TY - JOUR

T1 - Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras (with an Appendix by Andrey Krutov)

AU - Bouarroudj, Sofiane

AU - Grozman, Pavel

AU - Lebedev, Alexei

AU - Leites, Dimitry

N1 - Funding Information:
We are thankful to A. Krutov for his help, e.g., for computing several examples (Section 2.3). We are thankful to N. Chebochko and M. Kuznetsov for helpful discussions of their unpublished results pertaining to this paper; to A. Dzhumadildaev, P. Zusmanovich, and Sh. Ibraev for helpful discussions. We are very thankful to the referees, carefully selected by SIGMA, especially one of them, for very constructive criticism. S.B. and D.L. were supported by the grant AD 065 NYUAD. D.L. is thankful to MPIMiS, Leipzig, where he was Sophus-Lie-Professor (2004-07), when a part of the ideas of this paper were conceived, for financial support and most creative environment. The authors of the main text and A. Krutov, who wrote the Appendix, are grateful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi for the possibility to perform the difficult computations of this research. Andrey Krutov was supported by the GAČR project 20-17488Y and RVO: 67985840.
Funding Information:
S.B. and D.L. were supported by the grant AD 065 NYUAD. D.L. is thankful to MPIMiS, Leipzig, where he was Sophus-Lie-Professor (2004-07), when a part of the ideas of this paper were conceived, for financial support and most creative environment. The authors of the main text and A. Krutov, who wrote the Appendix, are grateful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi for the possibility to perform the difficult computations of this research. Andrey Krutov was supported by the GACˇR project 20-17488Y and RVO: 67985840.
Publisher Copyright:
© 2023, Institute of Mathematics. All rights reserved.

PY - 2023

Y1 - 2023

N2 - 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 super-algebras (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.

AB - 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 super-algebras (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.

KW - central extension

KW - derivation

KW - modular Lie superalgebra

UR - http://www.scopus.com/inward/record.url?scp=85161910964&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85161910964&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2023.032

DO - 10.3842/SIGMA.2023.032

M3 - Article

AN - SCOPUS:85161910964

SN - 1815-0659

VL - 19

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

M1 - 32

ER -