On the algebras obtained by tensor product

TL;DR

This paper characterizes a specific operad associated with a quadratic operad P, ensuring that the tensor product of a P-algebra and a ~P-algebra remains a P-algebra, thus advancing operad theory.

Contribution

It introduces an associated operad ~P that guarantees the tensor product of P-algebras and ~P-algebras is again a P-algebra, providing new insights into operad tensor products.

Findings

Identification of the operad ~P for any quadratic operad P

Proof that A ⊗ B is a P-algebra when A is P and B is ~P

Clarification of the algebraic structure preserved under tensor products

Abstract

Let P be a quadratic operad. We determine an associated operad ~P such that for any P-algebra A and any ~P-algebra B then the tensor product $A \otimes B$ is a P-algebra.