal subgroup H of a Lie group
G is the kernel of a matrix group homomorphism G → G/H. The problem
is that the quotient G/H of matrix groups is not necessarily a matrix group.
This is why we derived the relationship between normal subgroups and
ideals without reference to homomorphisms.
Nevertheless, some important normal subgroups are kernels of matrix
Lie group homomorphisms. One such homomorphism is the determinant
map G → C × ,whereC × denotes the group of nonzero complex numbers
(or 1 × 1 nonzero complex matrices) under multiplication. Also, any ideal
is the kernel of a Lie algebra homomorphism—defined to be a map of
Lie algebras that preserves sums, scalar multiples, and the Lie bracket—
because in fact any Lie algebra is isomorphic to a matrix Lie algebra.
An important Lie algebra homomorphism is the trace map,
Tr(A)= sum of diagonal elements of A,
for real or complex matrices A. We verify that Tr is a Lie algebra homomorphism
in the next section.
The general theorem about kernels is the following.
Kernel of a Lie algebra homomorphism. If ϕ : g → g ′ is a Lie algebra
homomorphism, and
is its kernel, then h is an ideal of g.
h = {X ∈ g : ϕ(X)=0}