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

$\mathrm{A}\mathrm{d}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{𝔤})\subset \mathrm{G}\mathrm{L}(\mathfrak{𝔤})$

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

$\mathrm{a}\mathrm{d}P=P{\times }_{\mathrm{A}\mathrm{d}}\mathfrak{𝔤}$

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

$[p\cdot g,X]=[p,{\mathrm{A}\mathrm{d}}_g(X)]$

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.

Restriction to a closed subgroup

Let G be any Lie group with Lie algebra $\mathfrak{𝔤}$, and let H be a closed subgroup of G. Via the (left) adjoint representation of G on $\mathfrak{𝔤}$, G becomes a topological transformation group of $\mathfrak{𝔤}$. By restricting the adjoint representation of G to the subgroup H,

$\mathrm{A}\mathrm{d}{\mathrm{|}}_{\mathrm{H}}:H\hookrightarrow G\to \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{𝔤})$

also H acts as a topological transformation group on $\mathfrak{𝔤}$. For every h in H, $Ad{|}_H(h):\mathfrak{𝔤}\mapsto \mathfrak{𝔤}$ is a Lie algebra automorphism.

Since H is a closed subgroup of the Lie group G, the homogeneous space M=G/H is the base space of a principal bundle $G\to M$ with total space G and structure group H. So the existence of H-valued transition functions $g_{ij}:U_i\cap U_j\to H$ 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 $\xi =(E,p,M,\mathfrak{𝔤})=G[(\mathfrak{𝔤},\mathrm{A}\mathrm{d}{\mathrm{|}}_{\mathrm{H}})]$ is a bundle of Lie algebras, with typical fibre $\mathfrak{𝔤}$, and a continuous mapping $\mathrm{\Theta }:\xi \oplus \xi \to \xi$ induces on each fibre the Lie bracket.

Properties

Differential forms on M with values in $\mathrm{a}\mathrm{d}P$ 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 $\mathrm{a}\mathrm{d}P$.

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

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

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, https://books.google.com/books?id=YQXtCAAAQBAJ&pg=PP1 . As PDF