Metabelian Associative Algebras



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 \(Xu = Yu\).


Metabelian algebras Congruence of bilinear forms Classification of algebras 

Mathematics Subject Classification

15A21 16D70 16Z05 



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


  1. 1.
    Agore, A.L.: Classifying complements for associative algebras. Linear Algebra Appl. 446, 345–355 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agore, A.L., Militaru, G.: Itô’s theorem and metabelian Leibniz algebras. Linear Multilinear Algebra (2014). doi: 10.1080/03081087.2014.992771
  3. 3.
    Agore, A.L., Militaru, G.: Hochschild products and global non-abelian cohomology for algebras. Applications, arXiv:1503.05364
  4. 4.
    Amberg, B., Franciosi, S., Giovanni, F.: Products of Groups. Oxford University Press, Oxford (1992)MATHGoogle Scholar
  5. 5.
    Baumslag, G., Mikhailov, R., Orr. K.E.: A new look at finitely generated metabelian groups, arXiv:1203.5431
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Drensky, V., Piacentini Cattaneo, G.M.: Varieties of metabelian Leibniz algebras. J. Algebra Appl. 01, 31 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fite, B.W.: On metabelian groups. Trans. Am. Math. Soc. 3, 331–353 (1902)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    De Graaf, W.: Classification of nilpotent associative algebras of small dimension, arXiv:1009.5339v1
  10. 10.
    Hochschild, G.: Cohomology and representations of associative algebras. Duke Math. J. 14, 921–948 (1947)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Horn, R., Sergeichuk, V.: Canonical matrices of bilinear and sesquilinear forms. Linear Algebra Appl. 428, 193–223 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Itô, N.: Über das produkt von zwei abelschen gruppen. Math. Z. 62, 400–401 (1955)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kruse, R.L., Price, D.T.: Nilpotent Rings. Gordon and Breach, New York (1969)MATHGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 15.
    Murray, W.: Bilinear forms on Frobenius algebras. J. Algebra 293, 89–101 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Riehm, C.: The equivalence of bilinear forms. J. Algebra 31, 45–66 (1974)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58, 141–163 (1936)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania

Personalised recommendations