- Adjoint bundle
In
mathematics , an adjoint bundle is avector bundle naturally associated to anyprincipal bundle . The fibers of the adjoint bundle carry aLie algebra structure making the adjoint bundle into analgebra bundle . Adjoint bundles has important applications in the theory of connections as well as ingauge theory Formal definition
Let "G" be a
Lie group withLie algebra mathfrak g, and let "P" be a principal "G"-bundle over asmooth manifold "M". Let:mathrm{Ad}: G omathrm{Aut}(mathfrak g)submathrm{GL}(mathfrak g)be theadjoint representation of "G". The adjoint bundle of "P" is theassociated bundle :mathrm{Ad}_P = P imes_{mathrm{Admathfrak gThe adjoint bundle is also commonly denoted by mathfrak g_P. Explicitly, elements of the adjoint bundle areequivalence class es of pairs ["p", "x"] for "p" ∈ "P" and "x" ∈ mathfrak g such that:pcdot g,x] = [p,mathrm{Ad}_g(x)] for all "g" ∈ "G". Since thestructure group of the adjoint bundle consists of Lie algebraautomorphism s, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over "M".Properties
Differential forms on "M" with values in Ad"P" are in one-to-one corresponding with horizontal, "G"-equivariant
Lie algebra-valued form s 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 Ad"P".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 transformation s 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.
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