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 ${\displaystyle {\mathfrak {g}}}$, and let P be a principal G-bundle over a smooth manifold M. Let

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

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

${\displaystyle \mathrm {ad} P=P\times ^{\mathrm {Ad} }{\mathfrak {g}}}$

The adjoint bundle is also commonly denoted by ${\displaystyle {\mathfrak {g}}_{P}}$. Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, x] for p ∈ P and x ∈ ${\displaystyle {\mathfrak {g}}}$ such that

${\displaystyle [p\cdot g,x]=[p,\mathrm {Ad} _{g}(x)]}$

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.

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 ${\displaystyle u\in G}$ , and ~${\displaystyle l\in L}$ , we have ,

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

where ${\displaystyle u\mapsto d\alpha _{u}}$ is the adjoint representation of G , ${\displaystyle u\mapsto \alpha _{u}}$ is a homomorphism of G into A which is an automorphism group of G and ${\displaystyle \alpha _{u}}$ is the mapping ${\displaystyle r\mapsto uru^{-1}}$ of G into itself. H is a topological transformation group of L and obviously for every u in H, ${\displaystyle d\alpha _{u}:L\mapsto L}$ 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 ${\displaystyle g_{ij}:U_{i}\cap U_{j}\rightarrow H}$ is assured where ${\displaystyle U_{i}}$ is an open covering for X. Then by the existence theorem there exists a Lie bundle ${\displaystyle \xi =(E,P,X,L,H)}$ with the continuous mapping ${\displaystyle \Theta$ :\xi \oplus \xi \rightarrow \xi } inducing on each fibre the Lie bracket.[3]

Properties

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

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 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.

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

Notes

1. Janyška, J. (2006). "Higher order Utiyama-like theorem". Reports on Mathematical Physics. 58: 93–118 See p. 96. Bibcode:2006RpMP...58...93J. doi:10.1016/s0034-4877(06)80042-x.
2. Kolář, Michor & Slovák 1993, pp. 161, 400
3. Kiranagi, B.S. (1984), "Lie algebra bundles and Lie rings", Proc. Natl. Acad. Sci. India A, 54: 38–44

References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, 1, Wiley Interscience, ISBN 0-471-15733-3
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry, Springer, pp. 161, 400, ISBN 978-3-662-02950-3. As PDF