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Algebraic geometry over free metabelian lie algebras. II. Finite-field case

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Abstract

This paper is the second in a series of three, the object of which is to construct an algebraic geometry over the free metabelian Lie algebra F. For the universal closure of a free metabelian Lie algebra of finite rank r ⩾ 2 over a finite field k we find convenient sets of axioms in two distinct languages: with constants and without them. We give a description of the structure of finitely generated algebras from the universal closure of F r in both languages mentioned and the structure of irreducible algebraic sets over F r and respective coordinate algebras. We also prove that the universal theory of free metabelian Lie algebras over a finite field is decidable in both languages.

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References

  1. Yu. A. Bahturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  2. G. Baumslag, A. G. Myasnikov, and V. N. Remeslennikov, “Algebraic geometry over groups. I. Algebraic sets and ideal theory,” J. Algebra, 219, 16–79 (1999).

    Article  MathSciNet  Google Scholar 

  3. N. Bourbaki, Elements of Mathematics. Commutative Algebra, Hermann, Paris; Addison-Wesley (1972).

    Google Scholar 

  4. C. C. Chang and H. J. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, 1973.

  5. E. Yu. Daniyarova, I. V. Kazatchkov, and V. N. Remeslennikov, “Algebraic geometry over free metabelian Lie algebras. I. U-algebras and universal classes,” Fund. Prikl. Mat., 9, No. 3, 37–63 (2003).

    Google Scholar 

  6. S. Lang, Algebra, revised third edition, Graduate Texts in Mathematics, Vol. 211, Springer, New York (2002).

    Google Scholar 

  7. A. I. Malcev, Basic Linear Algebra [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  8. A. G. Myasnikov and V. N. Remeslennikov, “Algebraic geometry over groups. II. Logical foundations,” J. Algebra, 234, 225–276 (2000).

    Article  MathSciNet  Google Scholar 

  9. V. A. Roman’kov, “On equations in free metabelian groups,” Sib. Mat. Zh., 20, No. 3, 671–673 (1979).

    MathSciNet  Google Scholar 

  10. A. Seidenberg, “Constructions in Algebra,” Trans. Amer. Math. Soc., 197, 273–313 (1974).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Omsk Branch of Institute of Mathematics (Siberian Branch of the Russian Academy of Science), Russia

    E. Yu. Daniyarova, I. V. Kazatchkov & V. N. Remeslennikov

Authors
  1. E. Yu. Daniyarova
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  2. I. V. Kazatchkov
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  3. V. N. Remeslennikov
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 65–87, 2003.

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Daniyarova, E.Y., Kazatchkov, I.V. & Remeslennikov, V.N. Algebraic geometry over free metabelian lie algebras. II. Finite-field case. J Math Sci 135, 3311–3326 (2006). https://doi.org/10.1007/s10958-006-0160-4

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