Algebraic Geometry over Lie Algebras

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 principle example is, of course, algebraic geometry over fields. The foundations of algebraic geometry over groups were laid by Baumslag, Myasnikov and Remeslennikov [4, 28]. The present paper transfers their ideas to the algebraic geometry over Lie algebras. Let A be a fixed Lie algebra over a field k. We introduce the category of ALie 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: