Algebraic geometry over Lie algebras

Abstract

Quasivarieties 21 11. Universal closure 24 12. Geometric equivalence 26 13. Algebraic geometry over free metabelian Lie algebra 27 13.1. The ∆-localisation and the direct module extension of the Fitting's radica 13.2. The case of a finite field 29 13.3. Main results 31 14. Algebraic geometry over a free Lie algebra 33 14.1. Parallelepipedons 34 14.2. Bounded algebraic sets and coordinate algebras 35 14.3. The correspondences between algebraic sets, radicals and coordinate algeb References 39 1. The category of A-Lie algebrasWe work with a fixed algebra A of coefficients and introduce a notion of an A-Lie algebra, a Lie algebra analogue of (associative) algebras over an associative ring.Definition 1.1. Let A be a fixed Lie algebra over a field k. A Lie algebra B over k is called an A-Lie algebra if it contains a designated copy of A, which we shall usually identify with A. More precisely, an A-Lie algebra B is a Lie algebra together with an embedding α : A → B. A morphism or A-homomorphism ϕ from an A-Lie algebra B 1 to an A-Lie algebra B 2 is a homomorphism of Lie algebras which is the identity on A (or, in a more formal language, α 1 ϕ = α 2 where α 1 and α 2 are the corresponding embeddings of the Lie algebra A into the A-Lie algebras B 1 and B 2 ).Obviously, A-Lie algebras and A-homomorphisms form a category. In the special case A = {0}, the category of A-Lie algebras is the category of Lie algebras over k. Note that if A is a nonzero Lie algebra then the category of A-Lie algebras does not possess a zero object.Notice that A is itself an A-Lie algebra. We denote by Hom A (B 1 , B 2 ) the set of all A-homomorphisms from B 1 to B 2 , and by ∼ = A the isomorphism in the category of A-Lie algebras (Aisomorphism). The usual notions of free, finitely generated and finitely presented algebras carry over to the category of A-Lie algebras.We say that the set X generates an A-Lie algebra B in the category of A-Lie algebras if the algebra B is generated by the set A ∪ X as a Lie algebra, i. e. B = A, X . We use notation B = X A .Notice that an A-Lie algebra B can be finitely generated in the category of A-Lie algebras without being finitely generated as a Lie algebra.

Undecidability of equations in free Lie algebras

“…, u n ∈ L and sometimes requiring characteristic zero of the field K). The general theory of equations in free Lie algebras is not developed yet, though the fundamentals of the algebraic geometry over arbitrary Lie algebras were outlined in the works of Daniyarova,Kazachkov,and Remeslennikov [6,7,12,8]. In particular, in the paper [8] systems of equations over free Lie algebras whose solution sets are contained in a finite-dimensional affine subspace were studied.…”

On Systems of Equations over Free Partially Commutative Groups

“…In [Rem89] and [MR00] A. Myasnikov and V. Remeslennikov established relations between universal classes of groups, algebraic geometry and residual properties of groups, see Theorems 2.3 and 2.4 below. We refer the reader to [MR00] and [Kaz07] for proofs.…”