In mathematics, an adjoint bundle [1] is a vector bundle naturally associated with any smooth 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.
, and let P be a principal G-bundle over a smooth manifold M. Let be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundle The adjoint bundle is also commonly denoted by
Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈
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 Lie algebra
, and let H be a closed subgroup of G. Via the (left) adjoint representation of G
, G becomes a topological transformation group
By restricting the adjoint representation of G to the subgroup H,
also H acts as a topological transformation group on
is a Lie algebra automorphism.
Since H is a closed subgroup of Lie group G, the homogeneous space M=G/H is the base space of a principal bundle
with total space G and structure group H. So the existence of H-valued transition functions
is an open covering for M, and the transition functions
form a cocycle of transition function on M. The associated fibre bundle
is a bundle of Lie algebras, with typical fibre
induces on each fibre the Lie bracket.
[2] Differential forms on M with values in
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
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
where conj is the action of G on itself by (left) conjugation.
has fibre in the general linear group
(either real or complex, depending on
This structure group has Lie algebra consisting of all
, and these can be thought of as the endomorphisms of the vector bundle
{\displaystyle \operatorname {ad} {\mathcal {F}}(E)\cong \operatorname {End} (E)}