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Veronese powers of operads and pure homotopy algebras

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Abstract

We define the mth Veronese power of a weight graded operad to be its suboperad 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 are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects such 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.

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Acknowledgements

Some of the key results of this paper were obtained in CINVESTAV (Mexico City); the first author and the second author wish to thank that institution for the excellent working conditions. The authors are also grateful to Murray Bremner for his interest in our work (he communicated to us [4] that arrived at the same definition of Veronese powers independently, when trying to define what an n-ary algebra over an operad is), to Eric Hoffbeck for comments on the first draft of the paper, and to Jim Stasheff for useful remarks on different definitions of n-ary Lie algebras.

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Authors and Affiliations

  1. School of Mathematics, Trinity College, Dublin 2, Ireland

    Vladimir Dotsenko

  2. Mathematical Institute of the Academy, Žitná 25, 115 67, Prague 1, Czech Republic

    Martin Markl

  3. Laboratoire de Mathématiques et Applications, Faculté des Sciences et Techniques, Université de Haute Alsace, 4, rue des Frères Lumière, 68093, Mulhouse cedex, France

    Elisabeth Remm

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  1. Vladimir Dotsenko
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  2. Martin Markl
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  3. Elisabeth Remm
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Correspondence to Martin Markl.

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This work was supported by the Eduard Čech Institute [P201/12/G028 to M.M.] and by a Grant GA ČR [18-07776S to M.M.]. The final revision was moreover supported by Præmium Academiae of Martin Markl.

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Dotsenko, V., Markl, M. & Remm, E. Veronese powers of operads and pure homotopy algebras. European Journal of Mathematics 6, 829–863 (2020). https://doi.org/10.1007/s40879-019-00351-6

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