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 - 2015 |
Keywords
- Characteristic 2
- Deformation
- Jurman algebra
- Kaplansky algebra
- Kostrikin-Shafarevich conjecture
- Lie algebra
ASJC Scopus subject areas
- Mathematics(all)