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

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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 (left) 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.

Restriction to a closed subgroup

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

 

also H acts as a topological transformation group on  . For every h in H,   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   with total space G and structure group H. So the existence of H-valued transition functions   is assured, where   is an open covering for M, and the transition functions   form a cocycle of transition function on M. The associated fibre bundle   is a bundle of Lie algebras, with typical fibre  , and a continuous mapping   induces on each fibre the Lie bracket.[2]

Properties

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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   where conj is the action of G on itself by (left) 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

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  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

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