Adjoint bundle

In mathematics, an adjoint bundle ^[1] 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 (left) 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.

Restriction to a closed subgroup

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

${\displaystyle \mathrm {Ad\vert _{H}} :H\hookrightarrow G\to \mathrm {Aut} ({\mathfrak {g}})}$

also H acts as a topological transformation group on ${\displaystyle {\mathfrak {g}}}$. For every h in H, ${\displaystyle Ad\vert _{H}(h):{\mathfrak {g}}\mapsto {\mathfrak {g}}}$ 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 ${\displaystyle G\to M}$ with total space G and structure group H. So the existence of H-valued transition functions ${\displaystyle g_{ij}:U_{i}\cap U_{j}\rightarrow H}$ is assured, where ${\displaystyle U_{i}}$ is an open covering for M, and the transition functions ${\displaystyle g_{ij}}$ form a cocycle of transition function on M. The associated fibre bundle ${\displaystyle \xi =(E,p,M,{\mathfrak {g}})=G[({\mathfrak {g}},\mathrm {Ad\vert _{H}} )]}$ is a bundle of Lie algebras, with typical fibre ${\displaystyle {\mathfrak {g}}}$, and a continuous mapping ${\displaystyle \Theta :\xi \oplus \xi \rightarrow \xi }$ induces on each fibre the Lie bracket.^[2]

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

References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, vol. 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