Adjoint bundle

<h2 id="formal-definition">Formal definition</h2>
Let G be a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> with <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> 
 
 
 
 
 
 g
 
 
 
 
 {\displaystyle {\mathfrak {g}}}
 
, and let P be a <a href="/facts/Principal_bundle/bbB2D94B">principal G-bundle</a> over a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> 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) <a href="/facts/Adjoint_representation/bWG4c2su">adjoint representation</a> of G. The adjoint bundle of P is the <a href="/facts/Associated_bundle/7JfKDcBJ">associated bundle</a>

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 <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> 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 <a href="/facts/Structure_group/SJ7Ek6cI">structure group</a> of the adjoint bundle consists of Lie algebra <a href="/facts/Automorphism/WnyxNFkQ">automorphisms</a>, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over M.

<h2 id="restriction-to-a-closed-subgroup">Restriction to a closed subgroup</h2>
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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="properties">Properties</h2>
<a href="/facts/Vector-valued_differential_form/GohcjkUh">Differential forms</a> on M with values in 
 
 
 
 
 a
 d
 
 P
 
 
 {\displaystyle \mathrm {ad} P}
 
 are in one-to-one correspondence with <a href="/facts/Tensorial_form/GohcjkUh">horizontal, G-equivariant</a> <a href="/facts/Lie_algebra-valued_form/ot9XbLEk">Lie algebra-valued forms</a> on P. A prime example is the <a href="/facts/Curvature_form/LWp7DMDM">curvature</a> of any <a href="/facts/Connection_(principal_bundle)/UchSqJ0C">connection</a> 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 <a href="/facts/Gauge_transformation/fSW3gggv">gauge transformations</a> 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) <a href="/facts/Conjugation_(group_theory)/AFuPIFNW">conjugation</a>.
If 
 
 
 
 P
 =
 
 
 F
 
 
 (
 E
 )
 
 
 {\displaystyle P={\mathcal {F}}(E)}
 
 is the <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a> of a <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> 
 
 
 
 E
 →
 M
 
 
 {\displaystyle E\to M}
 
, then 
 
 
 
 P
 
 
 {\displaystyle P}
 
 has fibre in the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> 
 
 
 
 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)}
 
.

<h2 id="notes">Notes</h2>

<ul><li>Kobayashi, Shoshichi; Nomizu, Katsumi (1996), <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry</a>, vol. 1, <a href="/facts/Wiley_Interscience/53g3LqJc">Wiley Interscience</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15733-3</li>
<li>Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), <a href="https://books.google.com/books?id=YQXtCAAAQBAJ&pg=PP1">Natural operators in differential geometry</a>, Springer, pp. 161, 400, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-662-02950-3. As <a href="https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf">PDF</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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 <a href="https://books.google.com/books?id=YQXtCAAAQBAJ&pg=PP1" target="_blank">https://books.google.com/books?id=YQXtCAAAQBAJ&pg=PP1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kiranagi, B.S. (1984), "Lie algebra bundles and Lie rings", Proc. Natl. Acad. Sci. India A, 54: 38–44 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Adjoint bundle open-in-new

Adjoint bundle