Adjoint bundle

In mathematics, an adjoint bundle ^{[1] [2]} is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory.

Formal definition

Let G be a Lie group with Lie algebra $\mathfrak g$, and let P be a principal G-bundle over a smooth manifold M. Let

$\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g)$

be the adjoint representation of G. The adjoint bundle of P is the associated bundle

$\mathrm{ad} P = P\times^{\mathrm{Ad}}\mathfrak g$

The adjoint bundle is also commonly denoted by $\mathfrak g_P$. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, x] for p ∈ P and x ∈ $\mathfrak g$ such that

$[p\cdot g,x] = [p,\mathrm{Ad}_{g^{-1}}(x)]$

for all g ∈ G. Since the structure group of the adjoint bundle consists of Lie algebra automorphisms, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over M.

Properties

Differential forms on M with values in $\mathrm{ad} P$ are in one-to-one correspondence with horizontal, G-equivariant Lie algebra-valued forms on P. A prime example is the curvature of any connection on P which may be regarded as a 2-form on M with values in $\mathrm{ad} P$.

The space of sections of the adjoint bundle is naturally an (infinite-dimensional) Lie algebra. It may be regarded as the Lie algebra of the infinite-dimensional Lie group of gauge transformations of P which can be thought of as sections of the bundle P ×^Ψ G where Ψ is the action of G on itself by conjugation.

Notes

1. J. Janyška (2006). "Higher order Utiyama-like theorem". Reports on Mathematical Physics 58: 93–118. doi:10.1016/s0034-4877(06)80042-x. [cf. page 96]
2. Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag page 161 and page 400

References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag page 161 and page 400