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2 results

  • Degenerations of Leibniz and Anticommutative Algebras

  • Part of
  • Nurlan Ismailov, Ivan Kaygorodov, Yury Volkov
  • Journal:
    Canadian Mathematical Bulletin / Volume 62 / Issue 3 / September 2019
    Published online by Cambridge University Press:
    29 January 2019, pp. 539-549
    Print publication:
    September 2019
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  • We describe all degenerations of three-dimensional anticommutative algebras $\mathfrak{A}\mathfrak{c}\mathfrak{o}\mathfrak{m}_{3}$ and of three-dimensional Leibniz algebras $\mathfrak{L}\mathfrak{e}\mathfrak{i}\mathfrak{b}_{3}$ over $\mathbb{C}$. In particular, we describe all irreducible components and rigid algebras in the corresponding varieties.

  • The Variety of Two-dimensional Algebras Over an Algebraically Closed Field

  • Part of
  • Ivan Kaygorodov, Yury Volkov
  • Journal:
    Canadian Journal of Mathematics / Volume 71 / Issue 4 / August 2019
    Published online by Cambridge University Press:
    16 October 2018, pp. 819-842
    Print publication:
    August 2019
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  • The work is devoted to the variety of two-dimensional algebras over algebraically closed fields. First we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of two-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.