Crossed products of 4-algebras. Applications

Abstract

A 4-algebra is a commutative algebra A over a field k such that (a2)2=0, for all a∈A. We have proved recently [22] that 4-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let A be a 4-algebra, E a vector space and π:E→A a surjective linear map with V=Ker(π). All 4-algebra structures on E such that π:E→A is an algebra map are described and classified by a global cohomological object GH2(A,V). Any such 4-algebra is isomorphic to a crossed product V#A and GH2(A,V) is a coproduct, over all 4-algebras structures ⋅V on V, of all non-abelian cohomologies Hnab2(A,(V,⋅V)), which are the classifying objects for all extensions of A by V. Several applications and examples are provided: in particular, GH2(A,k) and GH2(k,V) are explicitly computed and the Galois group Gal(V#A/V) of the extension V↪V#A is described.