Hochschild products and global non-abelian cohomology for algebras. Applications.

Let A be a unital associative algebra over a field k , E a vector space and π : E → A a surjective linear map with V = Ker ( π ) . All algebra structures on E such that π : E → A becomes an algebra map are described and classified by an explicitly constructed global cohomological type object G H 2 ( A , V ) . Any such algebra is isomorphic to a Hochschild product A ⋆ V , an algebra introduced as a generalization of a classical construction. We prove that G H 2 ( A , V ) is the coproduct of all non-abelian cohomologies H 2 ( A , ( V , ⋅ ) ) . The key object G H 2 ( A , k ) responsible for the classification of all co-flag algebras is computed. All Hochschild products A ⋆ k are also classified and the automorphism groups Aut Alg ( A ⋆ k ) are fully determined as subgroups of a semidirect product A ⁎ ⋉ ( k ⁎ × Aut Alg ( A ) ) of groups. Several examples are given as well as applications to the theory of supersolvable coalgebras or Poisson algebras. In particular, for a given Poisson algebra P , all Poisson algebras having a Poisson algebra surjection on P with a 1-dimensional kernel are described and classified. [ABSTRACT FROM AUTHOR]