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Veronese powers of operads and pure homotopy algebras

Veronese powers of operads and pure homotopy algebras

Authors :
Martin Markl
Vladimir Dotsenko
Elisabeth Remm
Trinity College Dublin
Institute of Mathematics of the Czech Academy of Science (IM / CAS)
Czech Academy of Sciences [Prague] (CAS)
Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))
Source :
European Journal of Mathematics, European Journal of Mathematics, Springer, 2020, 6, pp.829-863. ⟨10.1007/s40879-019-00351-6⟩
Publication Year :
2019
Publisher :
Springer Science and Business Media LLC, 2019.

Abstract

We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie $k$-algebras and Lie triple systems. In the case of Lie $k$-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg--Kapranov power series test is inconclusive.<br />Comment: 21 pages, comments are welcome

Details

ISSN :
21996768 and 2199675X
Volume :
6
Database :
OpenAIRE
Journal :
European Journal of Mathematics
Accession number :
edsair.doi.dedup.....f23f8294a40a595b889c6c5c1f534c7c
Full Text :
https://doi.org/10.1007/s40879-019-00351-6