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. 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. Let G be a Lie group with Lie algebra g {displaystyle {mathfrak {g}}} , 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 g P {displaystyle {mathfrak {g}}_{P}} . Explicitly, elements of the adjoint bundle are equivalence classes of pairs for p ∈ P and x ∈ g {displaystyle {mathfrak {g}}} 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. 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 u ∈ G {displaystyle uin G} , and ~ l ∈ L {displaystyle lin L} , we have , where u ↦ d α u {displaystyle umapsto dalpha _{u}} is the adjoint representation of G , u ↦ α u {displaystyle umapsto alpha _{u}} is a homomorphism of G into A which is an automorphism group of G and α u {displaystyle alpha _{u}} is the mapping r ↦ u r u − 1 {displaystyle rmapsto uru^{-1}} of G into itself. H is a topological transformation group of L and obviously for every u in H, d α u : L ↦ L {displaystyle dalpha _{u}:Lmapsto L} 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 g i j : U i ∩ U j → H {displaystyle g_{ij}:U_{i}cap U_{j} ightarrow H} is assured where U i {displaystyle U_{i}} is an open covering for X. Then by the existence theorem there exists a Lie bundle ξ = ( E , P , X , L , H ) {displaystyle xi =(E,P,X,L,H)} with the continuous mapping Θ : ξ ⊕ ξ → ξ {displaystyle Theta :xi oplus xi ightarrow xi } inducing on each fibre the Lie bracket. Differential forms on M with values in a d P {displaystyle 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 a d P {displaystyle mathrm {ad} P} .