The braided monoidal structures on the category of vector spaces graded by the Klein group

TL;DR

This paper classifies all braided monoidal structures on the category of vector spaces graded by the Klein group, explicitly computing relevant 3-cocycles, and constructs examples of quasi-Hopf and weak braided Hopf algebras.

Contribution

It explicitly determines all braided monoidal structures on vector spaces graded by the Klein group using 3-cocycles and abelian 3-cocycles, leading to new algebraic examples.

Findings

Explicit forms of 3-cocycles on C2×C2 with coefficients in k*

Explicit forms of abelian 3-cocycles on C2×C2 with coefficients in k*

Construction of quasi-Hopf and weak braided Hopf algebras from these structures

Abstract

Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras, out of the vector space $k[C_2\times C_2]$.