n-ary associative algebras, cohomology, free algebras and coalgebras

When n is odd, a cohomology of type Hochschild for n-ary partially associative algebras has been defined in Gnedbaye’s thesis. Unfortunately, the cohomology definition is not valid when n is even. This fact is found again in the computations of the n-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for n-ary partially associative algebras. First by reducing the space of cochains, secondly by using a graded version. Next we compute the free n-ary algebra, giving a basis of this algebra. At last we extend the notion of coalgebras to n-ary algebras. All algebraic objects will be considered over a commutative field K of characteristic zero. 1 Relations between n-ary partially associative algebras and Gerstenhaber products 1.1 Definition Let V be a K-vector space and consider C(V ) = HomK(V , V ), for any natural number k. By defintion a n-ary partially associative algebra is a pair (V, μ) where V is a K-vector space and μ a linear map μ : V ⊗n → V satisfying n