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March 2

Notation question

All the following notation is gleaned from WP. Given a differentiable manifold $M$ with tangent bundle $TM$ and cotangent bundle $T * M$ , the set of sections of $TM$ (also called vector fields) is denoted $Γ(TM)$ , and similarly of $T * M$ is denoted $Γ(T * M)$ . The exterior algebra on $Γ(T * M)$ is denoted $Ω(M)$ (the differential forms on $M$ ). Is there a suitable equivalent notation for the exterior algebra of $Γ(TM)$ , i.e., the dual of $Ω(M)$ ? —Quondum 13:45, 2 March 2021 (UTC)[reply]

I haven’t been able to find an article considering this exterior algebra. As I understood what I saw – this is far from my areas of expertise –

Ω(M)

is not so much a convenient alternative notation for the exterior algebra on

Γ(T * M)

, but it is the case that

Γ(T * M)

and

Ω 1 (M)

happen to be isomorphic.^[1] According to the Encyclopedia of Mathematics, entry Lie algebroid, the Lie algebra structure of

Γ(T * M)

is isomorphic to that of

Γ(TM)

. Is that a helpful fact? --Lambiam 11:52, 9 March 2021 (UTC)[reply]

It's more that I'm trying to fill in some detail in the occasional article such as Exterior calculus identities. Looking at the PDF that you linked,

Λ k (TM)

and

Λ k (T * M)

are used for the kth exterior power of

Γ(TM)

and

Γ(T * M)

, which in the style of the paper would suggest the notations

Λ * (TM)

and

Λ * (T * M)

for the exterior algebras of the sections (not used, though), the latter actually being denoted

Ω * (M)

. The EoM page denotes these

Γ(⋀ TM)

and

Γ(⋀ T * M)

, assuming an identification of

A

with

TM

. I find the detail of bundles and sections of bundles a little confusing, and it seems these notations are not entirely consistent, but out of this something like the EoM notation should suffice, and I can treat

Ω(M)

as an auxiliary notation. The "exterior algebras of sections of the cotangent bundle" –

Λ(Γ(T * M))

and

Λ(Γ(TM))

– and the "sections of the exterior algebra of the cotangent bundle" –

Γ(Λ(T * M))

and

Γ(Λ(TM))

– are effectively the same, so either choice should do.

I don't see the "not so much a convenient alternative notation" or "happen to be isomorphic" as much as a direct identification of the same thing. See Exercise 8.11: "The space of sections

Γ(T * M)

of the cotangent bundle of a manifold

M

is the space of 1-forms on a manifold

M

. That is,

Γ(T * M) = Ω 1 (M)

." This also fits with my understanding.

Following on from your isomorphism statement, the EoM article has as premise additional (Poisson) structure to define a Lie algebra on

Γ(T * M)

, so I think the isomorphism of Lie algebras that you mention is not in general canonical. Technically, this is another topic.

Thank you for digging this out. The links have helped me get a stronger handle on the notation, enough to be usable. —Quondum 22:29, 9 March 2021 (UTC)[reply]