Abstract: | We define the notion of admissible pair for an algebra A, consisting on a couple (Γ, R), where Γ is a quiver and R a unital, splitted and factorizable representation of Γ, 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 TAn:=T(Kn,A)mathcal{T}_A^n:=mathcal{T}(K^n,A). We describe all these representations in the case A = K m . |