Adjoint bundle

In mathematics, an adjoint bundle [1][2] 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 , and let P be a principal G-bundle over a smooth manifold M. Let

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

The adjoint bundle is also commonly denoted by . Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, x] for pP and x such that

for all gG. 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.

Example

Let G be any Lie group with a closed sub group H and let L be the Lie algebra of G. Since G is a topological transformation group of L by the adjoint action of G,that is , for every , and ~ , we have ,

  defined  by  

where is the adjoint representation of G , is a homomorphism of G into A which is an automorphism group of G and is the mapping of G into itself. H is a topological transformation group of L and obviously for every u in H, is a Lie algebra automorphism.

since H is a closed subgroup of a Lie group G, there is a locally trivial principal bundle over X=G/H having H as a structure group. So the existence of coordinate functions is assured where is an open covering for X. Then by the existence theorem there exists a Lie bundle with the continuous mapping inducing on each fibre the Lie bracket.[3]

Properties

Differential forms on M with values in 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 .

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 ×Ψ G where Ψ is the action of G on itself by conjugation.

If is the frame bundle of a vector bundle , then has fibre the general linear group (either real or complex, depending on ) where . This structure group has Lie algebra consisting of all matrices , and these can be thought of as the endomorphisms of the vector bundle . Indeed there is a natural isomorphism .

Notes

  1. Janyška, J. (2006). "Higher order Utiyama-like theorem". Reports on Mathematical Physics. 58: 93–118 See p. 96. Bibcode:2006RpMP...58...93J. doi:10.1016/s0034-4877(06)80042-x.
  2. Kolář, Michor & Slovák 1993, pp. 161, 400
  3. Kiranagi, B.S. (1984), "Lie algebra bundles and Lie rings", Proc. Natl. Acad. Sci. India A, 54: 38–44

References

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