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

Let $k$ be a field, $k^{\ast }=k\setminus \{0\}$ 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 $k^{\ast }$, 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 $k^{\ast }$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras, out of the vector space $k[C_2\times C_2]$.