Braided Frobenius Algebras from certain Hopf Algebras

TL;DR

This paper constructs braided Frobenius algebras from specific Hopf algebras using heap operations, demonstrating how these induce braidings compatible with Frobenius structures through diagrammatic proofs.

Contribution

It introduces a method to derive braided Frobenius algebras from certain Hopf algebras via heap operations and Yang-Baxter operators, expanding the algebraic framework.

Findings

Heap operations induce braidings in Hopf algebras.

Constructed braided Frobenius algebras satisfy compatibility conditions.

Diagrammatic methods verify the algebraic structures.

Abstract

A braided Frobenius algebra is a Frobenius algebra with braiding that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation $(x,y,z) \mapsto xy^{-1}z$, that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a braiding, by means of a Yang-Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between the braiding and Frobenius operations.