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adjoint bundle ：ウィキペディア英語版

adjoint bundle

In mathematics, an adjoint bundle 〔 (page 96 )〕〔 page 161 and page 400
〕 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.
==Formal definition==

Let ''G'' be a Lie group with Lie algebra

\mathfrak g

, and let ''P'' be a principal ''G''-bundle over a smooth manifold ''M''. Let
:

\mathrm: G\to\mathrm(\mathfrak g)\sub\mathrm(\mathfrak g)

be the adjoint representation of ''G''. The adjoint bundle of ''P'' is the associated bundle
:

\mathrm P = P\times^__{g^{-1}}(x)">)

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''.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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