# Veronese powers of operads and pure homotopy algebras

## TL;DR

This paper introduces the Veronese powers of operads, explores their properties, and connects them to homotopy algebras and Koszul duality, with applications to Lie k-algebras and Lie triple systems.

## Contribution

It defines Veronese powers of operads, analyzes their homological properties, and relates them to strongly homotopy algebras via Koszul duality, expanding understanding of operadic structures.

## Key findings

- Veronese powers of quadratic Koszul operads relate to homotopy algebras.
- Homological properties are not generally improved by Veronese powers.
- Certain operads lack good homotopy properties and are non-Koszul.

## 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.