Lie algebra-valued differential form

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.

## Contents

* 1 Formal definition
* 2 Wedge product
* 3 Operations
* 4 Forms with values in an adjoint bundle
* 5 See also
* 6 Notes
* 7 References
* 8 External links

## Formal definition

A Lie-algebra-valued differential k-form on a manifold, M, is a smooth section of the bundle (𝔤×M)⊗∧kT*M, where 𝔤 is a Lie algebra, T*M is the cotangent bundle of M and ∧k denotes the kth exterior power.

## Wedge product

Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie-algebra-valued forms can be composed with the bracket operation to obtain another Lie-algebra-valued form. For a 𝔤-valued p-form ω and a 𝔤-valued q-form η, their wedge product [ω∧η] is given by

[ω∧η](v1,…,vp+q)=1(p+q)!∑σsgn⁡(σ)[ω(vσ(1),…,vσ(p)),η(vσ(p+1),…,vσ(p+q))],

where the vi's are tangent vectors. The notation is meant to indicate both operations involved. For example, if ω and η are Lie-algebra-valued one forms, then one has

[ω∧η](v1,v2)=12([ω(v1),η(v2)]−[ω(v2),η(v1)]).

The operation [ω∧η] can also be defined as the bilinear operation on Ω(M,𝔤) satisfying

[(g⊗α)∧(h⊗β)]=[g,h]⊗(α∧β)

for all g,h∈𝔤 and α,β∈Ω(M,ℝ).

Some authors have used the notation [ω,η] instead of [ω∧η]. The notation [ω,η], which resembles a commutator, is justified by the fact that if the Lie algebra 𝔤 is a matrix algebra then [ω∧η] is nothing but the graded commutator of ω and η, i. e. if ω∈Ωp(M,𝔤) and η∈Ωq(M,𝔤) then

[ω∧η]=ω∧η−(−1)pqη∧ω,

where ω∧η, η∧ω∈Ωp+q(M,𝔤) are wedge products formed using the matrix multiplication on 𝔤.

## Operations

Let f:𝔤->𝔥 be a Lie algebra homomorphism. If φ is a 𝔤-valued form on a manifold, then f(φ) is an 𝔥-valued form on the same manifold obtained by applying f to the values of φ: f(φ)(v1,…,vk)=f(φ(v1,…,vk)).

Similarly, if f is a multilinear functional on ∏1k𝔤, then one puts[1]

f(φ1,…,φk)(v1,…,vq)=1q!∑σsgn⁡(σ)f(φ1(vσ(1),…,vσ(q1)),…,φk(vσ(q−qk+1),…,vσ(q)))

where q=q1+…+qk and φi are 𝔤-valued qi-forms. Moreover, given a vector space V, the same formula can be used to define the V-valued form f(φ,η) when

f:𝔤×V->V

is a multilinear map, φ is a 𝔤-valued form and η is a V-valued form. Note that, when

f([x,y],z)=f(x,f(y,z))−f(y,f(x,z)),(*)

giving f amounts to giving an action of 𝔤 on V; i.e., f determines the representation

ρ:𝔤->V,ρ(x)y=f(x,y)

and, conversely, any representation ρ determines f with the condition (*). For example, if f(x,y)=[x,y] (the bracket of 𝔤), then we recover the definition of [⋅∧⋅] given above, with ρ=ad, the adjoint representation. (Note the relation between f and ρ above is thus like the relation between a bracket and ad.)

In general, if α is a 𝔤𝔩(V)-valued p-form and φ is a V-valued q-form, then one more commonly writes α⋅φ=f(α,φ) when f(T,x)=Tx. Explicitly,

(α⋅ϕ)(v1,…,vp+q)=1(p+q)!∑σsgn⁡(σ)α(vσ(1),…,vσ(p))ϕ(vσ(p+1),…,vσ(p+q)).

With this notation, one has for example:

ad⁡(α)⋅ϕ=[α∧ϕ].

Example: If ω is a 𝔤-valued one-form (for example, a connection form), ρ a representation of 𝔤 on a vector space V and φ a V-valued zero-form, then

ρ([ω∧ω])⋅φ=2ρ(ω)⋅(ρ(ω)⋅φ).[2]

## Forms with values in an adjoint bundle

Let P be a smooth principal bundle with structure group G and 𝔤=Lie⁡(G). G acts on 𝔤 via adjoint representation and so one can form the associated bundle:

𝔤P=P×Ad𝔤.

Any 𝔤P-valued forms on the base space of P are in a natural one-to-one correspondence with any tensorial forms on P of adjoint type.

## See also

* Maurer–Cartan form
* Adjoint bundle

## Notes

1. ↑ Kobayashi–Nomizu, Ch. XII, § 1. harvnb error: no target: CITEREFKobayashi–Nomizu (help)
2. ↑ Since ρ([ω∧ω])(v,w)=ρ([ω∧ω](v,w))=ρ([ω(v),ω(w)])=ρ(ω(v))ρ(ω(w))−ρ(ω(w))ρ(ω(v)), we have that

(ρ([ω∧ω])⋅φ)(v,w)=12(ρ([ω∧ω])(v,w)φ−ρ([ω∧ω])(w,v)ϕ)
is ρ(ω(v))ρ(ω(w))φ−ρ(ω(w))ρ(ω(v))ϕ=2(ρ(ω)⋅(ρ(ω)⋅ϕ))(v,w).

## References

## External links

* Wedge Product of Lie Algebra Valued One-Form
* groupoid of Lie-algebra valued forms in nLab

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Lie_algebra-valued_differential_form&oldid=184221"