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.

Let *G* be a Lie group with Lie algebra , 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 on , G becomes a topological transformation group of .
By restricting the adjoint representation of G to the subgroup H,

also H acts as a topological transformation group on . For every h in H, 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 with total space G and structure group H. So the existence of H-valued transition functions is assured, where 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 , and a continuous mapping 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.

If is the frame bundle of a vector bundle , then has fibre the general linear group (either real or complex, depending on ) where . This structure group has Lie algebra consisting of all matrices , and these can be thought of as the endomorphisms of the vector bundle . Indeed there is a natural isomorphism .