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 (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 ${\displaystyle {\mathfrak {g}}}$, 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 ${\displaystyle {\mathfrak {g}}_{P}}$. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, x] for p ∈ P and x ∈ ${\displaystyle {\mathfrak {g}}}$ such that

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