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  • 1.
    Belitskii, Genrich
    et al.
    Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
    Dmytryshyn, Andrii
    Faculty of Mechanics and Mathematics, Kiev National Taras Shevchenko University, Kiev, Ukraine.
    Lipyanski, Ruvim
    Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
    Sergeichuk, Vladimir
    Institute of Mathematics, Kiev, Ukraine.
    Tsurkov, Arkady
    Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
    Problems of classifying associative or Lie algebras over a field of characteristic not 2 and finite metabelian groups are wild2009In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 18, p. 516-529, article id 41Article in journal (Refereed)
    Abstract [en]

    Let F be a field of characteristic different from 2. It is shown that the problems of classifying

    (i) local commutative associative algebras over F with zero cube radical,

    (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and

    (iii) finite p-groups of exponent p with central commutator subgroup of order  are hopeless since each of them contains

    • the problem of classifying symmetric bilinear mappings UxU → V , or

    • the problem of classifying skew-symmetric bilinear mappings UxU → V ,

    in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

  • 2.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Sergeichuk, Vladimir V.
    Ukrainian Acad Sci, Kiev, Ukraine.
    Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations2014In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 27, p. 1-18Article in journal (Refereed)
    Abstract [en]

    The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

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