On the computation of Manin products for operads

TL;DR

This paper introduces a computer program designed to compute Manin products of binary quadratic operads, facilitating calculations of operad compositions and their Koszul duals, with applications demonstrated on classical operads.

Contribution

The paper presents a novel computational tool for calculating Manin products and Koszul duals of binary quadratic operads, streamlining complex algebraic operations.

Findings

Successfully computed the white product of Lie and As operads, resulting in the magmatic operad.

The program automates routine calculations of operad products and duals.

Demonstrated the utility of the tool with classical operads in algebra.

Abstract

In the theory of binary quadratic operads, the white and black products of operads (called Manin products) play an important role. Given two such operads, the computation of either of their Manin products is a routine task. We present and describe a computer program that helps to compute white and black Manin products of binary quadratic operads. The same utility allows to find the Koszul-dual operads. In particular, we compute the white product of the operads Lie and As (governing the varieties of Lie and associative algebras, respectively). It turns out that the resulting operad is magmatic, i.e., defines the variety of all algebras with one bilinear operation.