Skip to main content
Log in

Abstract

Metabelian algebras are introduced and it is shown that an algebra A is metabelian if and only if A is a nilpotent algebra having the index of nilpotency at most 3, i.e. \(x y z t = 0\), for all x, y, z, \(t \in A\). We prove that the Itô’s theorem for groups remains valid for associative algebras. A structure theorem for metabelian algebras is given in terms of pure linear algebra tools and their classification from the view point of the extension problem is proven. Two border-line cases are worked out in detail: all metabelian algebras having the derived algebra of dimension 1 (resp. codimension 1) are explicitly described and classified. The algebras of the first family are parameterized by bilinear forms and classified by their homothetic relation. The algebras of the second family are parameterized by the set of all matrices \((X, Y, u) \in \mathrm{M}_{n}(k)^2 \times k^n\) satisfying \(X^2 = Y^2 = 0\), \(XY = YX\) and \(Xu = Yu\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We recall that for a vector space V, we denote \(V_0 = V\) with the abelian algebra structure.

References

  1. Agore, A.L.: Classifying complements for associative algebras. Linear Algebra Appl. 446, 345–355 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agore, A.L., Militaru, G.: Itô’s theorem and metabelian Leibniz algebras. Linear Multilinear Algebra (2014). doi:10.1080/03081087.2014.992771

  3. Agore, A.L., Militaru, G.: Hochschild products and global non-abelian cohomology for algebras. Applications, arXiv:1503.05364

  4. Amberg, B., Franciosi, S., Giovanni, F.: Products of Groups. Oxford University Press, Oxford (1992)

    MATH  Google Scholar 

  5. Baumslag, G., Mikhailov, R., Orr. K.E.: A new look at finitely generated metabelian groups, arXiv:1203.5431

  6. Daniyarova, E., Kazachkov, I., Remeslennikov, V.: Algebraic geometry over free metabelian Lie algebra II: finite field case. J. Math. Sci. 135, 3311–3326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drensky, V., Piacentini Cattaneo, G.M.: Varieties of metabelian Leibniz algebras. J. Algebra Appl. 01, 31 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fite, B.W.: On metabelian groups. Trans. Am. Math. Soc. 3, 331–353 (1902)

    Article  MATH  Google Scholar 

  9. De Graaf, W.: Classification of nilpotent associative algebras of small dimension, arXiv:1009.5339v1

  10. Hochschild, G.: Cohomology and representations of associative algebras. Duke Math. J. 14, 921–948 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  11. Horn, R., Sergeichuk, V.: Canonical matrices of bilinear and sesquilinear forms. Linear Algebra Appl. 428, 193–223 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Itô, N.: Über das produkt von zwei abelschen gruppen. Math. Z. 62, 400–401 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kruse, R.L., Price, D.T.: Nilpotent Rings. Gordon and Breach, New York (1969)

    MATH  Google Scholar 

  14. Lennox, J.C., Robinson, D.J.S.: The theory of infinite soluble groups, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2004)

    Book  Google Scholar 

  15. Murray, W.: Bilinear forms on Frobenius algebras. J. Algebra 293, 89–101 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Riehm, C.: The equivalence of bilinear forms. J. Algebra 31, 45–66 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58, 141–163 (1936)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Grant No. 88/05.10.2011.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Militaru.

Additional information

Communicated by Miin Huey Ang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Militaru, G. Metabelian Associative Algebras. Bull. Malays. Math. Sci. Soc. 40, 1639–1651 (2017). https://doi.org/10.1007/s40840-015-0157-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-015-0157-6

Keywords

Mathematics Subject Classification

Navigation