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 $$\mathfrak g$$, 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 (left) 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
 * $$[p\cdot g,X] = [p,\mathrm{Ad}_{g}(X)]$$

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 $$\mathfrak g$$, and let H be a closed subgroup of G. Via the (left) adjoint representation of G on $$\mathfrak g$$, G becomes a topological transformation group of $$\mathfrak g$$. By restricting the adjoint representation of G to the subgroup H,

$$\mathrm{Ad\vert_H}: H \hookrightarrow G \to \mathrm{Aut}(\mathfrak g) $$

also H acts as a topological transformation group on $$\mathfrak g$$. For every h in H, $$Ad\vert_H(h): \mathfrak g \mapsto \mathfrak g$$ 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 $$g_{ij}: U_{i}\cap  U_{j} \rightarrow H$$ 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 $$ \xi= (E,p,M,\mathfrak g) = G[(\mathfrak g, \mathrm{Ad\vert_H})] $$ is a bundle of Lie algebras, with typical fibre $$\mathfrak g$$, and a continuous mapping $$ \Theta :\xi \oplus \xi \rightarrow \xi $$ induces on each fibre the Lie bracket.

Properties
Differential forms on M with values in $$\mathrm{ad} P$$ 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 $$\mathrm{ad} P$$.

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 $$P \times_{\mathrm conj} G$$ where conj is the action of G on itself by (left) conjugation.

If $$P=\mathcal{F}(E)$$ 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 $$r\times r$$ matrices $$\operatorname{Mat}(r)$$, and these can be thought of as the endomorphisms of the vector bundle $$E$$. Indeed there is a natural isomorphism $$\operatorname{ad} \mathcal{F}(E) = \operatorname{End}(E)$$.