Adjoint Bundle - Formal Definition

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.