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The algebraic and geometric classification of noncommutative Jordan superalgebras

Hani Abdelwahab, Ivan Kaygorodov, Abror Khudoyberdiyev

Abstract

The algebraic and geometric classifications of complex $3$-dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of $3$-dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson-Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional anticommutative superalgebras and its principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative $\mathfrak{CD}$-, $\mathfrak{s}_4$-, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative $\big($rigid$\big)$ superalgebras.

The algebraic and geometric classification of noncommutative Jordan superalgebras

Abstract

The algebraic and geometric classifications of complex -dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of -dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson-Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of -dimensional anticommutative superalgebras and its principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative -, -, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative rigid superalgebras.
Paper Structure (22 sections, 29 theorems, 16 equations)

This paper contains 22 sections, 29 theorems, 16 equations.

Table of Contents

  1. The algebraic classification of superalgebras
  2. Preliminaries: the method for algebraic classification
  3. Noncommutative Jordan superalgebras of type $(1,2)$
  4. Anticommutative superalgebras of type $(1,2)$
  5. The classification Theorem A1
  6. The proof of Theorem A1
  7. Noncommutative Jordan superalgebras of type $(2,1)$
  8. Anticommutative superalgebras of type $(2,1)$
  9. The classification Theorem A2
  10. The proof of Theorem A2
  11. Corollaries
  12. Kokoris superalgebras
  13. Associative superalgebras
  14. Standard superalgebras
  15. Anticommutative superalgebras
  16. ...and 7 more sections

Key Result

Proposition 3

$({\rm A},\cdot)$ is a noncommutative Jordan superalgebra if and only if $({\rm A},\circ ,[\cdot,\cdot])$ is a generic Poisson--Jordan superalgebra.

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • Definition 4
  • Proposition 5
  • Definition 6
  • Lemma 7
  • Theorem 8
  • Theorem 9
  • ...and 41 more