Computer-Aided Study of Double Extensions of Restricted Lie Superalgebras Preserving the Nondegenerate Closed 2-Forms in Characteristic 2

Abstract

A Lie (super)algebra with a nondegenerate 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 nondegenerate 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.

Keywords:

• Characteristic 2
• double extension
• restricted Lie superalgebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 22E46
• 22E70
• Secondary: 81S99
• 51P05

Acknowledgments

For the possibility to perform the difficult computations of this research we are grateful to M. Al Barwani, Director of the High Performance Computing resources at New York University Abu Dhabi. We are thankful to J. Bernstein, P. Grozman, A. Krutov, A. Lebedev, and I. Shchepochkina for helpful advice.

Notes

1 The (left) Leibniz algebra L satisfies $[x,[y,z]]=[[x,y],z]+[y,[x,z]]$ for any $x,y,z\in L;$ if, moreover, L it is anti-commutative, it is a Lie algebra. Superization is immediate, via the Sign Rule.

Additional information

Funding

S.B. and D.L. were partly supported by the grant AD 065NYUAD.