[Paper Review] A dichotomy between twisted tensor products of bialgebras and Frobenius algebras

Pablo S. Ocal, Amrei Oswald|arXiv (Cornell University)|Nov 27, 2022

Algebraic structures and combinatorial models5 citations

TL;DR

This paper introduces a diagrammatic framework to endow twisted tensor products of bialgebras and Frobenius algebras with natural counit and comultiplication structures. It establishes that bialgebra structures are inherited only when the twisting map is trivial, while Frobenius algebra structures are always inherited if the twisted product is a counital coassociative coalgebra—enabling systematic construction of noncommutative symmetric Frobenius algebras and recovering quantum complete intersections as 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.

Motivation & Objective

• To determine under what conditions twisted tensor products of bialgebras inherit a bialgebra structure.
• To characterize when twisted tensor products of Frobenius algebras inherit a Frobenius algebra structure.
• To recover known results—such as quantum complete intersections being symmetric Frobenius algebras—within a unified framework.
• To construct new families of noncommutative symmetric Frobenius algebras from group algebras via non-trivial twisting maps.
• To extend the theory to separable and special Frobenius algebras, determining inheritance conditions.

Proposed method

• The authors use a purely diagrammatic approach in (braided) monoidal categories to define natural counit and comultiplication maps on twisted tensor products.
• They introduce a canonical comultiplication and counit derived from the coalgebra structures of the input algebras A and B.
• The key technical tool is verifying commutativity of specific diagrammatic identities involving the twisting map τ, Δ, and ε.
• They use the co-pairing β and its symmetry properties to determine when the Frobenius algebra structure is preserved.
• The method applies to general monoidal categories, ensuring broad applicability beyond vector spaces.
• They verify conditions on the twisting map τ via group-like elements and bicharacter-like relations to ensure coassociativity and counitality.

Experimental results

Research questions

• RQ1 Under what conditions does a twisted tensor product of bialgebras inherit a bialgebra structure?
• RQ2 When does a twisted tensor product of Frobenius algebras remain a Frobenius algebra?
• RQ3 Can quantum complete intersections be systematically shown to be symmetric Frobenius algebras via this framework?
• RQ4 What conditions on the twisting map ensure that the twisted product of group algebras is a symmetric Frobenius algebra?
• RQ5 When are separable or special Frobenius algebras preserved under twisted tensor products?

Key findings

• The twisted tensor product of two bialgebras is a bialgebra if and only if the twisting map τ is trivial.
• The twisted tensor product of two Frobenius algebras is a Frobenius algebra if and only if it is a counital coassociative coalgebra.
• Quantum complete intersections Λq,m^n are symmetric Frobenius algebras when qij is a root of unity of order dividing gcd(mi−1,mj−1), or in positive characteristic with m=(p,…,p).
• Noncommutative symmetric Frobenius algebras can be constructed from group algebras kG and kH via non-trivial strongly graded twisting maps satisfying specific λ-relations.
• Twisted tensor products of separable algebras are separable if the twisting map satisfies appropriate compatibility conditions.
• Twisted tensor products of special Frobenius algebras are special Frobenius algebras under the same conditions that preserve the Frobenius structure.