Algebraic Geometry over Free Metabelian Lie Algebra II: Finite Field Case

TL;DR

This paper develops algebraic geometry over free metabelian Lie algebras of finite rank over finite fields, describing their algebraic structures and proving the decidability of their universal theories.

Contribution

It provides a set of axioms, describes algebraic structures, and proves decidability for free metabelian Lie algebras over finite fields.

Findings

Axioms for universal closure of free metabelian Lie algebras over finite fields

Description of finitely generated algebras and irreducible algebraic sets

Decidability of the universal theory in both languages

Abstract

This paper is the second in a series of three, the aim of which is to construct algebraic geometry over a free metabelian Lie algebra $F$. For the universal closure of free metabelian Lie algebra of finite rank $r \ge 2$ over a finite field $k$ we find a convenient set of axioms in the language of Lie algebras $L$ and the language $L_{F}$ enriched by constants from $F$. We give a description of: * The structure of finitely generated algebras from the universal closure of $F_r$ in both $L$ and $L_{F_r}$ * The structure of irreducible algebraic sets over $F_r $ and respective coordinate algebras. We also prove that the universal theory of a free metabelian Lie algebra over a finite field is decidable in both languages.