| Title:
|
Operads for $n$-ary algebras – calculations and conjectures (English) |
| Author:
|
Markl, Martin |
| Author:
|
Remm, Elisabeth |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
5 |
| Year:
|
2011 |
| Pages:
|
377-387 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In [8] we studied Koszulity of a family ${t\mathcal{A}\it ss}^n_d$ of operads depending on a natural number $n \in \mathbb{N}$ and on the degree $d \in \mathbb{Z}$ of the generating operation. While we proved that, for $n \le 7$, the operad ${t\mathcal{A}\it ss}^n_d$ is Koszul if and only if $d$ is even, and while it follows from [4] that ${t\mathcal{A}\it ss}^n_d$ is Koszul for $d$ even and arbitrary $n$, the (non)Koszulity of ${t\mathcal{A}\it ss}^n_d$ for $d$ odd and $n \ge 8$ remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations. (English) |
| Keyword:
|
operad |
| Keyword:
|
Koszulity |
| Keyword:
|
minimal model |
| MSC:
|
18D50 |
| MSC:
|
55P48 |
| idZBL:
|
Zbl 1265.18015 |
| idMR:
|
MR2876941 |
| . |
| Date available:
|
2011-12-16T15:25:50Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141785 |
| . |
| Reference:
|
[1] Getzler, E., Jones, J. D. S.: Operads, homotopy algebra, and iterated integrals for double loop spaces.Preprint hep-th/9403055, March 1994. |
| Reference:
|
[2] Ginzburg, V., Kapranov, M. M.: Koszul duality for operads.Duke Math. J. 76 (1) (1994), 203–272. Zbl 0855.18006, MR 1301191, 10.1215/S0012-7094-94-07608-4 |
| Reference:
|
[3] Hanlon, P., Wachs, M. L.: On Lie $k$-algebras.Adv. Math. 113 (1995), 206–236. Zbl 0844.17001, MR 1337108, 10.1006/aima.1995.1038 |
| Reference:
|
[4] Hoffbeck, E.: A Poincaré–Birkhoff–Witt criterion for Koszul operads.Manuscripta Math. 131 (1–2) (2010), 87–110. Zbl 1207.18009, MR 2574993, 10.1007/s00229-009-0303-2 |
| Reference:
|
[5] Markl, M.: A cohomology theory for $A(m)$-algebras and applications.J. Pure Appl. Algebra 83 (1992), 141–175. Zbl 0801.55004, MR 1191090, 10.1016/0022-4049(92)90160-H |
| Reference:
|
[6] Markl, M.: Models for operads.Comm. Algebra 24 (4) (1996), 1471–1500. Zbl 0848.18003, MR 1380606, 10.1080/00927879608825647 |
| Reference:
|
[7] Markl, M.: Intrinsic brackets and the ${L_\infty }$-deformation theory of bialgebras.J. Homotopy Relat. Struct. 5 (1) (2010), 177–212. MR 2812919 |
| Reference:
|
[8] Markl, M., Remm, E.: (Non–)Koszulness of operads for n-ary algebras, galgalim and other curiosities.Preprint arXiv:0907.1505. |
| Reference:
|
[9] Markl, M., Shnider, S., Stasheff, J. D.: Operads in Algebra, Topology and Physics.Math. Surveys Monogr., vol. 96, Amer. Math. Soc., Providence, RI, 2002. Zbl 1017.18001, MR 1898414 |
| . |
