Page 161 - Curvature and Homology

fields in En(n-l)J2. We denote by TA,, (r < s, A = n + 1, ---, - 1)/2),
n(n
(n(n - 1)/2) - n orthonormal vectors in En(n-1)/2 orthagonal to the
vectors Tj,. Hence,

for i < j, k < I (S(il)(kl) = 1 if i = k, j = I and vanishes otherwise),
and so
n(n-l)/%
8 2 (2 Tij. + 2 (2 Tij~ fij)l = 2
s=l id A=n+l ici i <j

We may therefore conclude that

This completes the proof.
A straightforward application of theorem 3.2.4 shows that b2(G) = 0
by virtue of the lemma and formula (4.5.2).

Theorem 4.5.1. b3(G) 5 1.
For, the torsion tensor (4.3.1 1) defines a harmonic 3-form on G.
For more precise information on b3(G) the reader is referred to
(1V.B).

4.6. Determination of the betti numbers of the simple Lie groups

We have seen that a p-form on a compact semi-simple Lie group G
is harmonic, if and only if, it is invariant and therefore, in order to
find the harmonic forms /3 on G, it is sufficient to solve the equations
(4.4.2) for the coefficients Ba ap of 8.
A semi-simple group is the direct product of a finite number of simple
non-commutative groups. (A Lie group is said to be simple if there are
no non-trivial normal subgroups). Hence, in order to give a complete
classification of compact semi-simple Lie groups it is sufficient to
classify the compact simple Lie groups. There are four main classes
of simple Lie groups:
1) The group A, of unitary transformations in (1 + 1)-space of
determinant + 1 ;