Algebraic geometry over free metabelian Lie algebras. I. U-algebras and universal classes

This paper is the first in a series of three, the object 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 U-Lie algebra and establish connections between metabelian U-Lie algebras and special matrix Lie algebras. We define the $\Delta$-localization of a metabelian U-Lie algebra $A$ and the direct module extension of the Fitting radical of $A$ and show that these algebras lie in the universal closure of $A$.