On the structure and classification of Bernstein algebras

TL;DR

This paper characterizes Bernstein algebras by showing they are isomorphic to a specific semidirect product involving a commutative algebra with nilpotent elements and an idempotent endomorphism, providing a classification and automorphism group description.

Contribution

It introduces a new structural description of Bernstein algebras as semidirect products and classifies their types using linear algebra tools.

Findings

Bernstein algebras are isomorphic to a semidirect product involving a commutative algebra with nilpotent elements.

The set of types of Bernstein algebras is explicitly classified using linear algebra.

The automorphism group of Bernstein algebras is described as a subgroup of a canonical semidirect product.

Abstract

We prove that any Bernstein algebra $(A, \omega)$ is isomorphic to a semidirect product $V \ltimes_{(\cdot, \, \Omega)} \, k$ associated to a commutative algebra $(V, \cdot)$ such that $(x^2)^2 = 0$, for all $x\in A$ and an idempotent endomorphism $\Omega = \Omega^2 \in {\rm End}_k (V)$ 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 $(V, +) \ltimes {\rm GL}_k (V)$.