Integrable cocycles and global deformations of Lie algebra of type G 2 in characteristic 2

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

Section: Explicit Cocycles Vs Ecologymentioning

confidence: 99%

Deformations of Symmetric Simple Modular Lie (Super)Algebras

Bouarroudj¹,

Grozman²,

Leites³

2023

“…[13]. Therefore, in [74], the claims describing all deformations of svect(m; N ) should have been confined to m > 2 and, moreover, Tyurin's main theorem should only claim a complete description of non-isomorphic filtered deforms related to the standard Z-grading; for examples of filtered deforms of svect (1) (3; 1) h (1) (4; 1) corresponding to distinct Z-gradings, see [18].…”

Section: On Deforms Of Svect and H Quantizationsmentioning

confidence: 99%

Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

Bouarroudj¹,

Grozman

Lebedev

et al. 2020

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs-new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergencefree vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

“…solv, see (6.13) 1 1, 0 (01), (10), (11) vect (1) (1; 3) The following two conjectures concern vect (1) (1; n), where n ≥ 3.…”

Section: Summary Of Computer-aided Experimentsmentioning

confidence: 99%

On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

Krutov¹,

Lebedev²

2018

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finitedimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for o Π (8)). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height n > 1, there is an (n − 2)-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple 3|2-dimensional Lie superalgebra (new result).