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Formal definition

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

A d : G → A u t ( g ) ⊂ G L ( g ) {\displaystyle \mathrm {Ad} :G\to \mathrm {Aut} ({\mathfrak {g}})\subset \mathrm {GL} ({\mathfrak {g}})}

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

a d P = P × A d g {\displaystyle \mathrm {ad} P=P\times _{\mathrm {Ad} }{\mathfrak {g}}}

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

[ p ⋅ g , X ] = [ p , A d g ( X ) ] {\displaystyle [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 g {\displaystyle {\mathfrak {g}}} , and let H be a closed subgroup of G. Via the (left) adjoint representation of G g {\displaystyle {\mathfrak {g}}} , G becomes a topological transformation group g {\displaystyle {\mathfrak {g}}} . By restricting the adjoint representation of G to the subgroup H,

A d | H : H ↪ G → A u t ( g ) {\displaystyle \mathrm {Ad\vert _{H}} :H\hookrightarrow G\to \mathrm {Aut} ({\mathfrak {g}})}

also H acts as a topological transformation group on g {\displaystyle {\mathfrak {g}}} . For every h in H, A d | H ( h ) : g ↦ g {\displaystyle 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 → M {\displaystyle G\to M} with total space G and structure group H. So the existence of H-valued transition functions g i j : U i ∩ U j → H {\displaystyle g_{ij}:U_{i}\cap U_{j}\rightarrow H} is assured, where U i {\displaystyle U_{i}} is an open covering for M, and the transition functions g i j {\displaystyle g_{ij}} form a cocycle of transition function on M. The associated fibre bundle ξ = ( E , p , M , g ) = G [ ( g , A d | H ) ] {\displaystyle \xi =(E,p,M,{\mathfrak {g}})=G[({\mathfrak {g}},\mathrm {Ad\vert _{H}} )]} is a bundle of Lie algebras, with typical fibre g {\displaystyle {\mathfrak {g}}} , and a continuous mapping Θ : ξ ⊕ ξ → ξ {\displaystyle \Theta :\xi \oplus \xi \rightarrow \xi } induces on each fibre the Lie bracket.²

Properties

Differential forms on M with values in a d P {\displaystyle \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 a d P {\displaystyle \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 × c o n j G {\displaystyle P\times _{\mathrm {c} onj}G} where conj is the action of G on itself by (left) conjugation.

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

Notes

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 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. As PDF

References

1. Kolář, Michor & Slovák 1993, pp. 161, 400 - 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 ↩

2. Kiranagi, B.S. (1984), "Lie algebra bundles and Lie rings", Proc. Natl. Acad. Sci. India A, 54: 38–44 ↩