By explicit evaluation as in (3.3.37) we can also show that any left-invariant vector field commutes with any right-invariant one and vice-versa. Indeed we find:

The interpretation of these relations is indeed very simple. The left-invariant vector fields happen to be the infinitesimal generators of the right-translations while the right-invariant ones generate the left translations. Hence the vanishing of the above commutators just amounts to say that the left-invariant are indeed invariant under left translations while the right invariant are insensitive to right translations.

3.3.1.2 Maurer-Cartan Forms on Lie Group Manifolds

Let us now consider left-invariant (respectively right-invariant) one-forms on the group manifold. They were defined in (3.3.15). Starting from the construction of the left- (right-)invariant vector fields it is very easy to construct an independent set of n differential forms with the invariance property (3.3.15) that are in one-to-one correspondence with the generators of the Lie algebra G. Let us consider the explicit form of the T(L/R)A as first order differential operators:

According to the already introduced convention αμ are the group parameters and the square matrix:

whose entries are functions of the parameters can be calculated starting from the constructive algorithm encoded in (3.3.33). In terms of the inverse of the above

(l/r)

matrix denoted ΣαA(α) and such that:

116 3 Connections and Metrics

which by construction satisfy the relations:

From (3.3.46) follows that all the forms σL/RA are left- (respectively right-) invariant. Indeed we have:

which implies Lγ σ(L)A = σ A since both forms have the same values on a basis of sections of the tangent bundle as it is provided by the set of left-invariant vector fields T(L)B . A completely identical proof holds obviously true for the right-invariant one-forms σ(R)A defined by the same construction.

We come therefore to the conclusion that on each Lie group manifold G we can construct both a left-invariant σ(L) and a right-invariant σ(R) Lie algebra valued one-form defined as it follows:

where, just as before, {tA} denotes a basis of generators of the abstract Lie algebra G. One may wonder how the Lie algebra forms σ(L/R) could be constructed directly without going through the previous construction of the left- (right-)invariant vector fields. The answer is very simple. Let g(α) G be a running element of the Lie group parameterized by the parameters α which constitute a coordinate patch for

the corresponding group manifold and consider the one-forms:

It is immediate to check that such one-forms are left- (respectively right-) invariant. For instance we have:

What might not be immediately evident to the reader is why the left- (right-)invariant one-forms θL/R introduced in (3.3.50) should be Lie algebra valued. The answer is actually very simple. From his basic courses in Lie group theory the reader should recall that the relation between Lie algebras and Lie groups is provided by the exponential map: every element of a Lie group which lies in the branch connected to the identity can be represented as the exponential of a suitable Lie algebra element:

The Lie algebra element actually singles out an entire one parameter subgroup:

namely the left-invariant one-form associated with this parameter subgroup lies in the Lie algebra. This result extends to any group element, so that indeed the previously constructed Lie algebra valued left- (right-)invariant one forms σL/R and the theta forms defined in (3.3.50) are just the very same objects:

All the above statements become much clearer when we consider classical groups whose elements are just matrices subject to some defining algebraic condition. Consider for instance the rotation group in N-dimensions SO(N ). All elements of this group are orthogonal N × N matrices:

The elements of the orthogonal Lie algebra so(N) are instead antisymmetric matrices:

which proves that the left-invariant one-form is indeed Lie algebra valued.

3.3.1.3 Maurer Cartan Equations

It is now of the utmost interest to consider the following identity which follows immediately from the definitions (3.3.50) and (3.3.55) of the left- (right-)invariant one-forms:

To prove the above statement it is sufficient to observe that g−1 dg = −dg−1 g. The reason why we introduced a special name F[σ ] for the Lie algebra valued

two-form dσ + σ σ , which turns out to be zero in the case of left- (right-)invariant one-forms σ , is that precisely this combination will play a fundamental role in the theory of connections, representing their curvature. Equation (3.3.59) are named the Maurer Cartan equations of the considered Lie algebra G and they translate into the statement that left- (right-)invariant one-forms have vanishing curvature.