High Quality Content by WIKIPEDIA articles! In mathematics, an adjoint bundle 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 Let G be a Lie group with Lie algebra , and let P be a principal G-bundle over a smooth manifold M. Let be the adjoint representation of G. The adjoint bundle of P is the associated bundle, The adjoint bundle is also commonly denoted by . Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, x] for p P and x such that 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.

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