Varieties of initial dialgebras and some of their Koszul dual operads

TL;DR

This paper introduces a universal method for constructing initial dialgebra varieties from a given algebraic variety and provides bases for free initial Lie and associative dialgebras.

Contribution

It presents a universal algorithm for initial dialgebra varieties and constructs bases for free initial Lie and associative dialgebras.

Findings

Constructed a basis for free initial Lie dialgebras.

Constructed a basis for free initial associative dialgebras.

Provided a method to recover algebras from subvarieties of dialgebras.

Abstract

In this paper, for a given variety $\Var$, we present a universal algorithm for constructing a subvariety of $\Var$-dialgebras from which one can recover an algebra belonging to $\Var$. Such a subvariety is called the variety of initial $\Var$-dialgebras. In addition, we construct a basis of the free initial Lie and associative dialgebras.