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

Abstract

Let $A$ be a unital associative algebra over a field $k$, $E$ a vector space and $\pi : E \to A$ a surjective linear map with $V = {\rm Ker} (\pi)$. All algebra structures on $E$ such that $\pi : E \to A$ becomes an algebra map are described and classified by an explicitly constructed global cohomological type object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$. Any such algebra is isomorphic to a Hochschild product $A \star V$, an algebra introduced as a generalization of a classical construction. We prove that ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is the coproduct of all non-abelian cohomologies ${\mathbb H}^{2} \, \, (A, \, (V, \cdot))$. The key object ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ responsible for the classification of all co-flag algebras is computed. All Hochschild products $A \star k$ are also classified and the automorphism groups ${\rm Aut}_{\rm Alg} (A \star k)$ are fully determined as subgroups of a semidirect product $A^* \, \ltimes \bigl(k^* \times {\rm Aut}_{\rm Alg} (A) \bigl)$ 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.