### Abstract

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved ℤ/2-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every ℤ/2-graded simple Lie algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras and one of the two type-4 algebras are demystified as nontrivial deforms (the results of deformations) of the alternate Hamiltonian algebras. The type-1 Kaplansky algebra is recognized as the derived of the nonalternate version of the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form. Deforms corresponding to nontrivial cohomology classes can be isomorphic to the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly describe the Jurman algebra as such a "semitrivial"deform of the derived of the alternate Hamiltonian Lie algebra. This paper helps to sharpen the formulation of a conjecture describing all simple finite-dimensional Lie algebras over any algebraically closed field of nonzero characteristic and supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple finite-dimensional modular Lie algebras are either of "standard"type or deforms thereof. In characteristic 2, we give sufficient conditions for the known deformations to be semitrivial.

Original language | English (US) |
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Pages (from-to) | 353-402 |

Number of pages | 50 |

Journal | Mathematical Research Letters |

Volume | 22 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2015 |

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### Keywords

- Characteristic 2
- Deformation
- Jurman algebra
- Kaplansky algebra
- Kostrikin-Shafarevich conjecture
- Lie algebra

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*22*(2), 353-402. https://doi.org/10.4310/MRL.2015.v22.n2.a3