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]\displaystyle{ \mathfrak g }[/math], and let P be a principal G-bundle over a smooth manifold M. Let

[math]\displaystyle{ \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]\displaystyle{ \mathrm{ad} P = P\times_{\mathrm{Ad}}\mathfrak g }[/math]

The adjoint bundle is also commonly denoted by [math]\displaystyle{ \mathfrak g_P }[/math]. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈ [math]\displaystyle{ \mathfrak g }[/math] such that

[math]\displaystyle{ [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]\displaystyle{ \mathfrak g }[/math], and let H be a closed subgroup of G. Via the (left) adjoint representation of G on [math]\displaystyle{ \mathfrak g }[/math], G becomes a topological transformation group of [math]\displaystyle{ \mathfrak g }[/math]. By restricting the adjoint representation of G to the subgroup H,

[math]\displaystyle{ \mathrm{Ad\vert_H}: H \hookrightarrow G \to \mathrm{Aut}(\mathfrak g) }[/math]

also H acts as a topological transformation group on [math]\displaystyle{ \mathfrak g }[/math]. For every h in H, [math]\displaystyle{ 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]\displaystyle{ G \to M }[/math] with total space G and structure group H. So the existence of H-valued transition functions [math]\displaystyle{ g_{ij}: U_{i}\cap U_{j} \rightarrow H }[/math] is assured, where [math]\displaystyle{ U_{i} }[/math] is an open covering for M, and the transition functions [math]\displaystyle{ g_{ij} }[/math] form a cocycle of transition function on M. The associated fibre bundle [math]\displaystyle{ \xi= (E,p,M,\mathfrak g) = G[(\mathfrak g, \mathrm{Ad\vert_H})] }[/math] is a bundle of Lie algebras, with typical fibre [math]\displaystyle{ \mathfrak g }[/math], and a continuous mapping [math]\displaystyle{ \Theta :\xi \oplus \xi \rightarrow \xi }[/math] induces on each fibre the Lie bracket.^[2]

Properties

Differential forms on M with values in [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ P \times_{\mathrm conj} G }[/math] where conj is the action of G on itself by (left) conjugation.

If [math]\displaystyle{ P=\mathcal{F}(E) }[/math] is the frame bundle of a vector bundle [math]\displaystyle{ E\to M }[/math], then [math]\displaystyle{ P }[/math] has fibre the general linear group [math]\displaystyle{ \operatorname{GL}(r) }[/math] (either real or complex, depending on [math]\displaystyle{ E }[/math]) where [math]\displaystyle{ \operatorname{rank}(E) = r }[/math]. This structure group has Lie algebra consisting of all [math]\displaystyle{ r\times r }[/math] matrices [math]\displaystyle{ \operatorname{Mat}(r) }[/math], and these can be thought of as the endomorphisms of the vector bundle [math]\displaystyle{ E }[/math]. Indeed there is a natural isomorphism [math]\displaystyle{ \operatorname{ad} \mathcal{F}(E) = \operatorname{End}(E) }[/math].

Notes

References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, 1, Wiley Interscience, ISBN 0-471-15733-3 
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry, Springer, pp. 161, 400, ISBN 978-3-662-02950-3, https://books.google.com/books?id=YQXtCAAAQBAJ&pg=PP1 . As PDF

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