Nondegenerate Invariant Symmetric Bilinear Forms on Simple Lie Superalgebras in Characteristic 2

Abstract

As is well-known, the dimension of the space of non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the ground field is distinct from 2. We prove that in characteristic 2, the superdimension of the space of NISes can be equal to 0, or 1, or 0|1, or 1|1. This superdimension is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple restricted Lie algebra with a NIS (for examples of such Lie algebras, although mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations with both even and odd parameter of several simple finite-dimensional Lie superalgebras in characteristic 2.