Page 154 - Curvature and Homology

136 IV. COMPACT LIE GROUPS

Let /3 be an element of Ap(T,*). Then, /3 is a left invariant p-form on G,
and so may be expressed in the form = B %... , ual I\ ... A map
where the coefficients are constants. Applying lemma 4.2.2 we obtain
the formula

It follows from lemma 4.2.4 that

An element /3 of the Grassman algebra of G is said to be L-invariant
or, simply, invariant if it is a zero of every derivation O(X), XE L, that is,
if B(X)#3 = 0 for every left invariant vector field X. Hence, an invariant
differential form is bi-invariant.
Proposition 4.2.t. An invariant form is a closed fonn.
This is an immediate consequence of lemma 4.2.4.
Remark: Note that the operatol? B(X) of 3.5 coincides with the
operator B(X) defined here on forms only.

4.3. Local geometry of a compact semi-simple Lie group

From (4.1.2) it is seen that the structure constants are the components
of a tensor on T, of type (1,2). A new tensor on T, is defined by the
components
gag = CaoP CP;
relative to the base Xa(u = 1, ..a, n). It follows from (4.1.6) and (4.1.7)
that this tensor is symmetric. It can be shown that a necessary and
sufficient condition for G to be semi-simple is that the rank of the matrix
(g@) is n. (A Lie group is said to be semi-simple if the fundamental
bilinear symmetric form-trace ad X ad Y is non-degenerate). Moreover,
since G is compact it can be shown that (g4) is positive definite.
The tensor defined by the equations (4.3.1) may now be used to raise
and lower indices and for this purpose we consider the inverse matrix
(fl. The structure constants have yet another symmetry property.
Indeed, if we multiply the identities (4.1.7) by Coba and contract we
find that the tensor
Cagy = gyo C*' (4.3.2)
is skew-symmetric.