Adjoint bundle

In mathematics, an adjoint bundle 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

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.

References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. Vol. 1 (New ed.), Wiley Interscience, ISBN 0-471-15733-3 {{citation}}: |volume= has extra text (help)
• 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.