In mathematics, an adjoint bundle 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.

Formal definition

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

. 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.

Restriction to a closed subgroup

Let G, and let H. By restricting the adjoint representation of G to the subgroup H,

is a Lie algebra automorphism.

induces on each fibre the Lie bracket.

Properties

Differential forms on M 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.

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 where conj is the action of G on itself by (left) conjugation.

is the frame bundle of a vector bundle has fibre in the general linear group.

Notes

References

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