Restricted Lie (Super)Algebras in Characteristic 3

Bouarroudj, Sofiane; Krutov, Andrey; Lebedev, Alexei; Leites, Dimitry; Shchepochkina, Irina

doi:10.1007/s10688-018-0206-7

Cited by 8 publications

(8 citation statements)

References 7 publications

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“…From [16]: "This classification is implicit to this day when dealing with deforms: simple deforms of the divergence-free algebras [62] and of Hamiltonian type algebras [55,58] were classified only several years after [7] was published. In the divergence-free case, explicit formulas of the p-structure were obtained only recently, see [14]. ("The problem of restrictedness is approached.…”

Section: All Algebras But P ≥mentioning

confidence: 99%

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Bouarroudj¹,

Grozman²,

Lebedev³

et al. 2023

SIGMA

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Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

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Section: All Algebras But P ≥mentioning

confidence: 99%

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Bouarroudj¹,

Grozman²,

Lebedev³

et al. 2023

SIGMA

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“…It consists of the deforms of the simple Z-graded vectorial Lie algebras. Observe that the classification of simple restricted Lie algebras obtained in [8] (the final list) does not contain all explicit expressions of p-structures to this day: e.g., for the deforms of svect(m; 1|n), the description was only recently obtained, see [15]. For several families of Hamiltonian algebras their p-structures remain to be described: "The problem of restrictedness is approached.…”

Section: The Vague Part Of the Ksh-listmentioning

confidence: 99%

Deformations of Symmetric Simple Modular Lie (Super)Algebras

Bouarroudj¹,

Grozman²,

Leites³

2023

SIGMA

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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.

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“…Hereafter, the ground field K is of characteristic p > 2. Following [4,29], we say that a Lie superalgebra g has a p|2p-structure if there exists a mapping [p] : g0 → g0, a → a [p] such that (SR1) ad a [p] (b) = (ad a ) p (b) for all a ∈ g0 and b ∈ g. (SR2) (αa) [p] = α p a [p] for all a ∈ g0 and α ∈ K.…”

Section: Maximal-dimensional Solvable Lie Superalgebras With Model Ni...mentioning

confidence: 99%

Cohomologically rigid solvable Lie superalgebras with model filiform and model nilpotent nilradical

Bouarroudj,

Navarro

2021

Preprint

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In this paper, we find a family SL n,m , in any arbitrary dimensions, of cohomologically rigid solvable Lie superalgebras with nilradical the model filiform Lie superalgebra L n,m . Moreover, we exhibit a family of cohomologically rigid solvable Lie superalgebras with nilradical the model nilpotent Lie superalgebra of generic characteristic sequence. Both cases correspond to solvable Lie superalgebras of maximal dimension for a given nilradical. Contrariwise, we will show that the family of Lie superalgebras SL n,m can be deformed if defined over a field of odd characteristic.

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Restricted Lie (Super)Algebras in Characteristic 3

Cited by 8 publications

References 7 publications

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Deformations of Symmetric Simple Modular Lie (Super)Algebras

Cohomologically rigid solvable Lie superalgebras with model filiform and model nilpotent nilradical

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