Adjoint bundle
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 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 formula_1, 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 formula_4. Explicitly, elements of the adjoint bundle are equivalence classes of pairs ["p", "x"] for "p" ∈ "P" and "x" ∈ formula_1 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". Example. Let G be any Lie group with a closed sub group H and let L be the Lie algebra of G. Since G is a topological transformation group of L by the adjoint action of G ,that is , for every formula_7 , and ~formula_8 , we have , where formula_11 is the adjoint representation of G , formula_12 is a homomorphism of G into A which is an automorphism group of G and formula_13 is the mapping formula_14 of G into itself. H is a topological transformation group of L and obviously for every u in H, formula_15 is a Lie algebra automorphism. since H is a closed subgroup of a Lie group G, there is a locally trivial principal bundle over X=G/H having H as a structure group. So the existence of coordinate functions formula_16 is  assured where formula_17 is an open covering for X. Then by the existence theorem there exists a Lie bundle formula_18 with the continuous mapping formula_19 inducing on each fibre the Lie bracket. Properties. Differential forms on "M" with values in formula_20 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 formula_20. 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" × "G" where Ψ is the action of "G" on itself by conjugation. If formula_22 is the frame bundle of a vector bundle formula_23, then formula_24 has fibre the general linear group formula_25 (either real or complex, depending on formula_26) where formula_27. This structure group has Lie algebra consisting of all formula_28 matrices formula_29, and these can be thought of as the endomorphisms of the vector bundle formula_26. Indeed there is a natural isomorphism  formula_31.