Adjoint bundle explained

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

ak g , and let P be a principal G-bundle over a smooth manifold M. LetAd : G\to Aut ( ak g)\sub GL ( ak g) be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundlead P = P x _{Ad} ak g The adjoint bundle is also commonly denoted by ak g _{P} . Explicitly, elements of the adjoint bundle are equivalence classes of pairs [''p'', ''X''] for p ∈ P and X ∈ ak g such that[p ⋅ g,X] = [p, Ad _{g} (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.

Restriction to a closed subgroup

Let G be any Lie group with Lie algebra

ak g , and let H be a closed subgroup of G. Via the (left) adjoint representation of G on ak g , G becomes a topological transformation group of ak g . By restricting the adjoint representation of G to the subgroup H, Ad\vert _{H} : H \hookrightarrow G \to Aut ( ak g)

also H acts as a topological transformation group on

ak g . For every h in H, Ad\vert _{H(h):} ak g \mapsto ak g is a Lie algebra automorphism.

Since H is a closed subgroup of the Lie group G, the homogeneous space M=G/H is the base space of a principal bundle

G \to M with total space G and structure group H. So the existence of H-valued transition functions g _{ij} : U _{i} \cap U _{j} → H is assured, where U _{i} is an open covering for M, and the transition functions g _{ij} form a cocycle of transition function on M.The associated fibre bundle \xi = (E,p,M, ak g) = G[( ak g, Ad\vert _{H} )] is a bundle of Lie algebras, with typical fibre ak g , and a continuous mapping \Theta :\xi ⊕ \xi → \xi induces on each fibre the Lie bracket.

Properties

Differential forms on M with values in

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 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 x _{conj} G where conj is the action of G on itself by (left) conjugation.

If

P = l{F}(E) is the frame bundle of a vector bundle E\to M , then P has fibre the general linear group \operatorname{GL}(r) (either real or complex, depending on E ) where \operatorname{rank}(E) = r . This structure group has Lie algebra consisting of all r x r matrices \operatorname{Mat}(r) , and these can be thought of as the endomorphisms of the vector bundle E . Indeed there is a natural isomorphism \operatorname{ad} l{F}(E) = \operatorname{End}(E) .

References

• • . As PDF