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

Journal Contribution - Journal Article

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 GH 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 GH 2(A,V) is the coproduct of all non-abelian cohomologies H 2(A,(V,⋅)). The key object GH 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.

Journal: J. Pure Appl. Algebra
ISSN: 0022-4049
Issue: 2
Volume: 221
Pages: 366-392
Publication year:2017
CSS-citation score:1
Authors:International