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Veronese powers of operads and pure homotopy algebras
Veronese powers of operads and pure homotopy algebras
- 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
- Subjects :
- Pure mathematics
Koszul duality
Triple system
General Mathematics
Context (language use)
010103 numerical & computational mathematics
Algebraic geometry
Mathematics::Algebraic Topology
01 natural sciences
Quadratic equation
Mathematics::K-Theory and Homology
Simple (abstract algebra)
Mathematics::Category Theory
Mathematics - Quantum Algebra
FOS: Mathematics
Quantum Algebra (math.QA)
Category Theory (math.CT)
18D50 (Primary), 18G55, 33F10, 55P48 (Secondary)
[MATH]Mathematics [math]
0101 mathematics
Associative property
[MATH.MATH-CT]Mathematics [math]/Category Theory [math.CT]
Mathematics
Mathematics::Commutative Algebra
Homotopy
010102 general mathematics
Mathematics - Category Theory
K-Theory and Homology (math.KT)
Mathematics - K-Theory and Homology
[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT]
[MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA]
Subjects
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