In mathematics, an '''adjoint bundle''' <ref>{{harvnb|Kolář|Michor|Slovák|1993||pp=161, 400}}</ref> is a vector bundle naturally associated with any smooth 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 <math>\mathfrak g</math>, and let ''P'' be a principal ''G''-bundle over a smooth manifold ''M''. Let :<math>\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g)</math> be the (left) adjoint representation of ''G''. The '''adjoint bundle''' of ''P'' is the associated bundle :<math>\mathrm{ad} P = P\times_{\mathrm{Ad}}\mathfrak g</math> The adjoint bundle is also commonly denoted by <math>\mathfrak g_P</math>. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [''p'', ''X''] for ''p'' ∈ ''P'' and ''X'' ∈ <math>\mathfrak g</math> such that :<math>[p\cdot g,X] = [p,\mathrm{Ad}_{g}(X)]</math> 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 <math>\mathfrak g</math>, and let ''H'' be a closed subgroup of G. Via the (left) adjoint representation of G <math>\mathfrak g</math>, G becomes a topological transformation group <math>\mathfrak g</math>. By restricting the adjoint representation of G to the subgroup H,
<math>\mathrm{Ad\vert_H}: H \hookrightarrow G \to \mathrm{Aut}(\mathfrak g) </math>
also H acts as a topological transformation group on <math>\mathfrak g</math>. For every h in H, <math>Ad\vert_H(h): \mathfrak g \mapsto \mathfrak g</math> 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 <math>G \to M</math> with total space G and structure group H. So the existence of H-valued transition functions <math>g_{ij}: U_{i}\cap U_{j} \rightarrow H</math> is assured, where <math>U_{i}</math> is an open covering for M, and the transition functions <math>g_{ij}</math> form a cocycle of transition function on M. The associated fibre bundle <math> \xi= (E,p,M,\mathfrak g) = G[(\mathfrak g, \mathrm{Ad\vert_H})] </math> is a bundle of Lie algebras, with typical fibre <math>\mathfrak g</math>, and a continuous mapping <math> \Theta :\xi \oplus \xi \rightarrow \xi </math> induces on each fibre the Lie bracket.<ref>{{citation |first1=B.S. |last1=Kiranagi |title=Lie algebra bundles and Lie rings |journal=Proc. Natl. Acad. Sci. India A |volume=54 |year=1984 |pages=38–44}}</ref>
==Properties==
Differential forms on ''M'' with values in <math>\mathrm{ad} P</math> 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 <math>\mathrm{ad} P</math>.
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 <math>P \times_{\mathrm conj} G</math> where conj is the action of ''G'' on itself by (left) conjugation.
If <math>P=\mathcal{F}(E)</math> is the frame bundle of a vector bundle <math>E\to M</math>, then <math>P</math> has fibre in the general linear group <math>\operatorname{GL}(r)</math> (either real or complex, depending on <math>E</math>) where <math>\operatorname{rank}(E) = r</math>. This structure group has Lie algebra consisting of all <math>r\times r</math> matrices <math>\operatorname{Mat}(r)</math>, and these can be thought of as the endomorphisms of the vector bundle <math>E</math>. Indeed, there is a natural isomorphism <math>\operatorname{ad} \mathcal{F}(E) \cong \operatorname{End}(E)</math>.
==Notes==
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==References==
* {{citation | last1=Kobayashi|first1=Shoshichi|author-link=Shoshichi Kobayashi|last2=Nomizu|first2=Katsumi|author2-link=Katsumi Nomizu | title = Foundations of Differential Geometry|volume=1| publisher=Wiley Interscience | year=1996 |isbn=0-471-15733-3}} * {{citation|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan |title=Natural operators in differential geometry |url=https://books.google.com/books?id=YQXtCAAAQBAJ&pg=PP1 |year=1993 |publisher=Springer |isbn=978-3-662-02950-3 |pages=161, 400 }}. As [https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf PDF]
{{Manifolds}}
Category:Lie algebras Category:Vector bundles