adjoint bundle

Let G be a Lie group with Lie algebra $\backslash mathfrak\; g$, and let P be a principal G-bundle over a smooth manifold M. Let

:$\backslash mathrm\{Ad\}:\; G\backslash to\backslash mathrm\{Aut\}(\backslash mathfrak\; g)\backslash sub\backslash mathrm\{GL\}(\backslash mathfrak\; g)$

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

:$\backslash mathrm\{ad\}\; P\; =\; P\backslash times\_\{\backslash mathrm\{Ad\}\}\backslash mathfrak\; g$

The adjoint bundle is also commonly denoted by $\backslash mathfrak\; g\_P$. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈ $\backslash mathfrak\; g$ such that

:$[p\backslash cdot\; g,X]\; =\; [p,\backslash mathrm\{Ad\}\_\{g\}(X)]$

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.

Let G be any Lie group with Lie algebra $\backslash mathfrak\; g$, and let H be a closed subgroup of G.

Via the (left) adjoint representation of G $\backslash mathfrak\; g$, G becomes a topological transformation group $\backslash mathfrak\; g$.

By restricting the adjoint representation of G to the subgroup H,

$\backslash mathrm\{Ad\backslash vert\_H\}:\; H\; \backslash hookrightarrow\; G\; \backslash to\; \backslash mathrm\{Aut\}(\backslash mathfrak\; g)$

also H acts as a topological transformation group on $\backslash mathfrak\; g$. For every h in H, $Ad\backslash vert\_H(h):\; \backslash mathfrak\; g\; \backslash mapsto\; \backslash mathfrak\; g$ 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 $G\; \backslash to\; M$ with total space G and structure group H. So the existence of H-valued transition functions $g\_\{ij\}:\; U\_\{i\}\backslash cap\; U\_\{j\}\; \backslash rightarrow\; H$ 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 $\backslash xi=\; (E,p,M,\backslash mathfrak\; g)\; =\; G[(\backslash mathfrak\; g,\; \backslash mathrm\{Ad\backslash vert\_H\})]$ is a bundle of Lie algebras, with typical fibre $\backslash mathfrak\; g$, and a continuous mapping $\backslash Theta\; :\backslash xi\; \backslash oplus\; \backslash xi\; \backslash rightarrow\; \backslash xi$ induces on each fibre the Lie bracket.{{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}}

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

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 $P\; \backslash times\_\{\backslash mathrm\; conj\}\; G$ where conj is the action of G on itself by (left) conjugation.

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

{{reflist}}

• {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | 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