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

On the classification of twisting maps between $K^n$ and $K^m$