Veronese powers of operads and pure homotopy algebras
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Veronese powers of operads and pure homotopy algebras Vladimir Dotsenko1 · Martin Markl2
· Elisabeth Remm3
Received: 2 August 2018 / Revised: 12 June 2019 / Accepted: 24 June 2019 © Springer Nature Switzerland AG 2019
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 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. Keywords Operad · Veronese power · Homological purity · Koszul duality · Koszulness · Zeilberger’s algorithm Mathematics Subject Classification 18D50 · 18G55 · 33F10 · 55P48
ˇ ˇ This work was supported by the Eduard Cech Institute [P201/12/G028 to M.M.] and by a Grant GA CR [18-07776S to M.M.]. The final revision was moreover supported by Præmium Academiae of Martin Markl.
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Martin Markl [email protected] Vladimir Dotsenko [email protected] Elisabeth Remm [email protected]
1
School of Mathematics, Trinity College, Dublin 2, Ireland
2
Mathematical Institute of the Academy, Žitná 25, 115 67 Prague 1, Czech Republic
3
Laboratoire de Mathématiques et Applications, Faculté des Sciences et Techniques, Université de Haute Alsace, 4, rue des Frères Lumière, 68093 Mulhouse cedex, France
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V. Dotsenko et al.
1 Introduction Many examples of algebras with m-ary structure operations are “pure” versions of homotopy algebras in the following sense. Suppose that P is a binary quadratic Koszul operad, and P∞ = (P¡ ) is its minimal model. We use the term pure P∞ -algebras for algebras over the quotient of the operad P∞ by the ideal generated by all its generating operations except for those of arity m. (The space of generators of the thus obtained operad is homologically pure, hence the terminology.) Another (in many ways more classical) type of algebras with m-ary structure operations is obtained as follows. Let P, once again, be a binary quadratic operad. We consider the suboperad of P generated by all operations of arity m; experts in classical theory of identities in algebras would probably call algebras over this operad m-tuple systems of type P. (See, for instance, the work of Jacobson [24], who used the term “triple systems” for this construct
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