Adjoint bundle

In mathematics, an adjoint bundle ^[1]^[2] is a vector bundle naturally associated to any 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 ${\mathfrak {g}}$ , and let P be a principal G-bundle over a smooth manifold M. Let

{\mathrm {Ad}}:G\to {\mathrm {Aut}}({\mathfrak g})\subset {\mathrm {GL}}({\mathfrak g})

be the 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^{{-1}}}}(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.

Example ^[3]

Let G be any Lie group with a closed sub group H and let L be the Lie algebra of G. Since G is a topological transformation group of L by the adjoint action of G,that is , for every $u\in G$ , and ~ $l\in L$ , we have ,

 $G\times L\rightarrow L$ 
  defined  by   $(u,l)\mapsto d\alpha _{u}(l)$

where $u\mapsto d\alpha _{u}$ is the adjoint representation of G , $u\mapsto \alpha _{u}$ is a homomorphism of G into A which is an automorphism group of G and $\alpha _{u}$ is the mapping $r\mapsto uru^{-1}$ of G into itself. H is a topological transformation group of L and obviously for every u in H, $d\alpha _{u}:L\mapsto L$ is a Lie algebra automorphism.

since H is a closed subgroup of a Lie group G, there is a locally trivial principal bundle over X=G/H having H as a structure group. So the existence of coordinate functions $g_{ij}:U_{i}\cap U_{j}\rightarrow H$ is assured where $U_{i}$ is an open covering for X. Then by the existence theorem there exists a Lie bundle $\xi =(E,P,X,L,H)$ with the continuous mapping $\Theta :\xi \oplus \xi \rightarrow \xi$ inducing 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 ×^Ψ G where Ψ is the action of G on itself by conjugation.

Notes

↑ J. Janyška (2006). "Higher order Utiyama-like theorem". Reports on Mathematical Physics. 58: 93–118. Bibcode:2006RpMP...58...93J. doi:10.1016/s0034-4877(06)80042-x. [cf. page 96]
↑ Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag page 161 and page 400
↑ B.S. Kiranagi,,Lie algebra bundles and Lie rings, Proc. Natl. Acad. Sci. India 54(a),1984,38-44.

References

Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3
Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag page 161 and page 400.
B.S. Kiranagi, Lie algebra fibre bundles, Thèse de $3^{e}$ cycle, Universitè de Paris-Sud,Centre d'Orsay,France,1976.
B.S. Kiranagi,Lie algebra bundles and Lie rings, Proc. Natl. Acad. Sci. India 54(a),1984,38-44.
B.S. Kiranagi, B. Madhu, K.Ajaykumar,On Smooth Lie Algebra Bundles, International Journal of Algebra, Vol. 11, 2017, no. 5, 247 - 254

    https://doi.org/10.12988/ija.2017.7314.

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[1] J. Janyška (2006). "Higher order Utiyama-like theorem". Reports on Mathematical Physics. 58: 93–118. Bibcode:2006RpMP...58...93J. doi:10.1016/s0034-4877(06)80042-x. [cf. page 96]

[2] Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag page 161 and page 400

[3] B.S. Kiranagi,,Lie algebra bundles and Lie rings, Proc. Natl. Acad. Sci. India 54(a),1984,38-44.