Wikipedia:Reference desk/Archives/Mathematics/2021 March 2

= 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)


 * 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. 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)


 * 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, $Λk(TM)$ and $Λk(T∗M)$ 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, $Γ(T∗M) = Ω1(M)$."  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)