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.

## Contents

* 1 Formal definition
* 2 Restriction to a closed subgroup
* 3 Properties
* 4 Notes
* 5 References

## Formal definition

Let G be a Lie group with Lie algebra 𝔤, and let P be a principal G-bundle over a smooth manifold M. Let

Ad:G->Aut(𝔤)⊂GL(𝔤)

be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundle

adP=P×Ad𝔤

The adjoint bundle is also commonly denoted by 𝔤P. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈ 𝔤 such that

[p⋅g,X]=[p,Adg(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.

## Restriction to a closed subgroup

Let G be any Lie group with Lie algebra 𝔤, and let H be a closed subgroup of G. Via the (left) adjoint representation of G on 𝔤, G becomes a topological transformation group of 𝔤. By restricting the adjoint representation of G to the subgroup H,

Ad|H:H↪G->Aut(𝔤)

also H acts as a topological transformation group on 𝔤. For every h in H, Ad|H(h):𝔤↦𝔤 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 G->M with total space G and structure group H. So the existence of H-valued transition functions gij:Ui∩Uj->H is assured, where Ui is an open covering for M, and the transition functions gij form a cocycle of transition function on M. The associated fibre bundle ξ=(E,p,M,𝔤)=G[(𝔤,Ad|H)] is a bundle of Lie algebras, with typical fibre 𝔤, and a continuous mapping Θ:ξ⊕ξ->ξ induces on each fibre the Lie bracket.[2]

## Properties

Differential forms on M with values in adP 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 adP.

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 P×conjG where conj is the action of G on itself by (left) conjugation.

If P=ℱ(E) is the frame bundle of a vector bundle E->M, then P has fibre the general linear group GL⁡(r) (either real or complex, depending on E) where rank⁡(E)=r. This structure group has Lie algebra consisting of all r×r matrices Mat⁡(r), and these can be thought of as the endomorphisms of the vector bundle E. Indeed there is a natural isomorphism ad⁡ℱ(E)=End⁡(E).

## Notes

1. ↑ Kolář, Michor & Slovák 1993, pp. 161, 400
2. ↑ Kiranagi, B.S. (1984), "Lie algebra bundles and Lie rings", Proc. Natl. Acad. Sci. India A 54: 38–44

## 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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