Английская Википедия:Adjoint bundle
In mathematics, an adjoint bundle [1] 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 <math>\mathfrak g</math>, and let P be a principal G-bundle over a smooth manifold M. Let
- <math>\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g)</math>
be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundle
- <math>\mathrm{ad} P = P\times_{\mathrm{Ad}}\mathfrak g</math>
The adjoint bundle is also commonly denoted by <math>\mathfrak g_P</math>. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈ <math>\mathfrak g</math> such that
- <math>[p\cdot g,X] = [p,\mathrm{Ad}_{g}(X)]</math>
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.
Restriction to a closed subgroup
Let G be any Lie group with Lie algebra <math>\mathfrak g</math>, and let H be a closed subgroup of G. Via the (left) adjoint representation of G on <math>\mathfrak g</math>, G becomes a topological transformation group of <math>\mathfrak g</math>. By restricting the adjoint representation of G to the subgroup H,
<math>\mathrm{Ad\vert_H}: H \hookrightarrow G \to \mathrm{Aut}(\mathfrak g) </math>
also H acts as a topological transformation group on <math>\mathfrak g</math>. For every h in H, <math>Ad\vert_H(h): \mathfrak g \mapsto \mathfrak g</math> is a Lie algebra automorphism.
Since H is a closed subgroup of the Lie group G, the homogeneous space M=G/H is the base space of a principal bundle <math>G \to M</math> with total space G and structure group H. So the existence of H-valued transition functions <math>g_{ij}: U_{i}\cap U_{j} \rightarrow H</math> is assured, where <math>U_{i}</math> is an open covering for M, and the transition functions <math>g_{ij}</math> form a cocycle of transition function on M. The associated fibre bundle <math> \xi= (E,p,M,\mathfrak g) = G[(\mathfrak g, \mathrm{Ad\vert_H})] </math> is a bundle of Lie algebras, with typical fibre <math>\mathfrak g</math>, and a continuous mapping <math> \Theta :\xi \oplus \xi \rightarrow \xi </math> induces on each fibre the Lie bracket.[2]
Properties
Differential forms on M with values in <math>\mathrm{ad} P</math> 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 <math>\mathrm{ad} P</math>.
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 transformations of P which can be thought of as sections of the bundle <math>P \times_{\mathrm conj} G</math> where conj is the action of G on itself by (left) conjugation.
If <math>P=\mathcal{F}(E)</math> is the frame bundle of a vector bundle <math>E\to M</math>, then <math>P</math> has fibre the general linear group <math>\operatorname{GL}(r)</math> (either real or complex, depending on <math>E</math>) where <math>\operatorname{rank}(E) = r</math>. This structure group has Lie algebra consisting of all <math>r\times r</math> matrices <math>\operatorname{Mat}(r)</math>, and these can be thought of as the endomorphisms of the vector bundle <math>E</math>. Indeed there is a natural isomorphism <math>\operatorname{ad} \mathcal{F}(E) = \operatorname{End}(E)</math>.
Notes
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