# Quadratic Algebras arising from Hopf operads generated by a single element

## TL;DR

This paper explores Hopf operads generated by a single element of any arity, showing their dual spaces are quadratic and Koszul algebras, with detailed descriptions of their structure and bases.

## Contribution

It provides a detailed analysis of Hopf operads generated by one element, including their quadratic and Koszul dual structures, expanding understanding of their algebraic properties.

## Key findings

- Dual spaces are quadratic and Koszul algebras.
- Explicit generators and relations are described.
- A monomial basis for these algebras is constructed.

## Abstract

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.