Adjoint bundle

In mathematics, an adjoint bundle ^[1] 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[edit]

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}})\subset \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[edit]

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.^[2]

Properties[edit]

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 $P\times _{\mathrm {c} onj}G$ 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 $\operatorname {ad} {\mathcal {F}}(E)=\operatorname {End} (E)$ .