On the structure and classification of Bernstein algebras

Linear algebra tools are used to give a new approach to the open problem of the classification of Bernstein algebras. We prove that any Bernstein algebra (A, omega) is isomorphic to a semidirect product N CC(<middle dot>, Omega) k associated to a commutative algebra (N, <middle dot>) such that (x(2))(2) = 0, for all x is an element of N and an idempotent endomorphism Omega = Omega(2) is an element of End(k)(N) of N satisfying two compatibility conditions. The set of types of (1 + |I|) -dimensional Bernstein algebras is parametrized by an explicitly constructed classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups (N, +) ix GL(k)(N).