Universal R-matrices for finite Abelian groups -- a new look at graded multilinear algebra

TL;DR

This paper characterizes universal R-matrices for finite Abelian groups and applies these findings to develop a framework for graded multilinear algebra using triangular and cotriangular Hopf algebras.

Contribution

It provides a detailed description of universal R-matrices for finite Abelian groups and introduces a new approach to graded multilinear algebra via Hopf algebra structures.

Findings

Universal R-matrices for finite Abelian groups are explicitly determined.

A formulation of graded multilinear algebra using triangular and cotriangular Hopf algebras.

Recollection of key properties of quasitriangular and coquasitriangular Hopf algebras.

Abstract

The universal R-matrices and, dually, the coquasitriangular structures of the group Hopf algebra of a finite Abelian group (resp. of an arbitrary Abelian group) are determined. This is used to formulate graded multilinear algebra in terms of triangular or cotriangular Hopf algebras. For the convenience of the reader, in a separate section the definitions and basic properties of quasitriangular and coquasitriangular Hopf algebras are recalled.