A computational algebraic geometry approach to enumerate Malcev magma algebras over finite fields

Abstract

The set of Mn(K) of n-dimensional Malcev magma algebras over a finite field can be identified with algebraic sets defined by zero-dimensional radical ideals for which the computation of their reduced Gröbner bases makes feasible their enumeration and distribution into isomorphism and isotopism classes. Based on this computation and the classification of Lie algebras over finite fields given by De Graaf and Strade, we determine the mentioned distribution for Malcev magma algebras of dimension $n\leq 4$. We also prove that every 3-dimensional Malcev algebra is isotopic to a Lie magma algebra. For n=4, this assertion only holds when the characteristic of the base field is distinct of two.