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 $V=\mathrm{Ker}(\pi )$. 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 ${\mathrm{Aut}}_{\mathrm{Alg}}(A\star k)$ are fully determined as subgroups of a semidirect product $A^{\ast }⋉(k^{\ast }\times {\mathrm{Aut}}_{\mathrm{Alg}}(A))$ 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.

Continues arXiv:1308.5559, arXiv:1309.1986 and arXiv:1305.6022; restates preliminaries and definitions for sake of clarity. Final version to appear in J. Pure Appl. Algebra

References in corpus (1)

• Canonical matrices of bilinear and sesquilinear forms

Cited by in corpus (3)

• Metabelian associative algebras
• A differential graded Lie algebra approach to non abelian extensions of associative algebras
• On a type of commutative algebras