Weak Leibniz algebras and transposed Poisson algebras

TL;DR

This paper explores the properties of weak Leibniz algebras, showing their self-duality and non-Koszul nature, and establishes a duality between weak Leibniz and transposed Poisson algebras through polarization.

Contribution

It demonstrates the self-duality and non-Koszul property of the weak Leibniz operad and reveals a duality between weak Leibniz and transposed Poisson algebras via polarization.

Findings

Weak Leibniz operad is self-dual.

Weak Leibniz operad is not Koszul.

Polarization links weak Leibniz and transposed Poisson algebras.

Abstract

An algebra with identities $[a,b]c=2a(bc)-2b(ac), a[b,c]=2(ab)c-2(ac)b$ is called weak Leibniz. We show that weak Leibniz operad is self-dual and is not Koszul. We establish that polarization of any weak Leibniz algebra is transposed Poisson, and, conversely, polarization of any transposed Poisson algebra is weak Leibniz.