Timeline for Generators of a certain ideal
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 4, 2012 at 6:51 | vote | accept | Pierre-Yves Gaillard | ||
| Feb 3, 2012 at 21:43 | comment | added | Vladimir Dotsenko | Yes, I understand that - and that's partly because I am making my comment. In Mathieu's setting, the generators are in a sense dual to Poisson brackets \{x_i,x_j\} and hence are naturally antisymmetric. So it matches the story very well (unlike the Arnold's story, where there is symmetry, not skew-symmetry in i and j), except for the square zero condition which is absent here. | |
| Feb 3, 2012 at 21:21 | comment | added | Mariano Suárez-Álvarez | I wrote throughout the $Y_{i,j}$ with the convention that $Y_{i,j}=-Y_{j,i}$, too. If Pierre-Ives wants to work without making that identification, then one should add the relations $Y_{i,j}+Y_{j,i}$ along with the quadratic relations above to generate the ideal. | |
| Feb 3, 2012 at 20:54 | comment | added | Vladimir Dotsenko | @Mariano: in fact, Arnold's generators satisfy X_{ij}=X_{ji}, so the difference is more subtle. However, there is an even analogue of Arnold's relation (still with square zero though) for antisymmetric generators, it is described in a paper by Olivier Mathieu called "The symplectic operad" from a collection of articles for Gelfand's 80th birthday. The relations are precisely these ones. Now one just has to figure out what parts of this observation are coincidental and what are not. | |
| Feb 3, 2012 at 19:11 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 3.0 |
added 913 characters in body
|
| Feb 3, 2012 at 18:44 | comment | added | Pierre-Yves Gaillard | Dear Mariano: Thanks a lot! Your answer is very concise, but I'm sure it contains a lot of maths. I'll try to digest and assimilate it. | |
| Feb 3, 2012 at 18:38 | comment | added | Mariano Suárez-Álvarez | These are the "same" relations that occur in Arnold's presentation of the de Rham cohomology of the complement of the braid arrangement or, equivalently, the cohomology of the pure braid group with real coefficients. The only difference is that Arnold's generators anticommute and square to zero, while yours commute. | |
| Feb 3, 2012 at 18:20 | history | answered | Mariano Suárez-Álvarez | CC BY-SA 3.0 |