Algebraic Geometry over Groups II. Logical Foundations

Myasnikov, Alexei; Ремесленников, В. Н.

doi:10.1006/jabr.2000.8414

Cited by 119 publications

(128 citation statements)

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“…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.…”

Section: Introductionmentioning

confidence: 88%

“…Geometric equivalence B. Plotkin [29] introduced an important notion of geometrically equivalent algebraic structures. Myasnikov and Remeslennikov [28] discuss this notion in the case of groups and observe that all their results can be transferred to an arbitrary algebraic structure. In this section, we transfer their results to Lie algebras.…”

Section: A-ucl(b) = Ldis a (B)mentioning

confidence: 99%

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Algebraic geometry over Lie algebras

Kazachkov¹

2007

Surveys in Contemporary Mathematics

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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.

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Section: Introductionmentioning

confidence: 88%

Section: A-ucl(b) = Ldis a (B)mentioning

confidence: 99%

Algebraic geometry over Lie algebras

Kazachkov¹

2007

Surveys in Contemporary Mathematics

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“…The proof of the following Lemma can certainly be extracted from [16] but for completeness we provide a proof.…”

Section: Definition 25 a Group G Is Said H-pseudo-limit If G Is A Momentioning

confidence: 99%

Limit Groups of Equationally Noetherian Groups

Houcine

2007

Geometric Group Theory

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We collect in this paper some remarks and observations about limit groups of equationally noetherian groups. We show in particular, that some known properties of limit groups of a free group or, more generally, of a torsion-free hyperbolic group can be seen as consequences of the fact that such groups are equationally noetherian. Especially, such properties are still true for linear groups and finitely generated abelian-by-nilpotent groups.

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“…The reader may learn this theory from the papers [1,9]. Given a group G, we denote by GOEX the free product of G and a free group with basis X D ¹x 1 ; : : : ; x n º.…”

Section: 3mentioning

confidence: 99%

Presentations for rigid solvable groups

Romanovskiy

2012

Journal of Group Theory

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Abstract. A group G is said to be m-rigid if it has a normal seriesin which each factor G i =G i C1 is abelian and torsion-free as a ZOEG=G i -module. Denote by † m the class of all m-rigid groups and by † m .R/ the set of groups in † m generated by x 1 ; : : : ; x n that satisfy a given set of relations R. We say that a group in † m .R/ is maximal if it has no proper covering in † m .R/. It is proved that, for every R, the set † m .R/ contains only finitely many maximal groups. The set of relations R is said to be complete if † m .R/ contains a unique maximal group. It is shown that every finitely generated group in † m is completely finitely presented. We give a definition of a canonical presentation for a rigid group with the generators x 1 ; : : : ; x n . If such a presentation is given, the group at least has decidable word problem. Given a finite set of relations R D R.x 1 ; : : : ; x n /, we effectively construct a finite set of canonical presentations in the generators x 1 ; : : : ; x n for groups in † m .R/ among which all the maximal groups in † m .R/ are contained.

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Algebraic Geometry over Groups II. Logical Foundations

Cited by 119 publications

References 8 publications

Algebraic geometry over Lie algebras

Algebraic geometry over Lie algebras

Limit Groups of Equationally Noetherian Groups

Presentations for rigid solvable groups

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