On the structure and classification of Bernstein algebras

Abstract

We prove that any Bernstein algebra $(A,\omega )$ is isomorphic to a semidirect product $V{⋉}_{(\cdot ,\Omega )}k$ associated to a commutative algebra $(V,\cdot )$ such that $(x^2{)}^2=0$, for all $x\in A$ and an idempotent endomorphism $\Omega ={\Omega }^2\in {\mathrm{End}}_k(V)$ of $V$ satisfying two compatibility conditions. The set of types of $(1+|I|)$-dimensional Bernstein algebras is parametrized by an explicitely constructed (using linear algebra tools) classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups $(V,+)⋉{\mathrm{GL}}_k(V)$.

The final version will appear in Journal of Algebra and its Applications