Computer-aided study of double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms in characteristic 2

Abstract

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman's Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times.

arXiv comment22 pages

References in corpus (1)

• Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras

Cited by in corpus (7)

• Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras
• Deformations of Symmetric Simple Modular Lie (Super)Algebras
• Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2
• Duflo-Serganova homology for exceptional modular Lie superalgebras with Cartan matrix
• Double extensions of quasi-Frobenius Lie superalgebras with degenerate center
• On a Hidden Supersymmetry of Cosmological Billiards
• New Constructions of Exceptional Simple Lie Superalgebras with Integer Cartan Matrix in Characteristics 3 and 5 via Tensor Categories