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

SIGMA 19 (2023), 032, 73 pages      arXiv:1307.1858      https://doi.org/10.3842/SIGMA.2023.032

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

Sofiane Bouarroudj a, Pavel Grozman b, Alexei Lebedev c and Dimitry Leites ad
a) New York University Abu Dhabi, Division of Science and Mathematics, P.O. Box 129188, United Arab Emirates
b) Deceased
c) Equa Simulation AB, Stockholm, Sweden
d) Department of Mathematics, University of Stockholm, SE-106 91 Stockholm, Sweden

Received November 16, 2016, in final form February 02, 2023; Published online May 29, 2023

Abstract
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.

Key words: modular Lie superalgebra; derivation; central extension.

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