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 an algebra bundle. Adjoint bundles has important applications in the theory of connections as well as in gauge theory.

Contents

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 AdP are in one-to-one corresponding 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 AdP.

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.  [cf. page 96]
  2. ^ Kolář, Ivan; Michor, Peter; Slovák, Jan (1993) (PDF), Natural operators in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf  page 161 and page 400

References