[Paper Review] Deforms of Lie algebras in characteristic 2: Semi-trivial for Jurman algebras, non-trivial for Kaplansky algebras

Sofiane Bouarroudj, Alexei Lebedev+2 more|arXiv (Cornell University)|Jan 13, 2013

Advanced Topics in Algebra3 citations

TL;DR

This paper investigates deformations of Lie algebras in characteristic 2, identifying nontrivial deformations for Kaplansky algebras—particularly type-2 and type-4—via novel $\mathbb{Z}/2$-gradings that are nonlinear in roots. It confirms Grishkov's claim by showing the Jurman algebra is a semitrivial deformation of the derived alternate Hamiltonian algebra, advancing the classification of simple modular Lie algebras.

ABSTRACT

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved $\mathbb{Z}/2$-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every $\mathbb{Z}/2$-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, not an exterior one. 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.

Motivation & Objective

• To clarify the deformation theory of simple finite-dimensional Lie algebras in characteristic 2.
• To identify and classify nontrivial deformations of Kaplansky algebras, particularly type-2 and type-4.
• To confirm Grishkov's implicit claim that the Jurman algebra arises as a semitrivial deformation of the derived alternate Hamiltonian algebra.
• To support the conjecture that all simple modular Lie algebras are either of standard type or deformations thereof.
• To provide sufficient conditions for known deformations to be semitrivial in characteristic 2.

Proposed method

• Applying a method that assigns a simple Lie superalgebra to every $\mathbb{Z}/2$-graded simple Lie algebra in characteristic 2.
• Constructing seven new series of simple Lie superalgebras from $\mathbb{Z}/2$-gradings of Lie algebras.
• Using cohomological techniques to analyze deformations, particularly focusing on nontrivial cohomology classes.
• Analyzing the structure of type-1 Kaplansky algebras as derived algebras of nonalternate Hamiltonian Lie algebras preserving a tensorial 2-form.
• Establishing that certain deformations can be isomorphic to the original algebra, thus being semitrivial.
• Providing sufficient conditions under which deformations are semitrivial, based on cohomological data.

Experimental results

Research questions

• RQ1 Are there nontrivial deformations of Kaplansky algebras in characteristic 2, and if so, how can they be classified?
• RQ2 Can the Jurman algebra be realized as a semitrivial deformation of the derived alternate Hamiltonian Lie algebra?
• RQ3 What is the role of $\mathbb{Z}/2$-gradings that are nonlinear in roots in constructing new Lie superalgebras?
• RQ4 How do deformations of the nonalternate Hamiltonian Lie algebra relate to the type-1 Kaplansky algebra?
• RQ5 Under what cohomological conditions are deformations of Lie algebras in characteristic 2 semitrivial?

Key findings

• Type-2 and type-4 Kaplansky algebras admit previously unobserved $\mathbb{Z}/2$-gradings that are nonlinear in roots.
• Seven new series of simple Lie superalgebras are constructed via the $\mathbb{Z}/2$-grading method in characteristic 2.
• Type-2 algebras and one type-4 algebra are shown to be nontrivial deformations of the alternate Hamiltonian Lie algebra.
• The Jurman algebra is explicitly confirmed as a semitrivial deformation of the derived alternate Hamiltonian Lie algebra, validating Grishkov's implicit claim.
• The type-1 Kaplansky algebra is identified as the derived algebra of the nonalternate Hamiltonian Lie algebra preserving a tensorial 2-form.
• Sufficient conditions are provided for deformations in characteristic 2 to be semitrivial, based on cohomological data.