Circle products as restrictions of the square product

TL;DR

This paper introduces a new tensor product that generalizes existing circle products in symmetric and antisymmetric tensor algebras, revealing their algebraic relationships and properties using Hopf algebra techniques.

Contribution

It defines a unified tensor product that specializes to known circle products, and proves their algebraic isomorphism using Hopf algebra methods.

Findings

The new product generalizes circle products in tensor algebras.

Specializations recover Brouder's and Rota-Stein's circle products.

The classical and new tensor algebra structures are isomorphic.

Abstract

We equip the tensor algebra of a vector space $U$ over the real or complex field with an alternative product. The new product has the property that if we specialize it to the symmetric tensor algebra becomes the circle product introduced by Brouder while specialization to the antisymmetric algebra becomes the circle product introduced by Rota and Stein. We prove some interesting properties of this product analogous to the properties proved by the previous authors. We use Hopf algebraic methods to simplify our results. Finally we prove that the classical algebraic structure of the tensor algebra and the new one are homomorphically isomorphic.\bigskip