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

Abstract

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 Δ-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.