On the computation of Manin products for operads

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.