Adjoint bundle


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 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 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 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.

Example B.S. Kiranagi,,Lie algebra bundles and Lie rings, Proc. Natl. Acad. Sci. India 54(a),1984,38-44.

Let G be any Lie group with a closed sub group H and let L be the Lie algebra of G. Since G is a topological transformation group of L by the adjoint action of G,that is, for every , and ~, we have,
defined by
where is the adjoint representation of G, is a homomorphism of G into A which is an automorphism group of G and is the mapping of G into itself. H is a topological transformation group of L and obviously for every u in H, is a Lie algebra automorphism.
since H is a closed subgroup of a Lie group G, there is a locally trivial principal bundle over X=G/H having H as a structure group. So the existence of coordinate functions is assured where is an open covering for X. Then by the existence theorem there exists a Lie bundle with the continuous mapping inducing on each fibre the Lie bracket.

Properties

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 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 ×Ψ G where Ψ is the action of G on itself by conjugation.
If is the frame bundle of a vector bundle, then has fibre the general linear group 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.