Algebraic Geometry over Free Metabelian Lie Algebra I: U-Algebras and Universal Classes

TL;DR

This paper develops foundational concepts for algebraic geometry over free metabelian Lie algebras, introducing U-algebras and exploring their properties and connections to matrix Lie algebras.

Contribution

It introduces metabelian Lie U-algebras, defines their localizations and extensions, and links them to universal closures, laying groundwork for further algebraic geometry studies.

Findings

Established connections between metabelian Lie U-algebras and matrix Lie algebras.

Defined the $ riangle$-localisation and module extension for these algebras.

Showed these algebras lie in the universal closure of the free metabelian Lie algebra.

Abstract

This paper is the first in a series of three, the aim of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra $F$. In the current paper we introduce the notion of a metabelian Lie $U$-algebra and establish connections between metabelian Lie $U$-algebras and special matrix Lie algebras. We define the $\Delta $-localisation of a metabelian Lie $U$-algebra $A$ and the direct module extension of the Fitting's radical of $A$ and show that these algebras lie in the universal closure of $A$.