Introduction

What is algebraic geometry over algebraic systems? Many important relations between elements of a given algebraic system A can be expressed by systems of equations over A. The solution sets of such systems are called algebraic sets over A. Algebraic sets over A form a category, if we take for morphisms polynomial functions in the sense of Definition 6.1 below. As a discipline, algebraic geometry over A studies structural properties of this category. The principal example is, of course, algebraic geometry over fields. The foundations of algebraic geometry over groups were laid by Baumslag, Myasnikov and Remeslennikov. The present paper transfers their ideas to algebraic geometry over Lie algebras.

Let A be a fixed Lie algebra over a field k. We introduce the category of A-Lie algebras in Sections 1 and 2. Sections 3–7 are built around the notion of a free A-Lie algebra A[X], which can be viewed as an analogue of a polynomial algebra over a unitary commutative ring. We introduce a Lie-algebraic version of the concept of an algebraic set and study connections between algebraic sets, radical ideals of A[X] and coordinate algebras (the latter can be viewed as analogues of factor algebras of a polynomial algebra over a commutative ring by a radical ideal). These concepts allow us to describe the properties of algebraic sets in two different languages:

• the language of radical ideals, and

• the language of coordinate algebras.