  We define the notion of admissible pair for an algebra $A$, consisting on a
couple $(\Gamma,R)$, where $\Gamma$ is a quiver and $R$ a unital, splitted and
factorizable representation of $\Gamma$, and prove that the set of admissible
pairs for $A$ is in one to one correspondence with the points of the variety of
twisting maps $\mathcal{T}_A^n:=\mathcal{T}(K^n,A)$. We describe all these
representations in the case $A=K^m$.
