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 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 ∈ \mathfrak g 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.

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, also H acts as a topological transformation group on \mathfrak g. 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 G \to M with total space G and structure group H. So the existence of H-valued transition functions 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 is a bundle of Lie algebras, with typical fibre \mathfrak g, and a continuous mapping induces on each fibre the Lie bracket.

Properties

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