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
:$\mathrm: G\to\mathrm(\mathfrak g)\sub\mathrm(\mathfrak g)$
be the (left) adjoint representation of ''G''. The adjoint bundle of ''P'' is the associated bundle
:$\mathrm P = P\times_\mathfrak g$
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
: \cdot g,X= ,\mathrm_(X)/math>
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,

$\mathrm: H \hookrightarrow G \to \mathrm(\mathfrak g)$

also H acts as a topological transformation group on $\mathfrak g$. For every h in H, $Ad\vert_H(h): \mathfrak g \mapsto \mathfrak g$ 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 $g_: U_\cap U_ \rightarrow H$ is assured, where $U_$ is an open covering for M, and the transition functions $g_$ form a cocycle of transition function on M.
The associated fibre bundle \xi= (E,p,M,\mathfrak g) = G \mathfrak g, \mathrm) is a bundle of Lie algebras, with typical fibre $\mathfrak g$, and a continuous mapping $\Theta :\xi \oplus \xi \rightarrow \xi$ induces on each fibre the Lie bracket.

Properties

Differential forms on ''M'' with values in $\mathrm P$ 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 $\mathrm P$.

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 $P \times_ G$ where conj is the action of ''G'' on itself by (left) conjugation.

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

Notes

References

*
* . A
PDF

{{Manifolds

Lie algebras
Vector bundles