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Concept
Adjoint bundle
Summary
In mathematics, an adjoint bundle 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.
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 p ∈ P and X ∈ such that
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.
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.
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.
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