Quasi-bialgebra Structures and Torsion-free Abelian Groups

TL;DR

This paper classifies all quasi-bialgebra structures on group algebras over torsion-free abelian groups, showing they are all triangular and induce a unique braided monoidal structure on their representation categories.

Contribution

It provides a complete classification of quasi-bialgebra structures for these groups and links them to known braided monoidal categories related to Hom-structures.

Findings

All quasi-bialgebra structures are triangular.

These structures induce a unique braided monoidal category.

Application to Laurent polynomial algebra recovers known categories.

Abstract

We describe all the quasi-bialgebra structures of a group algebra over a torsion-free abelian group. They all come out to be triangular in a unique way. Moreover, up to an isomorphism, these quasi-bialgebra structures produce only one (braided) monoidal structure on the category of their representations. Applying these results to the algebra of Laurent polynomials, we recover two braided monoidal categories introduced in \cite{CG} by S. Caenepeel and I. Goyvaerts in connection with Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras).