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

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, with underlying vector space $k[C_2\times C_2]$.

Publication year:2011

Keywords:Braided monoidal category, Klein group, group cohomology, quasi-Hopf algebra