A diagram connecting Symmetric Leibniz algebras with Lie-admissible algebras

TL;DR

This paper explores the relationships between symmetric Leibniz algebras, dialgebras, Perm-algebras, and their Koszul duals, establishing new algebraic structures and proving their Koszul properties.

Contribution

It introduces new algebraic structures related to symmetric Leibniz algebras and computes their Koszul duals, demonstrating the Koszulity of associated operads.

Findings

Identification of Koszul duals for symmetric Leibniz-related algebras

Introduction of commutative and associative admissible algebras

Proof that all studied operads are Koszul

Abstract

In this paper we study symmetric Leibniz and related algebras, namely symmetric dialgebras and symmetric Perm-algebras. We also calculate their Koszul duals, if not known. This will give us Lie-admissible algebras and new types of algebras, which we call commutative and associative admissible algebras. Finally we prove that all operads describing the considered algebras are Koszul.