Gravity and Strings

Appendix A
Lie groups, symmetric spaces, and Yang–Mills fields
In this appendix we review some basic definitions and properties of Lie groups and algebras
and their use in the construction of homogeneous and symmetric spaces and in field
theory. Rigorous definitions and proofs can be found, for instance, in [527, 630], and in the
physicist-oriented [89, 221 (Volume I), 252, 267 (Volume II), 454].
A.1 Generalities
A Lie group G of dimension n is both a group and a differential manifold of dimension n:
the points of the manifold are the elements of the group and the maps
G × G → G,
(g1, g2) → g1g2,
G → G,
g → g−1 ,
(A.1)
are differentiable. For each element g ∈ G there are also two natural diffeomorphisms: left
and right translations by g, denoted by L g and Rg respectively, and defined by
L g :G→ G,
h → L g(h) ≡ gh,
Rg :G→ G,
h → Rg(h) ≡ hg.
(A.2)
The identity e is a naturally distinguished point. The tangent space at the identity T (1,0)
e
is the Lie algebra g of G. This name will be justified later. Each element v(e) ∈ g can be
extended to a vector field v(g) defined at all points g ∈ Gbytaking the push-forward of the
left- or the right-translation diffeomorphisms
vL(g) ≡ L g ∗ v(e), vR(g) ≡ Rg ∗ v(e). (A.3)
Sometimes we use the following notation for them:
L g ∗ v ≡ gv, Rg ∗ v ≡ vg. (A.4)
The vector fields defined in this way have the property of being, respectively, left- and
right-invariant, i.e. they satisfy
L g ∗ vL(h) = vL(gh), Rg ∗ vR(h) = vR(hg). (A.5)
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