Gravity and Strings

592 Appendix A
Similarly, we can define left- and right-invariant differential forms ω and η of any rank
using the pull-backs associated with the left- and right-translation diffeomorphisms:
L g ∗ ω(h) = ω(gh), Rg ∗ η(h) = η(hg). (A.6)
It is customary to work with left-invariant vector fields and 1-forms. We can always
construct a basis of left-invariant vector fields {eI (g)} using the above procedure starting
with a basis of vector fields at the identity {eI (e)} and a dual basis of left-invariant 1-forms
{e I (g)}. The Lie bracket of two left-invariant vector fields is another left-invariant vector
field that we can write as a linear combination of elements of the {eI (g)} basis. 1 Thus,
[eI , eJ ] =−f IJ K eK , (A.7)
where f IJ K =−f JI K =+( f IJ K ) ∗ are the structure constants. The Jacobi identity
Eq. (1.14) implies
f J[K I fLM] K = 0. (A.8)
The vector fields at the identity are thus an n-dimensional vector space endowed with an
antisymmetric, bilinear (but non-associative) product [·, ·] (the Lie bracket of the associated
left-invariant vector fields) that satisfies the Jacobi identity; that is, by definition, a Lie
algebra, which justifies our definition of g. Wewill denote a basis of g by {TI } and, by
convention,
eI (e) ≡−TI , ⇒ [TI , TJ ] = f IJ K TK . (A.9)
The dual left-invariant 1-forms satisfy the Maurer–Cartan equations
de I = 1
2 f JK I e J ∧ e K , (A.10)
and d 2 e I = 0isequivalent to the Jacobi identity.
The exponential map provides a local parametrization of G in a neighborhood of the
identity with coordinates σ I :
g(σ ) = e σ I TI . (A.11)
If the group is a connected and compact manifold, any of its elements can be expressed in
this way. With this parametrization it is easy to construct a basis of left-invariant 2 1-forms
by expanding the Maurer–Cartan 1-form V ,
V =−g −1 dg = e I TI , (A.12)
in terms of which the Maurer–Cartan equations are dV − V ∧ V = 0.
For matrix groups the generators TI are just matrices and the Lie bracket is just the
standard commutator. The left and right translations are just matrix multiplications from
the left, gT, orfrom the right, Tg.Wecan take different sets of matrices of different dimensions
or operators that satisfy the same commutation relations and provide different
1 The same is true for right-invariant vector fields. On the other hand, the Lie bracket of any left-invariant
vector field with any right-invariant vector field vanishes.
2 Right-invariant 1-forms are provided by dgg −1 .