A dichotomy between twisted tensor products of bialgebras and Frobenius algebras

TL;DR

This paper investigates the conditions under which twisted tensor products of bialgebras and Frobenius algebras retain their algebraic structures, using diagrammatic methods applicable in monoidal categories.

Contribution

It provides necessary and sufficient conditions for twisted tensor products to be bialgebras or Frobenius algebras, extending the understanding of their structural properties.

Findings

Characterized when twisted tensor products are bialgebras or Frobenius algebras

Identified conditions for twisted tensor products of separable and special Frobenius algebras

Constructed families of noncommutative symmetric Frobenius algebras

Abstract

We endow twisted tensor products with a natural notion of counit and comultiplication, and we provide sufficient and necessary conditions making the twisted tensor product a counital coassociative coalgebra. We then characterize when the twisted tensor product of bialgebras is a bialgebra, and when the twisted tensor product of Frobenius algebras is a Frobenius algebra. Our methods are purely diagrammatic, so these results hold for (braided) monoidal categories. As an application, we recover that some quantum complete intersections are Frobenius algebras, and we construct families of noncommutative symmetric Frobenius algebras. Along the way, we also characterize when twisted tensor products of separable algebras are separable, and we prove that twisted tensor products of special Frobenius algebras are special Frobenius.