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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 17th 2013
• (edited May 3rd 2014)

It just occurred to me that there is an immediate axiomatization of the Liouville-Poincaré 1-form (the canonical differential 1-form on a cotangent bundle) in differential cohesion.

In fact, it is the special case of a much more general notion: for $A$ any type in differential cohesion the total space $\underset{X}{\sum} \mathcal{O}_X(A)$ of the $A$-valued structure sheaf over any $X$ carries a canonical $A$-cocycle.

For $A = \Omega^1$ the sheaf of 1-forms and $X$ a manifold, this is the traditional Liouville-Poincaré 1-form on $T^* X$.

I made a quick note on that at differential cohesion – Liouville-Poincaré cocycle.

Thanks to a conversation with Owen Gwilliam I now also understand how that construction gives the antibracket in the BV-BRST complex. I still need to write that out. Not today though.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJan 17th 2013

Google does not give any hits with phrase Liouville-Poincaré.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 17th 2013
• (edited Jan 17th 2013)

All sources on that 1-form say it is called the Liouville form or the Poincaré-form. I thought if it’s really due to both, one should give credit to both.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJan 17th 2013

Thanks. I do know it as canonical form, indeed, http://en.wikipedia.org/wiki/Tautological_one-form. Liouville-Poincaré sounds OK a priori, but it may repel some of the outside users – one has to be careful with new mixed names as they are expected to be different from the one with one name; for example Euler equation (in fluid dynamics) is completely different from say Euler-Lagrange equation (in variational calculus). I was just told be a student yesterday that he does not read $n$Lab any more, because the $n$Lab is like machine – always gives a correct answer but unrecognizable and hi-brow even when he thinks he knows the notion. This was sad to hear because it was about a good student.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 17th 2013

Maybe somebody know or enjoys to dig out the original sources? Chances are that the idea is originally neither to Liouville nor to Poincaré! :-)

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeJan 17th 2013

Yes, likely :) it could be instructive, how they were lead there, it is always instructive when it is about such great minds :)