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 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 GH2(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 GH2(A,V) is the coproduct of all non-abelian cohomologies H2(A,(V,⋅)). The key object GH2(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 AutAlg(A⋆k) are fully determined as subgroups of a semidirect product A⁎⋉(k⁎×AutAlg(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.