Veronese powers of operads and pure homotopy algebras

Abstract

We define the mth Veronese power of a weight graded operad P to be its suboperad 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 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.Organisation of the paper. The paper is organised as follows. In Section 1, we recall the key relevant definitions of the theory of operads. In Section 2, we define Veronese powers of weight graded operads, prove some results about them, and provide counterexamples to some celebrated properties of Veronese powers of associative algebras. In Section 3, we relate our work to research of polynomial 2010 Mathematics Subject Classification. 18D50 (Primary), 18G55, 33F10, 55P48 (Secondary).Key words and phrases. Operad, Veronese power, homological purity, Koszul duality, Koszulness, Zeilberger's algorithm. The second author was supported by the EduardČech Institute P201/12/G028 and RVO: 67985840. 1 identities, both classical and recent. In Section 4, we define operads for pure homotopy P-algebras, and explain how they are related to Veronese powers by Koszul duality. Finally, in Section 5, we demonstrate that the ungraded versions of strong homotopy Lie algebras sometimes used in the literature possess worse homotopical properties than the strong homotopy Lie algebras obtained from the minimal model of the Lie operad, and show that those algebras provide an economic example where the positivity criterion for Poincaré series does not settle the matter of non-Koszulness.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 [5] that arrived at the same definition of Veronese powers independently, when trying to define what an n-ary algebra over an operad P is), and to Eric Hoffbeck on comments on the first draft of the paper.