Adjoint bundle

Formal definition

Let G be a Lie group with Lie algebra \mathfrak g, and let P be a principal G-bundle over a smooth manifold M. Let :\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g) be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundle :\mathrm{ad} P = P\times_{\mathrm{Ad}}\mathfrak g The adjoint bundle is also commonly denoted by \mathfrak g_P. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈ \mathfrak g such that :[p\cdot g,X] = [p,\mathrm{Ad}_{g}(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 \mathfrak g, and let H be a closed subgroup of G. Via the (left) adjoint representation of G \mathfrak g, G becomes a topological transformation group \mathfrak g. By restricting the adjoint representation of G to the subgroup H,

\mathrm{Ad\vert_H}: H \hookrightarrow G \to \mathrm{Aut}(\mathfrak g)

also H acts as a topological transformation group on \mathfrak g. For every h in H, Ad\vert_H(h): \mathfrak g \mapsto \mathfrak g is a Lie algebra automorphism.

Since H is a closed subgroup of Lie group G, the homogeneous space M=G/H is the base space of a principal bundle G \to M with total space G and structure group H. So the existence of H-valued transition functions g_{ij}: U_{i}\cap U_{j} \rightarrow H is assured, where U_{i} is an open covering for M, and the transition functions g_{ij} form a cocycle of transition function on M. The associated fibre bundle \xi= (E,p,M,\mathfrak g) = G[(\mathfrak g, \mathrm{Ad\vert_H})] is a bundle of Lie algebras, with typical fibre \mathfrak g, and a continuous mapping \Theta :\xi \oplus \xi \rightarrow \xi induces on each fibre the Lie bracket.

Properties

Differential forms on M with values in \mathrm{ad} P 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 \mathrm{ad} P.

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 \times_{\mathrm conj} G where conj is the action of G on itself by (left) conjugation.

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

Notes

References

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References

1. {{harvnb. Kolář. Michor. Slovák. 1993
2. (1984). "Lie algebra bundles and Lie rings". Proc. Natl. Acad. Sci. India A.